## Abstract

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.

- hyperosmolarity
- cell volume
- interstitial volume
- plasma volume expansion
- plasma osmolarity

physiological salt solutions, such as Ringer solution (with or without added macromolecules), are commonly used for and have proven effective in resuscitation after hemorrhage and other types of trauma. However, incidents of transient increase in intracranial pressure and occasional overhydration, as well as the possible occurrences of adult respiratory distress syndrome associated with the use of large volumes of these solutions, have directed efforts toward seeking alternative resuscitation protocols. In several experimental studies on animal models, resuscitation with hypertonic saline solutions has been investigated. The ability of hypertonic saline solutions to temporarily expand plasma volume up to two to three times the infused volume, even when given in relatively small amounts, may make these solutions efficacious in several clinical situations such as hemorrhage or burn shock. It is commonly agreed that expansion of plasma volume, as a result of the administration of hypertonic solutions, is caused primarily by fluid shifts based on osmotic differences between the vascular compartment and the interstitial extracellular and cellular reservoirs. However, it is still unclear whether the interstitium and the interstitial cells are directly responsible for the massive water mobilization into the vasculature (33) or, as previously reported by Mazzoni et al. (21), the capillary endothelial cells and the red blood cells are also significantly involved in plasma volume elevation. Clearly, a better understanding of the mechanisms underlying rapid plasma volume replacement during hypertonic resuscitation is required. Furthermore, such a quantitative description would aid in selecting optimal fluid replacement therapies.

As a complement to experimental studies of resuscitation protocols, there is a need to develop, as well as validate, mathematical models capable of predicting the fluid, protein, and small ion shifts that take place between the vascular, interstitial, and cellular compartments after the administration of a given type of solution. A number of mathematical models have emerged over the past few years that address the issue of body fluid distribution and transport with the particular goal of describing blood volume changes that follow hemorrhage. Significant modeling effort has been directed toward describing the mechanisms of blood restitution and, most importantly, the requirements of different resuscitation schemes (4, 21). However, before modeling mass transport in a highly perturbed state such as burn or hemorrhage, it would be prudent to create and validate an initial model based on the normal physiological behavior of the system, including the estimation of parameters that are difficult to measure or are unavailable from the literature.

In previous models of microvascular transport formulated by our group (1, 5), it was assumed and reasonably justified that, under specific conditions, the cellular compartments did not take part in the exchange process; i.e., the small ions and fluid in the cells were considered at all times to be at equilibrium with those in the plasma and interstitium. Nevertheless, important clinical situations, such as hemorrhage with its altered cellular activity and use of hypertonic resuscitants, require the consideration of a cellular component that plays an active role in exchange involving fluids and ions during short-term fluid regulation. The electrophysiological basis for this requirement was well established by Shires et al. (27) and Nakayama et al. (23). In some early models of fluid volume regulation (18, 30), the authors limited their descriptions to water and protein movement between plasma and interstitium. Although Wolf (33) and Mazzoni et al. (21) described the movement of water, protein, and crystalloids between plasma and interstitial fluid compartments and took into consideration the fluid exchange with cells, the authors did not account for the transport of small solutes associated with these intracellular compartments. The more complex model of Carlson et al. (4) added several perturbations hypothetically descriptive of hemorrhage to a nonvalidated precursor model. The magnitudes of these perturbations were adjusted to fit their experimental data; however, no true validation of this model was attempted.

The primary goal of the present study is to extend our previous modeling efforts (1, 5, 26, 35) as well as those of other authors (14,21, 32, 33) one step further in developing a compartmental model that describes fluid and solute exchanges between the plasma, interstitium, and cells. Furthermore, to maintain confidence in our description of the underlying physiological mechanisms that might contribute to effective plasma replacement, we will first test this model against experimental data that take into account solely the effect of different types of infusions, i.e., in the absence of any type of trauma such as hemorrhage or burn injury.

The mathematical model described in this work uses as building blocks two models previously reported in the literature: one dealing with transport in the microvascular exchange system described by Xie et al. (35) and the other considering the exchange that occurs at the cellular level according to concepts developed by Jakobsson (14). These models have been extended and modified to describe the simultaneous shifts of fluid, protein, and ions across the capillary membrane together with the exchanges of water and ions across the cellular membranes.

The first part of the present study, therefore, has as its aim the development of a conceptually up-to-date mathematical model that describes the fluid and solute (protein and ion) exchanges between circulating blood and the interstitium, taking into account the exchange occurring with the cells associated with these spaces. The companion paper (11) is concerned with*1*) the estimation of two parameters that govern the transport of small ions across the capillary wall,*2*) the validation of this model on the basis of comparisons of model predictions with available literature data, and *3*) an assessment of the magnitude and time course of fluid shifts across the microvascular barrier after hyperosmolar and isotonic saline solution administrations.

### Glossary

*A*- Cellular membrane surface area (cm
^{2}) - BW
- Body weight (kg)
- C
- Protein concentration (g/l)
*F*- Faraday’s constant (96,485.3 C/mol)
- [FA]
- Concentration of negative nondiffusible intracellular species FA
^{−}(mM) - [FC]
- Concentration of positive nondiffusible intracellular species FC
^{2+}(mM) - Hct
- Hematocrit (%)
- [Ion]
- Ion concentration (mM)
- (Ion)
- Ion concentration (meq/l)
*J*- Rate of fluid transfer (ml/h)
*k*_{F}- Fluid filtration coefficient (ml ⋅ mmHg
^{−1}⋅ h^{−1}) *k*_{Prot}- Proportionality constant (see
*Eq.ED5*) - LS
- Lymph flow sensitivity (ml ⋅ mmHg
^{−1}⋅ h^{−1}) - M
- Ion content (mmol)
- M˙
- Rate of ion transfer (mmol/h)
- MW
- Molecular weight (g/mol)
- Osm
- Total number of mosmoles (mosmol ≅ mmol)
*p*- Cellular membrane permeability (cm/s)
- P
- Hydrostatic pressure (mmHg)
*PS*- Permeability-surface area product (ml/h)
- Q
- Protein content (g)
- Q˙
- Rate of protein transfer (g/h)
*R*- Universal gas constant (8.314 × 10
^{3}J ⋅ mol^{−1}⋅ K^{−1}) - RP
- Rate of Na
^{+}-K^{+}pump (cm/s) - S
- Osmolarity (mosmol/l)
*t*- Time (h)
*T*- Temperature (K)
- V
- Volume (ml)
*V*_{m}- Cellular transmembrane electrical potential (mV)
*z*- Valence

### Greek Symbols

- ΔIon
_{D} - Capillary transmembrane concentration difference (meq/l)
- Δφ
- Dimensionless cellular transmembrane potential
- Φ
- Osmotic coefficient
- π
- Colloid osmotic pressure (mmHg)
- ς
- Reflection coefficient
- ρ
- Ratio of Na
^{+}-K^{+}pump

### Subscripts

- AV
- Available volume
- Cap
- Capillary
- Comp
- Compliance
- D
- Donnan effect
- Ex
- Excluded
- FA
- Negative nondiffusible intracellular species
- Ion
- Na
^{+}, K^{+}, C^{2+}, Cl^{−}, or A^{−} - I
- Interstitium
- L
- Lymph
- Norm
- Normal steady-state value
- Prot
- Protein
- Per
- Perspiration
- Pl
- Plasma
- RBC
- Red blood cells
- Res
- Infusion
- TC
- Tissue cells
- Ur
- Urinary loss
- W
- Cellular water shift

## METHODS

This section contains the overall description of the physiological system together with the main assumptions used in the formulation of the mathematical model, followed by a detailed presentation and discussion of the model equations and the numerical solution procedures employed.

### Overall Description

The model is composed of two interconnected homogenous extracellular fluid compartments, namely, the vascular and interstitial compartments, between which fluid, proteins, and small ions are exchanged across the capillary membrane and via the lymphatics (see Fig.1). Additionally, each of these compartments contains an embedded, cellular fluid compartment. Therefore, the interstitial compartment is composed of interstitial fluid with a volume V_{I} and interstitial cells with a fluid volume V_{TC}. The generic term “interstitial cells” describes cells with lumped properties that represent the diversity of cells included in the generalized interstitium. These are assumed to have the same properties as muscle cells. The vascular compartment is made up of plasma with a volume V_{Pl} and red blood cells with a fluid volume V_{RBC}. The red blood cell volume is related to the plasma volume via the systemic hematocrit (Hct), given by
Equation 1The plasma compartment is the only direct recipient of external fluid through infusion. Fluid is filtered from the plasma to the interstitium according to the Starling hypothesis, whereas the transport of proteins and small ions through the capillary membrane is governed by diffusive-convective mechanisms. Under normal physiological conditions, the small ions are distributed across the capillary membrane according to a Donnan equilibrium. Fluid and solutes (i.e., proteins and ions) within the interstitium are drained back into the circulation via the lymphatics by convection. Across the cell membrane associated with each of the two cellular compartments mentioned above, exchanges of water and small ions take place. The water movement across the cell membrane is governed by differences in osmolarity between the internal and external environment of the cell. The movements of small ions are determined in part by the electrochemical potential difference and the activity of a Na^{+}-K^{+}pump (i.e., active and passive transports) as well as by the cell membrane permeability properties. The model takes into account the interdependence of the membrane potential, cellular volume, and intra- and extracellular ionic concentrations as well as the internal and external osmolarity in describing these exchanges.

### Model Assumptions

The important assumptions that form the basis of this model, which includes microvascular transport as well as cellular exchange, are presented as follows.

All compartments including the cellular ones are considered to be well mixed with spatially constant descriptive parameters, i.e., mean values for descriptive properties (e.g., transport parameters) and dependent variables (e.g., concentrations for given solutes) are used at any given time. This assumption is consistent with the type of data normally reported, and its use is supported by several previous studies (13, 28, 29).

Similar to systems in other models (e.g., Ref. 4), all proteins in the system are assumed to have the same properties as albumin with an average molecular weight (MW_{Prot}) of 67,000 g/mol. These species generate oncotic pressures and are exchanged only between plasma and the interstitium (i.e., there is no protein exchange across the cell membrane). Additionally, they are responsible for generating a Donnan effect across the capillary membrane.

The same conductive pathways in the capillary membrane serve as sites for both fluid and protein transport between plasma and the interstitium.

There are five types of mobile ions accounted for in the extracellular compartments: Na^{+}, K^{+}, C^{2+}, Cl^{−}, and A^{−}. The cationic species C^{2+} represents all the positive ions other than Na^{+} and K^{+} present in plasma and the interstitium. Because most of the additional positive charge in these two fluid compartments is due to Mg^{2+} and Ca^{2+} (15), a charge of +2 was attributed to the C^{2+} species. A^{−} represents all anions other than Cl^{−} present in the compartments. A charge of −1 was assumed for this species. All these ions participate in transcapillary exchange. Additionally, they are distributed on either side of the capillary membrane according to a Donnan equilibrium.

The transport of small ions across the capillary membrane takes place through both convective and diffusive pathways.

Five types of ions are also accounted for in the intracellular compartments: Na^{+}, K^{+}, FC^{2+}, Cl^{−}, and FA^{−}. The only small ions transported across the cellular membrane between the intra- and extracellular compartments are Na^{+}, K^{+}, and Cl^{−}. All other intracellular cations besides Na^{+} and K^{+} are denoted as FC^{2+}, and all anions other than Cl^{−} are denoted as FA^{−}. These species are considered to be present in fixed amounts, i.e., they do not cross the cell membrane. Thus any changes in the concentrations of these species are due solely to cellular swelling or shrinking.

The transport of Na^{+} and K^{+} across the cell membrane occurs by both electrodiffusion and active transport (i.e., by means of a Na^{+}-K^{+}pump). As in a previously reported model (14), the Na^{+}-K^{+}pump is characterized by a constant ratio of Na^{+} to K^{+} transport [ρ = ([Na^{+}]_{out}/[K^{+}]_{in}) = 3/2] and a constant rate (RP). The flux of Na^{+} or K^{+} due to active transport is dependent on the pump rate and the intracellular Na^{+} and extracellular K^{+} concentrations.

Cl^{−} is transported across the cellular membrane by electrodiffusion in response to its electrochemical gradient. No active transport is ascribed to this ion. In the model, Cl^{−} transport is based on maintaining intercellular electroneutrality in the face of transmembrane Na^{+} and K^{+} exchanges.

Changes in the cell transmembrane potential (Δφ) take place instantaneously as a direct consequence of the charge separation and redistribution across the cell membrane.

As in several previous models of cell volume regulation (28, 32), the standard assumptions of bulk internal electroneutrality and osmotic transmembrane equilibrium were employed.

Changes in cell volume are directly related to the change in cellular water content and are assumed to occur instantaneously, i.e., the cellular membrane is assumed to be freely permeable to water (14,19, 20). Further discussion of this assumption is given at the end of*Intracellular compartments: red blood cells and tissue cells*. Water shifts across the cell membrane are imposed by the isotonicity condition between the extra- and intracellular environments.

The cellular parameters are taken to be different for red blood cells and interstitial cells according to experimental information (12, 15). It was assumed that the interstitial cells have the same characteristics as skeletal muscle cells for the following reasons:*1*) 65% of the total tissue mass available for capillary exchange is attributed to skeletal muscle tissue (17); and *2*) the muscle cells represent ∼40% of the total body weight and therefore are the main source of water mobilization as a result of an osmotic disturbance.

The properties of the cellular and capillary membranes are unaffected by the infusions modeled in this study.

Other assumptions more specific to the cases simulated are discussed where appropriate in the companion paper (11).

### Model Equations

To predict the interdependent fluid, protein, and small ion distribution and transport in the vascular, interstitial, and intracellular compartments, the model requires descriptions of the transcellular and transcapillary membrane exchanges.

#### Extracellular compartments: plasma and interstitium.

The dynamic behavior of the two extracellular compartments is based primarily on mass balance equations. Thus the fluid mass balances are
Equation 2
Equation 3the protein balances are
Equation 4
Equation 5and the small ion (Na^{+}, K^{+}, C^{2+}, Cl^{−}, and A^{−}) balances can be expressed as
Equation 6
Equation 7where V, Q, and M represent the compartmental fluid volume, protein content, and ion content, respectively, and*J*, Q˙, andM˙ represent rates of transport of fluid, protein, and small ions, respectively, into or out of the compartment. The subscripts Pl, I, RBC, and TC denote plasma, interstitial, red blood cell, and interstitial (tissue) cell compartments, respectively, and L indicates lymph, whereas Ion is a generic term used to describe any of the ionic species (i.e., Na^{+}, K^{+}, C^{2+}, Cl^{−}, or A^{−}). The subscript Res stands for resuscitation when a time-dependent resuscitation rate constitutes an input to the model; the subscripts Per and Ur indicate the loss of fluid through perspiration and urine production, respectively, for specific cases in which these losses are considered, i.e., when time-dependent rates of these latter fluids constitute known or predictable outputs from the interstitial and plasma compartments, respectively.

Fluid is filtered from the capillaries to the interstitium according to the Starling hypothesis. The filtration rate (*J*
_{I}) depends on the osmotic and hydrostatic pressures in both compartments and is given by
Equation 8where *k*
_{F} is the fluid filtration coefficient representing the hydraulic conductivity of the capillary membrane; P_{Cap} and P_{I} are the hydrostatic pressures in the capillary and interstitium, respectively; π_{Prot,Pl} and π_{Prot,I} are the colloid osmotic pressures exerted by the proteins and π_{Ion,Pl} and π_{Ion,I} are the osmotic pressures exerted by the small ions in the plasma and interstitium, respectively, whereas π_{Ion,D} indicates the osmotic contribution of the small ions restricted in their movement due to Donnan constraints. A more detailed description of the small ion contribution to the Donnan effect and the way in which this effect was accounted for in the model is given in appendix
. ς and ς_{Ion}are the average reflection coefficients for proteins (i.e., albumin) and small ions, respectively.

The rate of albumin transport across the capillary membrane (Q˙_{I}) is governed by the following equation developed by Bresler and Groome (3), which shows that protein transport from the circulation to the interstitium is nonlinearly coupled with the fluid exchange
Equation 9In *Eq. 9
*, C_{Prot,Pl} and C_{Prot,I,AV} are the protein concentrations in plasma and available interstitial volume as detailed in appendixes and
. *PS*is the protein permeability-surface area product of the capillary.

It was assumed in the present work that the transport rate of small ions across the capillary membrane (M˙_{Ion,I}) occurs through separate convective and diffusive pathways and therefore is described as
Equation 10where [Ion]_{Pl} and [Ion]_{I} are the ion concentrations in plasma and interstitium, respectively, and*PS*
_{Ion} is the capillary permeability-surface area product for small ions. For a given ion, ΔIon_{D} represents the concentration difference across the capillary membrane caused by the Donnan effect, which is described in appendix
. *Equation 10
* is used to represent the transcapillary transport of all the small ions present in the system except for A^{−}. It was assumed in the present study that the transport of species A^{−} occurs at a rate that is just sufficient to maintain an overall electroneutral transport across the capillary wall. Therefore
Equation 11where *z* is the charge of the ionic species.

According to the measurements reported by Crone and Christensen (6), the capillary endothelium has a very low electrical resistance. If no active transport is considered across the endothelial cells, the magnitude of the Donnan potential difference can be calculated as 1–2 mV, which is not expected to influence the ionic transport significantly. Hence, for simplicity, the transcapillary membrane potential was not accounted for, and the contribution of an electrical term in *Eq. 10
*, which describes the electrodiffusion of small ions, was ignored at this point.

Tissue fluid, proteins, and ions are drained back into the circulation by the lymphatic system at a rate*J*
_{L},Q˙_{L}, andM˙_{L}, respectively. It was assumed that no accumulation of material occurs in the lymphatics; consequently, the transport of fluid and solutes toward the vascular compartment by this mechanism takes place instantaneously.

The equations for lymph flow used in the model are based on those described by Chapple et al. (5) for humans. These equations assume that the lymph flow rate varies linearly with the interstitial hydrostatic pressure under both overhydrated and slightly dehydrated conditions but ceases when the interstitial pressure becomes equal to or falls below that of the excluded volume of the interstitium and are as follows
Equation 12
Equation 13
Equation 14where*J*
_{L,Norm} is the lymph flow rate under normal steady-state conditions corresponding to an interstitial hydrostatic pressure, P_{I,Norm}; LS represents the lymph flow sensitivity; and P_{I,Ex}denotes the hydrostatic pressure corresponding to the excluded interstitial volume, as previously described (5).

According to our past modeling practice (1, 2, 26), it is considered that the lymphatics transport albumin to the circulation only by convection Equation 15A similar approach is taken when describing the lymphatic transport of small ions. Thus Equation 16The hydrostatic pressure of the interstitial compartment is correlated with the interstitial volume through an interstitial compliance relationship. Such a relationship was previously developed and described for humans by Chapple et al. (5). A similar relationship is used in the current model and is given in appendix . As in the work by Chapple et al. (5), a linear compliance relationship between the hydrostatic pressure in plasma and the plasma volume was assumed (see appendix ).

Table 1 lists the values of the transport coefficients employed for the capillary membrane. Table TA1 inappendix
summarizes all of the auxiliary relationships required to describe protein and ion concentrations and osmotic pressures in the vascular and interstitial compartments. The protein and ion concentrations (*Eqs.EA1-EA5
* for the vascular compartment and*Eqs. EA18-EA23
* for the interstitial compartment) are based on the solute contents and volumes of the corresponding compartments. Plasma and interstitial protein concentrations in their distribution volumes are used to determine colloid osmotic pressures (*Eqs. EA6
*,*
EA7
*,*
EA24
*, and*
EA25
*); see alsoappendix
. The total osmolarity of the plasma and interstitium is calculated by assuming an ideal solution of protein and ions in each of these compartments (i.e., the osmotic contributions of these species are independently additive).

#### Intracellular compartments: red blood cells and tissue cells.

The behavior of the two intracellular compartments is described by a set of six time-dependent ordinary differential equations corresponding to each of the ions participating in cellular transport (i.e., Na^{+}, K^{+}, and Cl^{−}). Two additional algebraic equations account for water shifts to and from the cells in response to external changes in osmolarity. This approach of describing the cellular exchange follows that previously formulated by Jakobsson (14). Mass balances describe the cellular compartment embedded in the interstitium. The mass balance equation for intracellular Na^{+} is
Equation 17whereas the equation for intracellular K^{+} is
Equation 18and that for intracellular Cl^{−} is
Equation 19where M_{Na,TC}, M_{K,TC}, and M_{Cl,TC} represent the Na^{+}, K^{+}, and Cl^{−} contents of the cellular compartment; *A*
_{TC}is the membrane surface area of the tissue cell compartment;*p*
_{Na} and*p*
_{K} denote the permeabilities for Na^{+} and K^{+}, respectively; Δφ is the dimensionless cell membrane potential; RP is the rate of the Na^{+}-K^{+}pump; and ρ is the pump ratio. [Na], [K], and [Cl] are, respectively, the Na^{+}, K^{+}, and Cl^{−} concentrations for either intracellular medium (i.e., tissue cells, with subscript TC) or extracellular medium (i.e., interstitium, with subscript I).

The first term on the right-hand side of *Eqs.17
* and *
18
* represents transport due to electrodiffusion and establishes the interdependence among the ion permeabilities, the intra- and extracellular concentrations, and the transmembrane potential. This term represents the solution of the one-dimensional Nernst-Planck equation, assuming a linear potential profile across the cellular membrane. The second right-hand-side term of these two equations describes active transport and states that the transport rate associated with the Na^{+}-K^{+}pump through the term RP is a linear function of the intracellular Na^{+} concentration. The area term in *Eqs. 17
* and *
18
* represents the membrane area of the entire cellular compartment that is available for transport. This area is assumed to be constant (i.e., unaffected by cellular swelling or shrinking).

*Equation 19
* states that Cl^{−} crosses the membrane at a rate sufficient to maintain intracellular electroneutrality. Therefore, the mass balance equation for Cl^{−} implicitly includes the electroneutrality condition assumed for the internal environment of either type of cell.

According to the assumptions mentioned in the previous section, the other positive (FC^{2+}) and negative (FA^{−}) species are not transported across the cellular membrane. Consequently, the transport rates across the cell membrane for these two fixed species are zero, their intracellular contents are constant, and their concentrations are determined solely by the changes in cellular volume.

The algebraic equation describing the volume of water shifted at any instant into or out of the tissue cells (V_{W,TC}) has the following form
Equation 20where Osm_{I} and Osm_{TC} are the number of milliosmoles in the interstitium and tissue cell compartments, respectively. This equation is a direct consequence of the isotonicity condition, i.e., the external osmolarity equals the intracellular osmolarity, and shows that any disturbance in the extracellular osmolarity will cause an instantaneous water shift across the cellular membrane. V_{W,TC}becomes positive when the extracellular osmolarity increases, thereby causing the cells to shrink, or negative when the extracellular osmolarity decreases, causing the cells to swell. With each water shift, the interstitial volume (V_{I}) is updated instantly to (V_{I} + V_{W,TC}), independent of the terms on the right-hand side of *Eq. 3
*.

The tissue transcellular membrane potential (Δφ_{TC}) is described by a nonlinear algebraic equation employed initially by Jakobsson (14). This equation is a modified form of the Moreton (22) equation for transmembrane potential and can be written as followsThis dimensionless transmembrane potential can also be expressed as Δφ =*F* ⋅ *V*
_{m}/*RT*, where the ratio*RT*/*F*is 26.7 mV for mammalian cells at body temperature and*V*
_{m} is the dimensional calculated or measured cellular electric potential (see Table 3).

In summary, *Eqs. 17
* and *
18
* are mass balance equations that explicitly describe the exchange of Na^{+} and K^{+}, respectively, across the cell membrane. *Equation 19
*, the equation for intracellular Cl^{−}, incorporates the electroneutrality condition, whereas*Eq. 20
*, which describes the change in cellular volume, accommodates the isotonicity condition. These four equations must be solved simultaneously with the membrane potential equation, *Eq. 21
*, to obtain a complete description of the time-dependent behavior of the tissue cell compartment. A similar set of five equations describes the behavior of the red blood cell compartment.

This model of cellular behavior is based on two simplifications that are necessary to avoid dealing with a stiff set of ordinary differential equations. It assumes that all changes in the membrane potential and cellular volume take place instantaneously, i.e., at every instant the transmembrane potential and the cellular volume are in a quasi-steady state dictated by the ionic concentration differences across the cell membrane, the membrane permeabilities for small ions, and the magnitude of the active transport term. This simplification is justified, considering that a change in membrane voltage due to the infinitesimal separation of charges across the cell membrane occurs within milliseconds, whereas changes in intracellular concentrations take place on the order of seconds. Also, because the cellular membrane permeability for water in most cells is about five orders of magnitude higher than that for small ions (14), it can be considered that, to maintain isotonicity between the intra- and extracellular mediums, the water shifts take place instantaneously relative to small solute transport (19, 20). The result of this justifiable assumption is that the model does not predict the dynamics associated with the movement of cellular fluid based on differences in tonicity.

### Numerical Methods and Computational Procedure

The overall model, composed of four homogenous compartments, consists of 20 ordinary differential equations (accounting for balances of fluid volumes, ions, and proteins) coupled with two implicit nonlinear algebraic equations (corresponding to the cellular transmembrane potentials) together with two explicit algebraic equations (describing the changes in cellular volume). The connection between these equations is made through several auxiliary algebraic equations such as compliance relationships or equations for osmotic pressures (see TableTA1). The system of differential equations was integrated over time, employing a fifth-order Runge-Kutta method with Cash-Karp coefficients and adaptive error control (25). The nonlinear algebraic equations were solved by Brent’s method, which combines root bracketing, bisection, and inverse quadratic interpolation to converge from the neighborhood of a zero crossing (25). To obtain an accurate solution, the nonlinear membrane potential equations as well as the two algebraic equations for cell volume were solved simultaneously with the system of differential equations. This was achieved by calling the nonlinear equation solver, as well as the subroutine handling the cellular volumes updates, for each local time step of the Runge-Kutta method. Thus the membrane potential, together with the ion concentrations, albumin contents, fluid fluxes, and cellular volumes, were continuously updated until the integration converged over the imposed local time step.

## STEADY-STATE CONDITIONS

The normal steady-state conditions for a 70-kg, supine “reference” man are given in Table 1. The plasma and interstitial volumes as well as the hematocrit values are specified in the available literature (9, 10, 30). As in our previous modeling studies (1, 2, 26), it was assumed that proteins were excluded from occupying 25% of the total interstitial fluid volume. The volume of the red blood cell compartment was calculated on the basis of the hematocrit. For the “lumped” interstitial cell compartment, it was considered that all these cells bear the properties and characteristics of skeletal muscle cells and constitute ∼40% of the total body weight (17) (see also *Model Assumptions*).

Table 1 also summarizes the properties of the capillary membrane that separates the plasma and interstitial compartments. The transport properties of the capillary wall include the capillary filtration coefficient (*k*
_{F}), the capillary permeability-surface area products for proteins and small ions (*PS* and *PS*
_{Ion}, respectively), and the reflection coefficients for proteins and small ions (ς and ς_{Ion}, respectively). The fluid and protein transport coefficients, together with the normal lymph flow sensitivity (*J*
_{L}), were estimated by Xie et al. (35) by statistical comparison of model predictions with clinical data available for humans. On the basis of the studies reported by Yudilevich (36), it was assumed that the value for *PS*
_{Ion} is between 1,000 and 5,000 times higher than the corresponding permeability-surface area product for proteins, *PS*. Wolf and Watson (34) measured values of ς_{Ion} ranging from 0.1 to 0.5 for cat hindlimb, whereas Curry (7) obtained an average ς_{Ion} of 0.05 for frog mesentery. Carlson et al. (4) assumed ς_{Ion}values of 0.045 and 0.1 for Na^{+}and other small solutes, respectively, in their hemorrhage model. Thus it is anticipated that ς_{Ion}should lie in the range from 0.05 to 0.5, as listed in Table 1. Because these two capillary transport parameters are imprecisely known, one of the objectives of the companion paper (11) is to estimate*PS*
_{Ion} and ς_{Ion} using available experimental data.

The normal compartmental hydrostatic and colloid osmotic pressures for plasma and interstitium employed by Xie et al. (35) in formulating the human microvascular exchange model are also given in Table 1. All these compartmental pressure values are obtained from the available literature and correspond to a combined tissue compartment and the general circulation. The same table shows a typical value of the plasma protein concentration available from literature sources (15) as well as the calculated interstitial protein concentration (seeappendix ).

Table 2 shows how the small ions are normally partitioned between the intra- and extracellular compartments. The values selected for these ion concentrations as well as the properties ascribed to these cells are approximate, representative of a fairly large number of cells, but do not exactly characterize any of them. They are the result of combining experimental information with additional calculations. Table 2 was assembled on the basis of the following considerations.

#### Extracellular concentrations.

The extracellular ion concentrations for Na^{+}, K^{+}, and Cl^{−} in plasma were obtained from experimental measurements in rats (24). This reference was chosen to maintain the same source for initial conditions and for comparison with model predictions that are discussed in the companion paper (11). Similar concentrations have been reported for rabbits (16), dogs (29), and humans (15).

The extracellular values for C^{2+}and A^{−} in plasma were calculated by solving the following two algebraic equations
Equation 23
Equation 24 where [Ion]_{Pl} and*z*
_{Ion,Pl} are the molar concentration and charge, respectively, of the ionic species (i.e., Ion = Na^{+}, K^{+}, C^{2+}, Cl^{−}, or A^{−}) present in plasma, whereas C_{Prot,Pl},*z*
_{Prot}, and MW_{Prot} are the molar concentration, charge, and molecular weight of the protein in plasma, respectively. Φ and S_{Pl} are the osmotic coefficient and plasma osmolarity, respectively.*Equation 23
* represents the condition of electroneutrality for the plasma compartment, whereas*Eq. 24
* expresses the relationship for total plasma osmolarity. The unknowns in these equations are the molar concentrations for the A^{−} and C^{2+} species.

The value for the plasma osmolarity was obtained from experimental measurements for rats (24). In accordance with the modeling practice of previous authors (33), both intra- and extracellular media were assumed to be ideal solutions for which the van’t Hoff law applies; therefore, an osmotic coefficient Φ = 1 was assumed for all the solutes, i.e., ions and proteins, in plasma. The molar and normal protein concentrations given in Table 2 were calculated on the basis of the known mass concentration of protein in plasma and by considering for this species to have an average charge of*z*
_{Prot} = −17 and a molecular weight of MW_{Prot} = 67,000 g/mol (15). As previously mentioned, a charge of +2 was assumed for all positive ions other than Na^{+} and K^{+} and a charge of −1 was attributed to the A^{−}species.

The concentrations of small ions in the interstitium were calculated from Donnan considerations across the capillary membrane on the basis of the relationships presented in appendix . All of the extracellular ion and protein concentrations presented in Table 2 are in good agreement with corresponding values reported in the literature (e.g., Ref. 15).

#### Intracellular concentrations.

Values for the intracellular concentrations of Na^{+}, K^{+}, and the intracellular nondiffusible positive charges FC^{2+} were available in the literature (12, 15) for interstitial and red blood cells. The intracellular Cl^{−}concentration for red blood cells was calculated using the ratio [Cl^{−}]_{RBC}/[Cl^{−}]_{Pl}= 0.69, previously reported by Hoffman (12).

Numerous negative species, mainly proteins, constitute the internal environment of the cells. The generic term FA^{−} was used to describe these nondiffusible species. Its concentration and average charge were calculated by simultaneously solving two algebraic equations. The first equation is based on the bulk internal electroneutrality condition
Equation 25 where*z*
_{Ion,RBC} and [Ion]_{RBC} are the charge and the molar concentration, respectively, of the ionic species (Ion = Na^{+}, K^{+}, FC^{2+}, or Cl^{−}), whereas*z*
_{FA,RBC} and [FA]_{RBC} are the unknown average charge and molar concentration for the nondiffusible species FA^{−}. The second equation imposes the isotonicity condition between the cells and their surroundings as
Equation 26 where S_{RBC} represents the total osmolarity inside the red blood cells, with
Equation 27An approach similar to that for red blood cells was adopted to calculate the intracellular concentration and charge of FA^{−} for the interstitial cells. The only exception made was with respect to intracellular Cl^{−}, whose internal concentration was obtained from literature reports on skeletal muscle cells (15).

The above steps complete a simplified picture of the small ion partition between the intra- and extracellular mediums as well as between plasma and interstitium. They provide an initial set of numerical values that are in a reasonable physiological range compared with experimental data (15).

Four other cellular membrane parameters are required to completely describe the cellular exchange process. These parameters are the three membrane permeabilities, one for each of the ions transported across the cell membrane, i.e.,*p*
_{Na},*p*
_{K}, and*p*
_{Cl}, and the rate (RP) of the Na^{+}-K^{+}pump.

The cellular membrane permeabilities were obtained from published studies (12, 14). On the basis of known intra- and extracellular concentrations for a particular type of cell, i.e., RBC or muscle cell, the system of membrane transport equations given by*Eqs. 17
* and *
18
*, coupled with the nonlinear equation giving the membrane potential (*Eq.21
*), were solved for steady-state conditions (i.e., the left-hand-side accumulation terms of *Eqs.17
* and *
18
* were set to zero). The initial literature value for one of the permeabilities, either *p*
_{Na} or*p*
_{K}, together with the intracellular and extracellular concentrations presented in Table2, was assumed known. The unknowns were then the other permeability, either *p*
_{Na} or*p*
_{K}, as well as RP and Δφ (where Δφ =*F ⋅ V*
_{m}/*RT*).

The membrane transport parameters obtained by solving the set of nonlinear algebraic equations as well as the values calculated for the cellular transmembrane potential are shown in Table3. A comparison between the calculated and the literature values for these parameters is also given in the same table.

All the compartmental values and parameters presented here represent the steady-state values of the system. This requirement was satisfied by running the computer program associated with the transient model for an extended period of time and in the absence of any perturbation to the system until the changes in all dependent variables became insignificant.

### Conclusions

This first paper of a two-part study is concerned with the overall description and formulation of a compartmental model of whole body fluid volume, protein, and ion distribution and transport. Two main compartments, comprising the vascular and interstitial volumes, and two embedded cellular compartments, one for each extracellular compartment, represent the system. The model, based on 20 ordinary differential equations, 2 nonlinear cellular membrane potential equations, and a number of algebraic auxiliary equations, is capable of predicting both experimentally measurable variables (e.g., plasma volume, plasma osmolarity) and inaccessible or difficult-to-measure system variables (e.g., intracellular ion contents, cellular volume) for various types of infusions (e.g., hyperosmolar or Ringer-type solutions). The second part of this study, presented as a companion paper (11), deals with the validation and testing of the proposed mathematical model against a comprehensive range of experimental data available in the literature.

## Acknowledgments

We express appreciation to the Natural Sciences and Engineering Research Council of Canada and the Norwegian Council for Science for providing financial support for this study.

## Appendix

### Auxiliary Equations

The auxiliary equations describing the model are given in TableTA1. All of the symbols used in this model are presented in the glossary.

## Appendix

### Colloid Osmotic Pressure Relationships

As in the previous work of Chapple et al. (5), the following empirical linear relationship was established to correlate the total plasma protein concentration (using albumin as representative of all the proteins in the system) and the plasma colloid osmotic pressure Equation B1 where the units of colloid osmotic pressure are millimeters of Hg and those of protein concentration are grams per liter.

On the basis of the interstitial colloid osmotic pressure previously employed by this group for the human model (5, 35), and with the use of the same correlation coefficient as in *Eq.EB1
*, the protein concentration in the available interstitial volume was calculated as
Equation B2

## Appendix

### Interstitial Compliance

The interstitial compliance relationships were obtained from a previous model of the microvascular exchange system for humans (5). In the latter study the compliance relationship was separated into three regions: a “dehydration segment,” an “intermediate segment,” and an “overhydration segment.” The range of interstitial volume values and the compliance relationships corresponding to these segments are as follows. For the dehydration segment (V_{I} ≤ 8.4 × 10^{3} ml), the relationship is
For the intermediate segment (8.4 × 10^{3} ≤ V_{I} ≤ 12.6 × 10^{3} ml), the values for pressure are obtained by quadratic interpolation between experimental P_{I} and V_{I} data. For the overhydration segment (V_{I} ≥ 12.6 × 10^{3} ml), the relationship is
In all of the relationships for these segments, P_{I} is expressed in millimeters of Hg and V_{I} is expressed in milliliters.

## Appendix

### Donnan Constraints

The semipermeable characteristics of the capillary wall for proteins cause the retention of plasma proteins in the capillary lumen. Therefore, at any time, there is a protein concentration difference, Δ(Prot), between the plasma and the interstitium. The negatively charged proteins on each side of the membrane will associate with an equivalent number of positively charged species, namely, Na^{+}, K^{+}, and the remaining cations C^{2+}. These cations, which otherwise would cross the capillary freely, become effectively immobilized on one side of the membrane, thereby establishing a concentration difference in free small ions across the capillary. By considering Δ(Cat) as the sum of all the differences in cation concentrations across the capillary, one can write the steady-state condition as
Equation D1 where
Equation D2
Equation D3
Equation D4In *Eqs. ED2-ED4
*, Δ(Na)_{D}, Δ(K)_{D}, and Δ(C)_{D} are the Donnan transcapillary differences in free concentrations for Na^{+}, K^{+}, and the remaining cations, respectively. All the concentrations are expressed in milliequivalents per liter.

Because the presence of proteins is the determinant in creating a Δ(Cat) distribution across the capillary, it was assumed here that a proportionality constant*k*
_{Prot} relates the difference in the cationic distribution with the protein difference as follows
Equation D5 where the transcapillary protein concentration difference Δ(Prot), expressed in milliequivalents per liter, is
where (C_{Prot,Pl}) and (C_{Prot,I,AV}) represent the protein concentrations in the plasma and interstitium, respectively, expressed in milliequivalents per liter.

On the basis of the relationships in *Eqs.ED1
* and *
ED5
*,*Eqs. ED2-ED4
* can be rewritten for the non-steady state as follows. The relationship for the Donnan distribution for Na^{+}is
Equation D6whereas the Donnan distribution for K^{+} is
Equation D7and the Donnan distribution for C^{2+} is
Equation D8The relationships in *Eqs.ED6-ED8
* assume that the ability of the proteins in either the plasma or interstitium to associate with a given type of cation is proportional to the mass fraction of the respective cation in the plasma or the interstitium.

At steady state, the condition of electroneutrality across the capillary requires that the differences in positive charges across this membrane be equal to the differences in negative charges, as given by
Equation D9where Δ(Cl)_{D} and Δ(A)_{D} are, respectively, the transcapillary concentration differences for Cl^{−} and the anionic species A^{−}, expressed as milliequivalents per liter.

By substituting *Eq. ED5
* into*Eq. ED9
*, the following relationship, which relates the transcapillary anionic concentration differences with the protein concentration difference, is obtained
Equation D10Consequently, the Donnan distributions for the negative charges can be written as
Equation D11for Cl^{−} and as
Equation D12for A^{−}.*Equations ED6-ED8
*,*
ED11
*, and*
ED12
* express the contribution of the Donnan effect to the transport of small ions through the capillary membrane, and they were taken into account in describing the transcapillary fluxes of fluid and small ions, i.e., in*Eqs. 8
* and *
10
*, respectively, of*Model Equations*.

## Footnotes

Address for reprint requests and other correspondence: J. L. Bert, Dept. of Chemical Engineering, Univ. of British Columbia, Vancouver, BC, Canada V6T 1Z4 (E-mail: bert{at}chml.ubc.ca).

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