Sangren—Sheppard model

To develop the two-region Sangren-Sheppard model, consider a substance that traverses the endothelial cell clefts but does not enter endothelial cells (such as L-glucose, which is not taken up by cells.) This solute is assumed to exchange passively between the capillary and interstitial fluid (ISF) spaces. Applying a onedimensional approximation, the governing equation for solute concentration in the blood is the advection equation ... 

Equations (8.46) and (8.47) describe the Sangren-Sheppard model. While the equations are straightforward and can be thought of as the minimal model that captures the important biophysical phenomena of solute exchange along a capillary, this model represents nearly the maximal level of complexity that can be effectively analyzed without invoking numerical approximations to simulate it.6... 

Figure 8.7 Concentration profiles for the Sangren-Sheppard model following impulse injection of solute into the capillary. The solution predicted by Equations (8.52) and (8.53) is plotted at four different times, for parameter values indicated below. Capillary blood concentrations are plotted as solid lines and ISF concentrations are plotted as dashed lines. The impulse at z = vt is indicated by a vertical line and the relative strength e-ps /(vVs) is indicated in the plots. Parameter values are Vb = 0.05ml -g 1, V sf = 0.20ml-g 1, PS = 6.0ml-min 1 -g 1, F = 1.0 ml min-1 g-1, L = 500 qm, q0 = 10 3 mol g 1. The velocity is v = FL/Vb = 166.7 qm - s 1. Concentrations are plotted in Molar units.

Concentration profiles predicted by the Sangren-Sheppard model, following impulse injection of solution into the capillary, are illustrated in Figure 8.7. The figure plots capillary and ISF solute concentrations as a function of distance along the capillary at different times following the initial impulse. 

The governing equations for the multiple-region models such as that of Figure 8.6 are constructed analogously to those of the two-region Sangren-Sheppard model. For example, for this model the solute concentration in the blood is governed by the equation ... 

Explore the behavior of the Sangren-Sheppard model for different values of the unitless parameter PS/F. Replot the simulations illustrated in Figure 8.7 for the limiting case as PS/F - oo. Explain why the model behaves in the way it does. 

Actually, the distinction between analytically and numerically obtained model solutions is rarely clear. Ana-lytical solutions to governing differential equations are often expressed in terms of special functions such as exponentials, which must be approximated numerically. Here we will see that die solutions to die Sangren and Sheppard model are conveniently expressed in terms of a class of special functions called modified Bessel functions. 

Figure 8.6 shows a diagram of a general model of blood-tissue solute transport, used to analyze data on the transport of labeled solutes introduced in the blood or perfusate flow supplied to individual organs. The development and analysis of models of this sort to analyze solute transport in physiological systems is a field pioneered by Sangren and Sheppard [178], Renkin [172], and Crone [40], Optically detectable probes (such as Evans Blue dye bound to albumin) can be used in conjunction with model analysis to probe the intravascular transport of... 

Before examining how a model of the level of detail of the four-region model illustrated in Figure 8.6 is constructed, we first examine the two-region model analyzed in 1953 by Sangren and Sheppard [178], This model will allow us to explore the kinetics of blood-tissue exchange based on an analytically tractable set of governing equations. 

While it may be elegant to obtain analytic closed-form model solutions, such as Equations (8.52) and (8.53) (introduced by Sangren and Sheppard as solutions to their model governing equations [178]), modeling of transport in biological systems... 

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