John L. Stephenson

Theory of transport in linear biological systems: I. Fundamental integral equation

The conditions under which the output,γ b (t), of a biological system is related to the input,γ a (t), by an integral equation of the typeγ b (t) = ∫ 0 t γ a (ω)w(t−ω)dω, where ω(t) is a transport functioncharacteristic of the system, are analyzed in detail. Methods of solving this type of integral equation are briefly discussed. The theory is then applied to problems in tracer kinetics in which input and output are sums of exponentials, and explicit formulae, which are applicable whether or not the pool is uniformly mixed, are derived for “turnover time” and “pool” size.

Models of coupled salt and water transport across leaky epithelia

SummaryA general formulation is presented for the verification of isotonic transport and for the assignment of a degree of osmotic coupling in any epithelial model. In particular, it is shown that the concentration of the transported fluid in the presence of exactly equal bathing media is, in general, not a sufficient calculation by which to decide the issue of isotonicity of transport. Within this framework, two epithelial models are considered: (1) A nonelectrolyte compartment model of the lateral intercellular space is presented along with its linearization about the condition of zero flux. This latter approximate model is shown to be useful in the estimation of deviation from isotonicity, intraepithelial solute polarization effects, and the capacity to transport water against a gradient. In the case of uphill water transport, some limitations of a model of fixed geometry are indicated and the advantage of modeling a compliant interspace is suggested. (2) A comprehensive model of cell and channel is described which includes the major electrolytes and the possible presence of intraepithelial gradients. The general approach to verification of isotonicity is illustrated for this numerical model. In addition, the insights about parameter dependence gained from the linear compartment model are shown to be applicable to understanding this large simulation.

Solution of a Multinephron, Multisolute Model of the Mammalian Kidney by Newton and Continuation Methods

A method of numerically solving the differential equations specifying solute and water flow in a multinephron, multisolute model of the mammalian kidney by a combination of Newton and continuation techniques is described. This method is used to generate a connected component of the steady state solution manifold of the model. A three dimensional section of this manifold is shown to be convoluted, with upper and lower sheets of stable solutions connected by an unstable middle sheet. Two dimensional sections of this surface are followed from a trivial constant profile of concentrations in the nonconcentrating kidney to the profiles of the maximally concentrating kidney. Study of these sections shows that for a given choice of model parameters there may exist no solution, there may be a unique solution, or there may be multiple solutions. A study of the time dependent solutions shows that the dynamic transition from the lower to the upper state and return may be via a hysteresis loop.

Use of sparse matrix techniques in numerical solution of differential equations for renal counterflow systems

Abstract Mammals concentrate urine by intricate counterflow systems in their kidneys. The mathematical models of such systems involve the numerical solution of a system of differential equations. (A preliminary report of this work was given in Notices of the American Mathematical Society21, A498 (1974).) This leads to a system of nonlinear algebraic equations, that are solved by the Newton-Raphson method. The basic step in the Newton-Raphson method is the repeated solution of a system of linear equations with the Jacobian of the nonlinear equations as the coefficient matrix. It is shown that the zero-nonzero structure of the Jacobian can be predicted from the physical connectivity of the model. We exploit this zero-nonzero structure to permute the Jacobian to a bordered block triangular form, which is then used to compute the correction to the solution vector of the nonlinear equations. For this computation a modified form of partial pivoting is used. Results are given that show the high accuracy of the computed solutions, the optimum use of the internal computer storage, and the fast rate of computation.

Theory of transport in linear biological systems: II. Multiflux problems

If in a multiflux system theith flux is given by the integral equation,\(\gamma _i = \sum _j \int_0^t {w_{ij} (t - \omega )} \gamma _j (\omega )d\omega + m_i (t)\), the corresponding equation in the Laplace transforms is Γ i = Σ j W ij Γ j +M i -the entire system having the matrix formulaion, [I−W]Γ=M. The general solution of this equation and its physical interpretation are discussed. Explicit solutions are given for the general mammillary and catenary systems and for some capillary exchange problems. The theory is applied to the integrated from of the fundamental continuity equation to give equations for total quantity of material in the various “compartments.” If the compartments are uniformly mixed, the integral equation treatment is shown to be mathematically equivalent to the usual differential equation formulation.

Mathematical analysis of a model for the renal concentrating mechanism

Abstract A system of ordinary differential equations, designed to model the counterflow system in the renal medulla, is studied. An existence theorem for solutions of the model equations is obtained. An exact solution of the system is obtained in the limiting case of infinite water permeability. If there is diffusion in the core, evaluation of the exact solution leads to multiple stable solutions of the model equations. One solution has a large concentration ratio, which tends to a finite asymptotic limit as the pump strength tends to infinity.

Analysis of the transient behavior of kidney models

Non-steady-state equations for kidney models are stated. General conservation relations for these equations are derived. Transient equations for the central core model of the renal medulla are developed. Solution of the equations by Laplace transform methods for time invariant volume flows is discussed. The general theory of solving models with time dependent flows by finite difference methods is developed.

ICE CRYSTAL FORMATION IN BIOLOGICAL MATERIALS DURING RAPID FREEZING

In the rapid freezing of biological materials two closely related problems arise. The first is the dependence of the cooling rate of the sample on its size, shape, and thermal properties, on the velocity of immersion, and on the temperature and physical properties of the coolant. The second is the relation of the size and number of ice crystals to the thermal history of the sample. The primary complication in the analysis of these problems arises from the fact that the ice crystals, as they grow, release latent heat of crystallization. This heat release modifies the course of the cooling and, consequently, the temperature distribution in the sample. Conversely, the rate a t which a given ice crystal can grow depends on its size and the temperature distribution of the surrounding aqueous medium. Our present objective in the analysis of the situation is not to predict the thermal history of the sample from known values of the various physical parameters, but rather from measurements on the thermal history and various supporting data to gain information about the temperature dependence of ice crystal nucleation and growth rate. Two basic difficulties arise in doing this. The first is that, mathematically, we are forced to set up the problem as though we were going to compute the thermal history from basic constants and then go through a process of inversion. The second difficulty is that the amount of basic data a t our disposal is rather limited. In a typical experiment we can determine the total number and size distribution of ice crystals a t the end of cooling (insofar as they are of microscopic dimensions), and we can measure the temperature of one or several microthermocouples embedded in the sample. I n order to convert this information into basic physical information it is always necessary to make rather sweeping simplifications. Clearly, then, the actual experimental situation must be so designed that these simplifications are reasonable approximations. The first question is what relation the temperature of a thermal detector embedded in the sample has to the temperature of the surrounding sample. Clearly, any detector functions as an integrator of the thermal energy arriving a t its own surface. The average temperature of the detector then will have some lag with respect to its surface temperature. The temperature lag for a small thermocouple,2 20 to 40 p in diameter, is not serious; however, the averaging effect along its length is. Another complication is that, as soon as ice crystals begin to form throughout the sample, they act as numerous small heat sources, evolving heat as they grow in size. If these crystals grow with sufficient rapidity they can clearly cause local hot spots. I n regard to these local hot spots one would like to know both how hot they can become and how rapidly heat released by one of them can be dispersed throughout the sample. I€ there is to be no significant temperature variation in the neighborhood of a growing ice crystal, the rate a t which heat can be dispersed throughout the

A mathematical model of proximal tubule absorption

SummaryA previous model of the mechanisms of flow through epithelia was modified and extended to include hydrostatic and osmotic pressures in the cells and in the peritubular capillaries. The differential equations for flow and concentration in each region of the proximal tubule were derived. The equations were solved numerically by a finite difference method. The principal conclusions are: (i) Cell NaCl concentration remains essentially isotonic over the pressure variations considered; (ii) channel NaCl concentration varies only a few mosmol from isotonicity, and the hydrostatic and osmotic pressure differences across the cell wall are of the same order of magnitude; (iii) both reabsorbate osmolality and pressure-induced flow are relatively insensitive to the geometry of the system; (iv) a strong equilibrating mechanism exists in the sensitivity of the reabsorbate osmolality to luminal osmolality; this mechanism is far more significant than any other parameter change.

A test problem for kidney models

There is a need in the literature for standard problems with which investigators may validate the numerical schemes that they apply to solve renal models. We consider the six-tube vasa recta model first described in [I] and used by Farahzad and Tewarson [2] and by Lory in [3] to be a good candidate for inclusion for a number of reasons. First, this model is sufficiently complex to exhibit some of the characteristics of larger models such as discontinuous sources and small flows, while being small enough to be solved on any computer with a FORTRAN compiler. Secondly, it provides a test of an algorithm’s ability to conserve mass and water balance, which is required for accuracy (see [4]). Thirdly, a number of numerical methods have been used to solve it. All of these that express the difference equations in conservative form have obtained essentially the same solution, allowing for variation in the accuracy of each method. (Others have obtained solutions to the difference equations solved, but not to the differential equations.) In [l] a centered-trapezoidal-difference approximation was used for the space derivatives and the resulting nonlinear equations were solved simultaneously using Newton’s method. Farahzad and Tewarson in [2] used the same difference approximations and a sparse-matrix version of Newton’s method for solution of the nonlinear equations. Lory [3] has used a multiple-shooting scheme to solve the problem. We have also solved it using both a partitioning scheme described in [5] for a multinephron model and DDMAD,’ an adaptive finite-difference solver for two-point boundary problems [6,7].