A model of human microvascular exchange: parameter estimation based on normals and nephrotics.

A mathematical model is formulated and used to describe the distribution and transport of fluid and albumin in the human circulation, interstitium and lymphatics. Two transcapillary mass exchange mechanisms are investigated: a homoporous 'coupled Starling model', in which transcapillary albumin diffusion and convection occur within the same pathway, and a heteroporous 'plasma leak model', in which variations in structure and pressure are permitted along the length of the capillary. Parameters used in the transport models are determined based on statistical fitting of simulation predictions to experimental data from normal humans and nephrotic patients. The data consists of interstitial fluid volumes and interstitial colloid osmotic pressures as functions of plasma colloid osmotic pressure. Model validation is carried out based on comparison of (i) simulation predictions with experimental data used in parameter estimation, (ii) estimated transport parameters with experimentally determined values, and (iii) simulation predictions with a set of dynamic data from an albumin infusion study. While both models with their best-fit parameter estimates provide a good representation of experimental data, the drawbacks of the plasma leak model are three-fold: it requires more estimated parameters than the coupled Starling model, little experimental information exists with which to compare these parameters and, with the best fit values obtained, the plasma leak mechanism becomes insignificant. The model that employs a Starling-type exchange mechanism will therefore be favoured in future applications.

A model of fluid resuscitation following burn injury: formulation and parameter estimation.

A dynamic compartmental model is developed to describe the redistribution of fluid and albumin between the circulation and the intact and injured interstitia following burn injury in humans. Transcapillary fluid and albumin exchange is described by a coupled Starling mechanism, while the effect of the burn is represented by time-dependent perturbations to all three compartments. The unknown model parameters are determined for two groups of patients, having less than and greater than 25% total body surface area burns, by statistical fitting of model predictions to patient data from two sources. The parameters include the perturbations to the fluid filtration coefficients in uninjured and injured tissue, GkF,Tl and GkF,BT, respectively, the relaxation coefficient, r, which describes the exponential decay of the perturbations, and the exudation factor, EXFAC, which relates the protein concentration in the exudate to that in the injured tissue. Perturbations to other parameters, including the membrane permeability-surface area product and the albumin reflection coefficient in the injured and uninjured tissues, are determined based on interrelationships with GkF,Tl and GkF,BT. The values of GkF,BT, when corrected for tissue destruction and decreased post-injury perfusion, are in reasonable agreement with the limited experimental data available from the literature. The model and its parameters are further validated by comparing the simulated patient responses to the clinical data used in the parameter estimation as well as to data available from two additional sources.

A mathematical model of interstitial transport. I. Theory.

A generalized model is developed to describe the transport of fluid and plasma proteins or other macromolecules within the interstitium. To account for the effects of plasma protein exclusion and interstitial swelling, the interstitium is treated as a multiphase deformable porous medium. Fluid flow is assumed proportional to the gradient in fluid chemical potential and therefore depends not only on the local hydrostatic pressure but also on the local plasma protein concentrations through appropriate colloid osmotic pressure relationships. Plasma protein transport is assumed to occur by restricted convection, molecular diffusion, and convective dispersion. In a companion paper (D. G. Taylor, J. L. Bert, and B. D. Bowen, 1990, Microvasc. Res. 39, 279-306) a simplified version of the model is used to analyze steady-state fluid and plasma protein exchange within mesentery.

A mathematical model of interstitial transport. II. Microvascular exchange in mesentery.

A simplified version of the model of interstitial transport developed earlier (D. G. Taylor, J. L. Bert, and B. D. Bowen, 1990, Microvasc. Res. 39, 253-278) is used to investigate microvascular exchange of fluid and a single "aggregate" plasma protein species in mesenteric tissue. The interstitium is approximated by a rigid, rectangular, porous slab displaying two fluid pathways, only one of which is available to plasma proteins. The model is used to explore the effects of the interstitial plasma protein diffusivity, the tissue hydraulic conductivity, the restricted convection of plasma proteins, and the mesothelial transport characteristics on the steady-state distribution and transport of plasma proteins and flow of fluid in the tissue. The simulations predict significant convective plasma protein transport and complex fluid flow patterns within the interstitium. These flow patterns can produce local regions of high fluid and plasma protein exchange along the mesothelium which might be erroneously identified as "leaky sites."

Microvascular exchange and interstitial volume regulation in the rat: model validation.

A dynamic mathematical model is formulated and used to describe the distribution and transport of fluid and plasma proteins between the circulation, interstitial space of skin and muscle, and the lymphatics in the rat. Two descriptions of transcapillary exchange are investigated: a homoporous "Starling model" and a heteroporous "plasma leak model." Parameters used in the two hypothetical transport mechanisms are determined based on statistical fitting procedures between simulation predictions and selected experimental data. These data consist of interstitial fluid volume and colloid osmotic pressure measurements as a function of venous pressure for muscle and interstitial colloid osmotic pressure vs. venous pressure for skin. The values determined for the transport parameters compare well with data in the literature. The fully determined model is used to simulate steady-state conditions of hypoproteinemia, overhydration, and dehydration, as well as the dynamic response to changes in venous pressure and intravascularly administered protein tracers. Comparisons between the simulation predictions and experimental data for these various perturbations are made. The plasma leak model appears to provide a better description of microvascular exchange.

Protein adsorption in polysulfone hollow fiber bioreactors used for serum-free mammalian cell culture.

The recovery of serum-free medium proteins from poly-sulfone hollow fiber bioreactors (HFBRs) was investigated. More than 99% of the initial transferrin was adsorbed to the hydrophobic hollow fibers within 2 h of HFBR operation. A methodology to minimize transferrin adsorption by pre-adsorption of bovine serum albumin (BSA) was developed. BSA adsorption on suspended cut fibers was virtually complete within 1 h. BSA-coated fibers adsorbed only 5% of the transferrin within 10 days, whereas uncoated cut fibers adsorbed more than 99% of the transferrin within 1 h. An improved HFBR startup procedure, using a BSA-coating step before inoculation, resulted in substantially higher transferrin recovery. Additional factors influenced extracapillary space (ECS) transferrin concentrations. Pronounced downstream polarization of transferrin was observed in the ECS. In addition, the 30-kDa nominal molecular weight cutoff ultrafiltration membranes rapidly leaked transferrin from the ECS to the lumen.

Modeling transient exchange in mesentery.

In this paper, a mathematical model of interstitial transport and microvascular exchange within a rigid mesenteric tissue segment is employed to simulate the transient exchange of fluid and plasma proteins following two systemic disturbances: hypoproteinemia and venous congestion. In each case, the model system behavior is studied as a function of interstitial plasma protein transport mechanisms and mesothelial transport properties. Plasma protein washout was generally predicted in cases of hypoproteinemia. However, following venous congestion, the transient change in interstitial plasma protein content also depended on the relative sieving properties of the filtering and draining boundaries. When these boundaries display similar sieving characteristics, the interstitial plasma protein content increases following the disturbance. Such behavior may have some bearing on transient exchange in the hepatic microcirculation during venous congestion.

Microvascular exchange and interstitial volume regulation in the rat: implications of the model.

The present work uses and extends a dynamic mathematical model [J. L. Bert, B. D. Bowen, and R. K. Reed. Am. J. Physiol. 254 (Heart Circ. Physiol. 23): H384-H399, 1988] to investigate microvascular exchange and interstitial fluid volume regulation in the rat. Alternative concepts of transcapillary exchange as well as other parametric changes were incorporated into the model. In all cases, predictions resulting from these changes did not describe the available experimental information as well as the original model. A sensitivity analysis of the model showed the microvascular exchange system to be well regulated near its normal steady-state conditions through passive readjustment of the forces participating in the volume regulation. The transient rates of fluid and protein exchange were studied in order to determine the mechanisms inherent in the model that lead to fluid volume regulation during episodes of increased venous pressure and hypoproteinemia. In addition to interstitial compliance, lymph flow characteristics, and washdown of interstitial proteins, it was found that the magnitude and direction of reabsorption played an important role in the regulation process. Edema was always associated with a permanent reversal of the reabsorptive flow.

Mammalian cell retention devices for stirred perfusion bioreactors.

Within the spectrum of current applications for cell culture technologies, efficient large-scale mammalian cell production processes are typically carried out in stirred fed-batch or perfusion bioreactors. The specific aspects of each individual process that can be considered when determining the method of choice are presented. A major challenge for perfusion reactor design and operation is the reliability of the cell retention device. Current retention systems include cross-flow membrane filters, spin-filters, inclined settlers, continuous centrifuges and ultrasonic separators. The relative merits and limitations of these technologies for cell retention and their suitability for large-scale perfusion are discussed.

Microvascular exchange during burn injury: II. Formulation and validation of a mathematical model.

A mathematical model of microvascular exchange in the rat following a burn injury was developed by extending an existing model of normal microvascular exchange to include perturbations characteristic of burn injuries without fluid resuscitation. The changes anticipated for small (10% body surface area) and large (40% body surface area) burns are incorporated systematically into the model until there is no improvement in the statistical fit of the simulation predictions with the experimental data of Lund and Reed (Circulatory Shock 20:91-104, 1986). The "best fit" perturbations for the small burn include the experimentally measured changes in mean arterial pressure and injured tissue pressure as well as changes to plasma protein and fluid transport coefficients in the injured tissue. The larger burn "best fit" simulation required changes to the plasma protein transport coefficients in the intact tissues as well as all of the changes listed above. The simulation results are compared with the available experimental information on burn injuries as well as with the specific data of Lund and Reed (Circulatory Shock 20:91-104, 1986).

Microvascular exchange during burn injury: III. Implications of the model.

The present work investigates the implications of the predictions of a dynamic mathematical model of microvascular exchange following a nonresuscitated burn injury in a rat (Bert et al.: Circulatory Shock 28:199-219, 1989). Transport coefficients, transmicrovascular pressures, and the resultant fluid and protein fluxes were examined in order to assess their quantitative importance to the dynamic behavior of small (10% body surface area) and large (40% body surface area) burns. Edema accumulation in the injured tissue is dependent not only on events occurring in that tissue but is influenced strongly by interaction with the plasma and the noninjured tissue compartments.

Microvascular exchange during burn injury: IV. Fluid resuscitation model.

The present work is a continuation of studies concerned with mathematical modelling and simulation of microvascular fluid and protein exchange following burn injuries [Bert et al.: Circulatory Shock 28: 199-219, 1989: Bowen et al.: Circulatory Shock 28: 221-233, 1989]. The model has been extended to include the effects of different types of fluid resuscitation on the circulatory and microvascular exchange systems. The model and a statistical fitting procedure were used to find the ranges of fitting parameter values that best describe the changes in interstitial fluid volume and protein mass as well as transcapillary protein extravasation for three sets of experiments (no resuscitation, resuscitation with Ringer's or resuscitation with plasma). Typical changes in mass exchange related parameters postburn that resulted in simulation predictions which were a good fit to the experimental data include: an increase in the large pore pathway for protein of 100 times in the injured skin and 5 times in non-injured skin and skeletal muscle, an increase in fluid filtration coefficients in injured skin of 10 times and an instantaneous decrease of 50% in the area available for exchange in injured skin at the time of the burn.

A model of fluid and solute exchange in the human: validation and implications.

In order to understand better the complex, dynamic behaviour of the redistribution and exchange of fluid and solutes administered to normal individuals or to those with acute hypovolemia, mathematical models are used in addition to direct experimental investigation. Initial validation of a model developed by our group involved data from animal experiments (Gyenge, C.C., Bowen, B.D., Reed, R.K. & Bert, J.L. 1999b. Am J Physiol 277 (Heart Circ Physiol 46), H1228-H1240). For a first validation involving humans, we compare the results of simulations with a wide range of different types of data from two experimental studies. These studies involved administration of normal saline or hypertonic saline with Dextran to both normal and 10% haemorrhaged subjects. We compared simulations with data including the dynamic changes in plasma and interstitial fluid volumes VPL and VIT respectively, plasma and interstitial colloid osmotic pressures PiPL and PiIT respectively, haematocrit (Hct), plasma solute concentrations and transcapillary flow rates. The model predictions were overall in very good agreement with the wide range of experimental results considered. Based on the conditions investigated, the model was also validated for humans. We used the model both to investigate mechanisms associated with the redistribution and transport of fluid and solutes administered following a mild haemorrhage and to speculate on the relationship between the timing and amount of fluid infusions and subsequent blood volume expansion.

Preliminary model of fluid and solute distribution and transport during hemorrhage.

The distribution and transport of fluid, ions, and other solutes (plasma proteins and glucose) are described in a mathematical model of unresuscitated hemorrhage. The model is based on balances of each material in both the circulation and its red blood cells, as well as in a whole-body tissue compartment along with its cells. Exchange between these four compartments occurs by a number of different mechanisms. The hemorrhage model has as its basis a validated model, due to Gyenge et al., of fluid and solute exchange in the whole body of a standard human. Hypothetical but physiologically based features such as glucose and small ion releases along with cell membrane changes are incorporated into the hemorrhage model to describe the system behavior, particularly during larger hemorrhages. Moderate (10%-30% blood volume loss) and large (> 30% blood loss) hemorrhage dynamics are simulated and compared with available data. The model predictions compare well with the available information for both types of hemorrhages and provide a reasonable description of the progression of a large hemorrhage from the compensatory phase through vascular collapse.

A model of human microvascular exchange.

A compartmental model consisting of the circulation, a general interstitium, and the lymphatics, is formulated to describe the transport and distribution of fluid and plasma proteins (albumin) in the human microvascular exchange system. Transcapillary mass exchange is assumed to occur via a coupled Starling mechanism. Unknown or poorly quantified model parameters are estimated by statistical fitting of simulation predictions to five different sets of experimental data. The data consist of steady-state and transient plasma and interstitial volumes and colloid osmotic pressures measured under laboratory or clinical conditions for normal humans and for patients with nephrotic syndrome or mild heart disease. In all cases, it is assumed that the system response to perturbations imposed either artificially or through illness is due to changes in the Starling driving forces. The three best-fit parameters were found to be normal capillary hydrostatic pressure, Pc,o = 11.0 mm Hg; albumin reflection coefficient, sigma = 0.99; and lymph flow sensitivity, LS = 43.1 ml/mm Hg.hr. Three other parameters, which were unknown but related to the estimated parameters through steady-state mass balance equations, were determined to be fluid filtration coefficient, KF = 121.1 ml/mm Hg.hr; albumin permeability-surface area product, PS = 73.0 ml/hr; and normal lymph flow, JL,o = 75.7 ml/hr. The fully described model was validated by comparisons between (1) simulation predictions and data used in parameter estimation, (2) estimated transport parameters and available literature values, and (3) model predictions and an additional set of experimental data.

Transport of fluid and solutes in the body I. Formulation of a mathematical model.

A compartmental model of short-term whole body fluid, protein, and ion distribution and transport is formulated. The model comprises four compartments: a vascular and an interstitial compartment, each with an embedded cellular compartment. The present paper discusses the assumptions on which the model is based and describes the equations that make up the model. Fluid and protein transport parameters from a previously validated model as well as ionic exchange parameters from the literature or from statistical estimation [see companion paper: C. C. Gyenge, B. D. Bowen, R. K. Reed, and J. L. Bert. Am. J. Physiol. 277 (Heart Circ. Physiol. 46): H1228-H1240, 1999] are used in formulating the model. The dynamic model has the ability to simulate 1) transport across the capillary membrane of fluid, proteins, and small ions and their distribution between the vascular and interstitial compartments; 2) the changes in extracellular osmolarity; 3) the distribution and transport of water and ions associated with each of the cellular compartments; 4) the cellular transmembrane potential; and 5) the changes of volume in the four fluid compartments. The validation and testing of the proposed model against available experimental data are presented in the companion paper.

Transport of fluid and solutes in the body II. Model validation and implications.

A mathematical model of short-term whole body fluid, protein, and ion distribution and transport developed earlier [see companion paper: C. C. Gyenge, B. D. Bowen, R. K. Reed, and J. L. Bert. Am. J. Physiol. 277 (Heart Circ. Physiol. 46): H1215-H1227, 1999] is validated using experimental data available in the literature. The model was tested against data measured for the following three types of experimental infusions: 1) hyperosmolar saline solutions with an osmolarity in the range of 2,000-2,400 mosmol/l, 2) saline solutions with an osmolarity of approximately 270 mosmol/l and composition comparable with Ringer solution, and 3) an isosmotic NaCl solution with an osmolarity of approximately 300 mosmol/l. Good agreement between the model predictions and the experimental data was obtained with respect to the trends and magnitudes of fluid shifts between the intra- and extracellular compartments, extracellular ion and protein contents, and hematocrit values. The model is also able to yield information about inaccessible or difficult-to-measure system variables such as intracellular ion contents, cellular volumes, and fluid fluxes across the vascular capillary membrane, data that can be used to help interpret the behavior of the system.