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 ~270 mosmol/l and composition comparable with Ringer solution, and
3) an isosmotic NaCl solution with
an osmolarity of ~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.
hyperosmolarity; cell volume; interstitial volume; plasma volume
expansion; plasma osmolarity
 |
INTRODUCTION |
ONE OF THE FIRST PRIORITIES in the treatment of
patients undergoing traumas such as hemorrhage or burn injury is to
restore an adequate circulatory volume. The use of hypertonic saline
(HS) solutions for this purpose has received increasing interest during the past few years (1, 8-10). HS solutions have a potential advantage over reinfused blood or Ringer-type solutions because of
their ability to transiently restore the vascular volume at the expense
of the neighboring compartments, namely, the interstitial and cellular
fluid reservoirs. Although HS solutions have proven to be effective in
acute traumatic situations involving experimental animals, they can
cause hypernatremia, hyperosmolarity, and hypokalemia, conditions that
may impose limitations on their use as resuscitants. Therefore, a
considerable amount of experimental and modeling effort has been
directed toward determining the mechanisms and magnitudes of fluid and
solute redistributions between the different body compartments after HS
fluid infusions.
There are a multitude of interdependent factors that influence the
exchange of fluid and solutes between the vascular and interstitial
compartments as well as between the extracellular and cellular spaces.
Among these factors, the physicochemical properties of the
participating fluid compartments, the properties of the membranes that
separate them, and the chemical or electrochemical gradients governing
transport are just a few. The full interaction among these factors is
often difficult to assess from experimental studies, which are usually
able to investigate only a limited number of measurable variables at a
time. However, because of their ability to generate large quantities of
additional information, validated mathematical models can serve as
useful tools for better understanding the roles played by the many
factors implicated in physiological exchange processes.
Most of the mathematical models that have emerged over the past few
years have been concerned with the effect of resuscitation on blood
volume redistribution after hemorrhage. However, during hemorrhage, the
body responds to the trauma using a variety of physiological
compensatory reactions that represent a departure from the normal
state. Clearly, as a first step in developing an accurate resuscitation
model for traumatic conditions, it is important to establish that the
model is capable of yielding good predictions when compared with
the experimental data obtained from nontraumatic infusion experiments,
in which the pathophysiological changes associated with
hemorrhage are absent.
In the companion paper (6), a mathematical model describing the
transport and distribution of fluid, protein, and small ions in a
physiologically normal microvascular exchange system (MVES) was
formulated. Confidence in the ability of the proposed model to simulate
a real situation can only be based on the quality of the predictions it
can generate. The present work, therefore, has the following
objectives: 1) to estimate two
uncertain capillary wall exchange parameters by using one set of
available experimental data, 2) to
validate the resulting model by comparing its predictions with other
literature data, and 3) to assess
and interpret the dynamic behavior of the model system after the
administration of isosmolar and hyperosmolar resuscitant solutions.
Recently, a mathematical model similar in concept to ours was developed
by Carlson et al. (1). Their model was used to investigate possible
physiological mechanisms that occur during different degrees of
hemorrhage. Although more detailed than ours in some respects, e.g.,
with a more complex description of the Na+-K+
pump and a hypothetical mechanism of small solute distribution between the interstitial fluid and the tissue matrix, their model requires the adjustment of several additional parameters to fit the
experimental data.
Our approach was to extend our previous models of transport in the
microvascular system (3, 12, 14) by introducing plasma and interstitial
cellular compartments, incorporating small ion balances for all
compartments, and accounting for a Donnan effect across the capillary
membrane. Also, previously optimized (14) global transport parameters
for human fluid and protein exchange were scaled to the appropriate
animal values and used in the present model. Thus no parameters from
our previous models were adjusted and no arbitrary functions introduced
to force our model predictions to fit the experimental data. This
approach should give us better insight about what, if any,
parametric values have to be altered to accommodate the
pathophysiological changes associated with traumas such as hemorrhage.
Thus our approach to modeling mass distribution and exchange in
hemorrhage is a two-stage process:
1) modeling of the normal
system, followed by 2) modeling of
the changes associated with hemorrhage.
The primary concern of this work is the validation of the mathematical
model using data from several experimental studies in which the
dynamics of fluid redistribution from fluid overloading, with or
without an osmotic disturbance, has been studied. The ability of the
model to simultaneously predict the dynamic behavior of a variety of
system variables for each given experimental protocol then allows
us to hypothesize about the physiological mechanisms responsible for
fluid redistribution when no additional trauma is present.
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COMPARTMENTAL MODEL |
Stated briefly, the model consists of four distinct compartments:
plasma, interstitium, red blood cells (RBC), and tissue cells. All
compartments are assumed to be homogeneous and well mixed. Thus fluid
and solutes entering any compartment are instantaneously dispersed
throughout its entire volume. The parameters describing the behavior of
each compartment were obtained by averaging appropriately over the many
components that may have made up that compartment. As formulated in the
companion paper (6), the model is based on mass balance equations for
fluid and for each of the solutes (proteins and ions), combined with
auxiliary transport equations describing either mass-exchange rates or
the properties of the individual compartments (colloid osmotic
pressure, compliance, etc.). The simplifying assumptions of perfect
mixing and homogeneity allow the system to be represented by a set of
time-dependent ordinary differential equations. Once all of the
transport and compartmental properties are known, the model can be used
to predict the dynamic changes of fluid, protein, and ion contents of
the plasma, interstitium, and the two cellular compartments after a
perturbation to the system. In the current work, the perturbation takes
the form of an infusion to the plasma compartment of various resuscitants. The system is assumed to be otherwise normal (i.e., no
confounding pathophysiological effects, such as those occurring in
hemorrhage, are accounted for). A schematic diagram of the compartmental model as well as a detailed description of the set of
mass balance, transport, and other auxiliary equations that make up the
model are given in the companion paper (6).
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EXPERIMENTAL INFORMATION |
To estimate missing parameters and to validate the present model, its
predictions were compared with experimental data from previously
published reports by Manning and Guyton (7), Wolf (13), and Onarheim
(9). The experimental data were from nephrectomized dogs (7, 13) and
rats (9). These studies contain information about fluid and/or solute
redistribution between plasma and interstitium after fluid infusions of
either hyperosmotic or essentially isosmotic solutions in the absence
of any type of previous trauma.
Manning and Guyton (7) performed experiments in which dogs were infused
with lactated Ringer solution (RS; 270 mosmol/l) in amounts equivalent
to 5, 10, and 20% of the animals' body weight. The infusion times
ranged from 30 to 60 min. In this study, the changes in blood volume,
total extracellular volume, and plasma protein concentration were
followed over time.
In Wolf's study (13), normovolemic dogs were infused for 6 min with an
isosmotic saline solution. After a stabilization period of 45 min, the
animals were then infused with HS solution (2,030 mosmol/l). Plasma
volume and plasma osmolarity were measured continuously over a period
of 3 h. According to Wolf, all the infused volumes and times are
comparable with commonly used resuscitation protocols, i.e., short
infusion times and relatively low volumes infused.
In Onarheim's experiments (9), normovolemic rats were infused
continuously for a period of ~20 min with either acetated RS (270 mosmol/l) or HS solution (2,400 mosmol/l). The infused volumes were
selected such that the same amount of
Na+ was administered in both types
of infusion protocols. To allow better assessment of the alterations in
interstitial and intracellular fluid volumes after infusion, Onarheim
chose to administer larger amounts of fluid over longer infusion times
than are commonly used in clinical studies. Onarheim reported initial
and final values for plasma volume, total extracellular volume, plasma
Na+,
K+, and
Cl
concentrations,
hematocrit (Hct), and plasma osmolarity, which were measured before
administration of fluid and 1 h after the start of the infusion,
at which time an apparent steady state had been reached. Initial and
postinfusion intracellular volumes for different tissues were
calculated as the difference between the measured total tissue fluid
volumes (determined by drying) and tissue extracellular fluid
volumes (determined using tracers).
In all three studies, the animals were nephrectomized. Therefore, in
our model (6), the rate of urinary output was assumed to be zero in the
plasma compartment mass balance equations. For the short periods of
these experimental studies, i.e., between 2 and 4 h, it was assumed
that the fluid and ions lost with perspiration are negligible; thus the
perspiration rate in the interstitial mass balance equations was also
assumed to be zero.
| |
INITIAL CONDITIONS |
The initial values for the fluid compartments at the start of each
infusion experiment are given in Table 1.
The initial plasma and interstitial volumes were specified in the
experimental reports for dogs (7, 13) and rats (9). As was pointed out
in the companion paper (6), proteins were assumed to be excluded from
25% of the normal interstitial volume. The normal Hct values were also
reported in these experimental studies, whereas the corresponding
volumes for the RBC compartment were calculated on the basis of the
Hct. The interstitial cell volume was assumed to constitute ~40% of
the total body weight (6).
Table 1 also summarizes the properties of both the lymphatic system and
the capillary membrane that separates the plasma and interstitial
compartments. The majority of the parameters characterizing these two
exchange pathways were obtained by scaling the values obtained by Xie
et al. (14) for humans to the appropriate animal values. The basis for
this scaling was the assumption that the capillary density is about the
same for all species, and, hence, the area available for exchange is
proportional to the volume of the interstitium
(VI) such that
The
corresponding permeability-surface area products
(PSIon) used to
describe the rate of diffusive ion
(Na+,
K+,
C2+,
Cl, or
A) transfer across the
capillary membrane, are also shown in Table 1. These values were
estimated according to the procedures outlined in
Estimation of
PSIon and
Ion in this paper.
The initial values presented in Table 2
were maintained common for simulations of all resuscitation protocols
for all animals. The initial hydrostatic and oncotic pressures shown
are those previously employed by Xie et al. (14). These normal
steady-state conditions correspond to a generic tissue compartment and
the general circulation. Table 2 also shows a typical value of the plasma protein concentration that can be found in the literature (5)
and calculated values for interstitial protein concentrations (6). The
average reflection coefficients, (for proteins) and
Ion (for small ions), are,
respectively, the value reported by Xie et al. (14) and that estimated
by curve fitting in Estimation of
PSIon and
Ion. Table 2 also shows the
normal partition of Na+,
K+, and
Cl between the interstitial
and plasma compartments. The considerations, including the Donnan
equilibrium, taken into account in determining these concentration
values are described in the companion paper (6).
The interstitial compliance relationships corresponding to either dogs
or rats are given in the APPENDIX.
These relationships were obtained by scaling, on a weight basis, the
compliance relationships developed by Chapple et al. (3).
The determinations of the cellular membrane transport parameters, i.e.,
pNa,
pK, and
pCl, and the
contribution of the
Na+-K+
pump to the active transport term, RP, are also detailed in the companion paper (6).
All the values presented in Tables 1 and 2 are the steady-state values
obtained by running the computer program associated with our model in
the absence of any perturbation (i.e., fluid inputs) for a sufficient
length of time until none of these values changed beyond a small
preestablished error criterion. Hence, the modeled system was at
steady state before any disturbance due to external fluid and solute
infusion was simulated.
The infusion protocols reported for the three sets of experiments
examined constitute inputs to the model and are summarized in Table
3.
| |
RESULTS AND DISCUSSION |
Estimation of PSIon and
Ion
Because our earlier model study (14) was primarily concerned with fluid
and protein redistribution, optimized values for the
permeability-surface area product
(PSIon) and the
reflection coefficient (Ion),
which control the transport of small ions across the capillary barrier,
were not determined. Several authors have investigated these two
parameters, but, although ranges have been suggested, there is little
agreement about their precise values (1, 4, 6, 13). Thus, as part of
the current study, it was necessary to use a portion of the available
experimental data to estimate
PSIon and
Ion. The remainder of the data
was then employed to help validate the then fully defined MVES
transport model. To minimize the number of parameters to be determined, it was assumed that the same values of
PSIon and
Ion apply to all of the ionic
species (Na+,
K+,
C2+,
Cl, or
A) that are
exchanged between the plasma and interstitial compartments. Thus
the parameters obtained can be considered as average values for all
five ions.
Parameter estimation was carried out using the plasma volume and plasma
osmolarity data measured by Wolf (13). Wolf's experimental results
were selected for this purpose because
1) they include hyperosmotic
infusions where significant intercompartmental ion transfers occur and
2) the data are time dependent and,
hence, provide a more rigorous test of the model. For each trial, pairs of discrete values in the ranges of 1,000 
PSIon/PS 5,000 (where PS is the protein permeability-surface area
product) and 0.05 Ion 0.5 were specified as inputs to the simulation program. With all transport
parameters now defined, the ordinary differential equations governing
the model were integrated from t = 0 to t = 4 h, corresponding to the time
course of Wolf's replicated normal saline (NS) and HS infusion
experiments. In the simulations, the solution volumes and compositions,
as given in Table 3, were injected into the plasma compartment at a
constant rate over the reported infusion periods. On the basis of the
generated results, separate sums of squares of differences between the
predicted and experimental plasma volumes and osmolarities were
calculated. To obtain an overall sum-of-squares value for each pair of
parameters, the volume and osmolarity sums were first normalized with
respect to their minimum values over the ranges investigated, weighted by their respective number of data points, and then added together. These combined sum-of-squares values are plotted as a function of
PSIon/PS and
Ion in Fig.
1.
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Fig. 1.
Combined sum of squares (SSQ) of differences between model predictions
and Wolf's plasma volume and plasma osmolarity measurements (13) as a
function of the permeability-surface area product
(PSIon/PS)
and reflection coefficient
(Ion) for small ions, where
PS is the permeability-surface area for proteins. The
optimal solution is obtained for
PSIon = 3,000 and
Ion = 0.15.
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Figure 1 shows that the sum-of-squares surface is minimal and
relatively insensitive to the values of the two parameters over the
more constrained ranges of 2,000 PSIon/PS 4,000 and 0.1 Ion 0.2. Furthermore, there is a clear minimum at the point PSIon/PS = 3,000 and Ion = 0.15. Other
methods of weighting the plasma volume and plasma osmolarity data
in obtaining the combined sum of squares were attempted, but all
produced essentially the same results.
Figure 2 shows the experimental changes in
blood volume measured by Wolf (13) for an NS infusion followed, ~40
min later, by an HS infusion. Note that plasma volume increases in both
experiments by ~54% above control immediately after the
administration of NS at 303 mosmol/l, followed by a further
experimental increase of ~65 or 105% after HS administration at
2,030 mosmol/l. If the total infusate volumes had simply remained in
the plasma compartment in each case, the expected increases would have
been 58 and 23%, respectively. Thus, clearly, the HS resuscitant is
causing the recruitment of significant amounts of fluid from other
reservoirs such as the interstitium and the plasma and tissue cells.
However, the volume expansion is short-lived in both cases; as soon as the infusion ends, fluid begins leaking out of the plasma compartment such that, at least in the case of the HS infusion, a relatively stable
plasma volume is reached at ~1 h after fluid administration is
terminated. The model predictions, obtained with
PSIon/PS = 3,000 and Ion = 0.15 and
represented by the solid line in Fig. 2, are in good agreement with the
experimental data.
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Fig. 2.
Comparison of model predictions for plasma volume changes vs. time with
experimental data from Wolf (13). Solid line represents model
predictions for NaCl solution (NS) infusion followed by hyperosmolar
saline solution (HS) infusion;  and  represent 2 sets of
experimental data points for the same infusion protocol.
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Wolf's plasma osmolarity results are compared with the model
predictions in Fig. 3. Once again, the
agreement between the model results and the experimental measurements
is excellent. This is not surprising because both the plasma volume and
osmolarity data were used in the estimation of
PSIon and
Ion. Note that the plasma
osmolarity remains near its baseline value during the normal saline
infusion but increases rapidly during the hyperosmotic infusion. At the
end of the HS infusion, the osmolarity decreases sharply, as indicated
by both the predicted and experimental data. Additionally, both
experimental data and simulations reach an apparent steady state of
~10% above the control value within 0.5-1.0 h after the
infusion stops.
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Fig. 3.
Comparison of model predictions for changes in plasma osmolarity vs.
time with experimental data from Wolf (13). Dashed line represents
model predictions for NS infusion followed by HS infusion; and represent 2 sets of experimental data points for the same infusion
protocol.
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Model Validation
One important aspect of model validation is that the model
should provide a good representation of experimental data used in the
estimation of model parameters. This aspect was shown to be satisfied
in Estimation of
PSIon and
Ion. However, a more rigorous
validation requires that, with the same set of estimated parameters,
the model should be capable of predicting the results of other,
independent, experimental studies. On the basis of information available in the literature, the data generated in the infusion experiments of Onarheim (9) on normal rats and Manning and Guyton (7)
on normal dogs were selected for this purpose. These authors report the
changes to such variables as plasma volumes, Hct, osmolarities, and ion
and protein concentrations, as well as interstitial, extracellular, and
cellular volumes after infusions of either RS or HS solu. tions.
Thus we were interested in assessing the agreement, with respect to
both transient and steady-state behavior, between the model predictions
and these sets of available experimental data. Note that whenever
extensive properties (i.e., those that depend on the amount of a
particular variable) are compared within a simulation or between
simulations, they are compared on a relative (or percentage) basis.
Onarheim's experiments.
Onarheim (9) measured several system variables before and ~1 h after
the start of 18-min infusions with either HS solution (2,400 mosmol/l) or RS (270 mosmol/l). A comparison of the
model-predicted values of these variables with the
experimentally measured data is presented in Table
4. In the simulations, the infused
volume was added to the plasma compartment at a constant rate
over the infusion pe riod, in accordance with Onarheim's
experimental protocol.
One hour after the fluid administration is started, when an apparent
steady state is reported, the model predicts that the HS infusion
produces a 27% increase in plasma volume above the control value,
whereas the RS infusion, almost one order of magnitude larger than the
HS infusion, caused only a 35% increase. These results are in good
agreement with the experimental measurements, which indicate 28 ± 4% and 38 ± 5% plasma volume expansions, respectively. The
simulations also show that, for both types of solutions infused, the
plasma volume reaches its maximum expansion at the end of the infusion
period. At this intermediate time, the plasma volume almost doubles
with the simulated infusion of 10 ml/kg HS solution and increases
to only about three times the initial volume for the 100 ml/kg RS
infusion. If all of the infused fluid had simply been retained by the
plasma compartment, the expected expansions would have been ~40 and
390%, respectively. Thus, as was the case in Wolf's experiments
(13), Onarheim's HS infusion engenders additional fluid recruitment
far beyond the volume administered, and, after the infusion period for
both types of resuscitants, fluid rapidly leaks from the plasma
compartment such that an apparent steady state is reached in <1 h.
According to Table 4, blood Hct falls from 47% to a new steady-state
value of ~40% for both the RS and HS infusions. For the RS case, Hct
is predicted to drop continuously up to the end of the infusion period
because of an elevated plasma volume and a relatively constant RBC
volume. However, as the plasma volume returns toward its baseline value
(see above), Hct is partially restored to a final predicted value of
~40%. This end point, obtained by simulation, agrees closely with
the measured value of 38% reported by Onarheim. Good agreement between
the measured Hct value and the model prediction was also obtained for
the HS infusion. In this case, the final plasma volume is lower, but
the RBC volume is also reduced, leading to a similar steady-state Hct.
Table 4 indicates that, at 1 h postinfusion, the plasma osmolarity
increases for the HS infusion but decreases slightly for the RS
infusion. For the HS case, there is a model-predicted peak increase in
plasma osmolarity corresponding to ~40 mosmol/l occurring at the end
of the infusion period. This is largely due to the osmotic contribution
of the ionic species infused. After the HS infusion is terminated, the
plasma osmolarity decreases rapidly toward a new steady state. At
steady state, the model predicts a value of 338 mosmol/l, which
compares well with the 333 mosmol/l reported by Onarheim. For the RS
infusion protocol, a maximum decrease of ~6 mosmol/l in plasma
osmolarity is predicted during the infusion period. According to the
simulations, the minimum in plasma osmolarity corresponds to the peak
in plasma volume and occurs at the end of the infusion. This decrease
is due partly to dilution of plasma by the large volume of the slightly
hypotonic RS infused (270 mosmol/l in RS vs. 303 mosmol/l in plasma).
After the infusion stops, as the fluid and small ions are
reequilibrated between the plasma and interstitium, a slight increase
in overall plasma osmolarity toward the baseline value takes place. At
~0.5 h after the infusion has ceased, the model predicts a
steady-state osmolarity of 297 mosmol/l, which is in good agreement
with the experimental value of 294 mosmol/l. The plasma osmolarity
trends predicted for Onarheim's essentially isosmotic and hyperosmotic infusions are very similar to those measured by Wolf (13) (see Fig. 3).
For the RS infusion, the predicted plasma electrolyte concentrations
for Na+,
K+, and
Cl at 1 h postinfusion are
in good agreement with the values reported in Onarheim's experimental
study. In fact, for this case, neither the experiments nor the model
were expected to give concentration values that are significantly
changed from normal for any of the ionic species. For the HS infusion,
on the other hand, the model predicts that the plasma
Na+ increases from a baseline
concentration of 140 mM up to ~160 mM immediately after
infusion. This increase is a direct result of the high
Na+ content of the infused HS.
Once the concentration difference between the plasma and interstitium
begins to dissipate, the concentration of plasma
Na+ decreases gradually. When
steady state is achieved, the model indicates a plasma
Na+ concentration of 159 mM, which
compares well with the value of 155 mM reported experimentally. A
similar predicted trend is observed for
Cl, whose concentration
increases from an initial steady-state value of 106 mM up to 135 mM
immediately after infusion, followed by a return to 131 mM when the
system reaches its new steady state. This value is in good agreement
with the measured value of 127 mM. For
K+, the model predicts that, after
HS infusion, the concentration falls from the baseline value of 4.7 mM
to 4.4 mM immediately after the infusion stops and returns to a value
of 4.5 mM at the new steady state. As shown in Table 4, there is a
disagreement between the computed value and Onarheim's reported value
for K+ concentration for reasons
we cannot fully explain. Other experimental studies with HS/dextran
infusions after hemorrhage or burn show that serum
K+ often tends to decrease (11).
Onarheim (9) also measured the changes in total extracellular fluid
volume (ECV) after RS and HS infusions. The simulation of the HS case,
in close agreement with the experimental data, predicts a 25% increase
in ECV at 1 h postinfusion. However, for the RS infusion, the model
predictions underestimate the experimental values by ~15%; i.e., the
model-predicted change is 47% compared with the experimental value of
62%. If, as one would anticipate for an essentially isosmotic
infusion, little fluid exchange with the cells occurs, then the
increase in ECV would have been 50%, which is ~3% greater than the
change predicted by the model. Onarheim also found that, for both the
RS and HS infusions, ECV values measured at the end of the infusion
period were very close to the final steady-state values obtained at 1 h. The model predicts a similar ECV behavior: a linear increase during
the infusion period, followed by an almost instantaneous attainment of
steady state immediately after the infusion was completed. In contrast, the predicted plasma and interstitial volumes continue to change significantly after the infusion, only approaching their new steady states ~1 h after the infusion begins. These results were expected for the isotonic RS infusion, in which little transport to or from the
cells takes place, but they also offer experimental justification for
the assumption of a very rapid shift of fluid between the intra- and
extracellular spaces in the case of the HS infusion.
Finally, Onarheim (9) also provided information about the change in the
cellular fluid volume of skeletal muscle after RS and HS infusions. For
the RS case, the model predicts a small increase in the volume of the
tissue cell compartment because the administered fluid is slightly
hypotonic; the measurements corroborate this prediction, indicating
virtually no change in the tissue cell volume. According to the
simulations for the HS case, the tissue cells shrink gradually
throughout the HS infusion, reaching an apparent steady state
immediately after the infusion stops. At the final steady state, the
model predicts an ~10% decrease in the volume of these cells, a
value that is in good agreement with the measured 8.4% decrease
reported by Onarheim.
Manning and Guyton's experiments.
Figure 4 shows a comparison between the
simulated and experimental blood volume changes after RS infusions
according to the experimental protocol reported by Manning and Guyton
(7). Figure 4, A-C, corresponds to lactated RS infusions
equivalent to 5, 10, and 20% of body weight (BW), respectively. The
trends and the magnitudes of the predicted fluid volumes are in good
agreement with the experimental data, except near the end of the
infusion period for the 20% BW case. Note, however, that the maximum
measured blood expansion is essentially the same as for the 10% BW
case, even though the 20% BW infusion involved the addition of twice as much fluid over a 33% shorter period. The following values were
reported by the authors at 5 h after the infusion terminated: an
~14% increase for the 5% BW infusion, a 23% increase for the 10%
BW infusion, and a 25% increase for the 20% BW infusion. The corresponding simulated percentage increases at 5 h postinfusion are
19, 24, and 27% for the 5, 10, and 20% BW infusions, respectively. The simulations and the measurements both indicate an increase in blood
volume during RS infusion, followed by stabilization at an elevated
steady-state volume within ~1 h postinfusion.
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Fig. 4.
Comparison of model predictions for changes in blood volume vs. time
with experimental data from Manning and Guyton (7). Lines represent
model predictions whereas symbols represent experimental results for
5% body weight (BW) (A), 10% BW (B), and
20% BW (C) Ringer solution (RS) infusions.
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Manning and Guyton (7) provided information about plasma protein
concentration changes after RS infusions. Figure
5 shows their experimental results along
with our model predictions. For all three levels of infusion, there is
always a decrease in plasma protein concentration as a result of plasma
expansion. According to the simulations, the maximum decrease is
achieved immediately at the end of the infusion period. After the
infusion is completed, the plasma protein concentrations increase
slowly toward the control value. However, for the 5-h postinfusion
period studied, the protein concentrations remain well below the
initial value. At 5 h postinfusion, the experimental results indicate a
14, 20, and 28% decrease from the control level for the 5, 10, and
20% BW infusions, respectively. In good agreement with the
experiments, the model-predicted values at this time are 16, 21, and
27% below control, respectively, for the corresponding infusions.
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Fig. 5.
Comparison of model predictions for changes in plasma protein
concentration (CProt,Pl) vs.
time with experimental data from Manning and Guyton (7). Lines
represent model predictions whereas symbols represent experimental
results for 5% BW (A), 10% BW
(B), and 20% BW RS (C)
infusions.
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Manning and Guyton (7) also measured total ECV changes as part of their
infusion experiments with dogs. According to Fig. 6, the experiments and simulations both
show that, during RS infusions, the total ECV increases proportionally
with the fluid infused for the duration of the infusion period.
However, once the infusion is terminated, a new elevated steady-state
ECV level is almost immediately reached. A similar behavior was
observed by Onarheim (9) in his RS and HS infusion experiments.
The postinfusion predictions are in reasonable agreement with the
measured ECV values for all three of Manning and Guyton's
experiments.
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Fig. 6.
Comparison of model predications for changes in extracellular volume
(ECV) vs. time with experimental data from Manning and Guyton (7).
Lines represent model predictions whereas symbols represent
experimental results for 5% BW (A),
10% BW (B), and 20% BW RS
infusions (C).
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Summary of validation study.
The results presented above suggest that the model's predictions are
well supported by the experimental results for hyperosmotic as well as
isosmotic or essentially isosmotic solutions. The comparisons for fluid
volumes and solute concentrations proved to be satisfactory for the
first 1-5 h postinfusion, and no additional volume compensatory mechanisms needed to be accounted for in the mathematical formulation. The experimental data provided by Wolf (13) for NS and HS infusions, as
well as by Manning and Guyton (7) for RS infusions, provided information about the transient phase of plasma expansion in dogs. As
shown in Figs. 2 and 4, the time course of the predicted plasma volume
changes was in good agreement with the experimental data. Also, the
model provided excellent predictions of the steady-state plasma volume
changes in rats measured by Onarheim (9) for RS and HS infusions. These
results give us confidence that this improved four-compartment model
has the ability to provide reliable predictions of plasma expansion,
even for cases in which ionic resuscitants cause significant cellular
volume changes.
Fluid Infusions: Mechanisms and Implications of the Model
One of the main advantages of the validated model is its ability to
predict simultaneously a large number of both experimentally accessible
and difficult-to-measure (or experimentally inaccessible) variables.
This wealth of information can help contribute to a better
understanding of the phenomena occurring at both the microvascular and
cellular levels after resuscitation. On the basis of the predictions of
the model (some of which are shown in Model
Validation), the compartmental fluid and solute
changes that occur after the infusion of either HS or RS/NS are
discussed in more detail here.
Isosmotic solution (NS or RS) infusions.
Infusion with isosmotic solutions represents the less complex of the
two cases studied using the model. Because no osmotic disturbance is
present at the boundary between the extra- and intracellular
compartments, the cells play an essentially passive role and the
vascular and interstitial compartments are the only ones that undergo
significant changes in volumes and protein concentrations.
INFUSION PERIOD.
Corresponding to Wolf's experiments, the computed plasma volumes
increase during NS infusion to ~50% above the control value, a peak
that is slightly less than that predicted if all the infused fluid is
retained in plasma. On the basis of his transient experimental data,
Wolf reported a 91% vascular retention for NS when plasma expansion
was at its peak. According to the simulations, which indicate a similar
value of ~88%, the incomplete retention of infused fluid is due to a
continuous fluid shift from the vascular to the interstitial
compartment caused by an increased hydrostatic pressure and decreased
colloid osmotic pressure in the vasculature. As shown in Fig.
7 for this experiment, the model predicts
an increase of ~16 mmHg in the plasma hydrostatic pressure (Fig. 7A) and a decrease of ~8 mmHg in
the vascular colloid osmotic pressure (Fig.
7B). The model also predicts
(although not shown here) that, during NS infusion, the amount of
protein in plasma remains near the control value; however, the protein
concentration decreases due to plasma dilution. Manning and Guyton (7)
similarly reported that whereas the plasma protein content remained
essentially constant, a decrease in plasma protein concentration
occurred for each of the RS infusions they studied (see Fig. 5).
Simulations of their experiments also demonstrate that increased
hydrostatic pressures and decreased colloid osmotic pressures in plasma
favor fluid filtration across the capillary barrier toward the
interstitium. As shown in Fig.
8, which presents model simulations for the
relative transcapillary fluid flow corresponding to Manning and
Guyton's experimental conditions, fluid is shifted from the vascular
to the interstitial compartment at an increasing rate throughout the
infusion period. The model predicts that the fluid continues to be
filtered into the interstitium more rapidly than it can be removed by
the lymphatics. As shown in Fig. 8, the relative transcapillary flow
increases much more dramatically than does the lymphatic flow. The net
result is the accumulation of fluid in the interstitium. The
compliance-engendered increase in the interstitial hydrostatic pressure
causes slightly higher fluid and protein return rates through the
lymphatics.
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Fig. 7.
Model predictions performed according to experimental protocol
described by Wolf (13) for NS infusion. Lines represent predictions for
hydrostatic pressure (A and C) and colloid
osmotic pressure (B and D) vs. time for the
vascular (A and B) and interstitial
(C and D) compartments.
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Fig. 8.
Model predictions for relative transcapillary fluid flow into
interstitium
(JI/JI,Norm)
vs. time for 5% BW (solid line), 10% BW (dotted line), and 20% BW
(dashed line) RS infusions (A).
Simulations were performed according to experimental protocol described
by Manning and Guyton (7). Inset B:
relative lymph flow
(JL/JL,Norm)
vs. time, corresponding to each RS volume infused.
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POSTINFUSION PERIOD.
In agreement with the experimental results, the simulations for all the
cases considered show a near equilibration of fluid, proteins, and
small solutes between the plasma and interstitium within ~0.5-1
h postinfusion. When final steady states are achieved, the hydrostatic
pressures are increased and the colloid osmotic pressures are decreased
in both the vascular and the interstitial compartments. No experimental
information was available regarding the interstitial hydrostatic and
colloid osmotic pressure changes. The simulations for the NS infusion
case, based on the experiments by Wolf (13) and shown in Fig. 7,
C and
D, suggest, however, an increase of
~1.5 mmHg in interstitial hydrostatic pressure and a decrease of ~2
mmHg in the interstitial colloid osmotic pressure for the postinfusion
steady-state period. The increased interstitial hydrostatic and
decreased colloid osmotic pressures are, as mentioned earlier, a direct
consequence of fluid being shifted from the vasculature.
From Fig. 5, which showed comparisons between computed and experimental
plasma protein concentrations for Manning and Guyton's experiments
(7), it can be observed that, after a decrease during infusion, the
protein concentration returns toward its baseline value. However, both
the simulations and experiments demonstrate that the protein
concentrations remain 15-30% below control for the entire
postinfusion period, depending on the particular experimental protocol.
In all cases, the net effect of RS infusion is to produce hemodilution.
In accordance with experimental values reported by Manning and Guyton
(7), the model output shows that, at steady state, only ~10-20%
of the RS infused is retained in the vascular compartment. Similarly,
on the basis of data from Onarheim (9), a 10% isosmotic fluid
retention within the vasculature was calculated from his measurements
and also predicted by the model.
HS infusions.
For the HS infusions, according to the model predictions for both
infusion conditions described, namely, short term (13) and long term
with large infusion volume (9), the plasma volume increases up to a
maximum, which coincides with the end of the infusion period and
accounts for far more than the volume of fluid infused. This condition
is followed by a decrease in plasma volume with a significantly reduced
retention of infusate at steady state. In accordance with the transient
experimental data reported by Wolf (13) and presented in Fig. 2, the
model predicts an increase in plasma volume of ~70% at its peak, the
equivalent of about three times the infused volume. Furthermore, for
Onarheim's experiments (9) involving larger volumes and longer
infusion times, the simulations show a plasma volume elevation of more
than two times the infused volume at the end of the infusion period. At
this time, plasma osmolarity is increased by ~40 mosmol/l compared with controls (see Onarheim's
experiments).
INFUSION PERIOD.
During HS infusion, at least for the two experimental conditions
simulated in this work, the model suggests that the following events
account for fluid and solute shifts between compartments. The highly
hyperosmotic state created in plasma by the HS infusion has an initial
impact on RBCs as well as on the transcapillary transport of small ions
and fluid. The RBCs reduce their fluid volume to achieve an osmotic
balance with plasma. As shown in Fig.
9, the simulations
corresponding to Onarheim's experimental protocol (9) indicate that
the volume of these cells is rapidly reduced by ~15% from control
during the infusion. Additionally, the infused 7.5% NaCl solution
creates large ionic concentration differences across the capillary
wall. Measurements of plasma Na+
and Cl
concentrations during the transient infusion period were not reported in any of the experimental studies. For these experiments, however, the simulation predicts increases of ~20 and ~30 mM in the
plasma Na+ and
Cl concentrations,
respectively. The hemodynamic effects of HS infusion are short-lived
because the small ions leak rapidly through the highly permeable
capillary wall. The model predicts that these large differences in ion
concentration will begin to dissipate quickly (within seconds after the
beginning of infusion) by both an enhanced transcapillary transport of
small ions toward the interstitium by diffusion and a fluid absorption
from the interstitium due to the increased plasma osmolarity. As a
consequence of both of these effects, the interstitial osmotic pressure
increases and, in turn, has an osmotic impact on the tissue cells. As
shown by the model predictions in Fig. 9, fluid is also fairly rapidly mobilized from the tissue cells into the interstitial reservoir. The
simulations demonstrate that the shift of fluid from the tissue cells
takes place throughout the infusion period, independent of whether the
infusion is short term (13) or long term (9). The fluid mobilized from
these cells causes an increase in the interstitial volume throughout
this period. However, the interstitial fluid also participates in
elevating the plasma volume, mainly by absorption to plasma (driven by
increased transcapillary osmotic pressure differences) and, to a lesser
extent, through an enhanced lymphatic transport (due to elevation of
the interstitial hydrostatic pressure). Simulations of the
transcapillary fluid flow, for both Onarheim's HS experiment (9),
presented in Fig.
10A, and
Wolf's infusions (13), shown in Fig.
10B, show a continuous absorption of
fluid into plasma from the interstitium during HS infusion. According
to the simulations, the fluid absorption to plasma lasts as long as the
HS solution is infused, regardless of the duration of the infusion. The
progressive increase in the interstitial volume (not shown) paralleled
by a continuous fluid absorption into the vasculature suggests that the
two- to threefold plasma expansion relative to the infused volume,
noted at the end of the infusion period, is due to fluid
recruited from the cellular compartments and mainly from the
tissue cells.
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Fig. 9.
Model predictions for relative changes in cell volume vs. time after HS
infusion. Simulations shown are for red blood cells (solid line) and
tissue cells (dashed line) and were performed according to experimental
protocol described by Onarheim (9).
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Fig. 10.
Model predictions for relative transcapillary fluid flow
(JI/JI,Norm) vs. time
after HS infusion. Simulations were performed according to experimental
protocols described by Onarheim (9)
(A) and Wolf (13)
(B).
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POSTINFUSION PERIOD.
Both the experimental results and the model predictions demonstrate
that the redistribution of fluid and small ions between the plasma,
interstitium, and cells is essentially complete within 30 min
postinfusion. When the postinfusion steady state is achieved, the
plasma volume remains elevated compared with its preinfusion control
value. The experimental results for the two HS studies (9, 13), as
corroborated by the simulations, show an ~30% elevation in plasma
volume (see Fig. 2 and Table 4) at steady state. The data and the
simulations corresponding to Onarheim's experiments (9) indicate that,
when steady state is reached, the final plasma expansion is ~70% of
the infused fluid volume. Clearly, for this case, the HS resuscitant
results in a greater expansion of the vasculature than does an
essentially isosmotic infusion, which, as was mentioned earlier for
Onarheim's RS experiment, results in only 10% of the infused fluid
being retained in the plasma compartment. However, as presented in
Table 4, both simulation predictions and experimental studies show that
the plasma Na+ and
Cl concentrations, as well
as the plasma osmolarity, are also increased.
As shown in Fig. 9 for Onarheim's experimental protocol (9),
immediately after the HS infusion was terminated, the RBC volume
increases slightly up to ~10% below control, where it remains for
the entire postinfusion period. These changes reflect directly the
osmotic conditions in the plasma. At the final steady state, the model
also predicts a 10% volume decrease for the tissue cells. On the basis
of his experimental measurements (see Table 4), Onarheim reported an
~10% decrease in muscle cell volume. The experimental results as
well as the interpretation of the model predictions presented here
indicate that, after an HS infusion, the increase in the steady-state
plasma volume is caused by the recruitment of cellular fluid as well as
by the infused fluid.
Summary
Even though the present model may be less detailed in some respects
than other models in the literature, all of its assumptions and
simplifications are supported by basic physical and physiological information. In a first validation, for all the cases explored, the
model predictions compared well with experimental results for
compartmental fluid volumes as well as small ion and protein contents.
The experimental data for validation were chosen so that comparisons
with several variables were possible.
An important aspect of the overall exchange process, which the model
helps to clarify, is whether the increase in plasma volume that results
from an HS infusion is due to fluid transfer from the cellular
compartments alone or whether changes in interstitial volume are
involved as well. The model simulations suggest that, during infusion
with hyperosmolar NaCl solutions, all of the fluid that contributes to
an increase in plasma volume is recruited from the infusate and from
the cellular compartments, mainly from the tissue cells. As suggested
by other authors (8), it is possible that, in cases in which the
infusions involve colloidal compounds, water mobilization from the
interstitial fluid to plasma takes place as well, but these conditions
were not simulated in the present study.
The simulations predict a continuous absorption of fluid into plasma
throughout the HS infusion period, independent of its duration and the
volume of the infusate. In agreement with other studies (2, 8, 11, 13),
our model predicts that once the infusion period is over, the elevated
plasma volume rapidly declines such that, at the new steady state,
plasma retention of the infusate is poor. The model simulations
indicate that the transitory nature of plasma elevation is strictly
dependent on the duration of the infusion; i.e., a longer infusion
period results in a longer duration of fluid absorption into the
vasculature. Among other factors, this might be one of the reasons why
Onarheim et al. (10) reported hemodynamic improvements when lower
resuscitation rates and increased durations of HS infusion were used.
However, more experimental data and analyses are required to confirm
these results.
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APPENDIX |
Interstitial Compliance
The interstitial compliance relationships are based on the human
microvascular exchange model developed previously by Chapple et al.
(3). Different compliance relationships were formulated to accommodate
the experimental information for either dogs (7, 13) or rats (9). In
accordance with our past modeling practice, the compliance relationship
was separated into three regions: the "dehydration segment," the
"intermediate segment" for moderate hydration, and the
"overhydration segment." The range of interstitial volume values
and the compliance relationships corresponding to each of
these segments are presented in Tables
A1, A2, and
A3 for Onarheim's rat (9), Wolf's dog (13), and Manning and Guyton's
dog (7), respectively.
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.
The costs of publication of this
article were defrayed in part by the
payment of page charges. The article
must therefore be hereby marked
"advertisement"
in accordance with 18 U.S.C. §1734 solely to indicate this fact.
Address for reprint requests and other correspondence: J. L. Bert,
Dept. of Chemical Engineering, 2216 Main Mall, Univ. of British
Columbia, Vancouver, BC, Canada V6T 1Z4 (E-mail:
bert{at}chml.ubc.ca).
TOP
ABSTRACT
INTRODUCTION
