INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS STEADY-STATE CONDITIONS REFERENCES APPENDIX A APPENDIX B APPENDIX C APPENDIX D

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 (cm2)
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 FC2+ (mM)
Hct	Hematocrit (%)
[Ion]	Ion concentration (mM)
(Ion)	Ion concentration (meq/l)
J	Rate of fluid transfer (ml/h)
kF	Fluid filtration coefficient (ml · mmHg1 · h1)
kProt	Proportionality constant (see Eq. D5)
LS	Lymph flow sensitivity (ml · mmHg1 · h1)
M	Ion content (mmol)
	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)
	Rate of protein transfer (g/h)
R	Universal gas constant (8.314 × 103 J · mol1 · K1)
RP	Rate of Na+-K+ pump (cm/s)
S	Osmolarity (mosmol/l)
t	Time (h)
T	Temperature (K)
V	Volume (ml)
Vm	Cellular transmembrane electrical potential (mV)
z	Valence

Greek Symbols

IonD	Capillary transmembrane concentration difference (meq/l)
	Dimensionless cellular transmembrane potential
	Osmotic coefficient
	Colloid osmotic pressure (mmHg)
	Reflection coefficient
	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+, C2+, 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

TOP ABSTRACT INTRODUCTION METHODS STEADY-STATE CONDITIONS REFERENCES APPENDIX A APPENDIX B APPENDIX C APPENDIX D

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

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View larger version (22K): [in this window] [in a new window]   Fig. 1.   Schematic of general model of fluid, protein, and ion exchange between plasma, interstitium, and cells. Arrows indicate fluid (J), protein (), and ion () transport rates in relation to the compartments participating in mass exchange. See glossary and text for definition and explanation of terms.

Model Assumptions

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⁺, C²⁺, Cl, and A. The cationic species C²⁺ 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 Mg²⁺ and Ca²⁺ (15), a charge of +2 was attributed to the C²⁺ 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⁺, FC²⁺, 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 FC²⁺, 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 [ = ([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 () 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

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

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the protein balances are

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and the small ion (Na⁺, K⁺, C²⁺, Cl, and A) balances can be expressed as

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where V, Q, and M represent the compartmental fluid volume, protein content, and ion content, respectively, and J, , and 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⁺, C²⁺, 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.

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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

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whereas the equation for intracellular K⁺ is

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and that for intracellular Cl is

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where 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; is the dimensionless cell membrane potential; RP is the rate of the Na⁺-K⁺ pump; and  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).

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Numerical Methods and Computational Procedure

	STEADY-STATE CONDITIONS

TOP ABSTRACT INTRODUCTION METHODS STEADY-STATE CONDITIONS REFERENCES APPENDIX A APPENDIX B APPENDIX C APPENDIX D

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 ( and _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 _Ion ranging from 0.1 to 0.5 for cat hindlimb, whereas Curry (7) obtained an average _Ion of 0.05 for frog mesentery. Carlson et al. (4) assumed _Ion values of 0.045 and 0.1 for Na⁺ and other small solutes, respectively, in their hemorrhage model. Thus it is anticipated that _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 _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 (see APPENDIX B).

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).

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Intracellular concentrations. Values for the intracellular concentrations of Na⁺, K⁺, and the intracellular nondiffusible positive charges FC²⁺ 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).

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Conclusions