Transport of fluid and solutes in the body : a compartmental model approach

A mathematical model is formulated to study the transport and redistribution of fluid, proteins and small ions between the circulating blood, interstitium and cells. To achieve this task, the human/animal body was schematically divided into two distinct compartments, namely the plasma and interstitium. Two additional cellular compartments representing the red blood cells and generalized tissue cells were introduced as sub-compartments embedded in the two extracellular compartments. Two major sites of exchange are accounted for to characterize the movement of materials between these four fluid compartments. The microvascular exchange system (MVES) involves the movements of fluid, proteins and small ions from plasma into the interstitium across the capillary membrane as well as the return of these materials from the interstitial space back into plasma via the lymphatic system. Across the cellular membrane separating the intra- and extracellular compartments, there are dynamic exchanges of fluid and the three important ions, NA ⁺ , K ⁺ and CI⁻. These exchanges are assumed to occur by both passive and active mechanisms. The general model consists of a large set of time-dependent differential-algebraic equations that must be solved simultaneously to predict both clinically measurable data (e.g., plasma and blood volumes, plasma solute concentrations, and osmotic pressures) and experimentally difficult or impossible to measure variables (e.g., intracellular volumes and small ion concentrations, cellular transmembrane potentials, and transmembrane fluid shifts). The solution of these equations is carried out by the use of numerical methods. To describe mass exchange within the MVES and across the cell membranes, the transport characteristics of the principle resistances encountered by the exchanging materials must be known. The set of transport parameters needed to describe fluid and protein exchanges across the capillary membrane and within the lymphatic system were estimated previously from human data by other researchers in our group. As part of the present work, the transport parameters related to the movement of small ions across the capillary membrane (i.e., the reflection coefficient, OION, and the permeability-surface area product, PSION) were estimated using data from studies in which animals were successively infused with iso-osmolar saline (NS) and hyperosmolar saline (HS) solutions. Also, the transport parameters associated with cellular exchange (i.e., the cell membrane permeabilities for sodium, potassium and chloride, pNa, PK and pci, as well as the rate of the sodium-potassium pump, RP, were determined from the steady-state equations that describe cell volume regulation, together with the known normal distribution of ions between the intra- and extracellular fluids. Additional transport parameters required to accommodate external infusions of macromolecular species such as dextran were obtained from the literature. The validation of the model with these newly introduced parameters was carried out by comparing model-predicted results with experimental data from animals and humans that had undergone different resuscitation protocols (i.e., different rates and volumes of fluid administration using different types of infusates (NS, Ringer's solution (RS), HS or hyperosmotic saline/dextran solution (HSD)). Considering the physiological complexity of the body, the model-predicted results compared very well with the experimental data in the majority of cases simulated. As a subset of this study, mathematical expressions are developed to describe the excretion of fluid and small ions by the kidney. The formulation of this renal model is based on the physiological role of the kidney in maintaining the plasma volume and plasma sodium concentration at their normal values. Thus, it is assumed that the kidney responds via a negative feedback to any changes in these two values from their normal set-points. The generalized fourcompartment model that includes the 'kidney module' was tested using experimental animal and human data involving infusions of NS or HS solutions, or non-treated hemorrhages. The model predictions were generally in very good agreement with the measured results for all the cases simulated. Finally, the applicability of the model to the study of hemorrhagic shock was exemplified through a series of simulations that describe the distinct stages in the progression of shock. Empirical equations were proposed to characterize the release of glucose and other solutes that occur during the compensatory (hemodilution) phase of hemorrhage, as well as the disturbed cellular transport that takes place during the decompensatory (hemoconcentration) stage of shock. The weaknesses and strengths of the model to clarify certain mechanisms related to hemorrhagic shock were underlined.

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