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From the Department of Anesthesiology, Hyogo College of Medicine, Hyogo, Japan
Address correspondence and reprint requests to Tsuneo Tatara, MD, Department of Anesthesiology, Hyogo College of Medicine, 1-1 Mukogawa-cho, Nishinomiya, Hyogo 663-8501, Japan. Address e-mail to ttatara{at}hyo-med.ac.jp.
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Abstract |
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 Top Abstract IntroductionMETHODSRESULTSDISCUSSIONAPPENDIX AAPPENDIX BREFERENCES |
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METHODS: We developed a mathematical model describing the dynamic distribution and transport of fluid and proteins with the goal of quantifying the balance of fluid between intra- and extravascular compartments. Fluid volume changes in the plasma, interstitial and urine compartments were calculated for a simulated 4 h abdominal surgery in a 70 kg male. To validate the model, we compared the results obtained with those measured by segmental bioelectrical impedance on 30 patients undergoing elective abdominal surgery.
RESULTS: The model predicted that, compared to the normal state, surgical injury would result in the sequestration of 705 mL of interstitial fluid in injured tissue, whereas plasma volume would undergo a 356 mL decrease. During surgery, it was not possible to obtain a normal plasma volume, even with fluid replacement at a rate of almost 20 mL · kg–1 · h–1. Bias and limit of agreement on interstitial fluid volume changes in body segments between bioelectrical impedance and model prediction were –131 and 325 mL, respectively for limbs, and –157 and 834 mL for the trunk.
CONCLUSIONS: The model shows that increasing the fluid replacement rate above 10 mL · kg–1 · h–1 does not have the desired effect on plasma volume but instead increases the interstitial volume.
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Introduction |
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TopAbstractIntroductionMETHODSRESULTSDISCUSSIONAPPENDIX AAPPENDIX BREFERENCES |
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A mathematical model of microvascular exchange which incorporates basic scientific principles describing the dynamic distribution and transport of fluid and proteins provides a better understanding of time-dependent fluid dynamics and the impact of different fluid regimens (4–7). Such a model was applied here to predict fluid sequestrated in the interstitium of injured tissue during abdominal surgery. By incorporating an injured tissue component to the model, we estimated changes of interstitial fluid volume in limbs (i.e., arms and legs) and trunk during abdominal surgery. To validate the model, these fluid volume changes were compared with the available data on patients undergoing elective abdominal surgery. The model was also used to gain insight into the distribution of fluid in various compartments during abdominal surgery.
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METHODS |
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TopAbstractIntroductionMETHODSRESULTSDISCUSSIONAPPENDIX AAPPENDIX BREFERENCES |
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Modifications to the basic model were implemented to reflect changes of parameter values in response to surgical injury. We assumed that the conductivity portions of the fluid filtration coefficient (kF), the permeability-surface area product for protein (PS) as well as the reflection coefficient for protein (
) follow a simple decay curve having the general form 1 + (G – 1) · (1 – e–t), where G is the ratio of the value in the injured state to that in the normal state and t is time in h (8). Urinary dynamics were included in the model formulation.
Table 1 provides normal steady-state values for fluid and protein in the compartments and parameters related to capillary exchange, lymphatics, and kidney. The whole-body value of was assumed to be 0.875 which is the mean value of those for the arteriolar end of the capillaries and the venular end of the blood vessels (9).
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Mass Balance Equations for Extracellular Components
The mass balance equations for fluid and proteins are given with reference to the schematic diagram in Figure 1A.
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where the subscripts PL, IT, L, and U denote the value of the variable in the plasma, interstitial, lymphatic, and urinary compartments, respectively; d is change; t is time in h; V is compartment volume in mL; JINF and JHEM are fluid infusion rate and hemorrhage rate in mL/h; JIT,i, JL,i, and JU are rate of fluid transfer from plasma to interstitium, rate of fluid transfer from interstitium to lymphatics, rate of urine production in mL/h, respectively; JPER, JISL, and JISL,IJ are perspiration rate (i.e., 2.0 mL/h), insensible water losses from whole body (i.e., 40.0 mL/h), insensible water losses from injured tissue (i.e., 70 mL/h) (3), respectively; Hct is hematocrit in %.
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where Q is protein content in grams; 
IT,i and L,i are rates of protein transfer from plasma to interstitium and from interstitium to lymphatics in g/h, respectively; HEM is rate of protein transfer by hemorrhage in g/h.
The hematocrit was calculated by the mass balance of red blood cells associated with fluid infusion and hemorrhage. Details are given in Appendix A on how fluid and protein transports between plasma, interstitium, and lymphatics were calculated (4–6).
Differential equations of VPL, VIT,UN, VIT,IJ, QPL, QIT,UN, and QIT,IJ are solved with respect to time t using the Runge–Kutta method (10).
Determination of kF and PS in the Normal State
Because the original model (4–6) assumes the capillary membrane is relatively impermeable to water and protein (i.e., kF = 121.1 mL · mm Hg–1 · h–1, = 0.988), the kF and PS values for the whole body in the normal state were estimated by fitting the calculated time-course of plasma volume to the experimental data of Connolly et al. (11) using a nonlinear least-squares procedure (10). These data were obtained during saline infusion of 25 mL/kg over 20 min in conscious, spontaneously ventilating sheep.
Model Parameters for Injured Tissue
The fluid volume fraction of injured tissue in the whole body (FIJ) was set to 0.2, assuming that the surgical area incorporated approximately one-third of the trunk (i.e., thorax, upper, and lower abdomen, Fig. 1B) where 71% of total body water accumulates (12). The value of kF for injured tissue was assumed to be increased by 31% and that of to be reduced by 30% (i.e., 0.612) when compared with normal state values, according to data obtained for cat skeletal muscle during endotoxin-induced inflammation (13). As shown in Appendix B, the PS value in injured tissue was estimated from the relationship between solute diffusion and partition into cylindrical pores, assuming surface area for protein to be unchanged (14).
Simulation of Fluid Volume Changes During Abdominal Surgery
The time-course of fluid volume changes during abdominal surgery was simulated in a 70 kg male when crystalloid solution was administered at a constant infusion rate of 10 mL · kg–1 · h–1. Changes to VPL, VIT,UN, VIT,IJ, and VU over 4 h in the surgical state were compared with normal state values (i.e., permeability parameter values were not changed). In the normal state, VIT,UN and VIT,IJ denote interstitial volume of tissues at nonrisk and risk for injury, respectively. Additionally, percent changes of VPL, VIT,UN, and VIT,IJ over 4 h relative to preoperative volume as a function of fluid infusion rates were compared between surgical and normal states.
Comparison of Model-Predicted Fluid Volume Changes with Clinical Data
The fluid volume changes during abdominal surgery predicted by the mathematical model were compared with those obtained using segmental bioelectrical impedance analysis (BIA) (15). After approval by the local hospital ethics committee and obtaining informed written patient consent, 30 patients, aged 36–77 yr (mean age, 63 yr), who underwent elective surgery of radical intraabdominal cancer dissections were studied. During surgery, isotonic acetated Ringer’s solution was infused at a rate of 7–15 mL · kg–1 · h–1 (12.0 ± 2.5 mL · kg–1 · h–1) and blood transfusion was performed when indicated. Urine output was monitored. Segmental BIA was conducted preoperatively (i.e., before the induction of anesthesia) and postoperatively (i.e., after recovery from anesthesia). Impedance values were obtained for the right arm, for the left side of the trunk, and for the right leg. The extracellular fluid volume in each body segment (i.e., arms, trunk, and legs) was calculated by applying the equation derived from the cell suspension theory (16) to approximated conducting cylinders of arms, trunk, and legs (Fig. 1B). As suggested from the previous finding that bioelectrical impedance changes occurred 5–10 min later than the start of IV fluid infusion (17), the extracellular fluid volume estimated by BIA is relatively insensitive to plasma volume change and thus mainly reflects interstitial fluid volume change. Therefore, the changes of interstitial fluid volume in the limbs (i.e., arms and legs, 
VIT,LM) and trunk (VIT,TR) were obtained as the differences between pre- and postoperative values of estimated extracellular fluid volume in those body segments (15).
The time-course of changes to VPL, VIT,UN, VIT,IJ, and VU during surgery were predicted by the mathematical model using the model parameter values which were adjusted for body weight (56.0 ± 8.6 kg) when appropriate. Average values over operative time (4.1 ± 1.4 h) of infused fluid volume and net hemorrhage volume (i.e., hemorrhage volume minus blood transfusion volume, 429 ± 315 mL) were used as fluid infusion and hemorrhage rates in the model. The postoperative changes of VIT,UN (VIT,UN) and VIT,IJ (VIT,IJ) were converted to VIT,LM and VIT,TR assuming that fluid distribution was uniform between uninjured tissue. Namely,
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where FA, FT, and FL are the fluid volume fraction of arms (i.e., 0.07), trunk (i.e., 0.71), and legs (i.e., 0.22) in the whole body, respectively (12).
Statistical Analysis
Comparison of methods regarding VIT,LM and VIT,TR were made between BIA and model prediction by Bland and Altman plot, with values obtained by BIA as the criterion methods. Similarly, VU was compared between monitoring and model prediction. The model-predicted values of VIT,LM, VIT,TR and VU were subtracted from values obtained by the criterion methods, and the mean (the bias) and sd of these differences calculated. The 95% limits of agreement (mean difference ±2 sd) were expressed relative to the bias, which was adjusted to zero.
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RESULTS |
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TopAbstractIntroductionMETHODSRESULTSDISCUSSIONAPPENDIX AAPPENDIX BREFERENCES |
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Simulation of Fluid Volume Changes During Abdominal Surgery
Figure 3 shows a simulation of changes to plasma, interstitial (uninjured and injured tissues), and urine volumes during 4 h of abdominal surgery in a 70 kg male with fluid replaced at a rate of 10 mL · kg–1 · h–1. Compared with the normal state, surgical injury significantly decreased plasma volume and urine volume, despite the continuing fluid replacement. Interstitial volume in injured tissue increased gradually with time, reaching 705 mL in excess of that of the normal state. Interstitial volume in uninjured tissue increased similarly with time in surgical and normal states.
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As shown in Figure 4, surgical injury itself (i.e., no fluid infusion) increased interstitial volume in injured tissue by 19%, whereas plasma volume and interstitial volume in uninjured tissue were decreased by 16% and 6%, respectively. During surgery, fluid volume increased in all fluid compartments as fluid infusion rates increased. However, fluid administered at a rate more than 10 mL · kg–1 · h–1 made little difference in plasma volume, but contributed to the increase in the volume of the injured tissue compartment. It was also not possible to obtain a normal plasma volume, even with fluid replacement at a rate of almost 20 mL · kg–1 · h–1. In contrast, the normal state showed that fluid volume increased similarly in the interstitium of tissues at nonrisk and risk for injury and a normal plasma volume was obtained with fluid replacement at a rate of 4.6 mL · kg–1 · h–1.
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Comparison of Model-Predicted Fluid Volume Changes with Clinical Data
Bias, and limit of agreement between the criterion methods and model prediction for postoperative changes in the interstitial fluid volume of limbs and trunk and in urine volume in abdominal surgery patients were: –131 and 325 mL for VIT,LM; –157 and 834 mL for VIT,TR; and 91 and 297 mL for VU (Fig. 5). No significant systematic errors were found with respect to values for interstitial fluid and urine volume.
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DISCUSSION |
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TopAbstractIntroductionMETHODSRESULTSDISCUSSIONAPPENDIX AAPPENDIX BREFERENCES |
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). As a result, the calculated fluid volume changes could have been slightly exaggerated, given that the modified values were derived from parameter values used in an endotoxin-septic model (13). An advantage of the modified model is that, by subdividing the interstitium into uninjured and injured tissue compartments, we were able to evaluate interstitial fluid accumulation in each of those compartments.
Simulation of Interstitial Fluid Sequestration
Our results clearly showed that there was significant fluid accumulation in the interstitium of injured tissue, whereas plasma volume gradually decreased to the value below its preoperative level in spite of fluid infusion (Fig. 3). In contrast to injured tissue, the interstitial fluid volume in uninjured tissue was not significantly different between surgical and normal states. These findings may be attributed to albumin leaking from the intravascular space to the interstitium in injured tissue, which further increased transcapillary fluid flux toward the interstitium due to an increase of interstitial colloid osmotic pressure.
This scenario is supported by a significant decrease of plasma volume accompanying surgical injury itself (i.e., no infusion) as shown in Figure 4. Fluid administration in the presence of surgical injury increased the interstitial volume both in injured and uninjured tissues, but the magnitude of interstitial fluid accumulation was much greater in injured tissue when compared with uninjured tissue. This uneven interstitial fluid distribution between injured and uninjured tissues (95% vs 44% increase at 20 mL · kg–1 · h–1) suggests a greater tendency of the interstitium in injured tissue to accumulate fluid from the intravascular space. As a consequence, a normal plasma volume was not obtained even with fluid replacement at a rate of almost 20 mL · kg–1 · h–1.
Model Validity
Postoperative changes in the interstitial fluid volume of body segments and in urine volume were comparable to those obtained by bioelectrical impedance and urine output measurements in abdominal surgery patients (Fig. 5). It is not unreasonable that model-predicted fluid volume changes in the interstitium in limbs and trunk were larger than those calculated by BIA (bias = –131, –157 mL) considering that fluid volumes obtained by BIA should be evaluated functionally based on alternating current-flow characteristics in the human body, rather than anatomically (15). Additionally, the variation of differences of fluid volume changes (i.e., large scatter) between BIA and model prediction may be attributed in part to the differences among patients with respect to model parameter values such as surgical area and anthropometric data (i.e., fluid volume fraction of body segment). The fluid volume fraction of injured tissue in the whole body was set to 0.2. This value may not be an over-estimation because simulation showed that interstitial fluid accumulated in injured tissue at a rate of 3.5 mL · kg–1 · h–1 (Fig. 3) comparable to values obtained from the literature (i.e., 4–6 mL · kg–1 · h–1) (2). Moreover, it is conceivable that severe surgical trauma, such as radical intraabdominal cancer dissections, causes inflammation in an extensive area of the gastrointestinal tract due to tissue manipulation.
Another uncertainty in the mathematical model is insensible water loss from injured tissue (i.e., JISL,IJ), consisting mainly of evaporative losses from peritoneal surfaces exposed to ambient conditions. These water losses may vary substantially and are difficult to quantify. We used the lowest value of those from the literature (i.e., 1–4 mL · kg–1 · h–1) (3). For a 50 kg male at a fluid infusion rate of 10 mL · kg–1 · h–1 during 4 h-surgery, the increase of JISL,IJ by 1 mL · kg–1 · h–1 results in a decrease of approximately 150 mL in interstitial fluid volume in the trunk, which is comparable to the bias between BIA and the model prediction.
Clinical Implications
Our model has not been sufficiently validated to be used as a tool for predicting fluid balance for the individual patient. First, many parameter values in the model are derived from animal experiments, and thus may not be directly applicable to humans. Second, it is difficult to definitively validate the model, because no reference method is available to evaluate fluid volume changes in each body segment (i.e., arms, trunk, and legs) during surgery. Segmental BIA has been used to assess fluid shifts in the absence of a more reliable means to make that assessment. Finally, fluid volume changes predicted by the model were compared with the data from a previous clinical study where fluid infusion and hemorrhage rates were assumed to be constant throughout surgery. This assumption was a prerequisite for calculating fluid volume changes by using the mathematical model. However, it is unlikely that this assumption significantly affected the prediction of fluid volume changes because crystalloid solution was actually infused at a nearly constant rate during abdominal surgery of a single patient, and hemorrhage volume appeared to gradually increase with time. Given that the values of kF and PS are obtained in each patient by fitting the model-predicted time-course of plasma volume to the kinetic data of plasma volume during intravascular fluid administration (i.e., Fig. 2), this model may be applied more suitably for simulating a time-course of volume changes in fluid compartments in a single patient. A controlled, prospective study is required before the model can be considered fully validated.
The model did demonstrate that, compared with the normal state, when fluid infusion rate is gradually increased with time during surgery (reaching 20.4 mL · kg–1 · h–1 at 4 h) to maintain a normal plasma volume, there is massive fluid sequestration in the interstitium of injured tissue (i.e., 1050 mL). The model also shows that increasing the fluid replacement rate above 10 mL · kg–1 · h–1 does not have the desired impact on plasma volume but, instead, increases the volume in the interstitial compartment. Based upon the model results, rational fluid therapy during abdominal surgery would require limiting the maximum infusion rate of crystalloid, and recognizing that it is not possible to normalize plasma volume. Further clinical methods designed to reduce the amount of crystalloid administered during surgery will require techniques directed at preventing the accumulation of fluid in injured tissue or administration of factors that limit the tissue injury.
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APPENDIX A |
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TopAbstractIntroductionMETHODSRESULTSDISCUSSIONAPPENDIX AAPPENDIX BREFERENCES |
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The rate of fluid filtration, JIT, from the plasma to interstitial compartments is given by the Starling equation
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where the subscripts C and NL denote the value of the variable in the capillary and in the normal steady state, respectively; P is hydrostatic pressure in mm Hg; 
is colloid osmotic pressure in mm Hg.
The rate of transcapillary protein transport, IT, is governed by the following equation, which shows that protein transport from the circulation to the interstitium is nonlinearly coupled with the fluid exchange (20)
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where the subscript AV denotes the value of the variable in the available space.
Transport via the Lymphatics (4–6)
For the case of an overhydrated interstitium, the lymphatic return flow of fluid, JL, is described by the following relationship:
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where LS is lymph flow sensitivity in mL · mm Hg–1 · h–1.
For the dehydrated case, JL is given by
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where the subscript EX denotes the value of the variable when the interstitial fluid volume reaches the excluded volume.
The lymphatic transport of protein from the interstitium is given by
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where C is protein concentration in g/mL.
Renal Module (4–6)
The urinary fluid excretion equation assumes a first-order negative-feedback response of the kidney to change in plasma volume from its normal set-point, VPL,NL, i.e.,
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where kU = 168 (mL/h) (VPL < VPL,NL), 1250 (mL/h) (VPL 
VPL,NL).
Constitutive Relationships (4–6)
The capillary hydrostatic pressure, PC, is assumed to vary in proportion to the plasma volume, i.e.,
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where PC,COMP is the reciprocal of the circulatory compliance. For this we used the value of 7.29 x 10–3 (mm Hg/mL) inferred from known physiological data (21).
The interstitial compliance is separated arbitrarily into three segments. For the dehydration segment,
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For the overhydration segment,
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For the intermediate segment, the interstitial pressure is obtained by cubic spline interpolation of discrete PIT and VIT data as follows:
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Protein concentrations within the fluid compartments are expressed as
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The effective interstitial concentrations of proteins that govern exchange across capillary membranes, CIT,AV,i, are given by
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where VIT,AV,i = VIT,i – VIT,EX,i
Plasma and interstitial colloid osmotic pressures exerted by the protein are described by cubic functions of the protein’s concentration within a given compartment (22), i.e.,
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APPENDIX B |
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TopAbstractIntroductionMETHODSRESULTSDISCUSSIONAPPENDIX AAPPENDIX BREFERENCES |
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The permeability coefficient (P) for solute (i.e., protein) passing through a capillary membrane is described as
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where n is the density of pores of radius R, D is solute diffusion coefficient in pores, 
is solute partition coefficient in pores, and x is the membrane thickness (14). Assuming nR2 and x are constant, P is proportional to D · .
For cylindrical pores,
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where D0 is the free diffusion coefficient of solute in aqueous solutions, and 
is the ratio of solute radius to pore radius (14). The reflection coefficient for solute, , is described using , where
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By substituting values for normal (i.e., 0.875) and injured tissues (i.e., 0.612) into Eq. (A19), values for can be obtained. The use of these values in Eq. (A17) yields values for , which, by substituting into Eq. (A18), allows D/D0 to be obtained. Finally, we obtain P (injured tissue) = 13.7 · P (normal tissue).
The value of 13.7 is in the range of PS changes caused by histamine in dog paw (i.e., 7–20) (23).
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Footnotes |
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REFERENCES |
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TopAbstractIntroductionMETHODSRESULTSDISCUSSIONAPPENDIX AAPPENDIX BREFERENCES |
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This article has been cited by other articles:
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  T. Tatara, Y. Nagao, and C. Tashiro The Effect of Duration of Surgery on Fluid Balance During Abdominal Surgery: A Mathematical Model Anesth. Analg., July 1, 2009; 109(1): 211 - 216. [Abstract] [Full Text] [PDF]
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 T. Tatara, T. Tsunetoh, and C. Tashiro Crystalloid infusion rate during fluid resuscitation from acute haemorrhage Br. J. Anaesth., August 1, 2007; 99(2): 212 - 217. [Abstract] [Full Text] [PDF]
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