Chapter 1: General Synthetic and Physical Methods Free

Quyen N. Do, James S. Ratnakar and Zoltán Kovács*

Gd^III-based contrast agents for MRI are used in approximately 30% of MRI exams.¹  Although the FDA-approved contrast agents are among the safest drugs on the market, their core, the Gd^III ion, has a 50% lethal dose, (LD₅₀) around 0.1–0.2 mmol kg⁻¹.²  Therefore, for medical diagnostic applications, Gd^III ions must be chelated by ligands to prevent the metal ion from being released. Administration of complexes with insufficient kinetic inertness can result in the debilitating disease nephrogenic systemic fibrosis (NSF), deposition of metal in the brain, or other issues.^3–5  To develop safe and efficient Gd^III-based (Chapter 2) and lanthanide-based CEST agents (Chapter 3) for MRI, it is important to understand the role of the ligand in determining the relaxivity, thermodynamic stability, and kinetic inertness of complexes. The fundamentals described in this chapter are intended to assist with the design of future imaging agents for MRI.

Ligands play a critical role not only in reducing the toxicity of metal ions, but also in optimizing the parameters that determine the relaxivity or CEST-enhancing ability of metal complexes. These parameters include the number of inner-sphere water molecules (q), the residence lifetime of coordinated inner-sphere water molecules (τ_M), and the rotational correlation time (τ_R). Chelators sometimes contain moieties for biosensing or reactive functionalities as points of attachment for targeting vectors. All chelators currently used in clinically approved contrast agents for MRI are octadentate ligands based either on the open-chain ligand diethylenetriaminepentaacetic acid (DTPA) or the macrocyclic ligand 1,4,7,10-tetraazacyclododecane-1,4,7,10-tetraacetic acid (DOTA) (Figure 1.1). Most reported responsive and bifunctional ligands are also derivatives of these two chelators. With both of these ligands, the ninth coordination site of Gd^III is occupied by a rapidly exchanging water molecule that transfers the paramagnetic relaxation effect of the metal ion to the pool of bulk water. Properties of ligands, such as total basicity (sum of the protonation constants), basicity of the first protonation site, preorganization, and rigidity, influence the thermodynamic stability and kinetic behavior of the resulting metal complexes.

The bonding in lanthanide complexes is predominantly ionic, and coordination geometries are largely determined by the steric bulk of the ligands. Because Ln^III ions are considered to be hard Lewis acids, they favor hard donor atoms, such as fluoride, oxygen, and, to a lesser extent, nitrogen. Ln^III ions typically have coordination numbers of eight or nine, and they can form stable complexes with ligands that have matching denticities. Ln^III ions form relatively stable complexes with ligands that enable the formation of five-membered chelate rings; ligands that form six-membered chelate rings are far less stable.^6–12 

The thermodynamic stability of a complex is expressed as the equilibrium constant written for the reaction between the free metal ion and the fully deprotonated ligand [eqn (1.1)].^13–15 

The stability constant of the complex, K_LnL, is defined as follows:

In aqueous solution, the ligand can be fully or partially protonated depending on both the pH of the solution and the protonation constants (basicity) of the ligand. The formation of the metal complex is thus essentially a competition between protons and the metal ion for the donor sites of the ligand. Consequently, the true stability of a complex at a given pH is determined by its conditional stability constant, K$LnLC$, which takes into account the protonation of the ligand. Thus, for the formation of a lanthanide chelate with one inner-sphere water molecule, the stability of the complex is characterized by the following equilibrium.

In eqn (1.4), [L]_total is the total concentration of the free and protonated ligand species that are not bound to the lanthanide ion, and α_H is the total or equilibrium ligand concentration ratio.^14–17 α_H is expressed using the [H⁺] and the protonation constants of the ligand as:

For polyaminopolycarboxylate ligands, there is a nearly linear relationship between the basicity of the ligand, determined in terms of the sum of the protonation constants, Σlog K_i, and the stability of the corresponding Gd^III complexes (Figure 1.2). Deviations were reported for ligands that do not form five-membered chelate rings, for ligands that have non-coordinating peripheral groups that undergo protonation, and for non-polyaminocarboxylate ligands. For DTPA- and DOTA-type ligands, more basic ligands tend to form complexes with higher stabilities.^15,16 

According to the chelate effect, metal complexes with polydentate ligands are more stable than similar complexes with the same number of monodentate ligands. Macrocyclic ligands with the same number of donor atoms as linear ligands tend to be even more stable; this is known as the macrocyclic effect.^6,18–20  The chelate effect is driven by entropy because the number of particles increases as ligands displace coordinated water molecules during complex formation. There is also an enthalpic contribution to the formation of metal complexes that can be significant for some systems.^9,21  The origin of the macrocyclic effect is at least partially due to the increased preorganization of macrocyclic ligands relative to their linear analogues.^6,22  Therefore, less energy is needed for macrocyclic ligands to convert into the final conformation of the complex. A preorganized ligand is one whose metal-free conformation is similar to that of the metal-bound one.^23–26  Preorganized ligands undergo minimal reorganization upon complex formation. Such ligands are important in the design of improved lanthanide-based contrast agents for MRI.²³  Conformational preorganization is often achieved with rigid ligands.^7,27  Here, the term rigid is not limited to molecules that exist in a single minimum conformational energy. Ligands that are conformationally restricted by structural modification, most commonly either by hydrogen-bonding, by the introduction of bulky substituents, or by fusing of an aromatic or small aliphatic ring to the ligand backbone, are also rigid.^26,27  Polyazamacrocycles with pendant coordinating side arms, such as DOTA, are less pre-organized than cryptands, spherands, and coronands, which have nearly identical conformations in the free and bound states. Nevertheless, in spite of the larger work required to convert DOTA-type ligands from their free conformations into the final metal complexes, [Ln(DOTA)]⁻ complexes are more stable than corresponding cryptand complexes. The high stabilities observed for polyazamacrocyclic ligands with acetate pendant arms are largely owing to the steric efficiency of the carboxylate groups. The carboxylate C-atom is sp² hybridized with a trigonal planar geometry and the oxygens are not bonded to protons. Thus, upon coordination, carboxylates create minimal steric crowding around metal ions.⁷ 

If a ligand is preorganized in a conformation that is not favorable to the formation of a metal complex, then the stability of the latter will be low regardless of the basicity of the ligand. This is the case for dicyclohexyl DOTA, whose lanthanide complexes are about an order of magnitude less stable than their corresponding DOTA analogues because the cyclen ring of dicyclohexyl DOTA cannot adapt the square [3333] conformation.²⁸ 

The basicity and rigidity of a ligand also influence the kinetic inertness (lability) of the resulting metal complexes. Flexible open chain ligands such as ethylenediaminetetraacetic acid (EDTA) and DTPA form complexes with Ln^III ions rapidly. Rigid open chain ligands form complexes more slowly. In such cases, the kinetics for the formation of La^III–CDTA (CDTA=cyclohexanediaminetetraacetic acid) indicate the rapid formation of a protonated intermediate in which La^III is coordinated only by the acetate groups of the ligand. The final complex is formed upon rearrangement and concomitant deprotonation of the intermediate.²⁹  Lanthanide chelates of DOTA, 3,6,9,15-tetraazabicyclo[9.3.1]pentadeca-1(15),11,13-triene-3,6,9-triacetic acid (PCTA), and several other DOTA derivatives form via an analogous mechanism involving mono- or di-protonated intermediates. For these ligands, the rate-determining step for the formation of the metal complex involves deprotonation of the protonated intermediate. This step is followed by a rapid rearrangement in which the metal ion moves into the cavity of the ligand.^30,31  Because the rearrangement requires the removal of the last proton from the ligand, the rate constant for the base-catalyzed rearrangement of the protonated intermediates, k_OH, is inversely proportional to the log K_a of the most basic nitrogen of the macrocycle (Figure 1.3).^30,32 

Ligands fulfill two main roles in Gd^III-based contrast agents for MRI: maximizing relaxivity and minimizing dissociation. The r₁ and r₂ relaxivities (in units of mM⁻¹ s⁻¹) characterize the shortening effect of a contrast agent on the T₁ (longitudinal) and T₂ (transverse) relaxation times of the protons of bulk water (see Chapter 2.1). Relaxivity is influenced by a number of factors, and in this section the factors that can be modified by rational ligand design are briefly reviewed. The relaxation rate enhancement (shortening of the T₁ and T₂ relaxation times) of ¹H nuclei of the bulk water by Gd^III complexes originates from time-fluctuating dipolar interactions between the unpaired electrons of the Gd^III ion and the ¹H nuclear spins. Relaxivity has three components (inner-sphere, outer-sphere, and second-sphere) for Gd^III complexes that have at least one inner-sphere molecule of water. The inner-sphere contribution to relaxivity tends to be the most important and accounts for about 50–60% of the total relaxivity of most Gd^III complexes. A complete theoretical description is provided in Chapter 2.1. The inner-sphere relaxivity (r_i, i=1 for longitudinal or 2 for transverse) is determined by a complex interplay of several parameters, such as the number of inner-sphere water molecules (q), the bound-water residence lifetime (τ_M, the inverse of the exchange rate, k_ex), the distance between the Gd^III ion and the protons of the inner-sphere water molecule (r_H), the rotational correlation time (τ_R, the inverse of the tumbling rate of the complex), and the electronic relaxation times (T_1e and T_2e) of the metal ion.

Of these parameters, only q, τ_R, and τ_M can be optimized reliably by ligand design. The inner-sphere contribution increases linearly with the number of inner-sphere water molecules. Lanthanide ions in general have a coordination number of eight or nine in aqueous media. The number of inner-sphere water molecules is determined by the denticity of the ligand (number of donor atoms), the steric properties of the ligand, and the size of the lanthanide ion. All clinically approved contrast agents for MRI have eight donor atoms and one inner-sphere water molecule. Ligands with fewer than eight donor atoms often form Gd^III complexes with two or three inner-sphere water molecules. These complexes have higher relaxivities than complexes with one inner-sphere water molecule, but the kinetic inertness of complexes with fewer donor atoms is usually lower than that of monohydrated chelates. These include the hydroxypyridinone-based tripodal chelates, which have high relaxivities and near-optimal water-exchange rates at some magnetic field strengths.³³  Although kinetically labile, these complexes are highly stable and are minimally influenced by physiologically relevant cations or anions.^33–37  Some ligand systems with seven donor atoms form lanthanide complexes with satisfactory kinetic inertness. In particular, lanthanide chelates of PCTA and 6-amino-6-methylperhydro-1,4-diazepinetetraacetic acid (AAZTA) and their derivatives are more inert than DTPA derivatives.^30,38 

At the clinically relevant fields of 1.5 to 3 T, relaxivity will largely be determined by τ_M and τ_R (see Chapter 2.1). For low-molecular-weight complexes, such as [Gd(DOTA)]⁻ and [Gd(DTPA)]²⁻, inner-sphere relaxivity is limited by τ_R. An increase in τ_R has a strongly field-dependent effect on inner-sphere relaxivity with a maximum value that is dependent on τ_M and is usually around 1.5 T. Between 0.5 and 3 T, the gain in relaxivity is quite significant even when τ_M is not optimal. However, the effect levels off at higher fields, and above 3 T, r₁ decreases with increasing τ_R. At 9.4 T, the optimal τ_R is around 400 ps, corresponding to the tumbling rate of medium-sized rigid molecules.^15,39,40 

In practice, slowing the tumbling rate is usually achieved by binding Gd^III complexes covalently or non-covalently to macromolecules, such as polymers, dendrimers, proteins, viral capsids, gold and silica nanoparticles, or nanodiamonds, or by incorporating them into self-assembling systems.^{15,39,41–58}  A large number of Gd^III-based agents have been developed in which the Gd^III complex is covalently linked to a high-molecular-weight scaffold with free amino groups on the surface.^43,59,60  The synthesis involves the functionalization of amino groups with an amine-reactive bifunctional ligand (Figure 1.4)^61–63  followed by complexation with Gd^III ions.^44,58 

In addition to the peptide-coupling or protein-labeling functional groups shown in Figure 1.4, the azide–alkyne Huisgen cycloaddition reaction, commonly referred to as click chemistry, can also be used to functionalize biomolecules, polymers, and nanoparticles with metal complexes.^64–66  Several copper-catalyzed azide–alkyne cycloadditions as well as copper-free bifunctional chelators have been reported that contain alkyne or azide functionalities (Figure 1.5). This approach requires a complementary functional group to be synthesized into the targeting vector but offers site-selective labeling and mild reaction conditions.^67–81 

Noncovalent interactions have also been explored to form supramolecular adducts with slow tumbling rates. Such systems include micelles and liposomes formed with Gd^III complexes with one or more hydrophobic tails. These systems offer the additional benefits of being able to accumulate in tumors and delivering a large payload of Gd^III-based contrast agents to target sites. Similar approaches have been utilized to form CEST micelle agents (Chapter 3.5.2).

When a noncovalent interaction leading to a longer value of τ_R involves specific binding of a Gd^III complex to a particular target protein, the effect is known as receptor-induced magnetization enhancement.^82–84  Relaxation enhancement only occurs where the target receptor is present, thereby improving target-to-background ratios. One of the most successful adaptations of this concept exploits the reversible noncovalent binding of a complex to serum albumin.⁸³  Reversible binding confines the agent in the intravascular space (blood pool), increases its relaxivity, and results in high-quality vasculature angiographic images.^85,86  The first clinically approved blood-pool agent that binds non-covalently to albumin is gadofosveset (Ablavar) (Figure 1.6). The diphenylcyclohexyl group of this DTPA-based ligand was designed to bind human serum albumin. The reversible, noncovalent binding of the complex to albumin results in favorable pharmacokinetics because the complex has increased retention in blood. Yet, it is efficiently eliminated by the kidneys as a low-molecular-weight molecule when not interacting with albumin.

While albumin is an ideal target because of its abundance and binding properties, the receptor-induced magnetization enhancement strategy is not limited to this protein. Several other Gd^III-based agents have been developed that bind to other specific proteins.^87–89  The majority of these complexes contain peptide-targeting vectors, but peptoid, DNA aptamer, and small-molecule inhibitor conjugates have also been reported.^62,90–92  In general, the concept of receptor-induced magnetization enhancement is based on reversible binding. In some cases, however, a covalent bond can form between a complex and a protein. For example, derivatives of [Gd(DOTA)]⁻ with chloroalkane targeting moieties designed to interact with HaloTagged fusion proteins covalently bind their target protein (Figure 1.7). The HaloTag used to attach various labels to fusion proteins is a mutant bacterial haloalkane dehalogenase modified to form an ester between a chloroalkane and a specific Asp residue in the hydrophobic tunnel of a target protein. As a result of this conjugation, r₁ of the contrast agent increases from 3.8 to 22.0 mM⁻¹ s⁻¹ at 1.5 T.

For slowly tumbling Gd^III complexes, τ_M, the residence lifetime of an inner-sphere water molecule, also influences relaxivity (see Chapter 2.1). Based on Solomon–Bloembergen–Morgan theory, the optimal value of τ_m is field-dependent, but should be between 10 and 50 ns for a Gd^III complex to achieve maximum inner-sphere relaxivity at clinically relevant fields. Typical polyaminopolycarboxylate-based monohydrated Gd^III complexes have longer than optimal water residence lifetimes (τ_M around 200 ns), although some derivatives have faster water-exchange rates.⁹³  Unlike with Gd^III-based contrast agents, contrast agents for chemical exchange saturation transfer (CEST) require slow water-exchange rates (see Chapter 3.1).⁹⁴ 

The water-exchange rates of lanthanide complexes are influenced by both the steric and electronic properties of ligands.^15,95,96  Monohydrated Gd^III chelates generally have a dissociative water-exchange mechanism in which bound water dissociates before incoming water binds.⁹³  In these complexes, the exchange rate can be increased by increasing the steric crowding around the exchange site, which is known as steric compression.⁹⁷  This crowding can be achieved by inserting an extra methylene group into the polyamine backbone or the sidearm of the ligand, or by substituting the carboxylate groups with bulkier phosphonates.^15,95,96  Backbone substitution and increased negative charge also increase the water-exchange rate, although to a smaller degree.^95,96,98,99  Furthermore, steric compression around the site of water coordination can be different in different isomers. Lanthanide complexes of DOTA and similar ligand derivatives often exist in two interconverting diastereomeric isomers, one of which is square antiprismatic (SAP with an N4/O4 twist angle of approximately 39°) and the other twisted square antiprismatic (TSAP with a twist angle of around −29°).^100–102  In lanthanide complexes of DOTA-type ligands, the tetraazacyclododecane ring adopts a square conformation [3333]. All four ethylene groups have gauche conformations. Depending on the sign of the N–C–C–N torsion angle in the five-membered chelate rings, the conformation of each macrocyclic ethylene group is left-handed (λ, negative N–C–C–N torsion angle) or right-handed (δ, positive N–C–C–N torsion angle). The entire macrocyclic ring can have either a (λλλλ) or a (δδδδ) conformation in the complex. The helicity of the ligand pendant arms in the complex can be clockwise (Δ, positive N–C–C–O torsion angle) or counterclockwise (Λ, negative N–C–C–O torsion angle). Thus, two enantiomeric pairs of diastereoisomers exist, of which the Δ(λλλλ) and Λ(δδδδ) enantiomeric pairs adopt the SAP geometry, and the Λ(λλλλ) and Δ(δδδδ) enantiomeric pairs adopt the TSAP geometry.^103,104  The basal N4 and capped O4 squares are closer to each other in the SAP isomer than in the TSAP one, making the former more compact.

The inner-sphere water molecule occupies a capping position above the O4 square and experiences less steric compression in the more compact structure of the SAP isomer. Consequently, the bound-water residence lifetime is nearly two orders of magnitude longer in the SAP isomer than in the TSAP isomer.^105,106  Normally, these isomers interconvert by arm rotation (Λ↔Δ) and ring inversion [(λλλλ)↔(δδδδ)], forming an equilibrium mixture.¹⁰²  The SAP/TSAP ratio depends on the size of the Ln^III ion, the position of the steric bulk on the ligand, and other factors, including solvent and temperature.^{101,107–111}  For sterically non-demanding ligand systems, larger (lighter) lanthanide ions prefer the TSAP geometry, and the SAP isomer is preferred by the smaller (heavier) lanthanides. Towards the end of the lanthanide series starting from Er^III, the TSAP geometry without an inner-sphere molecule of water becomes the dominant structure. Alpha substitution of the acetate sidearms increases the ratio of the TSAP isomer.^109,110,112  Interestingly, the isomer ratio for the rigid DOTA derivative (SSSS)-(SSSS)-M4DOTMA (Figure 1.8) shows the opposite trend than that observed for [Ln(DOTA)]⁻ complexes: the lighter lanthanides (from Ce^III to Sm^III) almost exclusively exist in the SAP conformation, and the heavier ions prefer the TSAP geometry (Figure 1.9). The preference of SAP geometry by the early lanthanides in this case is explained by a repulsion that arises between the methyl groups present on both the sidearms and the cyclen backbone, leading to the destabilization of the more compact SAP geometry.¹¹¹  This example demonstrates the feasibility of stabilizing one isomer via the incorporation of steric bulk.^113,114 

This strategic placement of bulk can be achieved by chiral substitution on the alpha carbon of the acetate side arms and on the carbon atoms of the macrocyclic backbone. An RRRR configuration of the acetate alpha carbons bearing a methyl substituent generates the Λ pendant arm helicity, and an SSSS configuration leads to the Δ helicity. The conformation of the macrocyclic ring can also be locked with a single nitrobenzyl group.¹¹⁵  The Eu^III and Gd^III complexes of the (S)-(SSSS) and (S)-(RRRR) diastereomers of 2-(p-nitrobenzyl)-DOTMA exclusively adopt the TSAP (S)-(SSSS) and SAP (S)-(RRRR) geometries, respectively (Figure 1.10). Notably, the τ_M value for the TSAP (S)-(SSSS) isomer is one order of magnitude lower than that of the SAP (S)-(RRRR) isomer.

Because the interaction between the Ln^III ion and the bound water is predominantly ionic in character, the water-exchange rate is influenced by the electron deficiency around the lanthanide ion. Indeed, negatively charged carboxylates tend to increase water-exchange rates compared with neutral coordinating groups. It has been demonstrated that in Gd^III complexes of derivatives of DOTA in which a single coordinating sidearm was varied, the water-exchange rate decreases in the following order: phosphonate, phenolate>substituted acetate>acetate>hydroxamate>sulfonamide>amide, pyridyl, imidazole.^116,117 

Consecutive substitution of amides in place of carboxylates increase τ_M roughly three- to four-fold with each substitution. The exchange rate of the inner-sphere water in lanthanide DOTAM complexes is about three orders of magnitude slower than the corresponding complexes of DOTA.⁹⁴  The exchange rate in these tetraamide chelates can further be fine-tuned by adjusting the charge and polarity of the sidearms.^94,95  The values of τ_M in lanthanide tetraamides are strongly dependent on the size of the lanthanide ion, with a maximum value for Eu^III.¹¹⁸  As a result of the slow water-exchange kinetics and the favorable magnetic properties of Eu^III (negligible paramagnetic relaxation enhancement), [Eu(DOTA)]⁻ tetraamide complexes are effective chemical exchange saturation transfer agents (see Chapter 3.1).⁹⁴ 

Beyond the ability of a ligand to influence inner-sphere contributions to relaxivity, ligands also influence the contributions of second-sphere water molecules. If a ligand contains functional groups such as phosphonates and amides that can form strong hydrogen bonds, the contribution from the second-sphere water to relaxivity can be significant (see Chapter 2.1.3). Second-sphere water molecules are those that form hydrogen-bonds with the ligand on the hydrophilic side of the complex. Molecular dynamics simulations suggest that the residence lifetime of second-sphere water for Gd^III complexes of polyaminopolycarboxylate ligands is between 20 and 25 ps.¹¹⁹  For these complexes, second-sphere contributions to relaxivity are negligible because these residence lifetime values are too short for Gd^III to efficiently relax the protons of the second-sphere water.¹¹⁹  Polar groups such as phosphonates can stabilize the second coordination sphere, which increases the second-sphere relaxivity. For instance, [Gd(DOTP)]⁵⁻ (DOTP is 1,4,7,10-tetraazacyclododecane-1,4,7,10-tetramethylenephosphonic acid), which does not have inner-sphere water, has a slightly higher relaxivity than [Gd(DOTA)]⁻ because the negatively charged phosphonate groups in [Gd(DOTP)]⁵⁻ lead to a structured second hydration shell in which the hydrogen-bonded water molecules have relatively long residence lifetimes (56 ps). This long residence lifetime combined with the anisotropy of the second-sphere hydration shell (it is on the hydrophilic side of the complex) can adequately account for the relaxivity of [Gd(DOTP)]⁵⁻ at low magnetic fields.¹¹⁹ 

A related Gd^III complex with extended, non-coordinating phosphonate groups attached to DOTA via amide bonds displays pH-sensitive relaxivity. The value of τ_M in this tetraamide complex is extremely long, yet its relaxivity increases from 3.8 to 9.8 mM⁻¹ s⁻¹ as the pH goes from 8 to 6.^120,121  This pH range corresponds to the protonation of the phosphonate groups. The pH-sensitive relaxivity of this complex is likely due to the modulation of the inner- and second-sphere water exchange processes by protonation of the phosphonates. Second-sphere contributions to relaxivity cannot be controlled as predictably as those arising from inner-sphere contributions, largely because the number and residence lifetime of water composing the second coordination sphere is not well defined. Nevertheless, it was demonstrated that high relaxivity agents can rationally be designed and constructed by simultaneous optimization of τ_M, τ_R, and the second-sphere organization (Figure 1.11).^88,116,117 

The design platform in Figure 1.11 was based on the well-established DOTA framework containing an albumin-binding moiety linked to a cyclen backbone either via an α-substituted acetate or acetamide sidearm. Analysis of the parameters that affect inner-sphere relaxivity revealed that the two amide sidearms slow the water-exchange rate, and that this effect can be overcome by the incorporation of a phosphonate or phenolate sidearm. Another important observation is that the relaxivity of complexes with sidearms that contain two carboxylates per arm have substantial second-sphere contributions.

The development of new ligand systems for applications in MRI has remained an active area of research with major focuses on high relaxivity, targeted, and responsive agents. The main goal of targeted contrast agents is to deliver a large payload of paramagnetic centers to a target site.^{50,87,88,91,122–130}  Targeting is often combined with an amplification strategy in an attempt to overcome the low sensitivity of MRI.⁵⁶  Despite progress, the disparity between the detection limit of the best T₁-shortening agents (10–100 µM) and the concentration of target biomolecules, such as cell surface receptors (often in the nM range), remains a challenge.^125,131,132 

Responsive contrast agents are designed to have relaxivity or CEST effect that is dependent on a triggering molecule or event, such as the presence of a specific biomarker, a physiological process, a metabolite, an ion, pH, redox potential, temperature, or enzyme activity. The design of such probes is based on the modulation of one or more parameters that influence either relaxivity or the CEST effect (including q, τ_M, and τ_R). Responsive probes based on contrast agents for CEST MRI offer some advantages over Gd^III complexes because Gd^III-based T₁-shortening agents cannot be completely silent, even when they are turned off. In addition, changes in relaxivity in response to environmental stimuli are often only modest. With contrast agents for CEST MRI, saturation transfer can be selectively turned on and off by varying the presaturation frequency facilitating complete turning off of agents and the imaging of several agents in one experiment (see Chapter 3.1).¹³³ 

The stability of metal-based contrast agents for MRI is an important parameter for in vivo applications. This is particularly important in view of the rare occurrences of nephrogenic systemic fibrosis (NSF), a devastating disease linked to in vivo dissociation of Gd^III from its ligand in some patients with chronic kidney disease.^134,135  Moreover, recent research suggests that Gd^III deposition can occur in the brain and bones of patients with normal kidney function who received multiple injections of Gd^III contrast agents, although the medical consequences of these depositions are not yet clear.^2,136–142  It should be emphasized that no cases of NSF have recently been reported with macrocyclic agents.^4,143  The recent papers describing Gd^III deposition also strongly imply that open chain agents have lower stability in vivo than the macrocyclic ones.^2,144,145  These reports are in agreement with in vivo Gd^III retention and in vitro dissociation kinetic data.¹⁴⁶  The amounts of Gd^III released from clinically approved agents in human serum at 37 °C over several days decreases in the following order: DTPA bis(amide)s>DTPA and sidearm substituted DTPA>backbone substituted DTPA≫macrocyclic agents.¹⁴³  Thus, it appears that the in vivo stability of lanthanide-based contrast agents is determined both by their thermodynamic stability and their kinetic inertness. From a ligand design point of view, the kinetic inertness can in general be improved by lowering the basicity and increasing the rigidity of the ligand.^147,148 

This section focuses on general synthetic methods and selected reactions that are potentially useful to a wide range of readers. For more details, please refer to the following reviews on the synthesis of Gd^III-based contrast agents for MRI.^{16,61,62,149–164} 

[Gd(DTPA)]²⁻ was the first clinically approved contrast agent for MRI. Six approved agents are based on the same open chain ligand. Diethylenetriamine is a commercially available, inexpensive starting material for the synthesis of various N-substituted derivatives of DTPA. Pentaalkylation of diethylenetriamine and backbone-substitution of diethylenetriamine derivatives is fairly straightforward (Scheme 1.1). The parent ligand DTPA was originally prepared by reacting diethylene triamine with formaldehyde and sodium cyanide in the presence of NaOH.¹⁶⁵  Under these basic conditions, the cyanomethyl intermediate instantaneously hydrolyses to give the final product and ammonia.

On a small scale, it is more convenient to introduce the acetate sidearms on a polyamine backbone with chloroacetate in basic aqueous solutions or with haloacetic acid esters (tert-butyl bromoacetate) in an organic solvent, such as acetonitrile or dimethylformamide (DMF), in the presence of an inorganic (for example, K₂CO₃) or organic (for example, diisopropylethylamine) base.^166–168  The use of esters that are reactive towards nucleophilic acyl substitution, such as methyl or ethyl, can result in a competitive side reaction involving the formation of six-membered lactams between the acetate sidearm and backbone NH present in partially alkylated intermediates.^169–171  For example, the desired pentamethyl ester of DTPA could not be isolated when methyl bromoacetate was used as the alkylating agent. To suppress lactam formation, it is imperative that either a bulky ester, such as tert-butyl, or reactive alkylating functionalities, such as triflate, be used.¹⁶⁹  A lactam ring can also form in the intramolecular condensation between a free carboxylate and NH, and syntheses should be designed to avoid such intermediates.¹⁷²  The ethyl ester of DTPA can be obtained by the acid-catalyzed esterification of the free acid. Ammonolysis of the pentaethylester affords DTPA pentaamide in good yields.¹⁷³ 

Selective functionalization strategies of diethylenetriamine exploit the reactivity difference between primary and secondary amines. The more reactive primary amino groups can be protected in various forms, such as phthalimido,^174–177  tert-butyl carbamate,^{172,178–181}  trifluoroacetamide,¹⁸²  or p-toluenesulfonamide,¹⁸³  offering convenient access to selectively functionalized derivatives of DTPA. Starting from these compounds, derivatives in which the secondary N-atom is selectively protected can be obtained using protecting groups such as benzyl, tert-butoxycarbonyl or benzyloxycarbonyl, which can be removed with different conditions (Figure 1.12).^81,184–189  4-Benzyl diethylenetriamine is a valuable intermediate in the synthesis of various macrocyclic ligands, including selectively protected 1,4,7,10-tetraazacyclododecane derivatives.^183,190 

A number of DTPA analogs have been reported to have at least one pendant arm other than acetate. DTPA-bis(anhydride) (Figure 1.13) is a versatile intermediate for the synthesis of bis- and mono-amide derivatives as well as for the conjugation of DTPA to peptides and other biomolecules. Commercially available or easily prepared by the action of acetic anhydride or isobutyl chloroformate on DTPA in the presence of pyridine or triethylamine,^191–195  this compound reacts readily with two equivalents of a primary or secondary amine in DMF, dimethylsulfoxide, or water to afford the bisamides.^196–201 

DTPA bis(amides) form neutral complexes with Gd^III, thereby affording lower osmolality solutions than [Gd(DTPA)]²⁻. There are two Gd^III–containing DTPA bisamide chelates among the approved contrast agents (Omniscan and OptiMARK). Unfortunately, these agents are strongly implicated in the development of the disease NSF. It is also possible to obtain DTPA monoamides when excess DTPA bisanhydride is used, but the product of such reactions is usually a mixture of the monoamides and bisamides, which can be challenging to separate.^99,202  The monomethyl and monopropyl amide was also prepared through the mixed anhydride of DTPA formed with isobutyl chloroformate followed by ion-exchange purification.^99,203  The pentamethyamide was prepared from the N-hydroxysuccinimidyl (NHS) ester of DTPA, prepared in situ by reacting DTPA bisanhydride with hydroxysuccinimide, dicyclohexylcarbodiimide, and 4-dimethylaminopyridine.⁹⁹ 

Methylenephosphonate and phosphinate derivatives of diethylenetriamine are usually synthesized in a Mannich-related reaction (the Kabachnik–Fields reaction) with formaldehyde and an appropriate phosphorus compound (H₃PO₃, phosphite, or phosphinate esters) (Scheme 1.2).^204,205  Diethylenetriamine penta(methylenephosphonic acid) has been prepared by reacting the triamine with formaldehyde and orthophosphorous acid in strongly acidic solution.²⁰⁶ 

For special purposes the diethylenetriamine backbone itself can be built from suitable precursors. The tert-butyl or benzyl esters of N-(2-bromoethyl)iminodiacetic acid are useful reagents to introduce two or one iminodiacetic acid units by alkylating a mono- or di-amine starting material, most commonly a suitably protected ethylenediamine or amino acid derivative.^{80,99,169,207–218}  For example, DTPA derivatives with orthogonally protected carboxylate groups have been synthesized by alkylating a suitably functionalized ethylenediamine triacetic acid ester with a derivative of N-bromoethyl-iminodiacetic acid (Scheme 1.3).

DTPA derivatives bearing substituents on a carbon atom of the triamine backbone are usually synthesized through amide derivatives made from amino acids. Methyl esters of amino acids condense with ethylenediamine or 1,2-propylenediamine to form an amide when the diamine is used as a solvent. The large excess of the diamine speeds up the reaction and suppresses bisamide formation. The amide is reduced to the amine, usually with BH₃–tetrahydrofuran.^{98,167,219–221}  However, when the use of excess diamine is not feasible (for example, due to limited availability) or when the amide formation with the diamine is too slow (e.g. for steric reasons), activated amino acid esters or a peptide coupling agent can be used to form an amide bond between a protected amino acid and a selectively monoprotected or functionalized diamine.^171,222  This latter approach was used for the synthesis of bifunctional DTPA derivatives with a cyclohexyl rigidified backbone.^170,223  This methodology is outlined in Scheme 1.4 showing the synthesis of one of the four stereoisomers of 2-(p-nitrobenzyl)-trans-cyclohexyl–DTPA starting from N-tert-butoxycarbonyl l-p-nitrophenylalanine. Alternatively, the amino acid methyl ester can be alkylated followed by ammonolysis and reduction of the amide groups.²²⁴ 

The macrocyclic chelator DOTA is superior to the linear DTPA for biomedical applications in every respect except for complex formation kinetics with lanthanide ions. The parent cyclic tetramine (1,4,7,10-tetraazacyclododecane, cyclen) was first prepared by the reduction of the cyclic diamide 1,4-ditosyl-6,11-dioxo-1,4,7,10-tetraazacyclododecane obtained by the condensation of ethylenediamine and the tosyl-protected ethylenediamine N,N′-diacetic acid chloride.²²⁵  Shortly thereafter, it was also obtained by the general cyclization method of Richman and Atkins.^226,227  The original procedure reported by Richman and Atkins in 1974 employed the disodium salt of an N-tosylated polyamine and an appropriately protected bis-alkylating agent containing two leaving groups, such as tosyl, bromide, or mesyl. The reaction is usually performed in DMF. Surprisingly, high dilution conditions are not required because the restriction of the rotational freedom of the open-chain reactants by the bulky p-toluenesulfonyl groups favors macrocyclization over polymerization. Cations such as Na⁺ or K⁺ present in the reaction mixture do not induce a template effect.^226,228  The tosyl group is usually cleaved with hot sulfuric acid.²²⁹  The scope of this methodology is somewhat limited by the harsh conditions necessary to remove the tosyl protecting groups. However, the use of other protective groups has been reported, including 2-nitrophenylsulfonyl, β-trimethylsilylethanesulfonyl, and diethoxyphosphoryl, which are easier to remove than tosyl.^230–233  It was shown later that it is not necessary to use the preformed salts because the tosylamide can be deprotonated in situ with K₂CO₃ or Cs₂CO₃.²²⁹  Other variants include the use of LiOH as a base and running the reaction in a two-phase system with phase-transfer catalysts.^234,235  Another modification uses tritosylated diethanolamine and tosylamide mono sodium salt as starting materials and microwave irradiation in DMF. Interestingly, under microwave irradiation, macrocyclization is slightly favored over the formation of ditosyl piperazine, affording tetratosyl cyclen in about a 52% yield.²³⁶  However, the Richman–Atkins procedure is not amenable for scale-up, and consequently, several specific synthetic approaches were developed that are more convenient for industrial-scale production.^237–241 

Cyclen can be easily tetraalkylated with a variety of alkylating agents (Scheme 1.5).^{105,112,242–246}  By far, DOTA is the most significant cyclen derivative. It is usually prepared by reacting cyclen with four equivalents of bromo- or, more preferably, chloro-acetic acid in basic aqueous solution.^246–248  The pH of the reaction mixture is maintained above 10 with the addition of NaOH. The [Gd(DOTA)]⁻ complex (gadoteric acid, sold as the meglumine salt under the trade names Dotarem, Artirem, or Dotagita) is possibly the safest MR agent in clinical practice today.²⁴⁹  Tetraamide derivatives of DOTA have received a considerable amount of attention because their lanthanide complexes have markedly different exchange and kinetic properties from the corresponding DOTA chelates. Certain lanthanide complexes of DOTA tetraamides have found application in MRI as CEST agents (see Chapter 3).⁹⁴  DOTA tetraamides are usually synthesized by alkylating cyclen with bromo- or chloroacetyl amides in a dipolar aprotic solvent, most commonly acetonitrile or DMF in the presence of a base such as K₂CO₃ or i-Pr₂NEt. Alternatively, tetraamide derivatives of DOTA can be prepared starting from DOTA and activating the carboxylates with a peptide coupling agent (such as benzotriazole-1-yl-oxy-tris(dimethylamino)-phosphonium hexafluorophosphate or 2-(1H-benzotriazole-1-yl)-1,1,3,3-tetramethyluronium hexafluorophosphate followed by treatment with an amine. This latter approach can be useful when the haloacetyl amides are not available, but the yields can be less than satisfactory and the isolation and purification of the product can be difficult, especially when the product tetraamide is water soluble.⁹⁴ 

A number of tetrasubstituted derivatives of cyclen with methylenephosphonate and phosphinate sidearms have been reported.^{242,250–262}  The most important is DOTP, whose Tm^III complex is a popular in vivo shift reagent for MRI.^263,264  DOTP is prepared by reacting cyclen with excess phosphoric acid and formaldehyde in a strongly acid solution.²⁴² 

Selective functionalization of cyclen is particularly important because mono-, bi-, and tri-functionalized cyclen-based intermediates are frequently used in the synthesis of responsive and bifunctional derivatives. There are many reported synthetic routes to selectively N-functionalize cyclen. These routes have been described in several excellent reviews,^16,151–153  and this section focuses on only the most versatile methodologies (Scheme 1.6).

Interestingly, cyclen displays some preference for selective monoalkylation with certain alkylating agents in nonpolar, aprotic solvents, such as chloroform, in the absence of base. Yields in the range of 70 to 80% have been reported for some alkylating agents; however, purification using flash chromatography was necessary to remove over-alkylated derivatives.²⁶⁵  In the presence of weak bases, such as Et₃N, NaHCO₃, or NaOAc, selective 1,4-bis- and 1,4,7-tris-alkylations were observed with various alkylating agents, such as tert-butyl bromoacetate, benzyl bromide, or chloroacetamides.^266,267  The products are often isolated as monohydrobromide or chloride salts from which free bases can be liberated. The regioselectivity in these reactions arises from the formation of monoprotonated products that lower the nucleophilicity of unalkylated N-atoms.

Based on this approach, several procedures have been published for the synthesis of DO3A-tris(t-Bu ester), which is one of the most useful selectively functionalized derivatives of cyclen.²⁶⁸  For small-scale (<2 g) syntheses, cyclen can be reacted with three equivalents of tert-butyl bromoacetate in acetonitrile in the presence of sodium bicarbonate or Et₃N. The fully deprotonated DO3A-t-Bu ester is obtained after column chromatography.^{267,269–272}  On a larger scale (<100 g) DO3A-t-Bu can be prepared without the need for chromatographic purification by alkylating cyclen with three equivalents of tert-butyl bromoacetate using NaOAc as a base in dimethylacetamide.^273–275  The product is crystallized from the reaction mixture as the monohydrobromide salt. It can be deprotonated with aqueous potassium hydroxide and extracted into hexanes or ether. Removal of the solvent yields the free base as an oil that solidifies upon standing.²⁶⁸  Other esters of DO3A have also been prepared this way.^276,277 

The synthesis of partially functionalized cyclen derivatives by direct regioselective alkylation is extremely attractive because it can shorten the synthesis of cyclen-based ligands with mixed sidearms. However, regioselectivity is never 100%, and therefore the products usually need to be purified by chromatography. In addition, direct regioselective functionalization of cyclen is somewhat limited in scope because it works only with a few alkylating agents. With a few exceptions, 1,7-disubstituted derivatives usually are not accessible by this approach. These limitations prompted the development of selective protection-functionalization-deprotection strategies that provide access to cyclen derivatives that cannot be synthesized by direct functionalization. The topic has been reviewed before and, therefore, only two conceptually different approaches are described here. Cyclen can easily be converted with glyoxal into a tetracyclic bisaminal (perhydro-3,6,9,12-tetraazacyclopenteno[1,3-f,g]acenaphthylene) that reacts with various alkylating agents to form mono- or 1,7-bis-quaternary salts, depending on the reaction conditions. These can be hydrolyzed with aqueous sodium hydroxide or deprotected with hydrazine monohydrate, hydroxylamine, ethylenedi-amine, or o-phenylenediamine to produce mono- or 1,7-bisalkylated derivatives in excellent yields.^278–281  Mono- and 1,7-bisbenzyl cyclen can be obtained this way, and these are potentially useful intermediates (Scheme 1.7).

A versatile, selective functionalization of cyclen is based on the adaptation of orthogonal protection strategies used in peptide synthesis and involves the protection of one, two, or three macrocyclic N-atoms in the form of selectively cleavable carbamates. Since the introduction of benzyloxycarbonyl,²⁸²  carbamate-type protecting groups have been successfully used in peptide synthesis. Benzyloxycarbonyl can be cleaved using catalytic hydrogenolysis and forms a useful orthogonal pair with the acid-labile tert-butoxycarbonyl and tert-butyl ester functionalities. Depending on the reagents and reaction conditions, cyclen can be selectively protected on one, two (1,7-bis), or three N-atoms, as illustrated in Scheme 1.8.^283–289  The remaining N atoms can be functionalized or protected. 1,7-Bis-DO2A-t-Bu ester, a convenient starting material for mixed sidearm ligands, is usually prepared from 1,7-bis(benzyloxycarbonyl) cyclen.^285,290 

Selective 1,4-protection of cyclen is more challenging than 1,7-functionalization. A general synthetic route to 1,4-substituted cyclen proceeds through cyclenoxamide, accessible from cyclen in excellent yield, as shown in Scheme 1.9.²⁹¹ 

Backbone-substituted 1,4,7,10-tetraazacyclododecanes constitute an important class of cyclen derivatives. Bifunctional cyclen-based ligands, such as the widely used p-isothiocyanatobenzyl DOTA, have a reactive functionality attached to the macrocyclic backbone to avoid interference with the metal-binding face of the ligand. Backbone modification is also used to rigidify the resulting complex.

The Richman–Atkins cyclization remains one of the most versatile routes to polyaza macrocycles and can easily be adapted to synthesize backbone-substituted cyclen (Scheme 1.10). Good yields of the tetratosylated benzyl and nitrobenzyl cyclen were achieved in bimolecular [3+1] or intramolecular [4+0] cyclization using mesyl, tosyl, or triflate as leaving groups and Cs₂CO₃ as a base.^292–295  The nitrobenzyl derivative was detosylated with hot sulfuric acid. However, H₂SO₄ treatment of the benzyl derivative led to the formation of sulfonated side products, which were circumvented by deprotection with lithium aluminum hydride. Benzyl-functionalized cyclen was nitrated with HNO₃/H₂SO₄ to produce nitrobenzyl cyclen in about 60% yield.²⁹⁵ 

Backbone-functionalized cyclen can be synthesized via cyclic amide derivatives (Scheme 1.11). p-Nitrobenzylcyclen was obtained by alkylating p-nitrobenzyl ethylenediamine with ethylenediamine bis(bromoacetamide) to produce the cyclic diamide.²⁹⁶  Alternatively, macrocyclic amides were synthesized by acylating p-nitrobenzyl ethylenediamine or N-(2-aminoethyl)-p-nitrophenylalaninamide with a BOC-protected disuccinimidyl ester in dioxane to produce the cyclic products in about 40% yield (Scheme 1.12). These cyclizations lack the beneficial effect of the tosyl that favors macrocyclization, and therefore, they require the use of high dilution. The lower yields compared to the Richman–Atkins protocol are due to the formation of polymeric side products. The macrocyclic amide derivatives were reduced to amines with BH₃–tetrahydrofuran or RED-Al.³⁰²  Both the dialkylation and diacylation methodologies have been extended for the syntheses of derivatives of cyclen that have a cyclohexane²⁹⁸  or piperidine ring²⁹⁹  fused into the tetraazacyclododecane macrocycle.

An interesting and unusual route to cyclen and its backbone-substituted derivatives involve the cyclotetramerization of benzyl aziridines (Scheme 1.13). N-Benzylaziridine leads to good yields of N-tetrabenzyl cyclen when heated in ethanol in the presence of a catalytic amount of p-toluenesulfonic acid.³⁰⁰  It was shown that C-substituted benzylaziridines as well as a few other N-substituted aziridine derivatives tetramerize to produce tetraazacyclododecane derivatives under the right conditions. Yields are usually low and the products require extensive purification, but the reaction offers access to backbone substituted cyclen derivatives that would be difficult to obtain by other routes. The ring opening can be brought about by various catalysts, including p-toluenesulfonic acid, anodic oxidation, BF₃·Et₂O, Cu^II salts, and trialkyl aluminum compounds.^301–304  Chiral N-benzylaziridines, bearing a substituent on the carbon atom in position 2, undergo ring opening at the primary carbon. The resulting tetramer retains the configuration of the monomer at the secondary carbon.^305–307 

Finally, another route to obtain backbone-substituted cyclen involves Fe^III-templated condensation of glyoxals with the Fe^III complex of triethylenetetramine followed by reduction of the unsaturated intermediate with sodium borohydride (Scheme 1.14).^308,309 

The two heptadentate ligands PCTA and AAZTA form complexes with lanthanide ions that contain two inner-sphere molecules of water. These complexes have more favorable formation and water-exchange kinetics than the corresponding DOTA complexes and maintain sufficiently high thermodynamic stability and kinetic inertness for in vivo applications. PCTA contains a pyridine ring fused to a cyclen ring and can be considered to be a rigidified DO3A analog.³⁰  The parent macrocycle pyclen as well as backbone-substituted derivatives can be synthesized using the Richman–Atkins methodology (Scheme 1.15).^310–313 

AAZTA and its derivatives can be considered as crossover ligands between the open-chain ligands EDTA and DTPA and macrocyclic ligands such as DOTA or NOTA (1,4,7-triazacyclononane-N,N′,N″-triacetic acid.³¹⁴  Although AAZTA has lower denticity than DTPA or DOTA, the stability of the Ln^III complexes of AAZTA is comparable to that of [Ln(DTPA)]²⁻ complexes.¹⁶  The seven-membered cyclic triamine 6-amino-6-methylperhydro-1,4-diazepine can be prepared using a nitro-Mannich reaction³¹⁵  starting from N,N′-dibenzyl ethylenediamine, nitromethane, and formaldehyde (Scheme 1.16).^314,316 

DTPA, DOTA, and derivatives of those ligands can be purified by standard techniques used in organic chemistry; however, these compounds are quite hydrophilic, which can make purification challenging. Crystallization from aqueous solutions is a convenient and efficient way of purification of large quantities of ligands. Chelators such as DTPA, DOTA, or DOTP can successfully be crystallized as free acids.^242,247,317  Crystallization, however, does not always result in salt-free preparations, in which case, unwanted cations and anions can be removed by ion-exchange chromatography. The purification of DOTA is an example of how the combination of crystallization and ion-exchange chromatography can be used to obtain a high-quality product.³¹⁸  DOTA is usually synthesized by alkylating cyclen with chloroacetic acid in the presence of NaOH and is isolated by acidifying the reaction mixture below pH 3.²⁴⁷  The product that precipitates at low pH contains varying amounts of HCl and has an approximate composition of DOTA·2HCl when isolated at pH 0.5 or DOTA·HCl at around pH 3. Below pH 3, the ligand is positively charged, and when loaded onto a strongly acidic cation exchange resin, protonated ligand binds to the resin, enabling the anions to be removed with repeated washings. The ligand can be eluted by treating the ion exchange column with a solution of ammonium hydroxide because DOTA is negatively charged above pH 4. To remove undesirable cations, excess ammonia is removed and the product is loaded on an anion exchange column. The cations are removed with washing and the product is recovered by treating the resin with a dilute solution of formic acid.³¹⁸ 

A general anion-exchange chromatographic method has been reported for the purification of large amounts (up to 10 g) of p-nitrobenzyl-backbone-substituted DTPA ligands.³¹⁹  The crude products obtained by the acid hydrolysis of the penta-tert-butyl esters were loaded on a cation-exchange resin, eluted with ammonium hydroxide, and recovered by removing the water and ammonia. The products were loaded onto the chloroacetate form of AG1 anion-exchange resin, which is prepared by treating the hydroxide form of the resin with a solution of ClCH₂CO₂H (1 M). The column was washed with water to remove impurities that were not bound to the resin. Finally, the products were recovered by washing the column with a linear gradient of aqueous CICH₂CO₂H (from 0.0 to 1.0 M). Chloroacetic acid was chosen because it is a stronger acid than formic acid, which failed to elute DTPA from the resin. CICH₂CO₂H was removed from the desired products by extracting the aqueous solutions with ether.

On a smaller scale, gel filtration chromatography³²⁰  with Sephadex G-10 medium can be conveniently used for desalting. Analytically pure samples of most ligands can be obtained by preparative high-performance liquid chromatography (HPLC) on a C18 reversed-phase column using a linear gradient of water and acetonitrile containing trifluoroacetic acid, HCl, or some other buffer (0.1 to 0.3%).³²¹  An integrated HPLC–mass spectrometry platform composed of an HPLC system, detectors, columns, and mobile phases has been developed for the purification of DOTA-based targeted diagnostic and therapeutic agents.³²² 

The identity and purity of polyaminopolycarboxylate ligands are established by standard analytical techniques. Reverse-phase HPLC and ion-exchange chromatography are the preferred methods for assays because these methods are more sensitive than spectrophotometric and titrimetric techniques when a quality detector is used. Relatively simple polyaminopolycarboxylates, such as EDTA, DTPA, or DOTA, lack an easily detectable chromophore, and although HPLC methods based on the spectroscopic detection of the carboxylates (195–220 nm) have been developed,³¹⁸  the ligands are often complexed with Cu^II or Fe^III in a sample pretreatment step for easier detection in the UV–visible region of the electromagnetic spectrum.³²³  Derivatives with an aromatic chromophore, such as bifunctional chelators with a benzyl linker, can be detected around 250–280 nm.²²³ 

Other detection methods not limited by the lack of chromophores, such as evaporative light scattering, charged aerosol, or refractive index detection, have also been used in the analysis of derivatives of DTPA and DOTA. In particular, the extremely sensitive (ng to pg) evaporative light scattering detection³²⁴  works well in the characterization of conjugates of DOTA. Backbone substitution on DTPA and DOTA introduces an asymmetric center, and the stereochemistry has a significant influence on the in vivo stability of the resulting complexes.²²³  It is, therefore, important to determine the enantiomeric purity of such bifunctional ligands. Several chiral chromatographic methods have been explored for the chiral discrimination of the S and R enantiomers of nitrobenzyl–DOTA. A reversed-phase cyclodextrin-based column was found to give satisfactory separation of the two enantiomers of nitrobenzyl–DOTA, but chiral derivatization of the p-aminobenzyl derivatives with Marfey's reagent followed by chromatography on a C18 reversed-phase column failed to discriminate the resulting diastereomers.²⁹⁴ 

In a laboratory setting, elemental analysis is a satisfactory and cost-effective way of determining the composition of samples and provides useful information for the chelation reaction because the apparent molecular weight of polyaminopolycarboxylate ligands is usually much higher than the formal molecular weight due to protonation, water content, and inorganic salts.

NMR spectroscopy is a valuable tool for studying conformational, protonation and exchange processes of ligands. ¹H-NMR spectroscopy has been used extensively to study the protonation schemes of various polyaminopolycarboxylate ligands. In general, protonation of a functional group results in deshielding of the adjacent protons and, consequently, leads to a downfield shift of the resonance signal. The magnitude of the protonation shift depends on factors such as the nature of the protonated atom, its position relative to the observed proton, and the mole fraction of the protonated and deprotonated species at a given pH (assuming fast exchange). Therefore, the identity of the protonation sites can be established from the pH-dependence of the chemical shifts of the ¹H or ¹³C nuclei of the ligand (NMR titration).^{325–331  1}H-NMR studies of DTPA demonstrated that the first protonation occurred on the central nitrogen atom but the second protonation led to the formation of a di-protonated species in which only the two terminal nitrogen atoms were protonated.^331–333  The NMR spectra of relatively rigid macrocyclic ligands such as DOTA and its derivatives are often more challenging to interpret owing to slow exchange processes.^334–336  Below pH 1, the ¹H-NMR spectrum of DOTA has two broad acetate and four ethylene signals. At higher pH values (up to around 10), the spectrum consists of two broad peaks at 3.6 and 3.3 ppm assigned to the acetate and the ethylene protons, respectively, indicating faster molecular dynamics. The observed protonation shifts reflect the peculiar protonation scheme of this ligand (Figure 1.14).

At low pH, DOTA exists in its rigid, hexaprotonated form, [H₆DOTA]²⁺, in which all four carboxylates and two nitrogen atoms that are trans to one other are protonated (the fully protonated form would be [H₈DOTA]⁴⁺). Around pH 2, the two acetates attached the two protonated N-atoms deprotonate and, concomitantly, form strong H-bonds with the protonated nitrogens. At around pH 4 to 5, the acetate groups not participating in H-bonding undergo deprotonation. The prevalence of di-protonation at two macrocyclic N-atoms trans to one other in the pH range of 2 to 10 is a characteristic feature of DOTA and other cyclen-based ligands, and it influences the formation and dissociation kinetics of the metal complexes. Two-dimensional NMR techniques, such as homonuclear correlation spectroscopy (COSY), exchange spectroscopy (EXSY), nuclear Overhauser effect spectroscopy (NOESY), and heteronuclear multiple quantum coherence (HMQC) spectroscopy, have given helpful insights into the fluxional behavior of DOTA and related ligands.^306,336  The parent compound DOTA exhibits relatively fast dynamic processes at room temperature, except at low pH. However, substitution on the sidearm alpha carbon or on the macrocyclic backbone decelerates these dynamic processes. For example, variable-temperature NMR studies revealed that p-nitrophenyl substitution on the alpha carbon of one acetate sidearm slows the rotation of the arms of the ligand.¹¹³  Tetramethyl substitution on the macrocyclic backbone gives rise to two slowly exchanging species with elongated geometries where the methyl substituents are positioned either close to or away from the acetate sidearms, as evidenced by HMQC spectroscopy, COSY, and EXSY.³⁰⁶  Furthermore, NaOD or KOD might not be suitable bases to adjust pH when performing NMR titrations with DOTA and related ligands because Na⁺ and K⁺ can form weak complexes with these chelators and have an appreciable effect on the NMR spectra.³³⁶ 

The protonation constants of the ligands and the thermodynamic stability constants of their complexes are usually determined by pH-potentiometry. This technique is described in detail in Section 1.3.

Quyen N. Do, James S. Ratnakar and Zoltán Kovács*

In comparison to ligand synthesis, preparation of the complexes is relatively simple. The use of high-purity ligands with a known composition facilitates the synthesis, purification, and characterization of the resulting complexes. Therefore, purification and complete characterization of the ligand, including molecular weight determination prior to complexation, is highly recommended. The optimal conditions will largely depend on the structure of the ligand: in general, macrocyclic chelators may require harsher conditions for successful complexation than linear ones. To avoid toxicity problems or incorrect physical or chemical measurements, appropriate care must be taken to ensure that the desired complex is fully formed and that no uncomplexed metal or intermediary species remain. The most important factors to be optimized include the solvent, pH, temperature, reaction time, and the metal ion source.

Water is the most commonly used solvent. However, complexation can be performed in organic solvents such as methanol or acetonitrile if there are solubility concerns with either the reactants or products. pH is one of the most important parameters to be considered. On one hand, the ligand needs to be at least partially deprotonated to interact with metal ions. In addition, the formation of DOTA-like complexes proceeds through a protonated intermediate with a base-catalyzed rearrangement as the rate-determining step of complex formation. On the other hand, above pH 6, lanthanide ions start to hydrolyze, forming undesirable hydroxo-bridged polynuclear clusters and nanoparticles, and this hydrolysis slows the formation of the complex. Thus, the optimal pH is around 5 for linear ligands and is somewhat more basic, around 6, for macrocyclic chelators. However, the optimal pH might be different for individual ligands⁵⁷⁷  and speciation diagrams calculated using the pK_a values of the ligand and the log K values of the complex and its protonated species should be consulted to determine the dominant species at a given pH. Because the formation of a complex is accompanied by the release of protons from the ligand, pH is kept in the desirable range by the addition of an inorganic base such as NaOH or by the use of buffers or weak organic bases such as pyridine.^578,579  The progress of complexation can be monitored by the rate of change of pH. The time and temperature necessary to achieve full complexation depends on the nature of the ligand. Flexible open chain ligands such as EDTA or DTPA form lanthanide complexes nearly instantaneously at room temperature, but rigid macrocyclic ligands often require several days at elevated temperature.⁵⁸⁰  In most cases, the metal and ligand are in equimolar amounts, although excess metal can also be employed to ensure complete complexation of the ligand.^581,582  The ligand is rarely used in large excess except for in radiopharmaceutical preparations. Lanthanide salts like chlorides, trifluoromethanesulfonates, and acetates are convenient metal sources, but they generate an additional three equivalents of inorganic salts that might be a limitation for some applications where high osmolality is undesirable (e.g. liposomal encapsulation).⁵⁷⁹  Metal oxides (Gd₂O₃), hydroxides (Gd(OH)₃), or carbonates (Gd₂CO₃) can be used in place of chlorides, and the use of these starting materials will afford salt-free complexes if the ligand does not contain extra acid.^{198,348,482,529,583–588}  These starting materials work best when the ligand is in its protonated form and can be used in an equimolar ratio or in large excess followed by the removal of the undissolved oxide and the dissolved but uncomplexed metal. Complexations with metal oxides are relatively slow, and they are usually performed by refluxing the reaction mixture. The complexation is faster with lanthanide hydroxides, which should be freshly prepared by reacting the chlorides with NaOH followed by thorough washing with water.⁴⁸²  The final chelate preparations often contain a slight excess of ligand (1–5%) to suppress dissociation of the complex.

Uncomplexed lanthanide ions are toxic, and their presence interferes with most physicochemical or biological applications of the complex. In a laboratory setting, a colorimetric test with a metal indicator such as such as xylenol orange or Arsenazo III can be used to check for the presence of uncomplexed metal in solution.^588–593  This can be done visually as a qualitative test. For example, the change of yellow color to pink in a buffered aqueous solution of xylenol orange provide a visual indication of the presence of uncomplexed Gd^III ions in the micromolar range of concentration. The test also can be performed as a quantitative spectrophotometric method to determine the concentration of the free metal.⁵⁸⁹  An analytical HPLC method has also been developed for the detection of uncomplexed Gd^III in samples of ionic chelates (Gd^IIIEDTA, Gd^IIIDTPA, and Gd^IIIDOTA).⁵⁹⁴  It uses a second ligand (cyclohexanediaminetetraacetic acid, CDTA, in a buffered mobile phase) that forms a complex with uncomplexed Gd^III but does not equilibrate or react with the sample. Gd^IIICDTA can be separated from any of the other complexes using a C18 reversed-phase column equipped with a fluorescence detector (280 nm excitation and 310 nm emission wavelengths).

If a solution tests positive for uncomplexed metal then extra ligand can be added until complete complexation is achieved. If this is not a viable option, for example, due to unavailability of the ligand, then the uncomplexed metal must be removed. This removal can be done using several methods. The most common one is to raise the pH of the reaction mixture to about 8–9. The solution can then be centrifuged or filtered through a 0.2 mm syringe filter to remove precipitated hydroxide. Another frequently used method involves treating the complex solution with a chelating resin, Chelex 100 for example, at pH 6.^581,582  Uncomplexed metal ions bind to the available iminodiacetic acid moieties on the resin, and filtration of the resin followed by washing with water results in a solution with only metal complexes.

Other methods that remove inorganic salts include dialysis, ion exchange, and gel filtration, each of them having their own advantages and limitations. Dialysis in a laboratory setting is performed against water with commercially available dialysis tubing or dialysis cassettes with the appropriate molecular-weight cutoff. Dialysis is a membrane separation process: small metal ions pass freely through the membrane, and larger molecules cannot cross. The advantage of this technique is that it does not use organic solvents and the equipment setup is minimal. Commercially available Sephadex desalting columns are well-suited to removing salts and other small molecular weight impurities from lanthanide chelate–macromolecule conjugates.

Purification of lanthanide complexes can be done via a wide range of methods. Crystallization is a rapid and cost effective method of purification. Depending on the ligand structure, charge, and counter ion, lanthanide chelates have been crystallized from water, methanol, ethanol, and water/alcohol or water/acetone mixtures. However, crystallization is not always successful. Alternatively, complexes can be purified using preparative HPLC, which removes inorganic salts and other unwanted impurities.⁵⁹⁵  Occasionally, it might be easier to purify the lanthanide complex than the free ligand by HPLC. In general, it is a good idea to characterize the complex via analytical HPLC prior to purification on a large-scale HPLC system. Symmetry C18 or Delta-Pak C4 columns are two examples of column families that are designed for rapid scaling. Semi-preparative or preparative HPLC purification is most often done using a reversed-phase C18 column. Inorganic salts can be removed with 100% water, followed by a linear gradient of 0–95% acetonitrile to elute the complex. Other solvent systems, such as water/methanol can also be used. The addition of acids, such as or trifluoroacetic acid or hydrochloric acid (0.1%), to solvent systems can improve the eluting peak shape. Ion-exchange resins can also be used to remove salts.

Characterization of a metal complex starts with the analysis of its metal content. For biomedical applications, dosage is often determined based on metal content. Therefore, excess inorganic salts, ligands, and water present in the samples can increase the apparent molecular weight. The Gd^III content is typically measured using inductively coupled plasma atomic emission spectroscopy (ICP-AES) or inductively coupled plasma mass spectrometry (ICP-MS) (see Chapter 2.7 for details).

Lanthanide chelates of DOTA, DTPA, and related ligands exist as mixtures of interconverting coordination isomers. As described earlier, the water-exchange rates of these forms can be dramatically different, and so their ratio in the mixture can influence relaxivity. Therefore, it is fundamentally important to study the structure and interconversion of these coordination isomers. NMR spectroscopy is ideally suited for these studies. Because the dipolar field induced by paramagnetic Ln^III ions (other than the isotropic Gd^III ion) is distance- and direction-dependent, the structures of complexes can be determined from the pseudo-contact shifts of the ¹H-nuclei of ligands.^{27,94,147,337,338}  Yb^III is commonly used for this purpose because it induces large paramagnetic shifts with negligible line broadening, which facilitates the interpretation of NMR data. With lanthanide complexes of DOTA and related ligands, structural differences between SAP and TSAP isomers lead to dramatically different paramagnetic shifts of the axial protons of the macrocyclic ethylene bridge (H4 protons). The axial protons of the TSAP isomer lie farther away from the lanthanide ion and are not as shifted on a ¹H-NMR spectrum. For [Eu(DOTA)]⁻, the axial protons for the SAP isomer usually reside in the 31–38 ppm region, and the TSAP isomer protons are in the region of 10–14 ppm (Figure 1.15).¹⁰¹  For Eu^III–DOTAGly₄-based complex, the TSAP axial protons usually are around 11 ppm, compared to the ∼25 ppm for the SAP isomer. Integration of axial-proton ¹H-NMR peaks gives the isomeric ratio of the two species.

Occasionally, in the case of rigid chelators where interconversion between the two isomers is slow (several hours), it is possible to separate and characterize the different coordination isomers by HPLC. Separation of the SAP and TSAP isomers for [Ln(SSSS-SSSS-M4DOTMA)]⁻ (Figure 1.8) using HPLC has been reported.¹¹¹  The relative ratio of the two isomers was determined by the integration of the HPLC peaks. In this particular case, the interconversion was slow enough to enable NMR studies of the isomers.¹¹¹ 

The exchange processes involving coordination isomers are generally studied using variable-temperature NMR spectroscopy.^339,340  At low temperatures, isomeric exchange slows, enabling the observation of relatively sharp peaks of the exchanging isomers. For example, [Eu(DTPA)]²⁻ gives rise to 18 proton chemical shifts that are observable at −5 °C but only seven at 95 °C. This difference indicates an exchange occurring between two species (Figure 1.16).³⁴¹  Variable-temperature ¹³C- and ¹⁷O-NMR spectroscopy has also been used to investigate the solution-phase structure of lanthanide complexes of DTPA and DTPA-bisamide.^338,342  Variable-temperature NMR measurements enable determination of thermodynamic parameters that characterize exchange processes.^339,342  Although useful, one-dimensional ¹H-NMR data of lanthanide complexes of polyaminopolycarboxylate-based ligands can be complicated, and it can be difficult to resolve individual peaks. Two-dimensional NMR techniques such as homonuclear correlation spectroscopy (COSY) and exchange spectroscopy (EXSY) are often useful in studying dynamic exchange processes and identifying exchanging species. For example, the reorientation of acetate sidearms and ring inversions can be studied with two-dimensional NMR techniques (Figure 1.17).^100,343 

Two-dimensional NMR techniques such as COSY and EXSY can be used to assign specific resonances and are complimentary to the ¹H-NMR experiment.^100,102,113  Two-dimensional NMR experiments offer the advantage of sampling of the entire exchange matrix in one experiment. However, because of the shortened T₁ of paramagnetic complexes, if 1/T₁≫k_ex, the transverse magnetization can be decreased by T₁ relaxation before cross-peak intensity can accumulate. This problem can be at least partially eliminated by optimizing the mixing time.¹⁰²  In variable-temperature, two-dimensional NMR experiments, the optimal mixing time must be determined at each temperature.

The presence of coordination isomers can also be studied using luminescence spectroscopy provided that the exchange is slow on the Ln^III emission time scale. If there is a large enough difference in the excited state lifetimes of the isomers, then the observed excitation peaks can be assigned to different species applying a time delay that is longer than the lifetime of the shorter-lived species (see Chapter 2.4.3). Alternatively, the transition can be monitored at different emission wavelengths. If there is only one species present, then the excitation profile does not change. However, because different complexes might have different emission spectra, the dependence of the excitation profile on the emission wavelength is an indication of the presence of more than one species.³⁴⁴ 

Along with NMR data, X-ray crystal structures of lanthanide complexes are routinely obtained to study structural features in the solid state. Previously reported X-ray crystal structures of [Ln(DOTA)]⁻ complexes³⁴⁵  include those of Eu^III,³⁴⁶  La^III,³⁴⁷  Gd^III,³⁴⁸  Lu^III,³⁴⁹  Ce^III,³⁵⁰  Pr^III,³⁵⁰  Nd^III,³⁵⁰  Dy^III,³⁵⁰  and Tm^III.³⁵⁰  Although solid-state data are useful to understand structural differences between isomers, solution- and solid-state structures are not necessarily the same. In solution, the SAP and TSAP isomers coexist in equilibrium, with one often being dominant. However, in the solid state, complexes often adopt a single coordination geometry, which is not necessarily the dominant species in solution. For instance, ¹H-NMR solution studies show a 50 : 50 mix of TSAP/SAP isomers for [Nd(DOTA)]⁻, even though only SAP crystals can be grown.³⁵⁰  Preferential crystallization of the minor isomer indicates its coexistence with the major one in solution, even when its presence cannot be detected in solution. Complexes of DOTA-based ligands with bulky side arms preferentially crystallize as the TSAP isomers. Reported examples include La(DOTAAM) (DOTAAM=1,4,7,10-tetrakis(2-carbamoylethyl)-1,4,7,10-tetraazacyclododecane),³⁵¹  [Eu(THP)]³⁺ (THP=1,4,7,10-tetrakis(2-hydroxypropyl)-1,4,7,10-tetraazacyclododecane),³⁵²  Gd(DO3MA) (DO3MA=(1R,4R,7R)-α,α′,α″-trimethyl-1,4,7,10-tetraazacyclododecane-1,4,7-triacetic acid),³⁵³  and [Yb(DOTPBz4)]⁻ (DOTPBz4=1,4,7,10-tetraazacyclododecane-1,4,7,10-tetrakis(methylenebenzylphosphinic acid)).³⁵⁴ 

The X-ray crystal structures of several Ln^III complexes of DTPA have also been reported: Ba[Nd^III(DTPA)(H₂O)],³⁵⁵  Na₂[Gd^III(DTPA)(H₂O)],³⁵⁶  (NH₄)₂[Gd^III(DTPA)(H₂O)],³⁵⁷  and (guanidinium)₂[Gd^III(DTPA)(H₂O)].³⁵⁸  These structures display nine-coordinate metal ions bound to three nitrogen atoms and five carboxylate oxygen atoms of the DTPA ligand with the ninth coordination site occupied by a molecule of water. The coordination geometry is a distorted capped square antiprism. The solution structure deduced from the two-dimensional EXSY spectroscopy and lanthanide-induced relaxation enhancement of the ¹³C-nuclei of the Nd^III- and Eu^III-complexes are consistent with the X-ray crystal structures.

Although the solid-state structures of numerous lanthanide complexes of DOTA- and DTPA-like ligands with different side arms have been reported, it is not an easy task to obtain an X-ray quality crystal of such compounds. The easiest way is often through slow evaporation of solvents. Occasionally, crystals are obtained from NMR tube samples that have been left standing on the lab bench for several days. Obtaining an X-ray quality single crystal, however, requires both laboratory skills and luck. Diffraction-quality crystals usually need to be 0.1–0.3 mm in each dimension. Water or aqueous solutions with miscible organic solvents are normally used for crystallization. Occasionally, X-ray quality crystals were obtained from an organic solvent such as methanol.¹⁹⁸  Binary solvent systems, where (1) the two liquids are miscible and (2) the compound is soluble in only one liquid but practically insoluble in the other, can afford good quality crystals. Diffusion methods require only a small amount of material; water and organic solvents such as acetone or ethanol are often good starting points.^348,359  Regardless of the crystallization method, it might take several days or weeks to grow quality crystals.^347,360,361  The crystal structures of several lanthanide complexes can be accessed free of charge from the Cambridge Crystallographic Data Centre (CCDC) (Figure 1.18). NMR and X-ray crystal information are often used supplementary to each other. It is often necessary to consider both crystal structure and NMR data to understand the complexity of the structure of lanthanide chelates.³⁶² 

Gyula Tircsó,* Ferenc Krisztián Kálmán, Zsolt Baranyai, Zoltán Kovács, Ernő Brücher and Imre Tóth

A large number of open-chain, macrocyclic, and hybrid complexes have been reported as diagnostic, therapeutic, and theranostic agents. Lanthanide and transition-metal ions, which are Lewis acids, are used in biological systems in the form of complexes that do not dissociate in the body because the non-complexed (“free”) metal ions tend to be toxic. These include paramagnetic lanthanides (Ln^III) and transition metal ions (including Mn^II, Fe^III, Fe^II, Co^II, and Ni^II) used in T₁- or T₂-shortening contrast agents or chemical exchange saturation transfer agents for MRI, and metals used in radiopharmaceuticals, such as Cu^II, Ga^III, and In^III. Moreover, metal aqua ions readily hydrolyze at physiological pH forming metal hydroxides that can precipitate and are therefore not suitable for in vivo applications. The use of complexes in living systems necessitates the knowledge of their in vitro behavior. This behavior is generally characterized by physicochemical parameters such as stability constants, which in turn require knowledge of the protonation constants of the ligands, formation and dissociation rates of the complexes, and the structures of the complexes. The majority of these data are usually collected in vitro, but they are relevant to in vivo applications largely because biofluids (often referred to as competitive biological media) contain endogenous ligands and metal ions at concentrations that compete with the components of the administered complexes. Thus, knowledge of solution speciation and stability in vitro enables prediction of the in vivo fate of complexes and can yield useful information for the design of contrast agents.

Several reviews, book chapters, and books have been devoted to the physicochemical methods used to assess and refine the thermodynamic (stability) data that characterize the complexes that serve as contrast agents for MRI, including calorimetry, pH-potentiometry, and UV–visible and luminescence spectroscopies.^363–365  In particular, there are reviews and book chapters summarizing the thermodynamic data for an extraordinarily large number of Gd^III complexes formed with linear and macrocyclic ligands.^{156,366–368}  Detailed information on the factors affecting the stability of the complexes, such as the charge of the metal ion, hardness and softness of the metal ion and the ligand, the number of donor atoms, and the chelate and macrocyclic effects, have been discussed in numerous publications.^369–372  Information regarding the programs used to determine stability constants^373,374  and perform model calculations^375–379  is also available. In fact, the speciation diagrams generated from the stability constants provide a useful pictorial overview of the different species present in multi-component systems. This is nearly impossible to do just by looking at the numerical values of the stability constants alone. Thus, instead of discussing these data again, this chapter describes the constants that are used to characterize the strength of metal–ligand interactions and summarizes the selection of a proper equilibrium model for a given system. Additionally, more problematic cases are described that require extra attention owing to the complications associated with slow formation of complexes or the formation of stable complexes. This information is targeted at scientists interested in collecting high quality and reliable equilibrium data. The selected examples in this chapter are based on published data for Gd^III- (in some cases Eu^III-, which is used occasionally because of its luminescent properties), Cu^II-, and Mn^II-based systems.

Acid-dissociation constants (K_a) of the ligands (Lewis bases) are important physicochemical parameters that are frequently determined in the first step of equilibrium studies because they must be known to quantify the competition between metal ions and protons for the ligand, that is, to determine stability constants. Ligands used in biomedical applications tend to possess several basic sites, often referred to as polydentate chelators, and therefore, several protonated species can exist in solutions of these ligands. Each species can be characterized by protonation equilibria and constants corresponding to given equilibria. Stepwise equilibria are defined by the addition of a single proton at a time [eqn (1.6) and (1.7)].

where i=1, 2,…, n; and [H⁺], [H_i−1L], and [H_iL⁺] are the equilibrium concentrations of H⁺, H_i−1L, and H_iL⁺, respectively. Note that protonation constants are the inverse of acid-dissociation constants (K_a).

The equilibrium constant for the formation of H_nLⁿ⁺ from nH⁺ and L (formation of nth protonation species) is known as the overall protonation constant [β_01n, eqn (1.8) and (1.9)];

The total (net) basicity of a ligand, β_01n, is also used for comparative purposes when comparing the complexation properties of structurally similar ligands. For instance, in the case of ethylenediaminetetraacetic acid (EDTA), the value of β₀₁₄ is often used for this purpose, but for DOTA and its derivatives, the product of the first two protonation constants β₀₁₂ is frequently used for the same reason.

The formation equilibria of metal–ligand complexes are characterized by their stability constants, as defined by eqn (1.10) and (1.11):

where [ML], [M], and [L] are the equilibrium concentrations of the complex, the metal ion, and the deprotonated ligand, respectively.

Because most ligands used in contrast agents are multidentate, one or more donor atoms in the complex can be protonated at low pH. The complex might also exhibit acidic character, which in turn might lead to the coordination of hydroxide anions to produce ternary (also called mixed-ligand) complexes of the composition M(H₋₁L) or M(L)(OH). Protonated complexes are symbolized as M(H_iL) and the protonation constants of such complexes are defined as in eqn (1.12) and (1.13):

where i=1, 2,…, n; and [H⁺], [M(H_i−1L)], and [M(H_iL)⁺] are the equilibrium concentrations of H⁺, [M(H_i−1L)], and M(H_iL)⁺, respectively. In practice, all of these constants are derived from the overall or cumulative equilibrium constants (β_PQR) defined by eqn (1.14) and (1.15). These are usually the ones calculated by fitting programs.

Importantly, direct comparison of stability constants of structurally diverse complexes might lead to incorrect conclusions about their stability because these comparisons do not consider the competition of the ligands between protons and metal ions, which might differ considerably from ligand to ligand. To account for the protonation of the ligand occurring simultaneously with complexation, conditional stability constants, K$M(L)c$, were introduced. The term K$M(L)c$ reflects the apparent stability of a complex at a given pH [eqn (1.16)].³⁶³ 

Where:

Therefore:

In complex biological systems, a ligand L might or might not be protonated or form complexes with endogenous metal ions such as Ca^II, Mg^II, Zn^II, and Cu^II. Likewise, complexed metal ions might or might not interact with endogenous ligands present in biological fluids, such as citrate, phosphate, or carbonate (denoted below as A, B, etc.). In addition, metal complexes can be protonated, and such protonated complexes can also form ternary complexes with endogenous ligands. By taking into account all of these possible equilibria, a more general conditional stability constant (K*) can be defined as follows.

where:

and

The stability and protonation constants of complexes, the conditional stability constants (K*), and the concentration of the non-complexed metal ion ([M^z+]) can be calculated using the protonation constants of the ligand. Concentration is often included when reporting studies of complexes because it is often assumed that the toxicity of metal complexes is related to the concentration of “free” metal ions released from complexes, which can be expressed as the pM value.

where [M^z+] is the concentration of non-complexed metal ion. The more stable the complex, the lower the concentration of non-complexed metal ion, and the higher the pM. The pM values for Gd^III complexes are calculated for the special condition as proposed by Raymond and co-workers: [M]_t=10⁻⁶ M, [L]_t=10⁻⁵ M, physiological concentration of [Ca^II], [Zn^II], and [Cu^II], pH=7.4.³⁸⁰  Under these conditions, ligands that do not form complexes with M⁺ ions have a pM of 6.0. Tóth and co-workers suggested that pMn values could be calculated for complexes under slightly different conditions ([Mn]_t=[L]_t=10⁻⁵ M at pH=7.4) because ten-fold excess of ligand is not used in the formulation of contrast agents.³⁸¹  Thus, the pM values calculated using these conditions appear to be much smaller (in the range of 5–10) than those of Gd^III complexes (values calculated for the commercially available contrast agents are >15).

Although it is relatively easy to calculate pM values because software [such as Make Equilibrium Diagrams Using Sophisticated Algorithms (Medusa) or Hyperquad Simulation and Speciation (Hyss)] exists for the calculation of species-distribution curves, there are two issues that must be addressed for the sake of clarity. According to the original definition proposed by Raymond in 1979, the pM value reflects the concentration of the non-complexed, albeit fully solvated, metal ions. The use of pM values can lead, in some instances, to erroneous conclusions, as highlighted by Meyer and co-workers.³⁷⁷  Amphoteric metal ions, such as those belonging to group 13 (Al^III, Ga^III, or In^III), are typical examples of coordination compounds relevant to medical-imaging applications. These ions form hydroxido complexes near pH 7.4 (-ate complexes such as [Ga(OH)₄]⁻ that can exist at low concentrations). In the presence of ligands, high pM values can be computed for such systems, misleadingly suggesting a high metal-binding affinity for a ligand. It should be realized, however, that the competition of hydroxide anions for the metal ion can be so strong that the metal ion can be displaced from the ligand. It is obvious that the inefficiency of the chelator under such circumstances is overlooked if only the concentration of the aqua cation is considered as the only representative species of the non-complexed metal.

Conversely, in the absence of complexation (i.e. no affinity), the lower limiting pM value equals the total metal concentration chosen to perform the speciation calculation. To avoid confusion, the relative affinity criterion, denoted A_L/M, was introduced by Meyer and co-workers.³⁷⁷  Taking into account the fraction of all metallic species remaining unbound to the ligand of interest (L), A_L/M serves as a universal tool for the reliable assessment and comparison of the complexing power of any ligand [eqn (1.28)]. From a practical point of view, computation of relative affinities is performed in a straightforward manner using the Apparent Constant calculator implemented in the general speciation program Hyss2009, which is available free-of-charge from the Protonic Software website.³⁷⁵ 

Meyer and co-workers treated the problem of Ga^III complexation by calculating the speciation diagram for the [Ga(DOTA)]⁻ complex using published stability constants.³⁸²  In spite of the high pGa value of 21.6, the radiolabeling of DOTA and DOTA-type ligands with ⁶⁸Ga is often problematic and results in relatively low radiochemical yields.³⁸³  Moreover, in some cases the radiolabeling of DOTA and DOTA-type ligands was found to proceed with higher radiochemical yields at lower pH, which is rather unusual (coordination complexes are usually less stable under acidic conditions). As seen in Figure 1.19, Ga^III is fully complexed by the ligand DOTA at pH 7.4 at millimolar concentrations [Ga³⁺]_total=1×10⁻⁴ M and [Ligand]_total=1×10⁻³ M). However, [Ga(DOTA)]⁻ does not form near pH 7.4 at the more dilute conditions used in labeling ([Ga³⁺]_total=1×10⁻⁹ M and [Ligand]_total=1×10⁻⁸ M). These data indicate that high dilutions can become problematic for radiolabeling experiments. This problem compounds itself onto the slow complexation rates that are often a bottleneck for complex formation for labeling macrocyclic ligands with short-lived radioisotopes.

The first step in setting up a proper equilibrium model is a detailed investigation of the ligand, including the determination of its protonation constants and, potentially, a full characterization of the processes occurring at each protonation site. A difference between the real and estimated number of protonation steps or deviation from the actual values of the protonation constants of the ligand can significantly influence the calculated stability constants for the corresponding metal complexes. There are numerous practical methods for the determination of protonation constants in the pH range of 1.8–12.2. These methods include calorimetry, pH-potentiometry and other electrochemical methods, various spectrophotometric methods, and multinuclear NMR techniques. Problems can arise when the protonation constants of ligands do not fall into the pH range where the pH can be measured reliably with the use of combined glass electrode, for example with acidic protonation sites of phosphinates, basic protonation sites of ligands possessing phosphonate pendant arms, or basic nitrogen atoms of some cross-bridged macrocyclic polyamines.

This relationship between protonation and stability constants is demonstrated by the data reported for the first protonation constant of DOTA. This protonation constant is in the range of 10.14–12.72 and depends on the choice of the electrolyte used to set the ionic strength (the effect of the ionic strength on the protonation constants of the ligands is discussed in detail later). The reported values, however, differ by nearly 1 log unit, even for experiments performed with the same or similar ionic strengths. This discrepancy indicates the difficulty associated with the accurate measurement of such protonation constants. Performing pH-potentiometric titrations at high ligand concentrations (3.5–5.0 mM) helps resolve this problem because the equilibrium concentration of all the species present in solution are increased. Therefore, protonation and deprotonation cause larger effects that are more easily detectable by electrodes. When high concentrations are not practical, UV–visible spectroscopy or various NMR spectroscopic methods can be used as supporting techniques to determine pH values in samples from the total concentration of H⁺ or OH⁻ ions (c_analyte≪c_{H+ or OH−} and thus c_analyte can be neglected).^384–386  Noszál and co-workers developed a method that relies on the readout of pH from the chemical shifts of a set of indicator molecules with known protonation constants and chemical shift dispersions in the same sample as the molecule of interest. The results obtained for two biguanidine drugs, metformin and phenformin, indicate that a set of eight indicator molecules enables the precise determination of large values of log K with high accuracy and precision.³⁸⁷ 

The protonation scheme of the ligands can be characterized using NMR spectroscopy, as was demonstrated for open-chain³³¹  or macrocyclic ligands.^330,334  Spectrophotometric methods can be used to monitor protonation processes if the ligands contain chromophores close to the site of protonation and have measurable changes in their absorbance upon metal binding. For example, the protonation constants of a pyridine-based ligand were assigned using a combination of UV–visible and NMR spectroscopies.^388,389 

Finally, before starting equilibrium studies, the nature of the electrolyte used to set ionic strength should be carefully considered. Salts such as KCl or KNO₃ have frequently been used in different concentrations to maintain ionic strength. The use of K⁺ salts is a convenient choice because K⁺ tends to form weaker complexes with ligands designed for MRI applications than Na⁺. However, the protonation constants (especially the log K$1H$) determined in the presence of K⁺ are lower than those obtained in the presence of the tetramethylammonium cation. Tetramethylammonium would be the best ion to maintain ionic strength because interactions between ligands and tetramethylammonium can be neglected. However, the evaluation of protonation constants can be challenging because protonation equilibria are shifted to more basic pH regimes without the interaction between the cation and the ligand.^30,390–392  Thus, somewhat paradoxically, the advantages of using a salt that does not affect protonation of a ligand are outnumbered by the limitations of using one that weakly interacts with the ligand. However, it should be mentioned that the determination of a highly basic protonation constant (pK_a≥12) using NMR methods can be easily corrupted by the formation of Na⁺ or K⁺ complexes at high pH values, affecting chemical shifts. Nevertheless, to mimic in vivo conditions, characterization of ligands should be performed in a solution of NaCl (0.15 M). Despite some variability in the stability constants that results from differences in ionic strength, the use of NaCl does not translate into significant differences in the distribution of species because the values of protonation and stability constants are shifted in the same direction. Conditional constants and pM values measured at different ionic strengths are also usually similar for the same system.

The anions of the electrolyte can also have an effect on the outcome of characterization. For example, halogen ions are known to form stable complexes with In^III, Tl^III, and Bi^III ions, and thus their stability must be considered when calculating stability constants. The formation of these relatively weak complexes might actually aid the determination of stability constants because these complexes can compete with multidentate ligands at high concentrations, thereby shifting equilibria to higher pH values, where they can be followed by more common methods.^393–396 

There are several metal ion and ligand properties that have to be considered to create a proper equilibrium model for a metal–chelate system: (i) the number of donor atoms, which is the denticity of the ligand; (ii) the coordination number of the metal ion; (iii) the type of donor atoms of the ligand; and (iv) the hard–soft character of the metal ions. Along the lanthanide series, the coordination number of the +3 ions decreases from 9 to 8 with increasing atomic number.³⁹⁷  The coordination number of transition metals is often 6, for example, with Cu^II, Fe^II, and Fe^III complexes, but occasionally it can be 7, like with complexes of Mn^II.³⁹⁸  A comparison between the coordination number of the metal ion and the denticity of the ligand can yield useful conclusions: if the denticity is smaller than the coordination number, then the formation of ML₂ complexes, including the hydroxido species, is likely. This scenario was observed in the case of [Ln(EDTA)]⁻ complexes,^399,400  where the formation of [Ln₂(EDTA)₃]⁶⁻ species was evidenced in addition to the presence of Ln(L) and Ln(L)₂. The existence of ternary species was confirmed by spectrophotometric method in the Nd^III–EDTA system. The Nd^III aqua ion has a low-intensity band at 427.3 nm in the absorption spectrum corresponding to a ²P_1/2–⁴I_9/2 transition. This absorption is sensitive to the coordination environment of Nd^III, and changes in the coordination sphere lead to shifts in wavelength, making absorption spectroscopy a valuable tool to confirm the existence of different species in solution.^401,402  In the [Ln₂(EDTA)₃]⁶⁻ complex, each Ln^III ion is coordinated separately to one EDTA ligand and the third EDTA ligand acts a bridge between the two [Ln(EDTA)]⁻ units by binding each metal ion via an iminodiacetate group. The Ce^III ion is also useful for spectrophotometric studies because its 4fⁿ→4fⁿ⁻¹5d¹ allowed transition provides a high intensity band in the UV region of the electromagnetic spectrum. This transition is red-shifted when the number of coordinated donor atoms increases.^{30,386,403–405}  This transition has been used to obtain information about the structure of long-lived reaction intermediates involved in the formation of macrocyclic Ln^III complexes.

The formation of ternary complexes between contrast agents for MRI and endogenous ligands such as citrate, phosphate, and carbonate is an important issue because the formation of these complexes reduces the relaxation rate enhancement caused by paramagnetic complexes and can accelerate in vivo decomplexation of the agents.^406–408  The ternary hydroxido complexes of metal chelates usually form under basic conditions and in several cases lead to precipitation of metal hydroxides. The presence of mixed–ligand complexes in equilibrium can be confirmed by ¹H-relaxometric methods by following the relaxation rate (1/T₁, where T₁ is the longitudinal relaxation time) of samples as a function of substrate concentration at a fixed pH, like with the Gd^III complexes of phosphinate derivatives of propylene diamine tetraacetic acid.⁴⁰¹  The data from those studies revealed a decrease of relaxivity in basic solutions. The formation of ternary complexes can also be followed by spectrophotometry, as in the case of [Cu(DOTAgly)(OH)]³⁻ and [Gd(PTDITA)]⁻ (Figure 1.20).^148,389  When the formation of a ternary complex is favored, as is the case with complexes of tetra-, penta-, and hexa-dentate ligands with oxalate, malonate, citrate, or iminodiacetate,^{400,409–411}  pH-potentiometric titrations can be used to obtain reliable data about their formation and stability. The advantage of stability data is that they enable modeling of complexation (calculation of speciation diagrams) over wide ranges of pH values and ligand concentrations.

When the potential denticity of a ligand is larger than the maximum coordination number of a metal ion, the formation of dinuclear and protonated complexes is possible. Ligands designed for applications relevant to MRI usually contain fewer donor atoms than the maximum coordination number of lanthanide ions; thus, the formation of stable dinuclear complexes in equilibrium is rare. However, the relatively small coordination numbers of transition metal ions relative to lanthanides enables the existence of dinuclear species. Sometimes the formation of dinuclear complexes is the goal. For example, a chemical exchange saturation transfer agent candidate was reported to have a chemical exchange saturation transfer effect owing to the formation of a Cu^II-centered dinuclear species.⁴¹²  In this respect, electron paramagnetic resonance spectroscopy is a useful technique to study equilibria involving Cu^II species, including dinuclear complexes.⁴¹³ 

High relaxivity of contrast agents for MRI is a desirable property that can be obtained, for example, by designing contrast agents that contain more than one paramagnetic metal ion. This design feature can be achieved by multiplying the coordinative site, resulting in a paramagnetic oligomer such as (Gd-AAZTA)₂,⁴¹⁴  ditopic DO3A-based Mn^II complexes,⁴¹⁵  or [Mn₂(ENOTA)] (Figure 1.20).⁴¹⁶  Relatively flexible open chain ligands have higher propensities to form dinuclear complexes than rigid macrocyclic ligands.³⁹⁰  Furthermore, there is a special class of contrast agents for MRI designed to form dinuclear complexes with endogenous metal ions such as Ca^II, Cu^II, and Zn^II offering the possibility of determining the in vivo concentration of these ions.^417–419  These responsive probes are composed of a unit to coordinate the paramagnetic metal ion and another moiety to selectively bind the other metal ion, thereby activating the probe.^417–419  The characterization of these complexes has been performed with the techniques described earlier in this chapter.

Last but not least, the protonation of the metal chelates must be considered. Protonation of lanthanide(iii)–polyaminopolycarboxylate complexes such as [Ln(DTPA)]²⁻ or [Ln(DOTA)]⁻ and their derivatives occurs at the carboxylic pendant arms. Because these carboxylates are coordinated to the metal ions, the values of these protonation constants are generally low.^30,391  On the other hand, coordinated pendants such as phosphonates or amines have higher basicities, and therefore protonation is shifted to neutral or basic pH values. For example, [Gd(DOTP)]⁵⁻ has four protonation steps in the pH range of 4–8 related to the four phosphonate groups.⁴²⁰  Generally, the number of the protonation processes and the values of the protonation constants depend on denticity, the basicity of the donor atoms, and the coordination number of the metal ion. As mentioned above, protonation of transition–metal–ion complexes with DTPA- or DOTA-like ligands is also expected.³⁹⁰ 

In addition to the techniques mentioned above, NMR spectroscopy plays a role in the investigation of the equilibrium properties and confirmation of equilibrium models of metal chelates. Unfortunately, the study of paramagnetic metal complexes with NMR spectroscopy is often difficult or frequently impossible owing to line-broadening caused by paramagnetic metal ions. Therefore, in the case of lanthanides, diamagnetic or less paramagnetic surrogate ions such as Y^III, La^III, Lu^III, or Eu^III are often used. Similar approaches can be applied for the NMR studies of equilibria involving transition metal complexes: for example, the diamagnetic Zn^II ion can be substituted for paramagnetic Cu^II or Mn^II.^401,421,422 

In summary, the following items should be considered when selecting an equilibrium model to study contrast agents for MRI: The equilibrium model should be based on the formation of the ML species independently of the metal ion type. Furthermore, the presence of protonated M(H_iL) species and ternary ML(OH)_i hydroxo complexes should be assumed based on the denticity of the ligand and the coordination number of the metal ion. Protonation of ML complexes should be confirmed experimentally in samples containing the metal ion and ligand in a one-to-one ratio. Based on these data, the stability constants of ternary and dinuclear complexes can be determined by studying systems at various metal-to-ligand ratios.

Protonation and stability constants can be obtained from experimental data using various computer programs like HYPERQUAD, PSEQUAD, OPIUM, BEST, SUPERQUAD, and MINIQUAD. These equilibrium calculation programs are generally based on nonlinear least-squares fitting.^257,374,423  Among them, the commercially available HYPERQUAD is one of the most widespread,³⁷⁵  in part because of the regular software updates and maintenance. Although some versions of HYPERQUAD are limited to fitting pH-potentiometric data, other versions can simultaneously fit data obtained by different techniques. The company that produces HYPERQUAD also offers other programs that treat data collected by other methods. These include HypSpec2014 for fitting spectrophotometric data, HypNMR for fitting NMR chemical shift data (limited to the fast-exchange regime), and HypDH for obtaining stability constants and enthalpies from calorimetric data.³⁷⁵  Programs such as PSEQUAD or OPIUM can simultaneously fit data obtained by potentiometry, UV–visible spectroscopy, ¹H-relaxometry, and multinuclear NMR titrations.³⁷⁴  In addition to the software listed above, other programs are available for performing equilibrium calculations. Data obtained solely from UV–visible spectroscopy are often fit using SPECFIT/32TM, a sophisticated, multivariable data-analysis program for modeling and fitting equilibrium titration data.⁴²⁴  Analyses are also performed using SQUAD or the equilibrium routine of the ReactLab™ software.⁴²⁵  Multinuclear NMR data are frequently processed via HypNMR, PSEQUAD, or EQNMR.^{374,424,426–429} 

Generation of speciation diagrams is useful in the design of equilibrium and kinetic experiments. Although such diagrams can be generated using data-fitting software within the concentration limits of the calculation, there are specifically designed programs that create speciation diagrams from the H⁺–L–M^z+ equilibrium constants. These include the Medusa and HySS2009 freeware, which are also capable of handling redox equilibria and heterogeneous reactions.^375,376 

In each case, calculations are performed assuming a chemical model involving equilibria between components: the metals (M), ligands (L), and protons (H⁺), and complex species (M_pL_qH_r⁺). These equilibria are characterized by the cumulative formation constants β_pqr [see eqn (1.14) and (1.15)]. The software uses the proposed equilibrium model to fit measured data (titration volume, absorbance, or chemical shift) obtained from the concentrations of the components and the pH. During data fitting, the sum of squared residuals (the differences between the measured and calculated data) are minimized while values of β_pqr are varied. Some software programs calculate statistical data characterizing the goodness of fit (χ² or the standard deviation of fitted data) and the reliability of the estimated β_pqr constants. The infrastructure for computations that is commercially available enables calculations to be repeated with different equilibrium models within a few seconds, and the model that has the best statistical parameters is usually accepted.

The nature of the species involved in the equilibrium model should be supported by an independent method. For example, the formation of dinuclear species for Cu^II systems might be confirmed by electron paramagnetic resonance spectroscopy, mass spectrometry, or another analytical technique. Additionally, distribution curves for species can be compared: values calculated using the stability constants determined from one method can be compared with the absorbance values (for Cu^II), T₁ or 1/T₁ values (for Mn^II or Gd^III), or NMR-signal intensity data (for diamagnetic complexes). In some cases, however, when the coordination number of the metal ion is smaller than the potential maximum denticity of the ligand, protonation can occur at a donor atom located far from the coordination environment of the metal ion. For such systems, it is best to fit simultaneously the data collected by multiple techniques. For instance, the stability of Cu^II complexes formed with macrocyclic ligands can be estimated reliably only when the formation of the first species occurs at fairly low pH values. These species can be followed by UV–visible spectroscopy as a function of acid concentration. Once the constant for the formation of the copper complex under acidic conditions has been determined, then the stepwise deprotonation of this species can be determined by pH-potentiometry or UV–visible spectroscopy as a function of pH.

Individual stability constants (K corresponding to a given equilibrium) can be calculated from the log β_pqr values and the related protonation and equilibrium constants. The standard deviations calculated for stability constants originate from random experimental error and reflect only a part of the total uncertainty. The true errors in the calculated equilibrium constants can be evaluated by comparison of constants obtained by different methods or in independent studies.

A wide range of experimental methods have been proposed for the determination of equilibrium constants. The relative simplicity, low cost, and wide availability of the pH-potentiometric titration has made it one of the most commonly used methods for the determination of protonation and stability constants of ligands and metal complexes. Importantly, equilibria involving ligands without protonation sites cannot be studied by pH-potentiometry. The basicity of donor atoms and the competition reactions between metal ions and H⁺ for coordination by donor groups both affect the concentration of H⁺. Hence, both protonation of ligands and complex formation influence pH.

The instrumentation necessary to perform pH-potentiometric titrations is relatively simple and generally includes glass and reference electrodes, or a combined electrode, a voltmeter or a pH-meter, and an automatic burette. To determine the relationship between the measured electromotive force or pH values and the volume of H⁺ or OH⁻ added, both types of titration systems must be calibrated.

For potentiometric titrations, the electrode is calibrated by titrating a known amount of HCl or HNO₃ with a standardized solution of KOH, NaOH, or (CH₄)₄NOH at constant ionic strength (for example, KCl, KNO₃, NaCl, (CH₄)₄NCl, or (CH₄)₄NNO₃ electrolytes at 0.1 or 1.0 M). The relationship between the measured electromotive force and the concentration of H⁺ is expressed by the Nernst equation:

where the additive term E₀ contains the standard potential of the electrode system and the contribution of the liquid–junction potentials, S corresponds to the Nernstian slope, and K_W is the stoichiometric ionic product of water. The values of E₀, S, and K_W can be calculated from the titration data pairs (volume of base (mL)–pH or electromotive force) of the calibration titration curve. Taking into account the contribution of H⁺ and OH⁻ ions to the liquid–junction potentials, eqn (1.29) can be expressed in the following form:

where j₁ and j₂ are the coefficients characterizing the contribution of H⁺ and OH⁻ ions to the liquid–junction potentials in acidic and basic conditions, respectively. By using the related E₀, S, j₁, j₂, and K_w values, the concentration of H⁺ and pH are calculated from the electromotive force values obtained in the potentiometric titration experiments of ligands or ligands and metal ions.

For pH-potentiometric titrations, two or three standard buffer solutions are used for the calibration of the pH meter. The pH_a (pH_a=−log(H⁺)_a, where (H⁺)_a is the activity of the hydrogen ion), is expressed by:

where E, E⁰, and E_j are the measured electromotive force, standard potential, and sum of the liquid–junction potentials, respectively. If the sample contain salt, a constant ionic strength difference exists between the E_j of the sample and the buffer solution. The difference between the measured pH_r and pH_a can be taken into account by the following equation:

where ΔE_j is the difference between the liquid–junction potentials of the sample and the buffer solutions, pH is −log[H⁺], γ_± is the activity coefficient, A is a correction factor, and pK_W is the stoichiometric water ionic product. The calibration of the electrode involves determining the pH-independent correction factor (A) and the stoichiometric water ionic product (pK_W).⁴³⁰  The correction factor (A) and the stoichiometric water ionic product (pK_W) are then used to obtain the concentration of H⁺ from the pH_r values measured in the titration experiments of ligands or ligands and metal ions.

pH-potentiometric titrations can be used to determine the protonation constants of ligands and the protonation and stability constants of metal complexes. In these systems, equilibrium must be attained within a few seconds to minutes after the addition of the titrant. The metal-to-ligand concentration ratios are usually kept at 1 : 1. For some metal complexes, however, titrations are sometimes made with metal-to-ligand ratios of 2 : 1, 3 : 1, 1 : 2, or 1 : 3. The concentration of the ligand is generally kept in the range of 2–5 mM. Depending on the availability of the ligand, the volume of the sample is usually 3, 5, 6, 10, or 20 mL. The temperature of the samples must be kept constant during the titration (usually at 25 or 37 °C). Solutions are magnetically stirred with N₂ or Ar being bubbled through them to avoid the absorption of CO₂ during titrations. pH-potentiometric titrations are usually performed according to the calibration method described above in the pH range of 1.8–12.2.

Complex-formation reactions of polyaminopolycarboxylate-type macrocyclic ligands are generally too slow to monitor by direct pH-potentiometric titrations. Consequently, the stability constants of metal complexes of these macrocyclic ligands are often determined using an out-of-cell method. This is referred to as the batch method.⁴³¹  In this method, several separate samples are prepared in the pH range where complexation equilibria exist; each sample corresponds to a different point in the titration curve. The tightly closed samples are kept under an atmosphere of N₂ or Ar at 25 °C until equilibrium is reached. The time required to reach the equilibrium must first be confirmed by a supporting method, such as UV–visible spectroscopy for complexes of Cu^II and Ce^III, or by relaxometry for complexes of Mn^II or Gd^III. To obtain reliable stability constants, at least two or three parallel measurements should be performed.

The protonation and complexation properties of the ligands and the coordination of the metal ions can be studied by other techniques. If the metal ions and ligands have absorption bands in the UV–visible range of the electromagnetic spectrum, protonation and complexation can be monitored by UV–visible spectrophotometry. In this technique, absorbance can be expressed by the Lambert–Beer law:

where I₀ and I are the intensity of the incident light before and after, respectively, the absorbing solution; ε is the molar absorptivity; l is the path length; and c is the concentration of the absorbing species. Because the observed absorbance includes contributions from the absorption of each absorbing species in a sample, the absorbance for each wavelength can be expressed by the following equation:

where i=0, 1, 2,…, n; x_i is the molar fraction; l is the path length; and ε_i is the molar absorptivity of each involved species. The absorption bands of ligands generally occur in the UV–range. By taking into account of the total concentration ([L]_t=[L]+[HL]+[H₂L]+‥+[H_nL]) and the protonation constants (α_H=1+K₁^H[H⁺]+K₁^HK₂^H[H⁺]²+‥+K₁^HK₂^H…K_n^H[H⁺]ⁿ) of the ligand, eqn (1.34) can be expressed as eqn (1.35). Note that the term l can be omitted if the same cuvette is used for all measurements.

The protonation constants and the molar absorptivity of each protonated species of the ligand can be calculated by fitting the electromotive force or pH and the absorbance values with eqn (1.35). Similarly, the formation, protonation, and structure of metal complexes can be determined from the absorption bands of the ligands, metal ions, or complexes obtained from spectrophotometric titrations.

Stability constants of complexes are usually determined pH-potentiometrically because of the relative simplicity and precision of the method. Sometimes, complexation equilibria exist in pH ranges that are not easily measured. In those cases, competition reactions can be used to shift of the complexation equilibrium into a measurable pH range. Competition reactions can be either metal- or ligand-based. The appropriate competition partner is selected based on three criteria: (i) the relative simplicity of the system, which involves minimizing the number of species present in the equilibrium; (ii) the ease of monitoring the competition reaction; and (iii) prior knowledge of the stoichiometry and stability of the complex formed with the competition partner.

Metal and ligand competition reactions are described by eqn (1.36) and (1.37).

Such competition reactions can be monitored using the absorption bands of the exchanging metal ions, ligands, or both by spectrophotometry or by relaxometry if there is a difference in relaxivity between the two complexes (ML and ML′ or ML and M′L). For competition reactions that are monitored by the absorption band of M′L and ML′, the total absorbance of the system can be expressed by eqn (1.38) and (1.39), assuming that the absorbance of ML, L, and L′ can be neglected.

The stability constant of the metal complex ML can be calculated from the protonation constants of the ligands, the stability constants of the complexes formed with the competition partners, and the molar absorptivity of complexes by fitting the pH and absorbance of the systems. Note that the pH must be known because the protonation of the non-complexed ligand also affect the absorbance. The out-of-cell method is generally used for these reactions because competition reactions are often slow. The conditions of the competition reaction, including the concentration of each reactant, volume of samples, and wavelength, first need to be optimized by independent experiments. Note that the volume of the sample is generally limited by the pH measurements, the volume of the cuvette, or the availability and solubility of the ligand. To obtain reliable data, potentiometric or pH-potentiometric systems should be calibrated using the method described above to measure pH.

The optical properties of lanthanide complexes arise primarily from f–f electronic transitions. Perturbations of f–f transitions by coordinating donor atoms in Ln(L) tend to be small. Therefore, f–f transitions are usually characterized by small bandwidths and, due to their forbidden nature, low intensities (ε≤10 M⁻¹ cm⁻¹).⁴³²  Those transitions can nonetheless be used to monitor complex formation. For instance, the ²P_1/2 term of Nd^III is not split by crystal fields. Instead, the absorption bands corresponding to the ²P_1/2←⁴I_9/2 transitions (Nd(H₂O)₉: λ=427.3 nm) are shifted to longer wavelengths by the formation of Nd^III complexes. The ²P_1/2←⁴I_9/2 transitions of Nd^III can thus give information with respect to the number and crystal-field-strength of coordinating donor atoms.⁴³³  The ⁵L₆←⁷F₀ transition of Eu^III in the range of 390–400 nm shows a similar sensitivity to ligand binding.⁴³⁴  Because the shielding of 5d electrons by the outer shells is weak, the 5d←4f transitions of Ln^III ions can also be used to study the formation of Ln(L) complexes. For instance, the 5d←4f transitions of Ce^III, Pr^III, and Tb^III ions result in broad and intense absorption bands (ε≤1000 M⁻¹ cm⁻¹) in the UV region of the electromagnetic spectrum.

All Ln^III ions luminesce unless they have valence electronic configurations of f⁰ or f¹⁴. The luminescence of Ln(L) complexes is most commonly characterized by spectrofluorometry, that is, by measuring the emission spectra that result from excitation at an absorption wavelength associated with an emissive transition. The formation of Ln(L) complexes can generally be followed by spectrofluorometry in the concentration range of 10⁻⁶ to 10⁻⁸ M.^435,436  The relationships between the concentrations of species and the measured intensities is described by eqn (1.40).⁴³⁷ 

In eqn (1.40), I₀ is the intensity of the entering light, l is the path length of the cell, ε_i is the molar absorptivity of the species, c_i is the concentration of the species, and φ_i is the quantum yield of the species. Importantly, the presence of compounds that contain absorbing or fluorescent molecules might change the emission of Ln(L) owing to the inner-filter effect. The inner-filter effect refers to the absorbance or optical dispersion of light at the excitation or emission wavelength by a compound present in the sample.⁴³⁸ 

The luminescence properties of Eu^III (⁷F_0–4←⁵D₀) and Tb^III complexes (⁷F_0–6←⁵D₄) are often exploited to investigate the formation, structure, and number of inner-sphere water molecules in Ln(L) complexes (see Chapter 2.4.3).^439,440  The stability constants of several Eu^III complexes were determined by monitoring the excitation spectra of Eu^III–ligand systems with laser-induced luminescence spectroscopy in the pH range of 2–7.⁴⁴¹  The intensities of the ⁵D₀←⁷F₀ transitions of different Eu^III–ligand systems are shown in Figure 1.21. Note that the luminescence intensity increases with increasing pH owing to the formation of Eu(L). The measured luminescence intensity values can be expressed by the sum of the contribution of each species [eqn (1.41)].

In eqn (1.41), I_i and c_i are the molar intensity and the concentration of the species i, respectively. The values of c_i of the different species can be expressed in terms of the related equilibrium constants (such as K_EuL and K_EuHiL). The stability and protonation constants of Eu(L) can thus be calculated from I_meas and pH if the total concentration of each component ([Eu³⁺]_tot and [L]_tot) and the protonation constants of the ligand (K_i^H) are known. This is usually performed with equilibrium calculation software, as detailed above.

Multinuclear NMR spectroscopy has been used since the late 1950s to determine protonation and stability constants. This technique has numerous advantages relative to other techniques. For instance, it can be applied to both acidic and basic solutions. pH can be determined by using the measured concentration of acid or base. It can also be determined with indicator molecules whose chemical shifts and protonation constants are known.³⁸⁷  NMR titration data can be used to determine the order of protonation of the donor atoms, that is, the protonation sequence of the ligand. NMR titration data can also be used to evaluate microconstants in cases of simultaneous or overlapping protonation equilibria.^442–444  Advantageously, NMR titrations can be used to estimate protonation constants, even if ligands are not analytically pure.

When the pH-dependent acid–base equilibrium is fast on the NMR time scale, the chemical shift measured (δ_obs) is proportional to the chemical shifts of the HL and L species [eqn (1.42)]:

In eqn (1.42), δ_HL and δ_L are the chemical shifts of the protonated and deprotonated ligand, respectively; and χ_HL and χ_L are the molar fractions of the protonated and deprotonated ligand, respectively. Molar fractions can be expressed in terms of protonation constants (K_HL=[HL]/{[H⁺][L]}) and [H⁺] as follows:

Given that χ_HL+χ_L=1, eqn (1.42) and (1.43) can be combined to give eqn (1.44), which is an equation that enables determination of K_HL by nonlinear, least-squares refinement.

Although it is rare, slow exchange can sometimes be observed in protonation equilibria. This has been observed, for instance, in the protonation of DOTA–tetraamides or in the formation of the ternary hydroxido complex of [Ga(AAZTA)]⁻.^394,445  In the case of slow exchange, separate peaks corresponding to the same proton are observed in the NMR spectra of the protonated and deprotonated forms of the species of interest. The ratio of [HL]/[L] can then be calculated by the ratio of the integration of each peak. The concentrations of the protonated and deprotonated species can be calculated from this ratio and the total concentration of the ligand, [L]_tot. This, in turn, enables the determination of the protonation constants.

NMR spectroscopy is a good technique to determine protonation constants if the chemical exchange is slow enough that the signals for each species are resolved. For the multidentate ligands that are widely used in contrast agents for MRI, such as for macrocyclic systems, this condition is often satisfied, although the assignment of signals can be complicated. ¹³C-NMR spectroscopy is usually limited by its low sensitivity. If applicable, NMR spectroscopy based on nuclei that are more sensitive than ¹³C, such as ³¹P and ¹⁹F, is also viable for the determination of protonation constants. Similarly, equilibria of metal complexes can also be studied by monitoring the NMR signal of NMR-active metal isotopes. Detection of the metal ions can be useful in supporting data from ¹H-NMR titrations. Several metallic elements have I=1/2 spin isotopes, such as Y, Cd, and Tl. Others that are quadrupolar with I>½, such as Al, Ga, In, La, and Sc, can also be used in such experiments.^445–450  An advantage of directly observing metal ions is an enabling of the determination of stability constants for mixed ligand complexes, such as [Tl(DOTA)(CN)]²⁻.³⁹⁵  Quadrupolar nuclei are associated with broad signals that are sometimes immeasurable in non-symmetric species; therefore, only symmetric free aqua ions, such as Al(H₂O)₆³⁺, Al(OH)₄⁻, Ga(H₂O)₆³⁺, and Ga(OH)₄⁻, can be measured quantitatively. In most cases, M(L) complexes of these metals are difficult to detect. The further complication of NMR of metal nuclei is that the lone signal can be difficult to integrate. A standard, either internal or external, must therefore be used to determine the concentration of the metal species.

In general, NMR methods can be applied to diamagnetic complexes and many paramagnetic complexes. Gd^III complexes, however, cannot be investigated directly by ¹H-NMR spectroscopy because of the extreme line-broadening effect of the Gd^III ion. Complexes of Mn^II, Fe^III, or Gd^III can be studied through ¹H relaxometric studies by following the longitudinal (T₁) or transverse (T₂) relaxation times as a function of pH, exchanging metal, or ligand concentration (see Chapters 2.1 and 5.1 for further reading on the relaxivities of Gd^III and transition metal complexes, respectively). Paramagnetic metal ions and their complexes often display significant differences in r_1p and r_2p relaxivities^451–454  or the ratio of these relaxivities.⁴⁵⁵ 

The relaxivity of a sample containing paramagnetic species can be described as a weighted average of the contributions from [M(H₂O)_x] (x=6 for Mn^II and 8 for Gd^III) and each of the metal-ion-containing species as described in eqn (1.45) (for r_1p).

In eqn (1.45), χ is the mole fraction of [M(H₂O)_x] and MH_RL, r_1p,M is the molar relaxivity of the metal aqua ion (7920 M⁻¹ s⁻¹ for Mn^II and 13 170 M⁻¹ s⁻¹ for the Gd^III at 25 °C and 20 MHz), and r_1p,MpHRLQ is the molar relaxivity of the metal-ion-containing species. The formation constants of the metal complexes present in equilibrium can be determined from the protonation constants of each ligand species, which must be determined in advance by other methods. In the simplest case when complexes of 1 : 1 metal-to-ligand ratios are formed ([M]=[L]), this can be derived from eqn (1.46).

The reliability of the method can be increased by performing titrations at different metal-to-ligand ratios in a wide range of pH values or by supplementing the ¹H-relaxometric data with pH-potentiometric data. The simultaneous fitting of data obtained by multiple methods enables an estimation of the relaxivity of the different species, including those present at low concentrations or that overlap other species in solution.^120,456  One of the first examples of relaxometry to determine the stability of macrocyclic Gd^III complexes was for the study of triaza-triacetate macrocyclic ligands (Figure 1.22).⁴⁵⁷ 

The determination of stability constants of metal complexes by separation techniques such as high-performance liquid chromatography,^458,459  ion chromatography,⁴⁶⁰  and capillary electrophoresis^461–464  has limitations. First, information is obtained from analyses performed on a series of similar samples with varying composition as in the out-of-cell (batch) method used in pH-potentiometry described above. Second, capillary electrophoresis requires relatively expensive instrumentation that is not widely available.^460,465,466  These techniques, however, generally have high separation efficiency and relatively high speed of analysis, and require much less compound than other techniques discussed above. In addition, these measurements can be automated.

Generally, in capillary electrophoresis, the components of a test solution are introduced into the capillary via hydrodynamic injection, which is the most widely used injection method. The components move along the capillary with different velocities when voltage is applied at the ends of the electrolyte-filled capillary (Figure 1.23). The velocities depend on the charges and ionic radii of the species, such that the components of the samples reach the detection zone at different time points. The qualitative characteristic of the signal is the migration time (μ). Quantitatively, the heights and areas of the peaks are proportional to the concentrations of the components.

Practically, capillary electrophoresis has only been applied to the determination of stability constants for the simplest systems, such as for labile metal complexes which form in 1 : 1 metal-to-ligand stoichiometries.^467,468  In these cases, the metal ions are dissolved in the samples and the ligands are in the running buffer. Because the formation and dissociation of labile metal complexes is fast on the time scale of capillary electrophoresis, the resultant electrophoretic mobility (μ_obs) represents a weighted average of the electrophoretic mobility of the different species present in a sample [eqn (1.47)]. Stability constants are obtained from monitoring the electrophoretic mobility shift of samples upon changing the concentration of the ligand in the buffer.

In eqn (1.47), μ_M and μ_ML are the electrophoretic mobility of the metal ion and the ML complex, respectively, and x_ML is the molar fraction of the complex. K_ML, μ_M, and μ_ML can be calculated from eqn (1.47) once the molar fraction of ML is expressed as a function of [M]_t and K_ML (the stability constant of ML). These experiments require knowledge of [M]_t, which can be calculated by taking into account the injected volume of the test solution.

Measurements of stability constants become more straightforward when complexes remain intact, that is, when dissociation reactions of metal complexes are slow on the time scale of capillary electrophoresis experiments. In these cases, the peaks of the complexed and non-complexed ligands or metal ions are separated during electrophoresis.⁴⁶²  For samples already at equilibrium, a mixture of metal and ligand is injected into the capillary to detect the concentrations of the metal and the ligand as well as the concentration of the complex. The stability constants can then be assessed from the areas of the peaks because the areas of the peaks are proportional to the concentrations of the different species. These calculations require the knowledge of the molar area values of different species, which must be separately determined in experiments performed under identical conditions.

Calculations of stability constants, however, become more complicated when several species are present in equilibrium, such as when both protonated ligands and complexes are present. In these cases, the conditions used for capillary electrophoresis, including buffer, pH, and temperature, must first be evaluated to avoid further complications of equilibrium calculations. Capillary electrophoresis can be used to evaluate the stabilities of complexes either by direct formation, ligand-exchange reactions (ligand competition), metal-exchange, and double-exchange. It has been applied, for example, to determine the stability and the formation and dissociation rates of [Ln(DOTA)]⁻ complexes.⁴⁶⁵ 

Calorimetry, including isothermal titration calorimetry and isothermal titration microcalorimetry, was among the first techniques used to characterize equilibria and determine protonation and stability constants in dilute solutions. This technique is based on the measurement of the amount of heat that evolves as a result of physical or chemical processes taking place in the sample.⁴⁶⁹  This technique yields both stability constants and standard enthalpy changes for the equilibria.⁴⁷⁰  This time-consuming method is mostly limited to complexes of 1 : 1 stoichiometry and with relatively small formation constants. In the case of stable complexes suitable for medical applications, calorimetry has no routine application, especially for macrocyclic ligands with slow rates of complex formation.

Although the stability constants of metal–ligand systems are determined in equilibrium studies, the numbers alone cannot be used directly to visualize the chemical composition or speciation of solutions. The equilibrium constants characterizing a system—such as the protonation constants of ligands and the stability and protonation constants of complexes—enable calculation of the equilibrium distribution of a given system at any ratio of its components at any pH. Equilibria in solution are best illustrated by speciation diagrams, which are usually obtained by plotting the molar fraction of components or the logarithm of the equilibrium concentrations versus pH or the total concentration of a second component, while keeping the total concentration of other species constant. Calculations of distribution curves do not take a long time, mainly because they can be performed by computer programs like Medusa or HySS2009.^375,376  However, distribution curves that characterize equilibrium systems are most meaningful in the concentration, metal-to-ligand ratio, and pH ranges in which the equilibrium constants were determined. The mathematical characterization of complex equilibria is not covered in this chapter. Interested readers are referred to other books and papers detailing this subject.^6,364,471  In the following section, the benefits of using equilibrium distribution curves will be demonstrated using selected examples.

The simplest species distribution curves are obtained when two-component systems related to stepwise protonation processes of ligands are studied. The distribution curves obtained by plotting the molar fraction of protonated ligand species versus pH provide an overview of the protonation processes. These speciation curves help define buffer ranges and enable rapid comparison of protonation constants obtained by different techniques, such as pH-potentiometry and NMR titrations. The curves can also be used to assign the protonation sequence of ligands and calculate the fractions of different forms present in solution as a result of microspeciation.

Three-component distributions are often important when studying contrast agents, and these distributions generally involve components of metal ions, ligands, and protons. Distribution curves are usually not complicated for these systems because the equilibrium models of potential contrast agents tend to be relatively simple. Usually, the metal complex, its protonated form, and, in some cases, a ternary hydroxide form are the predominant species over a wide pH range. The situation might become more complicated with dinuclear or ternary complexes because, in such cases, the obtained distribution curves can be misleading if they are calculated incorrectly.^401,472  This problem is exemplified in Figure 1.24 for the Cu^II:TTHA⁶⁻:H⁺ system (Figure 1.20) in which Cu^II forms stable dinuclear complexes with TTHA⁶⁻, even at one-to-one metal-to-ligand ratios. In this case, two different speciation diagrams can be calculated according to two different methods. The occurrence of a dinuclear complex is explained by the potential denticity of TTHA, which is ten, being higher than the maximum coordination number of Cu^II, which is six. The mismatch enables coordination of two metal ions per ligand. The distribution curves in Figure 1.24 were created from published data.⁴⁷³  The distribution in Figure 1.24(A) was obtained by plotting the fractions of Cu^II as a function of pH, resulting in several extrema in the pH range of 1–10. Based on this distribution, it appears that complex formation is complete at pH 2. However, the plot of the fraction of TTHA⁶⁻ versus pH in (B) reveals that TTHA⁶⁻ is not completely metallated in acidic samples because of the formation of stable dinuclear complexes.

A similar phenomenon is observed with [Gd(EDDA)]⁺ (Figures 1.20 and Figure 1.25) and in many other systems in which the ligand of interest has a lower number of donor atoms than the coordination number of the Ln^III ion (8 or 9).⁴⁷⁴  The Gd^III ion has a propensity to form ML₂ or ternary MLL′ type complexes with the tetradentate EDDA²⁻ because the coordination number of the Gd^III (8 or 9) enables the coordination of two EDDA ligands. As shown in Figure 1.25(A), for this system, when a one-to-one metal-to-ligand ratio is used, the plot of EDDA²⁻ fractions versus pH shows that complex formation appears to be complete at pH 8. No non-complexed Gd^III appears to remain in solution after the formation of [Gd(EDDA)₂]⁻. The plot in Figure 1.25(B), however, shows that 10% of the Gd^III ions remain non-complexed in basic solutions as a result of the formation of a hydroxo species. Importantly, the speciation diagrams shown in Figure 1.25(A) and (B) were calculated using the protonation constants of EDDA²⁻ and the stability constants of the two complexes but omit metal-ion hydrolysis. These calculations assume that the non-complexed metal ions remain in solution as aqua ions, at least partially. However, hydrolysis of Gd^III does occur above pH 6, such that the actual equilibrium is shifted towards Gd(OH)₃, which precipitates. The distribution curves in Figure 1.24(A) and (B) thus give a false picture of speciation. The speciation diagram shown in Figure 1.25(C) incorporates the hydrolysis constants of Gd^III.⁴⁷⁵  In this case, Gd(OH)₃ precipitates at pH>7. Equilibrium can be shifted toward the formation of complex by adding excess ligand that prevents the formation of Gd(OH)₃. This system exemplifies some of the pitfalls in the interpretation of speciation diagrams.

Four-component systems are of a great value in both equilibrium and kinetic investigations and provide many benefits in the study of metal complexes. In some cases, determination of stability constants of metal complexes cannot be performed by direct methods because complex formation is complete even under acidic conditions. In such cases, ligand or metal competition reactions are used to obtain equilibrium constants. These reactions are usually performed at constant pH by varying the concentration of only one component while keeping the total concentration of the other components constant.^476,477  However, it is also possible to keep the concentrations of all components constant and vary the pH over a wide range of pH values. Ligands with large protonation constants (including many macrocyclic ligands) can be competed with less basic chelators (such as EDTA) in solutions of low pH because differences in protonation constants result in different conditional constants. The different conditional constants usually have different levels of dependence on pH. The large stability constants of Cu^II macrocyclic complexes are often determined by this method using EDTA or an EDTA-type ligand for competition.⁶  To determine stability constants by this method, the unknown system should be closely evaluated and compared to similar reported systems. This step is necessary to select the appropriate equilibrium systems for the determination of unknown constants.

The kinetic inertness of complexes is usually characterized by the rates of their metal- or ligand-exchange reactions. Some of these exchange reactions result in equilibrium instead of complete exchange, even in the presence of high excesses of competitors. In such cases, the equilibrium concentration of the species in each reaction can be calculated and taken into account using distribution curves.⁴⁷⁸ 

Complicated equilibrium calculations can be used to draw conclusions about the fate of injected complexes using artificial plasma models. Because of the difficulties associated with handling many species at low concentrations, simplified models are often used. One study of a simplified plasma model involved 20 components, including the most abundant ligands in biology and essential metal ions.⁴⁰⁷  This study contained more than 300 species and was created to study the in vivo fate of the linear, neutral complex Gd(DTPA-BMA). The distribution calculations showed that ∼17% of the intravenously injected complex dissociates at pH 7.4, Gd^III precipitates in the form of GdPO₄, and non-complexed DTPA-BMA³⁻ binds to other metal ions, including Cu^II, Zn^II, and Ca^II.

Because high thermodynamic stability is associated with many contrast agents for MRI, it is crucial to learn how to measure large equilibrium constants. With the most trivial case of a protonated ligand being a weak acid, the reversible chemical reaction can be described by eqn (1.48) and (1.49).

In eqn (1.49), charges are omitted for clarity. For large stability constants, the value of K is large because the equilibrium is shifted to the right, thereby making [M] and [H_nL] small. However, the accuracy of analytical methods for detecting extremely small concentrations is limited, resulting in large uncertainties in measured equilibrium constants. The most accurate measurements are expected in cases of comparable concentrations of ML, M, and H_nL. The equilibrium can be tuned by adjusting pH, but there are at least two limitations to this experimental trick. The first is that the concept of pH is not valid in extremely acidic media. The second is that the accuracy of pH measurements below 2 is not good enough to detect the small contributions of ML formation to the overall acidity. An often-used strategy, if ligands or metals are compatible with such measurements, is to measure [M], [ML], or both by spectrophotometry or NMR spectroscopy. Unfortunately, high-resolution NMR spectroscopy cannot be used for Gd^III complexes because of large paramagnetic broadening effects. Nonetheless, NMR spectra can be obtained for complexes of analogues diamagnetic ions or some paramagnetic lanthanide ions, such as Ce^III and Eu^III. Most NMR laboratories have tunable probes for such measurements; however, the sensitivity and natural abundance of the nuclei first need to be known. Measurement of quantitative intensities, necessary to obtain concentrations for calculations of K, requires proper acquisition parameters because T₁ values might differ considerably between M and ML. Additionally, large chemical shift differences associated with lanthanide ions often require adjustment of the spectral window to encompass all signals.

Extremely stable complexes exist for which even extended pH ranges (for example, −log[H⁺]<0) are not enough to shift equilibria to an optimal range of comparable concentrations of M and ML. In those situations, indirect measurements, also referred to as competition reactions, are used. In these experiments, secondary ligands (L′) might compete for binding to metal ions, and secondary metal ions (M′) compete for the ligand (L). Unfortunately, the competition method increases the experimental time because characterization of the subsystems, including the protonation constants of L′ and stability constants of ML′ and M′L, are needed for the evaluation of the stability constant of ML. Therefore, all of these values must be determined accurately in advance of studying ML. Some examples will be presented in the following section of this chapter. There are cases in which multiple competitions enable estimation of large stability constants. For example, in the following reaction, Tl^III competes with H⁺ and Na⁺, both present at high concentrations in a sample, and bromide competes with the DOTA ligand via the formation of relatively stable TlBr₃. The stability constant of the [Tl(DOTA)]⁻ is expected to be in the range of 50–60 log units, which is too high to be determined directly. Under acidic conditions and in the presence of large concentrations of NaBr, however, the stability constant can be determined according to eqn (1.50).³⁹⁵ 

Finally, kinetics cannot be ignored during the study of equilibria. Reversible equilibrium reactions might be slow, and the time required to reach equilibrium might be long, from minutes to months. For slow reactions, separated samples using the batch or out-of-cell method need to be prepared using sample holders that keep the system intact during the long experiment. Samples prepared in duplicates representing the extremes of the conditions in the batch samples, for instance at high and low pH or concentration of exchanging metal or ligand, must be measured periodically to ensure that the equilibrium in the samples is attained.

Published stability constants for a same metal complex can differ by several orders of magnitude depending on the experimental methods and conditions used as well as on the equilibrium models used for the fitting. There is no single method and set of conditions to follow to obtain good values; rather, the examples presented below are meant to help readers avoid selecting unsuitable methods for the determination of stability constants.

The challenges associated with the determination of stability constants of complexes with linear ligands differ from those of complexes with macrocyclic ligands. For linear complexes, the difficulty comes in choosing the correct methodology and equilibrium model for data fitting. For complexes with multidentate macrocyclic ligands, the preferred pH-potentiometry techniques can lead to erroneous results. As highlighted above, pH-potentiometry can only be used to determine equilibrium constants reliably for reactions that occur in the pH range of 1.8–12.2.

The complex [Cu(DO3A)]⁻ exemplifies how the choice of the equilibrium model for data fitting can influence the determination of the stability constants.⁴⁷⁹  A stability constant of log K([Cu(DO3A)]⁻)=23.1 is calculated if [Cu(DO3A)]⁻, [Cu(HDO3A)], and [Cu(H₂DO3A)]⁺ are the only copper macrocyclic species considered. However, log K([Cu(DO3A)]⁻)=26.49 if the triprotonated species [Cu(H₃DO3A)]²⁺ is also included in the model. Similarly, for a system containing Cu^II and DO3A-PIC⁴⁻, a stability constant log K([Cu(DO3A-PIC)]²⁻)=22.32 can be calculated from pH-potentiometric titration data if the only species considered are [Cu(DO3A-PIC)]²⁻, [Cu(HDO3A-PIC)]⁻, and Cu(H₂DO3A-PIC). Such a fitting yields an acceptable fitting parameter.⁴⁸⁰  The same data can also be fitted by including the [Cu(H₃DO3A-PIC)]⁺ species in the model. Including this triprotonated species worsens the fitting parameter and increases the standard deviation on the concentration of each species calculated during the fitting (by nearly three log units). These differences do not necessarily mean that either model is incorrect (although this is one of the possibilities that should be considered). As seen from the species distribution curves calculated under the conditions applied in the study, it is evident that the problem is that the experimental conditions lead to an absence of non-complexed metal ions at the beginning of the titration. In other words, the system was out of the range of pH-potentiometry. Therefore, this approach cannot distinguish the formation of [Cu(H₃DO3A-PIC)]⁺ followed by its deprotonation (a two steps process) from the one-step formation of the diprotonated complex Cu(H₂DO3A-PIC) from Cu^II and the protonated ligand. This problem was resolved with a UV–visible titration that indicated the complete formation of [Cu(H₃DO3A-PIC)]⁺ at pH 1.75. Deprotonation of the triprotonated species to form Cu(H₂DO3A-PIC) occurs in the pH range of 1–2.

Similar differences can be found for most Gd^III complexes. Table 1.1 contains maximum and minimum stability constants and pGd values of some of the most widely studied Gd^III complexes. These data highlight the large differences in stability constants reported for each complex. These differences are largely the result of the equilibrium models used for data fitting as well as the different experimental techniques employed (pH-potentiometric and UV–visible methods). The speciation diagrams for [Gd(DTPA)]²⁻ reveal that complexation is expected to be complete near pH 2.5 and that only 10% of Gd^III is present in solution in non-complexed form at pH 2.0, the starting point of most pH titrations. This case is an example when another supporting method must be considered to ensure that the stability constants calculated from pH-potentiometry are reliable. For Gd^III complexes, relaxometry is one of the best methods to check speciation in solution. Relaxometry can sometimes be used as a standalone technique for determining stability constants.⁴⁵⁷ 

There are other examples of open-chain Gd^III complexes for which even larger ranges of stability constants have been reported. For instance, the octadentate ligand OCTAPA (Figure 1.20) is often considered in radiochemistry for sequestering radioisotopes of Y^III, In^III, and Lu^III.⁴⁸⁶  It was also evaluated as a ligand for Gd^III for MRI applications.⁴⁸⁷  The stability constant of the Gd^III complex as determined by pH-potentiometry is log K([Gd(OCTAPA)]⁻)=15.1, whereas the stability constant for [Y(OCTAPA)]⁻ was reported to be 18.3.⁴⁸⁸  Furthermore, the stability of [Lu(OCTAPA)]⁻ was determined by competition titration with EDTA to be 20.08. The large difference between the stability constants of the complexes formed with intermediately sized Ln³⁺ ions (Gd^III) and smaller ones (Lu^III) suggests that OCTAPA could be used to separate Ln^III ions.⁴⁸⁹  A later study that used multiple methods, including ¹H-relaxometric and competition titrations with TTHA⁶⁻, demonstrated that this is not the case. There is little difference between the stability of the Gd^III and Lu^III OCTPA complexes; the stability constant of [Gd(OCTAPA)]⁻ is 20.23–20.39, which is in agreement with other reports.^477,488  The speciation diagrams calculated from the reported pH-potentiometry data⁴⁸⁷  and the ¹H-relaxometric titration data (Figure 1.26) indicate that the complexation of [Gd(OCTAPA)]⁻ is complete at pH 1.75. Thus, pH-potentiometry alone is not sufficient for studying this system. Instead, precise determination of speciation in solution requires that data from both pH-potentiometry and ¹H-relaxometry be treated simultaneously.

A similar situation was found with EDTMP (Figure 1.20), a linear tetraphosphonate ligand. The ¹⁵³Sm^III complex of EDTMP is used in the palliation of pain associated with metastatic bone cancer. The stability constants reported for the Gd^III complex of EDTMP vary widely between 14.87 [T=37 °C, NaCl (0.15 M)]⁴⁹⁰  and 21.80 [T=25 °C, KCl (0.1 M)].⁴⁹¹  Even larger differences were reported for the stability constants of [Y(EDTMP)]⁵⁻ they vary between log K=11.11 and 19.18.⁴⁹²  The first value was measured by pH-potentiometry [T=25 °C, KCl (0.10 M)];⁴⁹³  the second by competition with Cu^II in the presence of citrate monitored by UV–visible spectrophotometry [T=25 °C, NaCl (0.15 M)].⁴⁹²  The reported stability constants for the Ho^III complex of EDTMP vary by as much as 8.5 orders of magnitude.^490,491  These examples highlight the importance of the supportive data when determining the stability constants of complexes.

For macrocyclic systems, the problems of selecting the correct method and appropriate equilibrium model are compounded by the relatively slow kinetics of formation and dissociation. A final in-cage macrocyclic complex forms via a stable protonated, out-of-cage intermediate and its subsequent deprotonation followed by or concerted with structural rearrangements.^31,494  The absorption spectra of [Ce(DOTA)]⁻ shown in Figure 1.27 exemplify this stepwise binding.

As depicted in Figure 1.27, the formation of [Ce(DOTA)]⁻ is a slow process that can be monitored by UV–visible spectrophotometry. The formation of [Ln(DOTA)]⁻ complexes takes place via the formation of a stable diprotonated *[Ln(H₂DOTA)]⁺ intermediate that must deprotonate to form the final [Ln(DOTA)]⁻ complex. Water molecules act as Brønsted bases and OH⁻ ions assist the deprotonation in the rate-determining step.^31,495  Addition of Ce^III to H_xDOTA results in the immediate formation of the diprotonated *[Ce(H₂DOTA)]⁺, which is characterized by two new absorption bands at 260 and 296 nm. With time, the intensities of these absorption bands decrease, whereas the intensities of the absorption bands of the final [Ce(DOTA)]⁻ complex at 275 and 320 nm increase. The formation of [Ce(DOTA)]⁻ is fairly slow even at pH 4.46, and complete complexation requires a long time (6–12 weeks) at room temperature depending on the acidity of the samples (Figure 1.27). The most reliable experiments require the preparation of duplicate samples of the most acidic and most basic intermediates such that their absorbance (Ce^III and Eu^III), luminescence (Eu^III and Tb^III), or relaxivity (Gd^III) can be recorded as a function of time. One issue with this protocol is that the rates of complex formation depend on the size of the metal ions; data obtained using Ce^III might not be valid for Yb^III complexes.

The kinetics of complex formation is also a function of the ligand and its donor atoms (acetates in DOTA versus amides in DOTAM). For systems with slow kinetics of complexation, samples must be tightly sealed in vials so as to ensure that no solvent evaporates during the equilibration time. Sealed ampoules are preferable. For Ce^III complexes, photocatalytic oxidation of the metal to Ce^IV by visible light should also be considered. Such oxidation renders long-term studies with Ce^III complexes that require long equilibration times difficult to perform.⁴⁹⁶  Thus, it is easier to study Gd^III complexes and record in duplicate their relaxivity as a function of time. In this manner, reliable equilibrium data can be obtained with Gd^III even for slowly forming systems such as those involving DOTA^31,494  or DOTAM^404,445  if all of the suggestions in this chapter are taken into account. Several methods have been reported for determining stability constants of [Ln(DOTA)]⁻ complexes. These methods include direct techniques such as pH-potentiometry,⁴⁸⁵  capillary electrophoresis,⁴⁶⁵  and luminescence spectroscopy,⁴⁴¹  and indirect techniques, such as competitions titrations monitored by UV–visible spectroscopy.⁴³¹ 

Competition reactions can be used to determine the stability constants of macrocyclic complexes. However, the rates of complex formation are further decreased in the presence of competing partners, such as metal ions or ligands. These competitors reduce the concentration of the reaction intermediate (*[Ln(H₂DOTA)]⁺), thereby increasing the time necessary to reach equilibrium. The structure of the ligand might also affect the time required for equilibration and the determination of stability constants. For instance, the steric hindrance caused by the methyl groups in DOTMA slows the rate of complex formation by nearly two orders of magnitude. Similar decreases in rates of formation are found for some dipicolinates derived from cyclen (DODPA and Me2DODPA, Figure 1.20).^112,476  Moreover, in some cases, such as for dipicolinate derivatives of rigid, cross-bridged macrocycles like CB-TEDPA (Figure 1.20), there is no suitable method for the determination of stability constants because the Ln^III complexes of these ligands form and dissociate extremely slowly at room temperature.⁴⁹⁷  Computational methods such as density functional theory or quantitative structure–property relationships modeling might be useful when experimental determination of the stability constants is not feasible. Computational methods can also facilitate the design of experiments to measure stability constants.^498–501 

The authors thank the Hungarian Scientific Research Fund (NKFIH K-120224 project). The writing of this book chapter was supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences (G.T. and F.K.K.), the EU and co-financed by the European Regional Development Fund under the project GINOP-2.3.2-15-2016-00008 as well as the COST CA15209 “European Network on NMR Relaxometry” Action. Special thanks go to Prof. Michel Meyer (ICMUB, Dijon, France) for sharing the information concerning apparent constant and for reviewing the corresponding part in the chapter. F.K.K. and G.T. are also thankful for support from the Le Studium – Loire Valley Institute for Advanced Studies.

Gyula Tircsó,* Zsolt Baranyai, Ferenc Krisztián Kálmán, Zoltán Kovács, Ernő Brücher and Imre Tóth

Kinetic inertness is one of the most important parameters to predict the in vivo safety of metal complexes because the products of dissociation, the released metal ion and ligand, are potentially toxic. No metal–ligand complex has unlimited thermodynamic stability. Generally, the receiving biological media of a diagnostic or therapeutic agent contains a large number of compounds that can serve as ligands and many metal ions that can interact with the components of the administered agents. Therefore, biofluids are competitive media. Thus, metal complexes considered for in vivo use must be highly inert to prevent transmetallation or transchelation. High levels of inertness are required both for safety and for targeting purposes to ensure that intact complex is delivered to a target organ or tissue. Therefore, physico-chemical characterization of complexes intended for medical diagnosis and therapy should include kinetic studies, and knowledge of dissociation rates is particularly important.

Kinetic inertness is a better predictor of the safety of metal-based contrast agents than thermodynamic stability constants. This statement can be rationalized because complexes are expected to dissociate if any thermodynamic force for the dissociation exists. However, this does not necessarily mean that equilibrium is attained rapidly. Dissociation can be slow enough to enable safe applications of metal complexes with relatively lower thermodynamic stability constants. It is therefore important to know the expected rate of release of metal ions and ligands in living systems. Various approaches have been reported to characterize the kinetic inertness of metal complexes depending on their mechanism of dissociation. The mechanism is different for complexes formed with linear and macrocyclic ligands. Moreover, the rates of dissociation of complexes with linear ligands are usually faster than those incorporating macrocyclic ones. It should be emphasized, however, that the rates of dissociation of macrocyclic complexes are largely influenced by the size of the macrocycle. Rates of dissociation are faster when the macrocycle is either much smaller or much larger than the metal ion.

Dissociation of linear complexes is often catalyzed by other endogenous metal ions such as Cu^II via a mechanism that involves direct interaction between the complex and the exchanging metal ion. For Gd^III complexes of various 9- to 13-membered macrocyclic triaza and tetraaza ligands, acid-assisted decomplexation is the most important pathway leading to the release of Gd^III. Owing to these differences, the inertness of linear complexes is usually studied in metal-exchange reactions with biologically essential transition metal ions such as Cu^II and Zn^II or with other Ln^III ions such as Eu^III and Tb^III. The inertness of macrocyclic complexes, on the other hand, is normally evaluated by the acid-catalyzed dissociation of the complex.

The effect of biogenic ligands on the rates of dissociation has not been studied as much as dissociation catalyzed by metal ions. However, some studies indicate that the inertness of some neutral Gd^III complexes formed with linear ligands might be negatively affected by the presence of bioligands. Thus, the effect of these endogenous ligands needs to be considered in the design of new ligands for biomedical applications. The goal of this section is to present an overview of reported methods for obtaining dissociation kinetic data. Structural features of the ligands that influence dissociation kinetics are also summarized. These data are intended to aid in the design of inert complexes for future applications.

Metal complexes used as contrast agents in MRI should stay intact in vivo to avoid the dissociation of toxic metal ions. The half-life (t_1/2) of the excretion of small molecular weight lanthanide complexes through healthy human kidneys is about 1.5 hours.³⁶⁶  Therefore, 12 hours after intravenous administration, less than 1% of common contrast agents is expected to remain in the body. The retention times of complexes in vivo are lengthened in patients with compromised kidney functions, which increases the possibility of the contrast agent dissociating in vivo.

The in vivo dechelation of metal complexes can occur when endogenous metal ions—specifically Ca^II, Zn^II, or Cu^II—compete with the paramagnetic metal ion for the chelating ligand or when endogenous ligands—such as citrate, phosphate, or carbonate—compete with the chelating ligand for the paramagnetic metal ion. The dissociation mechanisms of metal complexes do not differ fundamentally from each other, and the dissociation pathways shown below for the lanthanide complexes can also be adapted to other complexes.

The kinetic inertness of a lanthanide complex is characterized by its rate of exchange with competing metal ions, often Cu^II or Zn^II, as indicated by eqn (1.51).

The stability constants of lanthanide complexes formed with endogenous ligands are generally orders of magnitude smaller than those of contrast agents. The mechanisms of ligand-exchange reactions are often associative and include the formation of ternary intermediate complexes. In this mechanism, some of the donor atoms of the exchanging ligand coordinate to the competing metal ion and others coordinate to the lanthanide ion. Overall ligand-exchange reactions are described by eqn (1.52).

Metal-exchange reactions can proceed through several pathways (Scheme 1.17). One such pathway is the exchanging metal-assisted mechanism, which involves direct attack of the metal ion at the complex, resulting in the formation of a dinuclear intermediate, [Ln(L)M]. In this intermediate, the functional groups of the ligand transfer stepwise from the Ln^III ion to the exchanging metal ion. Another possible mechanism involves the spontaneous or proton-assisted dissociation of the Ln^III ion from the complex followed by the reaction between the free ligand and the exchanging metal ion. The protonated complex can also be attacked by the exchanging metal ion resulting in a dinuclear protonated intermediate that dissociates to the exchanging metal complex and the Ln^III ion. Hydroxide-assisted pathways rarely play important roles in exchange reactions.

To obtain information regarding the kinetic inertness of metal complexes, metal-exchange reactions are frequently studied using at least 10-fold excess of the exchanging metal ion at different pH values. Under these conditions, reactions can be treated as pseudo-first-order processes, and thus, observed rate constants, k_obs, are also pseudo-first-order. In the presence of a large excess of the exchanging metal ion, the rate of exchange reactions can be expressed by eqn (1.53), where k_obs is the pseudo-first-order rate constant and [Ln(L)]_t is the total concentration of the Ln^III complex.

Considering all the possible pathways shown in Scheme 1.17, the concentration of Ln(L) can be given as the sum of the concentrations of all species comprising both L and M.

Eqn (1.53) and (1.54) can be combined to give the following rate equation:

The rate constants in eqn (1.55) characterize the rate of the spontaneous (k₀), proton-assisted (k_H and k$HH$), hydroxide-assisted (k_OH), hydroxide-metal-assisted (k$MOH$), metal-assisted (k_M), proton–metal-assisted (k$MH$), ligand-assisted (k_L), and proton–ligand-assisted (k$LH$) pathways.

Taking into account the different reaction pathways shown in Scheme 1.17 and the equations determining the K_Ln(HL), K_Ln(H2L), K_Ln(OH)(L), K_Ln(L)M(OH), K_Ln(L)M, K_Ln(HL)M, K_Ln(L)(L′), and K_Ln(HL)(L′) equilibrium constants, the pseudo-first-order rate constant (k_obs) can be expressed by eqn (1.56):

In eqn (1.56), K_LnHL=[Ln(HL)]/{[Ln(L)][H⁺]}, K_Ln(H2L)=[Ln(H₂L)]/{[Ln(HL)][H⁺]}, K_Ln(L)M=[Ln(L)M]/{[Ln(L)][M]}, K_Ln(HL)M=[Ln(HL)M]/{[Ln(HL)][M]}, K_Ln(OH)(L)=[Ln(OH)(L)]/{[Ln(L)][OH⁻]}, K_Ln(L)M(OH)=[Ln(L)M(OH)]/{[Ln(L)][M(OH)]}, K_Ln(L)(L′)=[Ln(L)(L′)]/{[Ln(L)][L′]}, K_Ln(HL)(L′)=[Ln(HL)(L′)]/{[Ln(HL)][L′]}, k₁=k_HK_Ln(HL), k₂=k$HH$K_Ln(HL) K_Ln(H2L), k₃=k_MK_Ln(L)M, k₄=k$MH$K_Ln(HL)M, k₅=k_OHK_Ln(OH)(L), k₆=k$MOH$K_Ln(L)M(OH), k₇=k_LK_Ln(L)(L′), and k₈=k$LH$K_Ln(HL)(L′). Eqn (1.56) takes into account all of the rational dissociation pathways and provides a general description for dissociation rates.

Kinetic data of slowly dissociating (macrocyclic) complexes can be obtained by performing dissociation reactions under conditions where the concentration of the complex is high (∼0.1 M) but the exchanging metal ion (for example, Cu^II) or its labile complex is present at a much lower concentration (pseudo-first-order conditions). Under these conditions, the rate of decomplexation (rate of formation of a CuL complex) can be followed by UV–visible spectrophotometry and expressed as follows:

In eqn (1.57), [Gd(L)]_t and [Cu(L)]_t are the total concentrations of the Gd(L) and Cu(L) complexes, and k_obs is the pseudo-first-order rate constant. The formation rates of Cu(L) complexes can be calculated from the slope of the absorbance versus time curves by accessing the concentration of Cu(L) complexes at various time points. The method requires knowledge of the molar absorptivity coefficients of the Cu^II or Cu(A) (where A is a ligand that binds Cu^II in a labile complex) and the Cu(L) complexes. This approach enables the rapid determination of the dissociation kinetic parameters; however, it also requires a relatively large amount of the complex Gd(L), even when micro-cuvettes are used.

Despite numerous publications describing the determination of rate constants (k₀) characterizing the spontaneous dissociation of metal chelates, there is no well-tested theory that satisfactorily interprets the molecular mechanism of this dissociation pathway. Based on the rate equations established by metal- or ligand-exchange reactions, the effect of the spontaneous pathway on the overall dissociation of a given complex must be independent of pH and of the concentrations of the exchanging partners. Analogously to the decay of radionuclides, spontaneous dissociation can be regarded as a monomolecular decay in which several coordinative bonds are broken simultaneously, a process that is induced by the fast internal motions of the molecule. These internal motions can be accelerated by molecular collisions with, for instance, solvent molecules. As expected, increasing the temperature increases the rate of spontaneous dissociation. Theoretically, spontaneous dissociation of GdL complexes could potentially be associated with the auto-dissociation of water, which results in the formation of H⁺ and OH⁻ ions, which in turn could induce decomplexation. Because this process would be independent of pH, it is unlikely.

The k₀ value of metal complexes is generally small, and therefore accurate determination of k₀ can be challenging. For example, the spontaneous dissociation of [Gd(DOTA)]⁻ was first determined to be <5×10⁻⁸ s⁻¹, which is within the limits of the experimental error.⁴⁸⁴  This value was later refined to be (5±2)×10⁻¹⁰ s⁻¹.³¹  Almost a decade later, the value of k₀ was re-determined for [Gd(DOTA)]⁻ to be (6.7±0.4)×10⁻¹¹ s⁻¹.⁴⁰⁸  For accurate determination of k₀, the kinetic inertness of a complex should be investigated at neutral or slightly basic conditions, where the role of proton-catalyzed dissociation is negligible or comparable with that of spontaneous dissociation. Considering that k₀ for [Gd(DOTA)]⁻ is near 10⁻¹⁰ s⁻¹, the half-life of such a reaction would be nearly 200 years, give or take a few decades. This timescale demonstrates why it is necessary to study the kinetic properties of the macrocyclic metal chelates in acidic conditions, where the relatively fast proton-assisted pathway is the major dissociation pathway. Under these conditions, the value of k₀ is frequently calculated to be zero or to have a negative value with large error, thus accounting for the discrepancies in the values of k₀ reported for [Gd(DOTA)]⁻. Although spontaneous dissociation rates of open-chain Ln^III complexes are orders of magnitude faster than those of macrocyclic complexes, k₀ can be neglected in most cases.³⁹¹ 

Complexes of ligands having highly basic donor groups, such as phosphonates or amines, can be protonated over a wide pH range. Frequently, investigation of kinetic inertness is performed in acidic solutions, where the basic donor groups are protonated. In these systems, k₀ characterizes the spontaneous dissociation of the protonated species Ln(H_xL), which means that information regarding dissociation of the completely deprotonated form, Ln(L), cannot be obtained from these measurements.^386,502  The presence of an OH⁻-assisted pathway in the exchange reaction of a complex can prevent the evaluation of k₀ because OH⁻-assisted dissociation becomes the dominant reaction pathway when the concentration of H⁺ decreases.^401,477,503 

Proton-assisted dissociation of metal complexes requires the formation of a protonated intermediate that can be either a thermodynamically stable species in the given pH range or a kinetically active but directly undetectable minor intermediate. The protonation of complexes usually occurs at pendant arms when they are dissociated from the metal ion. Following protonation, the proton from the pendant arm transfers to a nitrogen donor atom of the ligand backbone, which eventually leads to the dissociation of the metal ion from the complex. Proton transfer is generally a slow process because of the electrostatic repulsion between positively charged Ln^III ions and protons. The complexes formed between lanthanide ions and macrocyclic ligands are relatively inert to proton-assisted dissociation owing to the rigid structure of the coordination cavity. Proton-assisted dissociation of open-chain complexes is several orders of magnitude faster.^{30,386,391,403,404,407,484,504–509} 

Because of their high kinetic inertness, the dissociation reactions of macrocyclic complexes can in practice only be investigated in acidic or highly acidic conditions. Because macrocyclic lanthanide complexes are not stable thermodynamically in highly acidic solutions, their dissociation can be studied without the use of exchanging metal ions or ligands.^{31,250,386,403,404,477,484,510}  The determination of the rate constants (k₁ and k₂) characterizing the proton-assisted pathways is relatively straightforward. In some cases, however, the rate constants k₁ and k₂ cannot be determined from the kinetic data because the metal-exchange reaction involves competing metal- or hydroxide-assisted pathways. In such cases, an alternative method must be found to characterize the proton-assisted pathway independently. These alternative methods might require performing dissociation reactions under acidic conditions or ligand-exchange reactions in an appropriately chosen pH range.^477,503,511 

The metal-assisted pathways (k_M and k$MH$) play an important role in the metal-exchange reactions of complexes with flexible open-chain ligands but can be neglected in the dissociation of rigid macrocyclic complexes. In metal-assisted pathways, the formation of a relatively stable dinuclear intermediate is essential as it enables the donor atoms of the ligand to be transferred in a stepwise fashion to the incoming metal ion. Higher stabilities of dinuclear intermediates correlate with larger rate constants (k_M), which characterize metal-assisted dissociation. The direct interaction of the exchanging metal ion becomes important at pH values where the extent of proton- and metal-assisted dissociation are similar. Thus, the metal-assisted pathway is the dominant one for transmetallation under physiological conditions.^388,507,512  The exchanging metal ion can also interact with the protonated complex forming a protonated dinuclear intermediate.^391,506 

The hydroxide-assisted and hydroxide-metal-assisted pathways (k_OH and k$MOH$) rarely play a role in the exchange reactions of metal complexes. Nevertheless, the strength of the interactions between metal ions and coordinating ligands can be weakened by the coordination of a hydroxide ion due to a decrease of electrostatic attraction, resulting in faster dissociation.⁴⁰¹  When Cu^II is used as the exchanging metal ion, the dissociation of the complex depends on the concentrations of both OH⁻ and Cu^II.^477,503  The kinetic activity of [Cu(OH)]⁺ is greater than that of the Cu^II aqua ion in substitution reactions.⁵¹³ 

The rates of ligand-exchange reactions of contrast agents for MRI are rarely investigated. However, some data have been published on the role of ligand-exchange reactions in the dissociation of clinically approved and potential MR contrast agents. These studies revealed that the exchanging ligands can directly attack both the deprotonated and protonated form of the metal complex (k_L and k$LH$), forming ternary intermediates.⁵¹¹  Small endogenous ligands such as citrate, carbonate, or phosphate accelerate in vivo dissociation because the formation of ternary complexes between the contrast agents and these ligands eventually leads to the release of Gd^III ions.^407,514  These results demonstrate that the fast dissociation of ternary complexes formed with endogenous ligands can contribute to the in vivo dissociation of contrast agents. Previously, the dissociation assisted by endogenous metal ions was thought to be the most important dissociation pathway in spite of the low plasma concentration of these ions.

Last but not least, the protonation and stability constants of the intermediates formed during exchange reactions (K_Ln(HL), K_Ln(H2L), K_Ln(OH)(L), K_Ln(L)M(OH), K_Ln(L)M, K_Ln(HL)M, K_Ln(L)(L′), and K_Ln(HL)(L′)) often cannot be calculated because the concentrations of these complexes are low and their kinetic activity is high. With the exception of the protonated metal chelates, the independent determination of these constants is not possible because most of these intermediates are short-lived, reactive species. Values of K_Ln(HL) and K_Ln(H2L) can be determined separately and are often used as fixed constants in the calculations of kinetic parameters.

Several methods have been proposed to characterize the kinetic inertness of Gd^III complexes. These include optical methods such as UV–visible spectrophotometry,⁵¹⁵  luminescence spectroscopy, and luminescence-life-time measurements,⁵⁰⁴  direct NMR methods, T₁ or T₂ relaxometry,^408,516  capillary electrophoretic methods,⁵¹⁷  high-performance liquid chromatography methods, and inductively coupled plasma atomic emission spectrometry or inductively coupled plasma mass spectrometry coupled with other separation techniques.^518–520  Among these methods, UV–visible spectrophotometry is the most widespread owing to the availability of the instrument and applicability of the technique to a broad range of reaction times. Stopped-flow UV–visible spectrophotometry can be used for fast reactions, whereas conventional UV–visible spectrophotometry is applicable for reactions occurring on time scales of a few minutes to a few days. The rates of slow reactions can be followed by periodically measuring the absorbance of samples over a period of months. However, it should be kept in mind that these methods are limited to the study of complexes (or products of dissociation) that absorb either in the UV or in the visible range of the electromagnetic spectrum. Because this is often not the case for Gd^III complexes, Cu^II is usually used as a ligand scavenger when studying metal-exchange reactions; Cu^II usually rapidly forms complexes with ligands released from contrast agents. The ligand-to-metal charge transfer transitions of Cu^II complexes typically have strong absorptions in the UV–visible range of the electromagnetic spectrum. Thus, the dissociation of the Ln(L) complexes can be monitored indirectly by measuring the formation of the CuL complexes.

Studying the dissociation reactions of Eu^III complexes is another option. Because Eu^III has a nearly identical radius to Gd^III, Eu^III and Gd^III complexes are expected to have similar dissociation kinetics. The absorption bands of Eu^III that form broad shoulders in the range of 220 to 300 nm enable direct measurement of dissociation rates. Direct NMR (¹H-NMR spectra of La^III, Y^III, Eu^III, and Lu^III complexes) and relaxation techniques (T₁ or T₂ relaxometry) can be used to monitor reactions occurring on the half hour or longer time scales. This technique is more appropriate for slower reactions because each data points require seconds to tens of seconds (for the relaxation techniques) to be acquired. Experimental methods that can be used at moderately acidic or higher pH ranges, such as the ones based on capillary electrophoresis or high-performance liquid chromatography, are routinely used for studying dissociation reactions near physiological conditions as well as in more complex media such as artificial fluids that mimic biofluids or serum.

The kinetic stabilities of complexes are routinely characterized by the rates of their dissociation, k_d, first order-rate constants measured in HCl (0.1 M) or by the rates of transmetallation reactions occurring with Zn^II, Cu^II, or Eu^III ions in the pH range of 3 to 6 (25 °C and 1.0 M KCl).^507,512,521  Unfortunately, the rate constants obtained under such conditions cannot be directly used to estimate the kinetic behavior of complexes close to physiological conditions (pH 7.4, 37 °C, and 0.15 M NaCl). To understand and describe the kinetic properties of metal complexes, the rates of dissociation reactions should be studied as close as possible to physiologically relevant pH and other biological conditions.

It is generally assumed that dechelation of metal complexes in biological fluids takes place via transmetallation reactions with Zn^II and Cu^II. The formation of dinuclear complex intermediates is essential for metal exchange. Dechelation can also occur via ligand exchange reactions, whereby an endogenous ligand displaces the chelating agent of the Gd^III complex. Phosphate ions compete with the aminopolycarboxylate ligands for Gd^III in the presence of ZnCl₂ or CuCl₂ to yield the insoluble GdPO₄ (GdPO₄: K_sp=10^−20.53 at 37 °C in 0.15 M NaCl).⁴⁰⁷  In the absence of these metal salts, dissociation reactions do not proceed. The rate-determining step of transmetallation and ligand-exchange reactions is dissociation of the metal followed by fast reactions of the released metal ion and ligand with the exchanging ligand or metal ion, respectively.

Dissociation reactions of macrocyclic Gd^III complexes occur mainly via proton-assisted pathways.^{408,507,512,521}  Dissociation reactions of Gd^III complexes formed with derivatives of DTPA are affected by citrate, phosphate, and bicarbonate ions, which form labile ternary intermediates at physiological conditions.^407,408  The kinetic properties of the complexes formed with other Ln^III ions and Y^III are similar to those observed for Gd^III complexes.²⁵³ 

Several methods have been proposed to characterize the kinetic inertness of Gd^III complexes close to physiological conditions. These include (i) relaxometric;⁵²²  (ii) spectrophotometric;^407,408  and (iii) high-performance liquid chromatography,^523,524  inductively coupled plasma atomic emission spectroscopy,⁵²⁴  inductively coupled plasma mass spectrometry,⁵²³  and capillary electrophoresis^407,525,526  measurements.

• Relaxometric Measurements

  One relaxometric method determines the rates of dechelation of Gd^III complexes by the rate of formation of insoluble GdPO₄.⁵²²  In phosphate buffer at pH 7 and 37 °C, non-complexed Gd^III, released from Gd^III complexes in the presence of Zn^II, forms GdPO₄, which precipitates from solution. This results in a decrease of the relaxation rate of the solution. The experimental setup of this method is shown in Figure 1.28. The ratios of the relaxation rates measured at time t and at the start (t=0) of the reaction [R₁^P(t)/R₁^P(t=0)] for [Gd(DOTA)]⁻, [Gd(EOB–DTPA)]²⁻, [Gd(DTPA)]²⁻, and Gd(DTPA-BMA) are shown in Figure 1.29.

  This method enables rapid comparison of different complexes, but it cannot be used for kinetic studies because Zn₃(PO₄)₂ precipitate also forms during the reaction, decreasing the concentration of Zn^II over time. Additionally, an excess of phosphate increases the dissociation rates of Gd^III complexes owing to the phosphate-assisted mechanism.^407,408  Importantly, the kinetic properties of macrocyclic Gd^III complexes cannot be compared with this method because the relaxation rates of solutions of macrocyclic complexes do not change over time (Figure 1.29).

• Spectrophotometric Measurements

  Because of the broad and the intense absorption bands (ε≥1000 M⁻¹ cm⁻¹) of Cu^II complexes in the UV range of the electromagnetic spectrum, transmetallation reactions between Gd^III and Cu^II can be monitored by spectrophotometry at nearly physiological conditions.^407,408  This experimental set up is shown in Figure 1.30. Example absorbance versus time plots for the reaction of Gd(DTPA-BMA) with Cu^II at 25 and 37 °C in the presence of endogenous ions are shown in Figure 1.31.

  In Figure 1.31, the absorbance versus time curves are straight lines for about 7 minutes at 37 °C, after which time the absorbance values increase more rapidly because the formation of GdPO₄ precipitate scatters light. In this case, the excess citrate is not enough to prevent the formation of GdPO_4(s). From the slope of the straight lines, the k_d first-order rate constants characterizing the dissociation of the Gd(L) can be calculated using eqn (1.57), where ΔAbs is the increase of the absorbance during a time Δt; [Gd(L)]_t is the total concentration of Gd(L); ε_Cu and ε_CuL are the molar absorptivities of [Cu(Cit)H₋₁]²⁻ and the Cu(L) complex formed in the transmetallation reactions, respectively (e.g. [Cu(Cit)H₋₁]²⁻: ε_Cu(Cit)H−1=921 M⁻¹ cm⁻¹ and [Cu(DTPA-BMA)]⁻: ε_CuL=3293 M⁻¹ cm⁻¹ at 300 nm). The k_d values obtained for the dissociation of Gd(DTPA-BMA) at 25 and 37 °C are 1.2×10⁻⁵ s⁻¹ (t_1/2=15.9 h) and 3.8×10⁻⁵ s⁻¹ (t_1/2=5.1 h), respectively.⁴⁰⁷  The spectrophotometric method enables determination of dissociation rates of metal complexes at nearly physiological conditions. Moreover, the contribution of the different endogenous ions, including citrate, phosphate, carbonate, and lactate, to the dissociation rates of metal complexes can be determined by following the reactions in the presence of different concentrations of the ions of interest.

• High-performance Liquid Chromatography (HPLC), Inductively Coupled Plasma Atomic Emission Spectroscopy (ICP-AES), Inductively Coupled Plasma Mass Spectrometry (ICP-MS), and Capillary Electrophoresis (CE) Measurements

  Several experiments were performed with sensitive analytical methods to determine the kinetic inertness and the amount of Gd^III released in the dissociation reactions of Gd^III-based contrast agents for MRI in human samples.^{407,523–526}  The biodistribution of Omniscan was studied with inductively coupled plasma atomic emission spectroscopy (ICP-AES) and high-performance liquid chromatography (HPLC) methods in the blood and fecal samples of patients with severe renal insufficiency. The results indicated that ICP-AES and HPLC methods are suitable for detection of Gd^III in biological samples. Moreover, there was no evidence of metabolism of the chelator or transmetallation of Omniscan in samples collected seven days after the administration of the Gd^III-based contrast agent.⁵²⁴ 

  The dissociation rates of all marketed Gd^III-based contrast agents for MRI were examined at pH 7.4 and 37 °C in human serum samples.⁵²³  In these experiments, serum was spiked with Gd^III complexes to obtain a final concentration of 1 mmol L⁻¹. The assay mixtures were stored at 37 °C in closed vials. The samples were analyzed by HPLC-ICP-MS before the start of the incubation, 8 hours after the start, and after 1, 2, 3, 4, 5, 6, 7, 9, 11, 13, and 15 days of incubation. The results indicated that the dissociation of Gd^III-based contrast agents took place with the release of Gd^III in human serum samples. Macrocyclic Gd^III complexes showed higher kinetic inertness than the linear ones after 15 days at 37 °C in human plasma and in the presence of elevated phosphate levels (10 mM).⁵²⁴  In some experiments, the possible products of transmetallation of Gd^III-based contrast agents have been examined by capillary electrophoresis.^407,525,526  In the blood plasma of healthy volunteers, the products of transmetallation reactions of [Gd(DTPA)]²⁻ with Cu^II, Zn^II, and Fe^III could not be detected using capillary electrophoresis–electrospray ionization time-of-flight mass spectroscopy.^525,526  However, the dissociation of Gd(DTPA-BMA) and the formation of [Ca(DTPA-BMA)]⁻ in serum samples spiked with Omniscan (2 mM) at pH 7.4 and 37 °C was detected by micellar electrokinetic capillary chromatography measurements.⁴⁰⁷ 

To assess the amount of Gd^III released in body fluids, the equilibrium, kinetic, and pharmacokinetic properties of Gd^III-based contrast agents must be taken into account. Gd^III complexes are administered intravenously in doses of 0.1–0.3 mmol Gd^III kg⁻¹ body weight. These Gd^III complexes are distributed rapidly in the extracellular space of the body and the elimination occurs primarily through the kidneys via glomerular filtration. The half-life of elimination is generally 1.3–1.6 hours in patients with normal kidney function.^527,528  In patients with severe renal insufficiency, the rate of glomerular filtration is slow, and the half-life of elimination of Gd^III complexes can be in the range of 30–90 hours. During such a long residence time, Gd^III complexes might partially dissociate, with release and subsequent accumulation of Gd^III.⁵²⁹  The extent of release of Gd^III can be expressed by the stability constants of different complexes formed in body fluids. Considering the known stability constants, the equilibrium of competition reactions can be predicted between Gd^III and endogenous metal ions, in particular, Cu^II, Zn^II, and Ca^II, for a ligand.

Importantly, the amount of non-complexed Gd^III accumulating in the body is strongly influenced by the rate of dissociation and the rate of glomerular filtration. The latter is much faster for patients with normal kidney function than those with end-stage renal disease.¹  To predict the amount of the Gd^III released in the body fluids of patients with normal kidney function and with renal insufficiency, an open two-compartment model was developed by considering the equilibrium and the kinetic and the pharmacokinetic properties of Gd^III complexes (Figure 1.32).⁴⁰⁷ 

In Figure 1.32, k_int=k_−int=ln2/t_1/2α, k_el=ln2/t_1/2β. The rate constants k_int, k_el, and k_d characterize the distribution, elimination, and dissociation, respectively, of Gd^III complexes. The half-lives of distribution (t_1/2α) and elimination (t_1/2β) are 0.062 and 1.28 h, respectively. The rate constant k_f corresponds to the reformation of Gd^III complexes, and this rate constant can be calculated from the relationship K_e=k_f/k_d. The equilibrium constant K_e can be calculated from the dissociation equilibrium.

The time dependence of the concentration of Gd(DTPA-BMA) has been simulated during its distribution in the vascular and interstitial spaces, and the elimination from the extracellular space of patients with normal renal function and with renal insufficiency. The simulation was performed by taking into account the rates of distribution and elimination determined by the pharmacokinetic studies, the rates of reformation and the dissociation obtained by the equilibrium and kinetic studies, as well as all the possible pathways of distribution, dissociation, and elimination shown in Figure 1.32. The data obtained are shown in Figure 1.33.

The data calculated for patients with normal kidney function using this the model (Figure 1.33A) are in agreement with clinical observations, indicating that about 95% of the contrast agent is eliminated from the body of patients with normal kidney function in about 48 hours.^530,531  The open, two-compartment model enables calculation of the amount of the metal ion released from any metal complexes in the body of patients with different levels of renal insufficiency.

As outlined above, a variety of kinetic methods are available to study and characterize the kinetic properties of complexes relevant to MRI. The best approach would be the comparison of the dissociation kinetic parameters obtained at or near physiological conditions. However, because the amount of such data is limited, the rate of acid-catalyzed dissociation is normally used for comparison.

The structure of ligands—including the nature and number of the donor atoms, geometry, preorganization, and rigidity—affects the thermodynamic and kinetic parameters of complexes formed with the ligands. Prior to data analysis, modifications applied to basic ligand structures should be defined. Common ligand structures relevant to MRI include open-chain (also known as linear or acyclic chelators), tripodal and macrocyclic ligands, and ligands consisting of both acyclic and cyclic units (for example, AAZTA, NETA, NPTA, and DEPA).^{299,532–537}  Ligands are modified for various reasons, such as lowering the overall charge of the complex, incorporating a chemically reactive group for linking or targeting purposes, or accelerating the rate of complex formation. These changes to the structures of ligands require consideration because they can induce changes to the properties of their corresponding complexes, including changes in equilibrium, formation and dissociation kinetics, and relaxivity.

Modification of ligand structures might be performed either on the sidearm of the ligands^{78,172,207,293,297,538–548}  or on the backbone of the ligand (Figure 1.34).^{70,114,313,549}  Modifications made to the backbone of the ligands are often preferable because they maintain or improve the inertness of the resulting complexes. This increase in inertness is expected given that demetallation of complexes requires structural rearrangements. These rearrangements tend to proceed more slowly for complexes formed with more constrained ligands, and backbone modifications usually rigidify ligands.

The first marketed contrast agent for MRI was [Gd(DTPA)]²⁻ in the early 1980s (Figure 1.35).⁵⁵⁰  The rates of acid- (k₁=0.58 M⁻¹ s⁻¹) and metal-ion-catalyzed dissociation (k₃^Cu(II)=0.93 M⁻¹ s⁻¹, k₃^Zn(II)=0.056 M⁻¹ s⁻¹ and k₃^Eu(III)=4.9×10⁻⁴ M⁻¹ s⁻¹) were evaluated by studying the metal-exchange reactions of [Gd(DTPA)]²⁻ occurring with Zn^II, Cu^II, and Eu^III ions.⁵⁰⁷  From these data, it is evident that among the metal ions available in vivo, Cu^II has the greatest catalytic effect on the dissociation of [Gd(DTPA)]²⁻. The rate constants of acid- and Cu^II-catalyzed dissociation are often comparable, and the dissociation via Zn^II generally occurs one to two orders of magnitude slower than the reaction catalyzed by Cu^II. This trend holds for nearly all open-chain contrast agents for MRI. The removal of the central acetate (DTTA),^551,552  replacement of the central nitrogen atom with oxygen (OBETA),^514,553  and substitution of more basic phosphonate (DTTAP) or phosphinate groups (DTTAP^Ph) for the central carboxylate increase the lability of the Gd^III complexes (see Figure 1.35 for ligand structures). For complexes of these modified ligands, k₁ is (3–6)×10³ times greater than that of the parent [Gd(DTPA)]²⁻.⁵⁵⁴ 

Replacing the acetate arm attached to the central nitrogen with a pyridine nitrogen, such as in the pyridine-based DTTA derivatives with an extra methoxy or triazole ring systems, yields relatively inert Ln^III complexes (Table 1.2). The inertness of these derivatives is comparable to that of [Gd(DTPA)]²⁻. This observations indicates that the loss of inertness due to removing a single acetate arm can be compensated by rigidifying the molecule.^{388,555–558} 

Replacing one or two carboxylates by amides, such as in DTPA-NMA, DTPA-N′MA, DTPA-BMA, and DTPA-BMEA, yields charge-neutral Gd^III contrast agents that lower the osmotic pressure of the injectable solutions (see Figure 1.35 for ligand structures). These complexes are, however, less inert than [Gd(DTPA)]²⁻. Moreover, these modifications further increase the dissociation rate of the Gd^III complex in the presence of bioligands. The rate of dissociation is somewhat slower for Gd^III–DTPA-tris(amides) derivatives compared to Gd(DTPA-BMA) because the rates of acid- and metal-ion-assisted dissociation are slower for the tris-amide ligands. The rates of acid-catalyzed dissociation are also slower for 15-membered macrocyclic ligands such as 15-DTPA-EAM (Figure 1.35). Dissociation is also slower for Gd^III complexes of propionates derivatives of DTPA and for derivatives incorporating a propylene diamine chain as opposed to the ethylene diamine one of DTPAs.

The second generation of commercially available contrast agents, including the liver-specific agents [Gd(BOPTA)]²⁻ and [Gd(EOB-DTPA)]²⁻ and the albumin binding MS-325, each have modifications on their side arms or backbone relative to the parent ligand. As a result of these changes, the Gd^III complexes are more inert: k₁=0.41, 0.12, and 0.029 M⁻¹ s⁻¹ for [Gd(BOPTA)]²⁻, [Gd(EOB–DTPA)]²⁻, and [Gd(MS-325)]³⁻, respectively. [Gd(MS-325)]³⁻ (Ablavar) is the most inert commercially available linear-based contrast agent.

Gd^III complexes of the cyclohexanediamine derivative of DTPA likely have slower rates of dissociation than [Gd(DTPA)]²⁻. This speculation is based on the limited number of a studies reported with the Y^III analogue. Because Y^III and Gd^III have similar sizes, the corresponding complexes are expected to have similar dissociation rates. The methods used to obtain the kinetic data of these Y^III complexes, however, only yield conditional rate constants. Thus, data obtained this way can be used to make conclusions only for systems investigated by the same method. Nevertheless, these studies indicate that complexes of rigidified DTPA derivatives CHX–DTPA, CHX-A, and CHX-B dissociate 200, 300, and 3000 times more slowly than [Y(DTPA)]²⁻ (Figure 1.35).⁵⁷²  These studies also reveal that the incorporation of methyl or nitrobenzyl groups on the backbone of DTPA slows the dissociation of the corresponding complexes.⁵⁷² 

CDDADPA and CyAAZTA are derivatives of the octadentate ligand OCTAPA and the hexadentate ligand AAZTA (Figure 1.35).^38,562  [Gd(CDDADPA)]⁻ is among the most inert Gd^III complexes incorporating a linear ligand. Its dissociation rate is 1 to 3 orders of magnitude slower than that of the clinically approved Gd(DTPA-BMA), Gd(DTPA-BMEA), [Gd(DTPA)]²⁻, [Gd(MS-325)]³⁻, [Gd(BOPTA)]²⁻, and [Gd(EOB–DTPA)]²⁻. [Gd(CyAAZTA)]⁻ also displays slow dissociation rate relative to other bisaquated (q=2) complexes. High kinetic inertness was observed for a trinuclear Gd^III complex of thiacalix[4]arene-p-tetrasulfonate (TCAS) in which three Gd^III ions are chelated by two TCAS ligands.⁵⁶³  The k₁ value for [Gd₃(TCAS)₂]⁷⁻ (8.1×10⁻³ M⁻¹ s⁻¹) compares favorably to that of approved contrast agents, and is similar to that of [Gd(CyAAZTA)]⁻, although it is still faster than that of macrocyclic contrast agents.

The forefather of macrocyclic-based contrast agents, [Gd(DOTA)]⁻, is the most inert contrast agent. The kinetic inertness of this complex originates from the rigid, preorganized structure of the ligand. Gd^III and other metal ions are deeply buried in the coordination cage defined by the four macrocyclic nitrogen atoms and four carboxylate oxygen atoms. The coordination sphere of Gd^III in [Gd(DOTA)]⁻ is inaccessible to other ligands, except for the one inner-sphere water molecule, which is essential to relaxivity. The cavity of the ligand can be occupied by an exchanging metal ion only when the Gd^III ion leaves. Thus, complexes of DOTA and DOTA-type ligands dissociate via acid-catalyzed pathways and competing metal ions or bioligands do not accelerate dissociation rates (Figure 1.36).^{148,403,404,408,479,484,505,564,565,567}  Thus, the kinetic inertness of DOTA-type complexes is often characterized only by the rates of acid-catalyzed dissociation. [Gd(DOTA)]⁻ is the gold standard when comparing the inertness of Ln^III complexes.

Gd^III complexes formed with derivatives of DOTA tend to remain inert even when the coordination environment of the metal ion is affected by the nature of donor atoms. When one sidearm is removed completely as in the case of Gd(DO3A), however, the rate of acid-catalyzed dissociation increases dramatically by a factor of ∼10⁴.^479,564  Data published on a sulfonamide-based pH sensor and a pH-responsive chelate possessing an aminoethyl sidearm indicate that the rate constant of acid-catalyzed dissociation increases with modifications to DO3A. This should be considered when using the DO3A platform to design responsive contrast agents.^564,565 

DO3A type ligands can be rendered more inert by incorporating a pyridine moiety, which rigidifies the macrocycle, as in PCTA. A detailed dissociation kinetic study involving several Ln^III complexes of PCTA demonstrated that this rigidification slows dissociation relative to DO3A (k₁=5.1×10⁻⁴ M⁻¹ s⁻¹ for Eu(PCTA)).³⁰  Other studies including various bifunctional and rigidified derivatives of PCTA confirm these results.^{313,573–575}  Ln^III complexes of the bifunctional p-NO₂-Bn-PCTA chelator have similar dissociation kinetics as those of complexes formed by the parent PCTA ligand. This indicates that the attachment of the p-nitrobenzyl substituent to the rigid PCTA backbone has little effect on rates of dissociation.³¹³  Although the kinetic inertness of complexes of heptadentate ligands can be increased by rigidifying the ligand backbone, they remain more labile than [Gd(DOTA)]⁻.

Octadentate DOTA-type ligands obtained by enlargement of the macrocycle or replacement of one or more acetates with moieties containing alcohols or amines are more labile than [Gd(DOTA)]⁻. Replacing one carboxylate group by an acetamide does not affect the dissociation kinetic properties of the resulting complexes.^403,570  However, replacing an acetamide arm by a propionamide arm decreases the rate of acid-catalyzed dissociation of Ln(L) complexes by three order of magnitude.⁴⁰³  Gd^III complexes of macrocyclic ligands possessing an alcohol have somewhat higher rates of acid-catalyzed dissociation than Gd^III chelates of DOTA-monoamides.

Replacement of all four acetate sidearms in DOTA by acetamides leads to DOTA-tetraamides that are capable of inducing contrast via chemical exchange saturation transfer (see Chapter 3.1). Lanthanide complexes of DOTA-tetraamides are the most inert complexes of all DOTA-type ligands studied to date. The exceptional kinetic inertness of Ln^III complexes formed with DOTA-tetraamides can be rationalized in terms of the low basicity of the amide oxygen of DOTAM compared to the carboxylate oxygen of DOTA. Protonation and subsequent proton transfer to a ring nitrogen occurs is unlikely. The amide functionally offers another approach to fine tune the kinetic inertness of the resulting complexes because the nature of the amide group (primary, secondary, or tertiary) can also affect the rates of dissociation. Among the DOTA-tetraamides, the dimethylamide complexes have the highest kinetic inertness because the methyl groups hinder the transfer of protons to ring nitrogens, slowing dissociation.^105,404 

Lanthanide chemical exchange saturation transfer agents based on the 3,6,10,13-tetra-aza-1,8(2,6)-dipyridinacyclotetradecaphane macrocyclic platform containing four hydroxyethyl pendant arms were found to be extraordinarily inert.⁵⁷⁶  The rates of acid-catalyzed dissociation of complexes formed with Ln^III ions were not quantitatively evaluated in the this study because the complexes were so inert that the extent of dissociation of the [Eu(L)]³⁺ complex in competitive media (e.g. 1 M HCl) over a period of months at room temperature was negligible as studied by ¹H-NMR spectroscopy. Another astonishingly high kinetic inertness was observed for Ln^III complexes formed with a rigid cross-bridged cyclam derivative containing two picolinate pendant arms. This complex does not undergo dissociation under harsh conditions, such as 2 M HCl; under these conditions, no release of metal ion was detected for at least 5 months.⁴⁹⁷  The absence of a bound water molecule in this complex, however, prevents its application as a T₁-shortening contrast agent. Regardless, this compound demonstrates the possibility to further rigidify macrocyclic ligands and increase the kinetic inertness of Ln^III complexes.

The authors thank the Hungarian Scientific Research Fund (K-120224 project). The writing of this book chapter was supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences (G.T. and F.K.K.), the EU and co-financed by the European Regional Development Fund under the project GINOP-2.3.2-15-2016-00008 as well as the COST CA15209 “European Network on NMR Relaxometry” Action. F.K.K. and G.T. are also thankful for support from the Le Studium – Loire Valley Institute for Advanced Studies. Special thanks go to Prof. Dr Sophie Laurent for granting access to Figure 1.29.