Mathematical Model of HSPG Influence on Growth Factor Movement

While the in vivo pharmacokinetics and pharmacodynamics of growth factors will relate to a large number of processes within complex whole-organism systems, it is clear that the role of ECM interactions will need to be understood quantitatively. Hence, a system to measure growth factor transport through isolated basement membranes has been developed (14,182), and subsequent studies revealed that FGF-2 binding to HS within the basement membrane was the major determinant of growth factor transport rate. Consequently, mathematical models of this transport process have been generated so that the full range of this effect can be evaluated under a variety of conditions that might relate to various tissues and target sites for growth factor treatment. The initial approach treats the transport of growth factors through the ECM as a diffusion-with-reaction process where the reversible binding of growth factor to HSPG FIGURE 3 Schematic description of GF diffusion and reaction with ECM. The movement of heparin-binding GF was modeled as being governed by diffusion and reversible binding to HSPG sites within the ECM. Binding of GF to HSPGs is described by mass action reversible binding where Kd represents the equilibrium dissociation binding constant. The concentration of unbound (free) GF at any time is described by the equation in the box where Deff represents the effective diffusion coefficient and xthe distance from the initial GF source. Abbreviations: HSPG, heparan sulfate proteoglycan; GF, growth factor; ECM, extracellular matrix.

FIGURE 3 Schematic description of GF diffusion and reaction with ECM. The movement of heparin-binding GF was modeled as being governed by diffusion and reversible binding to HSPG sites within the ECM. Binding of GF to HSPGs is described by mass action reversible binding where Kd represents the equilibrium dissociation binding constant. The concentration of unbound (free) GF at any time is described by the equation in the box where Deff represents the effective diffusion coefficient and xthe distance from the initial GF source. Abbreviations: HSPG, heparan sulfate proteoglycan; GF, growth factor; ECM, extracellular matrix.

resident within the ECM is controlled by the affinity of this interaction and the numbers of binding sites (Fig. 3). An example of how these kinds of models can be developed and utilized is described below.

System Description

To develop a mathematical model to examine the effect of HSPG on growth factor movement through the ECM, some simplifying assumptions were made about the physical system. The ECM was considered to be a large volume of isotropic material with a homogenous distribution of stationary HSPG sites. A region of tissue where growth factor is introduced was represented as a plane surface of finite width that is placed in the ECM. At time t = 0, growth factor is instantaneously released from this region and diffuses through the surrounding medium. As free growth factor (F) moves through the matrix, reversible binding occurs with HSPG sites (H), resulting in fixed and nondiffusing growth factor (FH). The importance of this interaction on growth factor movement can be evaluated by knowing how the concentration of diffusing growth factor varies in response to changes in HSPG binding. In mathematical terms, this means that the concentration of free growth factor must be defined in terms of position and time through the ECM.

Mathematical Development

The simplified physical system described in the preceding section was simulated by using a mathematical model based on one-dimensional diffusion with reaction in an infinite slab (see relevant equations in Appendix 1) (183,184). This model is derived by writing mass balances for both free growth factor and bound growth factor. Each mass balance relates the accumulation of the component in the volume to the net amount that diffuses in through the surface plus the net amount that is produced by reaction within the volume. After some manipulations, substituting Fick's law for the diffusive term and assuming a constant diffusion coefficient, expressions are obtained for the change in each component with respect to time. In equation (1) for free growth factor (Appendix 1), [F], is the concentration, Deff is the effective diffusion coefficient for diffusion in a porous medium, rF is the rate of production per volume, x is the distance, and t is the time. Similarly, in equation (2) for bound growth factor, [FH] is the concentration and rFH is the rate of production per volume. It is obvious that equation (2) does not have a diffusive term because the component is bound to the ECM and does not diffuse.

The reaction terms in equations (1) and (2) are evaluated by writing the reversible binding reaction between free growth factor and each HSPG site [equation (3), Appendix 1], where kon and koff are the rate constants for association and dissociation, respectively. The binding sites are assumed to be homogeneous, with 1:1 binding and no cooperativity. They are constrained by the mass balance in equation (4), which states that the initial concentration of sites [H]0 is equal to the concentration of unoccupied sites [H] plus the concentration of occupied sites represented by the concentration of bound growth factor [FH]. From the binding reaction in equation (3), the value of rF is set equal to the rate of dissociation minus the rate of association, and after substitution for

[H] from equation (4), the expression for rF becomes equation (5). Since rFH = —rF [equation (6)], appropriate substitution of these expressions into equations

(I) and (2) gives the working equations for the model in the form of equations (7) and (8). These equations are subject to the initial conditions and boundary conditions specified by equations (9) and (10), respectively. The initial conditions [equation (9)] indicate that the instantaneous pulse of free growth factor is located in a region of width 2w with an initial concentration of [F]0. There are no HSPG sites within this region. The concentration of bound growth factor is zero everywhere. The boundary conditions [equation (10)] impose an infinite sink on the system, forcing both components to zero at the extremes. Equations (7) to (10) form the mathematical basis for the model of this system, and the solution gives the distributions of free growth factor and bound growth factor through the matrix as a function of time.

Numerical Implementation

The solution to the system defined by equations (7) to (10) is obtained by using a numerical method in which the partial derivatives are replaced by finite differences (185). In this case, the second-order partial derivative is substituted with a central-difference approximation, and the first-order partial derivatives are replaced with forward-difference approximations [equations (11-13)]. As a result of these transformations, equations (14) and (15) become the new working equations for the model. These equations allow for marching along the axes of a grid in the x, t plane, where i represents a node on the x-axis (distance) in increments of Ax and m represents a node on the t-axis (time) in increments of At. At time t = 0, the values of [F] and [FH] are known at all xi from the initial conditions. At time t = At, the first increment of time, new values of [F] are calculated implicitly from equation (14) by generating a set of linear equations for all nodes along the x-axis on the basis of values from the previous time step and the boundary conditions. If these equations are written in matrix-vector form, the coefficient matrix is tridiagonal and diagonally dominant and the equations can be solved by Gaussian elimination. For [FH] at this time step, new values are determined explicitly from equation (15) using values from the previous time step. By repeating this procedure for as many time steps as desired, concentration profiles for [F] and [FH] are produced as a function of time. Because both implicit and explicit formulas are used in the approximation of the concentrations, this technique is called a semi-implicit finite difference method. The numerical implementation of this method can be efficiently carried out by computer (e.g., a program written in Fortran 90/95 using a double-precision IMSL (International Mathematical and Statistical Libraries) routine called DLSLTR for solving a tridiagonal linear system by Gaussian elimination). Results from the semi-implicit finite difference method are in good agreement with the analytical solution to the simpler problem of one-dimensional diffusion in an infinite slab.

Model Results

Values of model parameters. Figure 4 shows the output of the diffusion-with-reac-tion model for various conditions of growth factor diffusion through the ECM. Many of the values of the model parameters are based on information from previous studies (14,70,81,186). For Figure 4A and B, the y-axis is centered on a 20-pm-wide region that contains growth factor at an [F]0 of 26 pM, and the concentration profiles [F]/[F]0 appear as symmetric curves around this region after instantaneous release. The x-axis represents the perpendicular distance from the central plane of this region outward through the ECM. HSPG sites are at an [H]0 of 10 pM in the ECM. Growth factor diffuses with a Deff of 1 x 10"7 cm2/sec and binds to HSPG with a Kd of 0.1 nM. Because binding is much faster than diffusion, components of free growth factor, HSPG, and bound growth factor are at local equilibrium.

Effect of HSPG on growth factor diffusion. Comparison of Figure 4A and B clearly shows the effect of the presence of HSPG in the ECM on the dissipation of growth factor. The curves in each figure represent the concentration distributions of free growth factor at successive times (0.05,0.15,0.5, and 5 minutes) from the initial release. In Figure 4A, there is no binding to HSPG in the matrix, and the results reflect typical distributions for diffusion with no reaction. After five minutes, there is still more than 4% of the initial concentration of growth factor within 100 pm of the release site. If a receptor for this growth factor has a Kd on the order of 10"11 M, then a functional response (50% receptor occupancy) is possible from a cell that is as far out as 400 pm. When HSPG binding is added to the matrix (Fig. 4B), growth factor binds to the proteoglycan as it diffuses outward, reducing the local concentration of free growth factor and providing the drive for more to diffuse into the area. As a result, when HSPG binding is present, growth factor is cleared more rapidly from the release site and at the same time prevented from establishing a wide range of influence. For example, after 30 seconds, a 4% concentration of free growth factor reaches its greatest extent at about 24 pm from the release site, and after five minutes, the concentration drops to less than 0.1% everywhere. A cell with receptors for this growth factor (Kd & 10"11 M) is unlikely to produce a functional response at a distance greater than about 37 pm. FIGURE 4 Simulation results for one-dimensional diffusion-with-reaction model of GF movement through ECM after instantaneous GF release from region of width 2w. (A) Concentration-distance curves for nonbinding GF at 0.05, 0.15, 0.5, and 5.0 minutes. (B) Concentration-distance curves for HSPG-binding GF at 0.05, 0.15, 0.5, and 5.0 minutes. (C) Concentration-time curve for nonbinding GF at 100 mm from release site. (D) Concentration-time curve for HSPG-binding GF at 100 mm from release site. Model parameters: [F]/[F]0 = relative concentration of GF; [F]0 = 26 mM; [H]0 = 10 mM; Deff = 1 x 10~7 cm2/sec; Kd = 0.1 nM; koff = 0.01/sec; kon = 1 x 10s/M-sec; w = 10 mm; Ax = 2 mm; At = 0.0001 to 0.001 second. Abbreviations: HSPG, heparan sulfate proteoglycan; GF, growth factor; ECM, extracellular matrix.

FIGURE 4 Simulation results for one-dimensional diffusion-with-reaction model of GF movement through ECM after instantaneous GF release from region of width 2w. (A) Concentration-distance curves for nonbinding GF at 0.05, 0.15, 0.5, and 5.0 minutes. (B) Concentration-distance curves for HSPG-binding GF at 0.05, 0.15, 0.5, and 5.0 minutes. (C) Concentration-time curve for nonbinding GF at 100 mm from release site. (D) Concentration-time curve for HSPG-binding GF at 100 mm from release site. Model parameters: [F]/[F]0 = relative concentration of GF; [F]0 = 26 mM; [H]0 = 10 mM; Deff = 1 x 10~7 cm2/sec; Kd = 0.1 nM; koff = 0.01/sec; kon = 1 x 10s/M-sec; w = 10 mm; Ax = 2 mm; At = 0.0001 to 0.001 second. Abbreviations: HSPG, heparan sulfate proteoglycan; GF, growth factor; ECM, extracellular matrix.

Another way to view this effect is illustrated in Figure 4C for a location in the ECM that is 100 mm from the site of growth factor release. Without HSPG binding in the intervening space, a substantial pulse of growth factor is experienced at this location, peaking at about 5% after 10 minutes and delivering at least 1% for eight hours. With the addition of HSPG binding to the space, a strong pulse does not form, and growth factor cannot be detected at this location (Fig. 4D). Thus, the presence of HSPG has enormous consequences on the signaling potential of growth factors in the ECM.

Modification of growth factor diffusion through ECM. The base model describing growth factor movement through ECM can be used to evaluate a wide range of parameters (Fig. 5). For example, growth factor transport can be related to the variable levels of HSPG in the ECM in various tissues and in disease states or to the affinity of the specific growth factor (or engineered mutant) for HSPG. These molecules can also be used to evaluate how soluble heparin, HS, or HSPGfs might alter growth factor distribution. This is especially interesting as codelivery or cotreatment with heparin is often considered for growth factor-based angiogenic therapies. It is also important to recognize that areas of tissue injury or inflammation generally contain high levels of extracellular proteases, which would lead to the release of HSPGfs from the ECM.

In simulating the presence of HSPGfs, the central region of the model represented an area injured by protease attack possibly associated with inflammation or disease (Fig. 5B). The local damage to the ECM would result in the generation of HSPGfs, which would retain their ability to bind growth factor. These growth factor-HSPGf complexes would then be able to diffuse from the site of growth factor administration. Because growth factor-HSPGf complexes are larger and less globular than growth factor itself, the value of Deff was assumed to be smaller by a factor of 10 (10~8 cm2/sec vs. 10~7 cm2/sec for growth factor only), and the complexes diffuse at a slower rate than growth factor alone. However, since the growth factor is bound to a proteoglycan fragment, it is less likely to bind to HSPG residing in the uninjured ECM as it diffuses away from the site of administration. As a result of these combined factors, the concentration profiles of growth factor-HSPGf complexes begin to approach those observed for nonbinding growth factor through ECM. On the basis of these results, the model suggests that growth

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