 1.1 Introduction
 1.2 SteadyState Diffusion
 1.2.1 SteadyState Diffusion Within the Native ECM
 1.2.2 SteadyState Diffusion Within Synthetic Hydrogels
 1.3 Soluble Factor Generation and Consumption
 1.3.1 Soluble Factor Generation and Consumption Within the Native ECM
 1.3.2 Soluble Factor Generation and Consumption Within Synthetic Hydrogels
 1.4 Matrix Interactions
 1.4.1 Interactions with ECM Components
 1.4.2 Interactions with Hydrogel Components
 1.5 Temporal Dependencies
 1.5.1 Temporal Dependencies During In Vivo Transport
 1.5.2 Temporal Dependencies During Transport in Synthetic Hydrogels
 1.6 Convection
 1.6.1 In Vivo Transport by Convection
 1.6.2 Transport in Synthetic Hydrogels by Convection
 1.7 Future Directions and Concluding Remarks
 References
Chapter 1: Soluble Molecule Transport Within Synthetic Hydrogels in Comparison to the Native Extracellular Matrix

Published:18 Mar 2014

Series: Soft Matter Series
M. Parlato and W. Murphy, in Hydrogels in CellBased Therapies, ed. C. J. Connon and I. W. Hamley, The Royal Society of Chemistry, 2014, ch. 1, pp. 130.
Download citation file:
Soluble molecule signalling and gradient formation are of known biological importance and direct many biological processes. Because there are many similarities between synthetic hydrogels and the native extracellular matrix (ECM), synthetic hydrogels can serve as model systems for the study of soluble molecule signalling and gradient formation in biological systems. Additionally, drug delivery technologies, bioactive peptides, and degradable polymer chemistries have all been incorporated into hydrogels to recapitulate many of the aspects of soluble transport that are found in the native ECM. Though transport within hydrogels and the native ECM have both been extensively reviewed, the purpose of this chapter is to compare and contrast the two. How does transport of soluble molecules within synthetic hydrogels compare to that in the native ECM, and how can synthetic hydrogels be made to more closely mimic the signalling of the native ECM? In this chapter, wellknown, fundamental models of molecular transport are used to introduce and discuss parameters that influence soluble molecule signalling and gradient formation, with a particular emphasis on emerging bioinspired approaches to mimic the natural ECM.
1.1 Introduction
Soluble factor signalling and gradient formation are of known biological importance and direct processes such as stem cell differentiation,^{1,2 } cellular migration,^{3–6 } limb bud development,^{7–9 } and neural tube development.^{7 } Soluble transport within the in vivo environment is complex, involving spatiotemporal interactions and molecular recognition between soluble molecules and extracellular matrix (ECM) components.^{2–16 } Because of such complexity, what is known and what can be studied about soluble transport in vivo is limited. Therefore, the use of welldefined in vitro experimental platforms is an attractive option. Because of the similarity of hydrogels to the native ECM, synthetic hydrogels can serve as model systems for the study of soluble transport and gradient formation within the ECM.^{17–19 } Synthetic hydrogels are also useful because of their biocompatibility and adaptability for use with a variety of chemistries.^{17,19–26 }
The hydrated polymer chains of synthetic hydrogels slow solute movement just as the macromolecules within the ECM do, thus assisting in the formation of concentration gradients.^{27 } Furthermore, drug delivery technologies have been incorporated into synthetic hydrogels that serve as welldefined soluble factor sources and sinks within the hydrogel.^{28–31 } Other experimental approaches seek to incorporate the ability of the native ECM to specifically bind and release soluble molecules into synthetic hydrogels by the incorporation of proteoglycans^{21,26,32–34 } or peptides that have high binding affinities for specific soluble molecules.^{31,35–37 } Many methods also exist that exert temporal control over transport within synthetic hydrogels by allowing the hydrogel to degrade over time, be remodelled by cellsecreted enzymes, or respond to external cues such as temperature or pH.^{20,24,30,38–47 }
There are many articles and reviews that discuss the first principles of transport within the native ECM and synthetic hydrogels separately;^{27,29,48–52 } however, the purpose of this chapter is to compare and contrast the two. We endeavour to address some of the critical questions that arise during development of synthetic hydrogels to mimic natural signalling gradients in the ECM, such as: (1) how does transport and gradient formation of soluble molecules within synthetic hydrogels compare to that within the native ECM? (2) What aspects of signalling within the native ECM have been mimicked within synthetic hydrogels and what aspects remain to be explored? and (3) what are the potential consequences of these differences, and how can the synthetic hydrogels be made to more closely mimic the signalling of the native ECM? This chapter is divided into five sections based on the following parameters that influence molecular transport in natural or synthetic ECMs: steadystate diffusion, soluble factor generation and consumption, matrix interactions, temporal dependencies, and convection. Each of these sections is divided into two subsections. The first subsection discusses the topic with regard to the native ECM and the second with regard to synthetic hydrogels. Finally, the chapter concludes with a short discussion of the future directions for synthetic hydrogels that seek to recapitulate various aspects of signalling in the native ECM.
1.2 SteadyState Diffusion
1.2.1 SteadyState Diffusion Within the Native ECM
Soluble factor gradients within the ECM are generated through a variety of mechanisms, but to begin the discussion, a simple case with defined ‘source’ and ‘sink’ regions is discussed. In the simplest scenario, defined source and sink regions occur due to one group of cells producing large amounts of soluble molecules while another nearby group does not produce these molecules and instead consumes them. The goal of this section is to understand the basic mechanisms by which soluble factor gradients may form within the native ECM. Additionally, we examine what fundamental properties of both the soluble factor and the native ECM affect this gradient formation.
Within this section, all sources and sinks are assumed to exist at a single point in space to facilitate mathematical descriptions. Therefore, they are referred to as ‘pointsources’ and ‘pointsinks’. They are also assumed to produce or consume molecules instantaneously and without limits. Due to these properties, they are referred to as ‘perfect sources and sinks’. Furthermore, we assume that the ECM in which the molecules are diffusing is homogeneous and that all parameters are constant with time (i.e. steady state). Notably, biological scenarios do not feature perfect, pointsources and pointsinks, but this is a useful and widely utilized starting point in discussions of transport in the native ECM.^{12,15,53 } A diagram of this problem is shown in Figure 1.1a. A summary of these assumptions is as follows:
All regions are homogeneous.
The source region is a perfect, point source.
The sink region is a perfect, point sink.
All parameters are constant with time (‘steady state’).
To analyse this situation, Fick's first law of diffusion is applied (eqn (1.1)) This law states that the mass flux, J, of a solute through a region of space is proportional to the rate of change of the solute's concentration with respect to position, dC/dx. We assume that at the position x = 0, the solute is produced in a way that maintains a constant concentration, C_{0}, and this assumption is used to develop the boundary condition (eqn (1.1), BC 1).
Eqn (1.1) shows the diffusion and gradient formation within a homogenous region from Fick's first law. Here C is the concentration of the soluble factor, J is the mass flux of the solute, x is position, and D is the diffusion coefficient. Here a linear relationship is demonstrated that is dependent on initial concentration of the molecules at the source (C_{0}), mass flux (J), and the solute diffusion coefficient (D). The ratio of J/D is the slope of the linear concentration gradient (Figure 1.1b), and C_{0} is the yintercept. This problem can also be solved by Fick's second law of diffusion. However, this law is applicable with nonsteady state problems because it assumes nonconstant mass flux, so it is discussed in Section 1.4.1 where temporal dependencies are discussed in detail.
In general, solutes that are small in size when compared to pores in the ECM diffuse quickly, and those that are large diffuse more slowly, because that larger solutes interact with the matrix more often than smaller solutes.^{51,54,55 } This slowing of molecular movement is referred to as ‘sieving action’. Many models exist to approximate the size of a molecule, and a commonly utilized model is the Stokes–Einstein relationship (eqn (1.2)). This model assumes a spherical molecule with a density close to that of water (1 g mL^{−1}).^{54 } Molecular radii calculated from this relationship are referred to as hydrodynamic radii (represented by the variable a) because this model is based on hydrodynamic diffusion theory.^{27 } Though derived for a solute diffusing freely in a homogenous solution, this model can also be related to diffusion coefficients of solutes in hydrogels.^{27,54,55 } One of the bestknown models was derived by Brinkman, which relates diffusion coefficients in a gel, such as the native ECM (represented by the variable D_{ECM}), to that in water (represented by the variable D_{0}) based solely on hydrodynamic radius, a, and a fitted system parameter, κ.^{54 }
Eqn (1.2) shows the Stokes–Einstein relationship for relating the diffusion coefficient, D, to molecular radius, a. In this relationship, an approximately spherical molecule is assumed with a density, ρ, which is close to that of water (1 g mL^{−1}). The notation used here is as follows: R is the universal gas constant, T is temperature, µ is the solution viscosity, N is Avogadro's number, and MW is the molecular weight. Diffusion coefficients within the ECM, D_{ECM}, can be compared to those found in dilute water solutions, D_{0}, through this relationship developed by Brinkman where κ is a fitted system factor and a is again molecular radius.
Eqn (1.2) presents a method predicting how various biomolecules of different hydrodynamic radii diffuse through the ECM (Figure 1.4c and d). Even when all other variables are held constant, molecular size alone can dramatically affect the slope of the gradient.
Many examples of Fick's first law are found in in vivo situations, and studies of the transport of solutes into cancerous tissue provide illustrative examples of such relationships.^{56–58 } For instance, some types of tumours are known to retain a large amount of fluid, increasing their interstitial volume and thus increasing the diffusion coefficients of solutes.^{54 } The change occurs because the increased volume lowers the concentration of macromolecules that compose the ECM, thus allowing for faster movement of solutes because there are fewer solute interactions with the matrix. In contrast, within other types of tumours diffusion is severely hindered because of increased ECM protein secretion around the tumour tissue.^{56,57 } Creation of more material around a site increases the concentration of the macromolecules that comprise the ECM, and transport by diffusion is slowed because of an increased number of interactions between solutes and matrix. Additionally, digestion of the tumour ECM by enzymes enhances diffusion into the tumour tissue, due in part to the corresponding decrease in ECM concentration.^{58 } From these studies, it is clear that transport within the native ECM is dependent on ECM network parameters in addition to solute properties.
1.2.2 SteadyState Diffusion Within Synthetic Hydrogels
As in the native ECM, gradients can form within synthetic hydrogels through multiple mechanisms; however, numerous examples of gradients forming primarily through sieving action can be found in the literature.^{29,42,52,59–61 } When analysing gradient formation in synthetic hydrogels, the same assumptions that were made in the previous section are made here. Notably, the assumption that the hydrogel is a homogeneous, porous network of inert macromolecules is more appropriate for synthetic hydrogels than for the native ECM.^{17,27,42 }
Prediction of the diffusion coefficients of solutes within synthetic hydrogels has been modelled using hydrodynamic theory and the Stokes–Einstein relationship (eqn (1.2)).^{27 } Specifically, the diffusion coefficient within hydrogels, D_{gel}, versus that of free solution, D_{0}, is commonly modelled with the following relationship:
Eqn (1.3) shows the Stokes–Einstein relationship specifically for the hydrogel diffusion case. Here a is the hydrodynamic radius, B is a system coefficient, and c is the polymer concentration of the hydrogel (expressed in units of volume fraction). D_{gel} is the diffusion coefficient within the hydrogel and D_{0} is the diffusion coefficient of the solute in free solution.^{27 }
The Stokes–Einstein relationship is a simple representation of the sieving action of a synthetic hydrogel, but this model is limited based on its dependence on a fitted parameter, the system coefficient B. A more broadly useful and predictive model minimizes its dependence on fitted parameters and maximizes its dependence on parameters that can be measured by the experimenter. Based upon thermodynamics first principles of hydrogel networks, the molecular weight between crosslinks, M_{c}, can be predicted based upon average polymer molecular weight, M_{n}, and volume fraction of polymer in the network, v_{2,s} (eqn (1.4)):^{17 }
Eqn (1.4) shows the molecular weight between crosslinks of a hydrogel based upon thermodynamic first principles. Here M_{c} is the molecular weight between crosslinks, M_{n} is the average polymer molecular weight, ν is the specific molar volume of the polymer, V is the molar volume of water, χ is the Flory interaction parameter between the polymer and the solvent, v_{2,s} is the swollen volume fraction of polymer, and v_{2,r} is the relaxed volume fraction of the polymer.^{17 }
Once the molecular weight between crosslinks is known, the average physical distance between crosslinks, ξ, can be determined based on known polymer parameters and the molecular weight between crosslinks, M_{c}.
Eqn (1.5) shows the distance between crosslinks, ξ, based on the swollen volume fraction of the polymer, v_{2,s}, the Flory characteristic ratio, C_{n}, the molecular weight between crosslinks, M_{c}, the average distance of one bond in the polymer backbone, l, and the molecular weight of polymer repeat unit, M_{r}.^{17 }
This physical length can be used to determine the diffusion coefficient of a solute through the hydrogel network. Comparing the molecular radius, a, to the distance between crosslinks in the hydrogel, ξ, one can calculate the amount by which the movement of the solute is slowed. Lustig and Peppas incorporated this principle into a solute diffusion model based on free volume theory:^{27,62 }
Eqn (1.6) shows the ratio of the diffusion coefficient within a hydrogel, D_{gel}, to the diffusion coefficient within free solution, D_{0}, based on molecular radius, a, and the average distance between crosslinks, ξ. Here, Y is the lumped free volume parameter, and v is the volume fraction of the polymer. This relation simplifies at low volume fractions since the exponential term becomes approximately 1.^{27 }
An example of diffusion from a source within synthetic hydrogels was described by Peret et al.^{59 } when proteinreleasing microspheres were encapsulated within a small section of a hydrogel to create a localized source (Figure 1.2). In this work, the rest of the hydrogel acted as a sink for molecules released from the localized source.^{59 } The gradients that formed in these hydrogels were dependent upon source concentration and the hydrodynamic radii of the diffusing molecules.^{59 } Both of these dependencies can be predicted from equations described thus far in this chapter. Another study by Cruise et al.^{60 } demonstrated trends in the diffusion of proteins through hydrogel networks that were similar to those seen by Peret et al. In this work, the permeability of poly(ethylene glycol) (PEG) hydrogel networks to proteins of various sizes was controlled solely by adjustments to hydrogel network properties.^{60 }
1.3 Soluble Factor Generation and Consumption
1.3.1 Soluble Factor Generation and Consumption Within the Native ECM
Though the relationships discussed in previous sections can be used to generally explain gradient formation in the ECM, they make unrealistic assumptions about soluble factor generation and consumption. For instance, cells are only capable of producing soluble molecules at specific rates,^{12,15,53 } which Savinell et al. established to be on the order of thousands of molecules per cell per second.^{63,64 } Additionally, the number and distribution of cells producing a particular soluble factor is also a matter to consider, as well as any feedback mechanisms that may influence the production rate. The same statements can be made about soluble factor consumption. Although there are in vivo situations where soluble molecules are produced or consumed regardless of all other variables, this is rare.^{65 } Therefore, a much better description of soluble factor production and consumption are the following functions (eqn (1.7)), which account for cell number at the source, N, as well as soluble molecule concentration, C, and time, t:
Eqn (1.7) shows that the soluble factor concentration is dependent on the rate at which the soluble factor can be produced (top) or consumed (bottom) by the pointsources and sinks. P′ and E′ represent functions that relate soluble factor production and consumption to cell number, N; concentration of that factor, C; and time, t, to soluble factor concentration.
With the assumption that the rate of production is independent of concentration and time, this relationship is incorporated into eqn (1.1) as the yintercept (eqn (1.8)). Of note is that if one assumes a timedependence on this source, then eqn (1.1) is no longer valid as it was derived under a steady state assumption.
Eqn (1.8) shows Fick's first law with a more realistic term, P′, than C_{0} for the soluble factor production term. Eqn (1.8) is fundamentally the same as eqn (1.1), except that now it includes a more realistic soluble factor production term. Variables such as the number of cells at the source can now be introduced. Since production is not necessarily constant over time as has been assumed here, the implications of temporal dynamics is discussed further in the following sections.
Besides including soluble factor production parameters, this equation can be further modified to include the fact that cells are not all condensed to a point. In many biologic examples, the source should not be approximated as a point and is actually distributed over a small area.^{12,15,53 } Including position dependence in the production term now changes the mathematical description to a new version of Fick's first law:
Eqn (1.9) shows Fick's first law when the production term is dependent on position, x. Here P(x) is equal to the integral of the P′(x) function. The relationship between concentration and position is now defined in part by the production term, P. Depending on the nature of the relationship between soluble factor production and position, many different solutions are possible. Several hypothetical cases are shown in Figure 1.3a. The gradient is no longer necessarily linear as it is in all cases discussed thus far in the chapter.
Similar to the assumptions of a perfect pointsource, the assumption of the perfect pointsink is not realistic, either. In fact, diffusion through a decellularized region to a perfect, pointsink is an unusual situation in vivo since it is understood that cellular consumption of soluble molecules is the principal method by which these molecules are cleared.^{65 } For instance, many soluble proteins bind to membranebound receptors, which are then internalized into the cells.^{66,67 } Other solutes, such as steroid hormones and other lipidsoluble molecules, are able to diffuse through the cell membrane and bind to intracellular components.^{66,67 } Cellular nutrients, such as O_{2}, are also consumed internally within the cells through various metabolic processes.^{66,67 } As a result, cells within the transport region should be included in the model, and it is assumed that these cells consume the soluble factor. Including this relationship into the original application of Fick's law gives:
Eqn (1.10) shows Fick's first law when the production and elimination terms are dependent on position, x. The concentration of soluble factors eliminated is represented by C_{e}. Here P(x) is the integral of P′(x) and E(x) is the integral of E′(x). Also, C_{0} is lumped into the P(x) function. This relationship shows a balancing act between the production, elimination, and diffusion terms. The dependence on position of the production and elimination terms, as well as the diffusion coefficient, determines the properties of the gradient (Figure 1.3b).
As discussed previously in this section, solutes are often consumed via interaction with cellular components.^{66,67 } Taking interaction kinetics into account can bring important improvements to the description of soluble factor consumption by cells.^{65 } We assume that the solute in question, referred to as the ‘ligand’, interacts with some type of cellular component, referred to as the ‘receptor’, and that this interaction is dependent on the concentration of both components. Many models have been derived that describe the kinetics of solutes binding to cellular components, and the general mathematical representation of such kinetics is:^{68,69 }
Eqn (1.11) is a typical equation relating the number of receptor–ligand complexes, CR, over time, t, to the concentration of the receptor, R, soluble factor concentration, C, the binding constant, k_{on}, and the dissociation constant, k_{off}.^{68,69 }
Eqn (1.11) is derived for a soluble protein binding to extracellular receptors;^{68,69 } however, this general expression can be adapted for most situations of solutes binding to cellular components. Combining eqn (1.10) and (1.11) can be accomplished with the assumption of steady state and the assumption that position dependence is small relative to the concentration dependence. Another assumption made is that all terms (aside from the concentration terms) are not dependent on time or position.
Eqn (1.12) is the function for solute concentration, C, in terms of position, x, and it shows a dependence on solute concentration at the source, C_{0}, the binding and unbinding constants, k_{on} and k_{off}, the diffusion coefficient, D, the receptor concentration, R, and the concentration of receptor–ligand complexes, CR. Though incorporation of realistic source and sink terms can increase the complexity of the solution, these models remain in the form of linear differential equations and therefore can be solved without great difficulty. With the removal of perfect pointsink and source assumptions, a more realistic model of transport in the native ECM can be created. Additionally, the important assertion that these gradients are not necessarily linear is also made.
1.3.2 Soluble Factor Generation and Consumption Within Synthetic Hydrogels
Like sources and sinks within the native ECM, most sources and sinks within synthetic hydrogels are not accurately modelled as perfect, point entities, either. Additionally, since cells may serve as sources or sinks within synthetic hydrogels, the geometry of their placement can affect transport and gradient properties just as in the native ECM. Synthetic constructs such as microparticles, microfluidic channels, and biochemically functionalized regions of the hydrogel can also serve as important source or sink regions.
Since the geometry of the production and consumption regions can greatly affect soluble factor gradients (Section 1.2.1), there is a need to spatially pattern cellular populations within hydrogels. Technologies to pattern cells into complex geometries within a hydrogel have been utilized in several studies, and nearly all have utilized microfluidic or lithographic processes.^{19,25,70 } Other approaches involve encapsulating the cells into microgels, which can then be assembled into larger hydrogels.^{19 } More complex methods exist to pattern cells within synthetic hydrogels; however, some methods involve fabrication steps that have proven to be toxic to cells.^{19 }
Microfluidic constructs within hydrogels have also been utilized as synthetic sources and sinks. Basic applications of microfluidic channels within hydrogels have been used as nutrient sources and waste sinks, but more recently microfluidic channels have been used to model the in vivo vasculature.^{19 } The advantage of microfluidic approaches is that parameters such as soluble factor concentration and flow rate can be used to adjust gradient parameters within the hydrogel. The relationship between soluble factor concentration, flow rate, and soluble factor gradients is a wellknown transport problem, involving both the Reynolds and Schmidt numbers.^{54,71 } Acting as a sink carrying soluble molecules away from the construct, a microfluidic channel behaves as follows:
Eqn (1.13) describes the mass flux, J, through the walls of a tube of flowing fluid in terms of the Reynolds number (Re) and the Schmidt number (Sc). Here R is the radius of the tube, D is the diffusion coefficient of the solute, C_{0} is the concentration of the solute in the fluid, L is the length of the tube, v is the velocity of the fluid, µ is the fluid viscosity, ρ is the fluid density, and Γ is the gamma function.^{54,71 } Equations similar to eqn (1.13) can be derived for most situations involving the flux of mass in or out of a tube of flowing fluid.^{54,71 } Eqn (1.13) demonstrates that as either the concentration of the solute at the tube boundary or the fluid flow velocity increases, the mass of solutes removed from the hydrogel increases.
Another method of creating soluble factor sources and sinks within hydrogels is the incorporation of microparticles. Microparticle incorporation has many advantages, including the ability to pattern them into complex arrangements.^{29 } Technologies exist that allow for the tailoring of molecule release from microparticles so that several different release rates and profiles are possible.^{28 } Some microparticle formulations are able to release molecules over extended periods of time (days to weeks) and maintain relatively constant release rates over this time frame.^{28 }
Functionalization of regions in the hydrogel network to deliver or sequester soluble molecules is yet another widely utilized approach. Significant work has been done on polymer functionalization, and many chemistries exist to bind biological molecules to synthetic polymers.^{30 } As the molecules are liberated from the polymer, they diffuse down their concentration gradients. Many of the release mechanisms employed are enzymatic or hydrolytic in nature,^{30 } but other methods have been established that utilize reversible binding of molecules to the hydrogel matrix.^{31 } These methods are discussed in detail in later sections.
1.4 Matrix Interactions
1.4.1 Interactions with ECM Components
Thus far in the chapter, all presented equations have assumed that diffusing solutes do not interact with the ECM components in any way other than ‘sieving action’. This is inaccurate for many solutes, and ECM components routinely interact with diffusing solutes through charge interactions and molecular recognition.^{14,72 } Additionally, recent studies have implicated proteoglycans as crucial elements in the transport mechanisms of some soluble molecules due to charge interactions and sequestering.^{73 }
The simplest case of matrix–solute interactions is that of charge interactions (attraction and repulsion) between diffusing solutes and ECM components. Since most of the ECM is negatively charged because of the presence of proteoglycans,^{11,16 } molecules with a strong positive charge can have their movement retarded by their attraction to ECM components of opposite charge.^{51 } Molecules with a negative charge can also have their movement retarded by repulsive interactions.^{51 } Because the ECM can slow molecules due to sieving action and electrostatic interactions, Lieleg et al. described the ECM as an ‘electrostatic bandpass’.^{14 }
Beyond electrostatic attraction and repulsion, many solutes exhibit binding regions that allow them to specifically bind to various ECM components.^{72,74,75 } Such mechanisms have been elucidated for solutes such as fibroblast growth factor (FGF), transforming growth factor beta (TFGβ), plateletderived growth factor (PDGF), and vascular endothelial growth factor (VEGF).^{72,74 } Because of this binding, the ECM can then act as a reservoir of soluble molecules that can be released due to specific cues or to influence fluctuations in the concentration of soluble molecules.^{72 } Evidence also exists that sequestering of soluble molecules within the ECM may serve as an important method of regulation for soluble factor signalling and gradient formation.^{72,74 }
The contribution of the binding and release of soluble molecules from the ECM to mass transport models is an extension of the elimination and production functions that were defined earlier in the chapter (Section 1.2.1). The function defining the matrix binding and release of soluble molecules is dependent on the concentration of soluble molecules (C), the concentration of matrix binding sites (MR), and the distribution of the ECM components with respect to position (x). Modification of the equations proposed in Section 1.2.1 with an ECM binding and release term yields:
Eqn (1.14) shows the addition of a matrix binding term to a model of soluble factor transport in the ECM. Here M′(C, MR, x) is a function that relates the matrix binding of molecules from solution to soluble factor concentration, C, to the concentration of free binding sites in the matrix, MR, and position, x.
This reservoir of soluble molecules that accumulates in the ECM can be released by proteolytic activity degrading the ECM and thus freeing the bound molecules.^{72,74 } This can be accounted for by the use of a step function, u, which has a value of 1 when proteolytic elements are present and 0 when these elements are not present.
Eqn (1.15) is the incorporation of the ‘proteolytic liberation’ term into eqn (1.14). This allows one to account for molecules freed from the matrix when proteolytic elements are present. Here u is the unitstep function and P$Matrix\u2032$(x) is the production term of molecules liberated from the matrix as a function of position.
Thorne et al. provided an example of the effect of ECM binding on soluble factor transport (Figure 1.4). The diffusion of two proteins in the brain ECM was investigated: lactoferrin (Lf) and transferrin (Tf). Lf is known to bind to heparin sulfate proteoglycans whereas Tf does not.^{75 } Performing in vivo experiments, the authors observed retarded diffusion of Lf within the ECM compared to the diffusion of Tf (Figure 1.4).^{75 } From this, the importance of the effects of matrix interactions on diffusing solutes is clear, as well as the fact that such interactions must be accounted for to accurately model solute diffusion in the native ECM.
1.4.2 Interactions with Hydrogel Components
Solute interactions with the synthetic polymers used in hydrogels are different from interactions with the native ECM for many reasons. Obviously soluble biological molecules do not have binding domains for synthetic polymers, and many synthetic polymers (such as poly(ethylene glycol)) typically have no specific interactions with diffusing solutes.^{76 } As a result, synthetic hydrogels do not recapitulate the specific binding and release mechanisms of the ECM.
However, interactions with ECM components can be incorporated into synthetic hydrogels through the use of functionalized biological molecules and peptides.^{21,26,32–45,77–80 } For example, in a study by Benoit et al. poly(ethylene glycol) hydrogels were modified with a chemically modified form of the glycosaminoglycan heparin (Figure 1.5a), an ECM component that has the ability to bind many soluble biological molecules.^{21,26 } The heparinfunctionalized hydrogels caused increased osteogenic differentiation of mesenchymal stem cells relative to nonfunctionalized controls.^{26 } Other work by Benoit et al. showed that heparinfunctionalized hydrogels were capable of binding soluble FGF and releasing it over a 5 week time frame.^{21 } Additionally, other studies have introduced the concept of using a peptide to bind heparin to biomaterials rather than incorporating it directly.^{32–34 }
Though a molecule like heparin can bind large quantities of soluble biological molecules, it cannot bind only specific biological molecules of interest. To accomplish the goal of binding specific molecules, work has been done with binding peptides that have been modelled after the heparin binding^{35 } or phagedisplay against specific molecules.^{36 } Though the binding capabilities of these peptides are well established, the extent of specificity of these peptides is less clear. An intriguing approach is the design of peptides based on cellular receptors,^{37 } as receptor interactions with soluble biological molecules are typically high affinity and highly specific. Such peptides were recently utilized in poly(ethylene glycol) hydrogel microspheres that were shown to bind and release VEGF, as well as inducing a biologic effect by sequestering or releasing VEGF during in vitro cell culture (Figure 1.5b and c).^{31 }
While the use of both peptides and functionalized ECM components shows promise, how these components affect transport and gradient formation within the hydrogel network remains to be fully investigated. For instance, few studies examine the effects of binding peptides on gradient formation between a welldefined source and sink within a synthetic hydrogel.^{72,74 } Additionally, though the peptide and glycosaminoglycan components of the synthetic hydrogels discussed here are susceptible to proteolytic degradation, the effect of degradation on soluble factor binding and release have yet to be characterized.
1.5 Temporal Dependencies
1.5.1 Temporal Dependencies During In Vivo Transport
Thus far in this chapter, all equations have been derived at steady state. While such an approach is not invalid for all in vivo scenarios,^{12,15,53,81 } most in vivo situations involve time dependency. For instance, the reactiondiffusion models that have been used to describe many situations in developmental biology rely on chemical reactions between two or more species,^{81 } which occur at specific rates that are not necessarily constant with time. Another case where steadystate assumptions do not hold is the establishment of a soluble gradient from one group of cells to another. The cells functioning as the source cannot suddenly put forth a gradient of soluble molecules; it must be established over time as the soluble factor is produced.^{12,15,53,63,64,81 }
To begin the discussion, we examine Fick's second law of diffusion. This law states the concentration, C, changes not only with position, x, but also with respect to time, t:
Eqn (1.16) is Fick's second law of diffusion, where C is concentration, D is the diffusion coefficient, t is time, and x is position.^{71 } Fick's second law returns to Fick's first law when steady state is assumed. Returning to eqn (1.7), it is reasonable to assert that both the production, P′, and consumption, E′, terms are dependent on time:
Eqn (1.17) is (1.7) reformulated with a dependence on time assumed on both the production, P″, and elimination, E″, terms. Eqn (1.17) is more complex than all previously proposed equations because the production and elimination terms show a dependence on position, x, as well as time, t. The focus of this chapter is not on solving partial differential equations, so the reader should consult other references for the solutions to such problems.^{82 } To simplify eqn (1.17), the production terms are neglected as well as any dependence on time or position for the elimination function (Figure 1.6a). This leaves the following wellstudied problem of reaction and diffusion from a constant source:^{65 }
Eqn (1.18) shows reaction and diffusion from a constant source. Here C is concentration, x is position, t is time, D is the diffusion coefficient, and k is the rate constant of the elimination reaction; erfc denotes the complementary error function and exp is the exponential function.^{65 } Eqn (1.18) is represented graphically in Figure 1.6b. A more realistic version of this problem would not only have a sink whose concentration is increasing over time but a source whose concentration falls over time. Such a situation is diagrammed in Figure 1.6c. The modelling of this situation is complex, so it is not discussed here further; however, it is important to understand that this type of complexity can occur in both natural and synthetic systems when temporal dynamics are taken into account.
Experiments using fluorescence recovery after photobleaching (FRAP) techniques offer insight into the temporal dynamics of gradient formation in vivo.^{13 } Using FRAP in conjunction with fluorescently tagged proteins, experimenters can track protein transport and gradient formation within an organism. Kicheva et al. used FRAP to evaluate the kinetics of decapentaplegic (Dpp) gradient formation in Drosophila (Figure 1.6d–f).^{13 } Their experimental approach involved the photobleaching of a chosen area next to the source region and observing the return of fluorescence to the region.^{13 } Based on data obtained from the FRAP technique, they were able to estimate the various parameters involved in Dpp gradient formation: the diffusion constant, the soluble factor degradation rate, and production rate at the source.^{13 }
Most models of transport in the native ECM do take into account temporal dynamics since nearly all in vivo processes are time dependent.^{81 } For instance, the rate of soluble factor production and consumption, unless perfectly matched, cause a system to move away from steady state. Other complications include changes in the number of cells that are consuming, producing, or otherwise altering soluble factor concentration; enzymatic effects on the soluble molecules themselves, on cellular receptors, or on ECM molecules; and secretion of additional ECM by cells. While incorporation of all of these aspects into a mathematical model may greatly increase its complexity, the overall point is clear that the most realistic representation of in vivo transport includes temporal dynamics.
1.5.2 Temporal Dependencies During Transport in Synthetic Hydrogels
As for steadystate assumptions in the native ECM, assumptions of steady state are not entirely invalid for synthetic hydrogels; however, the most realistic mathematical models of soluble factor diffusion in synthetic hydrogels take temporal dynamics into account. While many synthetic hydrogels have been shown to be stable when chemically crosslinked,^{38–41,83 } the hydrogel environment is not necessarily static. Cells encapsulated in the hydrogel can secrete their own ECM, proliferate, and alter their rates of soluble factor production and consumption. Furthermore, hydrogel components can be engineered to change over time via hydrolytic, enzymatic, or external mechanisms (such as pH or temperature).
A widely used method of altering synthetic hydrogels over time involves hydrolytic degradation. Many chemistries have been employed,^{30,42–44 } and many of these allow the experimenter to modulate the rate of degradation. The space freed by the hydrolytic degradation can lead to an increase in cell spreading and viability within a hydrogel.^{30,38,39,42–44 } Hydrolytic degradation has also been linked to increased cellular production of ECM proteins within the hydrogel.^{30,38,39,42–44 }
Besides altering cellular viability and ECM production, hydrolytic degradation also affects soluble molecule transport within the hydrogel. Returning to eqn (1.6), the permeability of the hydrogel to a solute is dependent on the average distance between crosslinks, ξ. As the hydrogel degrades, ξ increases. Additionally, a degrading hydrogel changes its concentration (i.e. volume fraction) of hydrogel components over time as well as changes in the system parameters. Alterations to the distance between crosslinks alters the diffusion coefficients of solutes within the hydrogel (eqn (1.6) and (1.19)), and thus provides an opportunity to modulate soluble cell–cell signalling and gradient formation within these hydrogels.
Eqn (1.19) is the restated Lustig–Peppas model of the diffusion constant within a hydrogel where the hydrogel properties are functions of time. Eqn (1.19) is complex since several variables are related to time, such as the distance between crosslinks, ξ, the polymer volume fraction, v, and the free volume theory diffusion parameters, Y. However, work by Anseth et al. on hydrolytically degrading PEGbased hydrogels used an approximation that is valid for highly swollen gels (i.e. the volume fraction can be approximated as zero):^{39 }
Eqn (1.20) is the simplified relation of the diffusion constant in highly swollen hydrogels to hydrogel parameters. Here Q is the mass swelling ratio of the hydrogel (wet mass over dry mass).^{39 } Eqn (1.20) relates the changing diffusion coefficient of a solute in the hydrogel to the molecular weight between crosslinks, M_{c}, and ultimately to the mass equilibrium swelling ratio of the hydrogel, Q.^{39 } A comparison of this model's predictions to experimental data on the release of proteins from the hydrogel was done to validate this model.^{39 } A potential limitation of this approach, however, is that the radius of the solute is neglected. Furthermore, this model is only valid for highly swollen hydrogels; if this is not the case, then the modelling of these relations returns to eqn (1.19).^{39 }
Hydrolytic degradation as a means to modulate soluble factor transport within hydrogels is not without drawbacks. The most obvious is the difficulty in matching hydrolytic degradation rates to the timescale of biologic events,^{45 } such as cellular migration within a hydrogel.^{46 } As a result, significant research has been done to develop hydrogels that degrade via cellmediated mechanisms. Many cellmediated degradation strategies involve the crosslinking of the hydrogel with molecules that are susceptible to enzymatic cleavage.^{30,39–44,83 } These studies demonstrate cell viability, proliferation, and ECM production by cells within these hydrogels.^{40,41 }
Two important differences between enzymatic and hydrolytic degradation mechanisms are:
Degradation is dependent on time and on enzyme concentration.^{39,83 }
The kinetics of the enzymatic degradation are dependent on several variables including concentration of the degradable unit, halflife of the enzyme in the active form, cellular production of the enzyme, and the rate at which the degradable unit can be cleaved by the enzyme.^{39,83,84 }
Rice et al. derived a model to describe the degradation of a synthetic hydrogel in the presence of an exogenously delivered enzyme.^{83 } During the course of this degradation (concentration of degradable crosslinker is represented by the variable C), the enzyme (whose concentration is represented by the variable E) is involved in the following reactions:
Eqn (1.21) shows the chemical reactions involved in the enzymatic degradation of a synthetic hydrogel. Here E is enzyme concentration, C is the degradable crosslinker concentration, D is the concentration of degradation products, E_{inactive} is the concentration of inactivated enzyme, and E*C is the enzyme–crosslinker complex. The rate constants k_{1}, k_{2}, and k_{d} are for reaction 1, reaction 2, and the enzyme inactivation reaction respectively.^{83 }
The following differential equations can be solved to yield an expression for the ratio of intact crosslinker molecules at a given time to the original number of crosslinker molecules, N/N_{0}:
From eqn (1.21), the relations given in eqn (1.22) can be constructed and solved for the ratio of intact crosslinker at a given time to the original amount of crosslinker, N/N_{0}. Note that rate constants denoted with an ‘r’ indicate that they are the reverse rate constant and k* indicates lumped enzyme kinetic parameters.^{83 }
Though not done by the authors of this study, the amount of intact crosslinker may be related to solute diffusion coefficients through eqn (1.20). Notably, this derivation assumes that the degradation reaction, and not the diffusion of the enzyme, is the limiting factor in the rate of hydrogel degradation.^{83 }
While there are numerous studies that discuss hydrogel degradation via hydrolytic or cellmediated mechanisms for controlled release applications, there is a relative lack of studies using these technologies to study cellular signalling within the hydrogel environment. Additionally, for studies that do incorporate cells into hydrolytically and enzymatically degrading constructs, it is routinely noted that the cells secrete their own matrix.^{30,38–44 } How this secreted matrix affects the transport properties within the hydrogel has yet to be investigated.
Aside from degradation via hydrolytic and enzymatic mechanisms, work has also been done to develop ‘responsive hydrogels’, which allow for reversible changes to occur in the hydrogel based upon changes in the polymers that comprise the hydrogel. Such changes ultimately affect the distance between crosslinks, ξ, and thus affect the transport of solutes within the hydrogel. These strategies revolve around changes to solution pH, solution ionic strength, temperature, or the presence of a specific chemical compound.^{30 } All of these approaches combine the possibilities of experimentercontrolled changes, cellularmediated changes, and reversibility of these changes.
The most ubiquitous of these designs are pH and ionic strengthresponsive hydrogels.^{30 } These hydrogels can be described on the basis of changes to the Gibbs free energy:^{30 }
Eqn (1.23) shows the Gibbs free energy of swelling for hydrogels in terms of the free energies from mixing, elastic retraction of the polymer chains, osmotic pressures, etc.^{17,30 }
Several models can predict the response of hydrogels designed to respond to pH or ionic strength changes.^{30 } A model derived by Peppas and colleagues relates the changes to diffusion coefficients within a hydrogel swelling in response to pH or ionic strength changes:
Eqn (1.24) is the relation of diffusion coefficient, D_{gel}, in a pH or ionic dependent hydrogel based on the initial diffusion constant (D_{0}), the polymer volume fraction (v), and the water–polymer interaction parameter (α).^{30 }
For hydrogels that change with respect to temperature (and not pH or ionic strength), an alternative expression has been developed:
Eqn (1.25) shows the diffusion coefficient of a solute in a hydrogel, D_{gel}, swelling in response to temperature changes. Here D_{0} is the original diffusion coefficient of the solute in the gel and v is the polymer volume fraction.^{30 } Since volume fraction of polymer within the hydrogel, v, is easily measured as a function of temperature, this is a convenient relation.^{30 } Though designs do exist that combine the ideas of pH, ionic strength, and temperature changes, mathematical representations of them remain to be developed.^{30 } Furthermore development of hydrogels that respond to specific chemical compounds remains particularly challenging, and although some recent approaches have shown limited ‘bioresponsiveness’, there are not yet generic mechanisms for response to a chemical compound of interest. Indeed, controlled bioresponsiveness represents a grand challenge in design of synthetic hydrogels.
1.6 Convection
1.6.1 In Vivo Transport by Convection
Convective flow is an aspect of in vivo transport that is neglected in many models. This is because convection is considered to be only a small component of transport in the ECM due to the slow flow velocities.^{85 } For instance, in the lymphatic system, flow velocities of 0.1 to 1.0 µm s^{−1} have been observed.^{85 } However, evidence exists that these small interstitial flows serve to make important alterations to gradients in vivo, such as in developing embryos, lymphangiogenesis, and capillary morphogenesis.^{85 }
To incorporate convective flow into transport equations, Fick's second law (eqn (1.16)) is modified as follows:^{71 }
Eqn (1.26) is a modified transport equation to incorporate mass transport due to convective flows.^{71 } In eqn (1.26), a new parameter has been added, the flow velocity, V. As a result, flows in the ECM must be known in order for eqn (1.26) to be utilized. Flow in the ECM can be modelled via the Brinkman equation for flow through porous media:^{85,86 }
Eqn (1.27) is the Brinkman equation for flow within a porous media. Here µ is the fluid viscosity, K is the media permeability, P_{x} is the derivative of the pressure function with respect to x, and V_{xx} is the second derivative of the velocity distribution with respect to x.^{86 }
Eqn (1.27) is a wellstudied relationship and many solutions are available in the literature. A study by Fleury et al. utilized known solutions to eqn (1.27) and found that even slow interstitial flow rates (0.1–6.0 µm s^{−1}) impart significant bias to a soluble gradient in the ECM (Figure 1.7).^{85 } Their model included ECM binding of, cell production and consumption of, and proteasemediated liberation of molecules from the ECM.^{85 } They noted that high levels of ECM sequestering of solutes did not overcome this effect, as both the soluble molecule diffusing to matrix binding sites and the protease to cleave it from such sites were biased by the convective flow.^{85 } Such studies demonstrate that while convective flows may not be the principle component creating soluble factor gradients in the native ECM, they may alter the shape of these gradients.
1.6.2 Transport in Synthetic Hydrogels by Convection
As in the native ECM, convection within synthetic hydrogels is assumed to be negligible except in special circumstances, such as the presence of macroscale pores within the hydrogel or forced flow through the hydrogel.^{70 } However, to determine whether or not the convection is significant in any situation, eqn (1.27) can be used to assess the flow velocity based upon the permeability of the hydrogel, K. Many models have been derived to determine K in polymeric materials,^{87 } and for the purpose of this section, we use the general model proposed by Jackson and James.^{87,88 } However, the reader should note that more specific models exist.
Eqn (1.28) is the permeability within a synthetic hydrogel matrix based on v, the volume fraction of the polymer, and r, the average radius of the polymer molecules that compose the matrix.^{87 } Eqn (1.28) shows a relation between the polymer volume fraction, v; the radius of the polymer molecules that compose the matrix, r; and the network permeability, K. As the volume fraction of the polymer increases, the permeability of the network decreases. Incorporating eqn (1.28) into (1.27) yields:
Eqn (1.29) is a modified form of eqn (1.27) introducing the known dependencies of permeability on polymer molecule radius and volume fraction.
A further simplification is that the velocity is relatively constant throughout the hydrogel, thus making V_{xx}, the second derivative of velocity, equal to zero:
Eqn (1.30) is a simplified form of eqn (1.29) when the velocity is nearly constant. Eqn (1.30) shows that decreasing permeability causes the pressure needed to maintain a given velocity to increase. Conversely, increasing permeability causes a decrease in the pressure needed to maintain a given velocity.
Kapur et al. have measured the permeability of polyacrylamide hydrogels.^{89 } The authors of this study forced water through the hydrogel with pressurized nitrogen and measured the velocity of the water as it moved through the hydrogel.^{89 } Permeability was dependent on the polymer volume fraction,^{89 } which is predicted by eqn (1.28). An alternative technique, proposed by Lin et al., involved estimating hydrogel permeability based upon mechanical properties of the hydrogel network.^{90 } Despite this work, however, convective transport in hydrogels is typically ignored, and its effects on transport within the hydrogel remain to be fully investigated.
1.7 Future Directions and Concluding Remarks
Soluble transport within the native ECM is complex, involving multiple populations of cells, matrix interactions, temporal dependencies, and convective flows. These complexities make the study of transport mechanisms in vivo a difficult task. Therefore, there is a great demand for in vitro methods to study these transport mechanisms. To address this demand, many different synthetic hydrogel technologies have been developed that aim to recapitulate aspects of in vivo soluble transport.
Future work will likely involve more intricate characterization of transport within these hydrogel environments, since many of these technologies have only been investigated from a standpoint of delivering solutes to the environment outside of the hydrogel. For instance, hydrogels that are engineered to degrade over time have been extensively explored from a drug delivery standpoint, but few studies have examined how the increasing distance between crosslinks allows for changes in soluble transport and gradient formation within the hydrogel. Another example is the use of hydrogels containing peptides that can bind specific soluble molecules. Although these hydrogels can deliver solutes to locations outside the hydrogel, little attention has been given to how these peptides affect soluble transport and gradient formation within the hydrogel.
Beyond characterizing the direct effects that many of these technologies have on transport within synthetic hydrogels, some fundamental aspects remain to be fully studied. For instance, although hydrogels have been designed respond to multiple types of stimuli at once, mathematical models to describe and ultimately predict their behaviour do not yet exist.^{30 } Convective flows within hydrogels are also typically ignored, and their effect on soluble transport within hydrogels has yet to be fully characterized.^{70 } Finally, patterning of source and sink regions (e.g. cell populations) within hydrogels into complex arrangements remains difficult, as many synthetic techniques are toxic to cells.^{19 } Therefore, although much progress has been made in the development of synthetic hydrogels to recapitulate aspects of native ECM soluble transport, much research remains to be done.