Acta Anaesthesiol Scand 2003; 47: 37–45 Copyright C Acta Anaesthesiol Scand 2003Printed in Denmark. All rights reserved

ACTA ANAESTHESIOLOGICA SCANDINAVICA

0001-6349

New mathematical model for the correct prediction ofthe exchangeable blood volume during acutenormovolemic hemodilution

J. MEIER1, M. KLEEN2, O. HABLER2, G. KEMMING1,2 and K. MESSMER1

1Institute for Surgical Research, 2Clinic of Anesthesiology, Klinikum Großhadern, Ludwig-Maximilians-University Munich, Germany

Background: The blood volume that has to be exchanged forcrystalloids and/or colloids during acute normovolemic hemo-dilution (ANH) in order to reach a preset target hemoglobinconcentration (hb) is usually predicted by the Bourke and Smithformula developed in 1974. This formula systematically over-estimates the ‘true’ exchangeable blood volume (EBV), a factthat may potentially endanger patients because the target hbwill be missed and the normovolemic anemia might turn out tobe more severe than a priori intended. Our objective was to de-velop a more accurate mathematical model of hemodilution kin-etics and to validate this new model in animals and in patientsundergoing ANH.Methods: Twenty-two anesthetized beagle dogs and 18 patientsunder balanced anesthesia underwent isovolemic hemodilutionwith hydroxyethyl starch (HAES 6%, 200000) to a target hb of 7gdlª1 or 9gdlª1, respectively. Exchangeable blood volume pre-dicted by use of the different mathematical models was com-pared with the blood volume actually exchanged to meet thepreset target hb.Results: Calculation of EBV by the Bourke and Smith formula

ACUTE normovolemic hemodilution (ANH) is an es-tablished procedure for the avoidance of allo-

genic blood transfusions in elective surgery. Betweeninduction of anesthesia and start of surgery, freshwhole blood is withdrawn from the patient and sim-ultaneously replaced by the identic volume of a col-loid solution or the triple volume of a crystalloid solu-tion (1). Presuming that normovolemia is maintained,hemodilution entails a reduction of red blood cells(RBC), and thus of hemoglobin concentration (hb).However, despite the reduced arterial oxygen contentand systemic oxygen delivery, systemic oxygen con-sumption remains constant over a large hb range as aresult of different compensatory mechanisms (2),mainly the increase of cardiac output, and the increaseof tissular oxygen extraction.

From the clinical point of view ANH offers severaladvantages: (1) in hemodiluted subjects blood lost

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(EBVBπS) systematically overestimated the volume actually ex-changed (overestimation: dogs 15%, patients 20%), whereas ournew iterative model predicted EBV (EBViterative) more reliably(overestimation: dogs 1%, patients 8%). In both cases EBVBπS

differed significantly from the EBViterative.Conclusion: Exchangeable blood volume is predicted more ac-curately by the new iterative model than by the Bourke andSmith formula. The iterative model leads to an improvement inpatient safety and provides a physiologically adequate basis forfuture studies investigating the efficacy of ANH in reducingallogenic blood transfusions.

Accepted for publication 17 June 2002

Key words: exchangeable blood volume; hemodilution; math-ematical model.

c Acta Anaesthesiologica Scandinavica 47 (2003)

during surgery contains fewer RBC per milliliter (3–5); (2) in case of transfusion need, the patient receivesfresh autologous whole blood, containing coagulationfactors and platelets in physiological concentrations;and (3) transfusions may be effectively avoided withapplication of ANH, presuming that a low target hbis accepted (6). Hemoglobin concentration declines ex-ponentially during ANH. This was described for thefirst time 1974 by Bourke and Smith (7).

EBV Ω BV ¿ ln hb0

hbt [1]

This is the original formula of Bourke and Smith (7).EBV Ω exchangeable blood volume, BV Ω initial (i.e.pre-ANH) blood volume, hb0 Ω initial hb, hbt Ω targethb.

Equation 1 is also used in clinically practiced ANH

J. Meier et al.

to predict the exchangeable blood volume (EBV)necessary to reach a target hb (hbt). Simultaneously itconstitutes the mathematical basis for theoretical con-siderations concerning the efficacy of ANH in avoid-ing allogenic transfusions.

This formula is a simplified version of the plasmaexchange equations presented by Wallerstein et al. (8)and Veall et al. (9) for replacement transfusion as aform of treatment for hemolytic disease of the new-born. Although developed for another therapeuticalconcept these equations are related to the kinetics ofthe decline of hb during hemodilution.

Calculation of EBV with the Bourke and Smith for-mula presents three major weaknesses:

1. The formula has been developed on the basis ofan integration process based on the assumption ofcontinuous and simultaneous substitution ofblood with a colloid. This is not always the case inthe clinical setting.

2. The formula has been developed on condition thatthe RBCs are distributed homogeneously through-out the body, a fact that can not be assumed.

3. One prerequisite for validity of the Bourke andSmith formula is constancy of blood volumethroughout the entire hemodilution process.

In the original publication the formula was validatedin patients using calculated instead of measured bloodvolumes. Therefore validation of equation 1 withthese data is difficult.

It has been repeatedly reported that the Bourke andSmith formula systematically overestimates EBV (11–13). This implicates that exchange of the calculatedblood volume leads to a lower hb than targeted, a factthat might endanger the patient.

The reasons for overestimation of EBV by the Bour-ke and Smith formula are not completely clear; poss-ible explanations are a systematic error arising fromestimation of blood volume; incorrect description ofthe decline of hb during ANH by this simplified for-mula itself; or changes of the actual total blood vol-ume during the hemodilution process.

Therefore the first aim of the present study was tofind out whether a more extensive mathematicalmodel can describe ANH kinetics more accuratelythan the original integrative model presented byBourke and Smith and whether the new model en-ables a more precise prediction of EBV. It is undis-puted that the formula of Bourke and Smith is math-ematically correct. However it is supposed that it doesnot sufficiently take into account all physiologic fun-damentals involved in clinically practiced ANH.

The second aim of this study was to validate our

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new iterative model in an animal model and in ourclinical setting.

Methods

Theoretical dilution modelThe equation provided by Bourke and Smith corre-sponds to an isovolemic, continuous and simultaneoussubstitution of homogeneously distributed red blood cellswith a colloid solution. This is not necessarily the waya clinical hemodilution procedure takes place. We de-veloped a model that corresponds to an intermittenthemodilution procedure considering inhomogeneous redblood cell distribution throughout the body.

Clinical performance of ANH requires the followingpreconditions:

P Knowledge of patients prehemodilution hb (hb0);P Definition of the posthemodilution target-hb (hbt);P Measurement or estimation of a patient’s actual

blood volume (BV);P Determination of the modality of the hemodilution

procedure (continuous vs. intermittent substitution,sequence of blood removal and substitution).

The following terms are used in the derivation of ournew iterative mathematical model: BV Ω total bloodvolume, E Ω volume exchanged during each single di-lution step, EBV Ω total exchangeable blood volume,i.e. blood volume to reach the target post-ANH hb.

It has to be noted that RBC are not homogeneouslydistributed within the vasculature, a fact that has notbeen taken into account for development of equation1. Hemoglobin concentration within large vessels(large vessel hb Ω hblv) is higher than hb within smallvessels (small vessel hb Ω hbsv) (14–16). The wholebody hb (hbwb) depends on hblv and hbsv, and canonly be measured by dilution kinetics. It has to benoticed that hb values are usually measured by takingblood samples from an arterial or venous line placedin a large vessel. The hb concentration therefore corre-sponds more likely to hblv than hbwb.

For the development of the new iterative model thefollowing terms are used:

P hblv n Ω large vessel hb before each single dilutionstep n (can be measured from arterial or venousblood samples);

P hbwb n Ω whole body hb before each single dilutionstep n (can not be measured easily, but correspondsto the number of RBC in the whole body).

Therefore hb0 and hbt of the original formula corre-spond to hblv 0 and hblv t.

ANH: prediction of exchangeable blood volume

Iterative modelThe first step towards the derivation of a new iterativemodel is to determine that hb arising after a givenblood volume is replaced in one single dilution step.‘One step’ in this context is defined as the sequentialprocedure of withdrawing a given amount of wholeblood and infusing a suitable volume of a colloidal orcrystalloid solution. This is one possible way ANH isperformed in clinical practice. It corresponds to the‘intermittent substitution formulas’ presented by Wall-erstein et al. (8) for blood substitution in the treatmentof erythroblastosis fetalis. For homogeneous distri-bution of RBC throughout the body the following for-mula for this intermittent procedure has been de-scribed (17):

YY0

ΩV ª xV n

[2]

This is the original dilutional equation from McCul-lough et al. (17) referring to the observations of Wall-erstein et al. (8): Y Ω final concentration; Y0 Ω initialconcentration; V Ω plasma volume, ¿ Ω size of aliquotremoved; n Ω number of aliquots removed.

It has been shown that hb declines faster whenblood removal is performed before replacement of thevolume removed takes place (17). Even though equa-tion 2 simulates a discontinuous procedure it does notconsider the fact that the RBCs are not distributedequally throughout the whole body, a fact that hasbeen considered for replacement transfusion as a formfor treatment for hemolytic disease in the newborn byVeall et al. (9) and for hemodilution kinetics by Hahn(10). We propose a combination of these two concepts:

The volume of RBC that is withdrawn in any of thedilution steps n of a discontinuous hemodilution pro-cedure is hblv n ¿ E. The volume of RBC remaining is:(hbwb n ¿ BV) – (hblv n ¿ E). The new resulting wholebody hb (hbwbn π 1) reads:

hbwb n π 1 Ω(hbwb n ¿ BV) ª (hb lv n ¿ E

BV[3]

Using this equation it is possible to obtain each fol-lowing hbwb by using the actual hbwb and hblv. Dur-ing clinical ANH it is impossible to measure hbwb foreach dilution step. Therefore hbwb has to be calcu-lated from hblv. The relationship between hbwb andhblv is known to vary within small limits (15). Al-though there might be a dependency on the actualhb level (16), it can be assumed that the relationshipbetween hbwb and hblv is a fairly constant quotient(dogs 0.9, humans 0.9), which only slightly varies

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with changes of hb, temperature, arterial pressure, etc.(14).

hbwb

hblv 0.9 ; hbwb 0.9 ¿ hblv [4]

Represents the relationship of hbwb to hblv in regardto Gibson et al. Similar values for dogs and humanshave been reported (14, 15).

Using Equation 4 it is possible to obtain the newlarge vessel hblvn π 1.

hblv n π 1 Ωhbwb n π 1

0.9[5]

Thus an algorithm has been found enabling calcu-lation of the new hblv (hblvn π 1) after each singlehemodilution step. The exchangeable blood volumecan be obtained as the sum of volumes exchanged ineach single hemodilution step.

EBV Ω a

x Ω 1Ex Ω E ¿ a [6]

The exchangeable blood volume can be obtained asthe sum of volumes exchanged in each single hemo-dilution step. a Ω number of dilution steps

To determine EBV it is necessary to choose E and torepeat the previously described steps until hbt isreached (Fig. 1). This can be automated using a com-puter program. Therefore an algorithm has beencoded in Visual Basic (Visual Basic 6, Microsoft Cor-poration, Redmond, WA), enabling calculation of EBVbased on hblv 1, BV and E. This program is freewareand may be obtained from the authors via e-mail(jens.meier/firemail.de). Although the new iterativeapproach differs from the integration process de-veloped by Bourke and Smith, it can be demonstrated

Fig.1. The two steps of the iterative model. The volume of blood to beexchanged reads as the sum of the blood volumes replaced in the differ-ent hemodilution steps. This algorithm can easily be implementedusing Microsoft Excel, or can be coded in any programming language.A version for Microsoft Windows and PalmOS may be obtained fromthe authors. Hb, hemoglobin concentration; BV, blood volume; wb,whole body; lv, large vessels; E, volume exchanged in one dilution step.

J. Meier et al.

Fig.2. The relation of exchangeable blood volume (EBV) calculatedwith the original formula of Bourke and Smith (EBVB πS) and EBVusing the computer program (EBViterative). An initial hemoglobin con-centration (hb) of 15g/dl has been assumed. X-axis: different targethb, Y-axis: EBV calculated as a fraction of blood volume. BViterative isalways lower than EBVB πS, indicating that using the original formularesults in an overestimation of EBV. Hb, hemoglobin concentration;BV, blood volume.

that the two formulas correspond to each other, pre-suming that three conditions are fulfilled:

1. Homogenous distribution of RBC within the vas-culature.

2. Infinitely small E (i.e. removal of blood and in-fusion of the substitute take place simultaneouslyand slowly, otherwise homogeneous distribution ofRBC can not be warranted).

3. Constant BV.

The complete confirmation of equivalency and the ex-planation of the three assumptions is given in the Ap-pendix section. However the above-mentioned re-quirements are not fulfilled during clinical ANH.Therefore the original formula of Bourke and Smith,although mathematically correct, systematically over-estimates EBV. This is because of the fact that theunderlying mathematical preconditions are physiolo-gically not realizeable.

Comparison of EBV obtained by the Bourke andSmith formula or by the iterative modelIn contrast to the Bourke and Smith formula, EBV cal-culated with the new iterative model accounts for theinhomogeneity of the distribution of RBC within thevasculature and for the fact that the removal of bloodand the infusion of the substitute do not take placeslowly and simultaneously. No known mathematical

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ANH model as yet takes these two facts into consider-ation at the same time.

In order to investigate whether EBV calculated withthe new iterative model deviates from EBV calculatedwith the Bourke and Smith formula and whetherusing the iterative model enables a better predictionof EBV, EBV has been calculated proportionally toblood volume either with the Bourke and Smith for-mula or with the iterative model for different targethb values. For this purpose E has been chosen verysmall (this corresponds to a slow and simultaneousexchange of blood with the substitute). An overviewof this comparison is given in Fig. 2.

Then the influence of E on EBV was investigated.The results are shown in Fig. 3. An overview of thederivation of this figure is given in Appendix B.

Validation of the iterative hemodilution model indogsIn a large experimental protocol, basically investigat-ing the effects of hyperoxemia on oxygen transportand tissue oxygenation during extreme hemodilution(18,19), 22 healthy beagle dogs of either sex, weighing15.4 ∫ 1.9 kg were diluted with hydroxyethylstarch(HAES 6%, MW 200 000/0.5%, Braun, Melsungen,Germany) in one step from their initial hb of 12.2 ∫2.7 g dlª1 to hbt Ω 7 g dlª1. The dogs had been splenec-tomized at least 8 weeks before entering the study toexclude changes of the red cell mass during ANH in-duced by splenic contraction (20).

Fig.3. The relation of exchangeable blood volume (EBV)iterative andEBVBπS depending on the aliquot removed in a single hemodilutionstep. The aliquot removed is given as a part of the total blood volume(E/BV). If the aliquot removed in a single hemodilution step is notlarger than 10% of BV then the difference of EBViterative andEBVBπS is not more than 5% (see Appendix B). Hb, hemoglobin con-centration; BV, blood volume; E, volume exchanged in one dilutionstep.

ANH: prediction of exchangeable blood volume

Measurement of hb and BVHemoglobin concentration was measured immedi-ately before (hb0) and after ANH (hbt) in bloodsamples withdrawn from the aorta using a CO-Oxy-meter (CO-Oximeter 682, Instrumentation Laboratory,Lexington, MA).

Before and after ANH the animals’ BV was meas-ured using indocyanine green (ICG) dilution kinetics(21,22). The BV measured before ANH was used forthe calculation of EBVB π S and EBViterative. After injec-tion of 0.25 mg kgª1 BW indocyanine green, the mono-exponential decay of the blood ICG concentration wasdetermined intermittently for at least 5 min every 30s. Extrapolation of the decay to the time point of injec-tion enabled calculation of the ICG dilution and thusblood volume:

BV ΩICG doseinjected

ICG concentrationt Ω 0[7]

Represents BV calculated by indocyanine green di-lution kinetics.

Deviation from the Bourke and Smith formulaIn order to quantify the difference of the quantity ofwhole blood exchanged to reach hbt (Vexchanged), andthe exchangeable blood volume calculated with theBourke and Smith formula (EBVB π S) the followingratio was defined:

ratio1 ΩVexchanged

EBVB π S[8]

Here, the volume actually exchanged is comparedwith the volume calculated by the Bourke and Smithformula.

Ratio1 equals 1 if EBV and Vexchanged are identical. Ifthe ratio is less than 1, a smaller volume of blood thanpredicted by the original formula of Bourke andSmith could be finally exchanged. If the calculatedblood volume would have been entirely withdrawn,hb would have exceeded the targeted value. A similarratio is used to determine the quantity of whole bloodto be exchanged to reach hbt (Vexchanged), and the ex-changeable blood volume was calculated with ouriterative computer model (EBViterative).

ratio2 ΩVexchanged

EBViterative[9]

Here, the volume actually exchanged is comparedwith the volume calculated by the computer model.

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Validation of the iterative hemodilution model inpatientsPresuming that EBV can be accurately predicted usingour new iterative model with blood volume havingbeen measured, it still has to be demonstrated that EBVcan also be accurately predicted when the blood vol-ume is estimated on the basis of a mathematical model.In the latter case any calculation of EBV depends oncorrect estimation of BV. Therefore our iterative modelwas also validated in patients undergoing ANH, inwhom blood volume was estimated.

For that purpose specific data obtained in a clinicalprotocol including normovolemic hemodilution wereretrospectively analyzed (23). In the present analysiswe refer to 18 patients who were contributed to themulticenter study by our own center. After inductionof anesthesia, the patients (weight 72.94 ∫ 15.7 kg)were diluted with hydroxyethylstarch (HAES, 6%,MW 200 000/0.5%, Braun, Melsungen, Germany) fromhb0 Ω 11.2 ∫ 1.0 g dlª1 to target hbt Ω 9 g dlª1.

Estimation of BV and measurement of hbBlood volume was estimated using the following for-mulas:

Men: BV Ω 0.417 ¿ T3 π 0.045 ¿ BW ª 0.03Women:BV Ω 0.414 ¿ T3 π 0.0328 ¿ BW ª 0.03[10]Estimationof blood volume for men and women. T Ω height inm, BW Ω body weight in kg (24).

Hemoglobin concentration in arterial blood wasmeasured using a CO-Oxymeter (HemoCue, AB LeoDiagnostics, Helsinborg, Sweden) before and aftereach hemodilution step.

The ratios presented were the same as describedearlier for the animal model.

Statistical analysisA Wilcoxon signed-rank test was used to assess statis-tically significant differences between ratio1 and ratio2

in both studies. P 0.05 was considered statisticallysignificant.

Results

Figure 2 depicts the comparison of the data obtainedby the Bourke and Smith formula and by the new iter-ative model for very small E (E»0). Exchangeableblood volume/BV calculated by the new iterativemodel is approximately 10% lower than the values ob-tained using the Bourke and Smith formula. Thereforethe new iterative model suggests systematically lowervalues for EBV than the Bourke and Smith formula,and an overestimation of EBV is less likely to occurusing the new iterative model.

J. Meier et al.

Fig.4. Values of equation 7 and equation 8 obtained in the animalmodel. ANH, acute normovolemic hemodilution.

The influence of E on EBV can be seen in Fig. 3. Ifblood is withdrawn quickly (i.e. E is not infinitelysmall) homogeneous distribution of RBC is notachieved during the blood exchange and Hb will de-cline more rapidly. The bigger the E that is chosen,the lower the EBV will become as calculated by thecomputer program (EBViterative). If 10% of the totalblood volume of an animal is withdrawn and subse-quently (not simultaneously) replaced (E/BV Ω 0.1),then the ratio of EBViterative/EBVB π S will be morethan 0.95. Hence, even if in the clinical setting singledilution steps amount to 10% of the total blood vol-ume (E Ω 0.1 ¿ BV), the ratio EBViterative/EBVB π S re-mains still greater than 0.95. Therefore the error im-posed by large exchange volumes may explain ap-proximately 5% of the deviation of the Bourke andSmith equation. In a clinical setting blood withdrawaland infusion of the substitute usually take place sim-ultaneously. Therefore the time necessary for distri-bution of the substitute is less than is needed for in-fusion of 10% of the blood volume, so this error mighteven be smaller. A smaller E corresponds to a betterprognosis of EBV by the Bourke and Smith formulaas the influence of E on EBV will diminish. Neglectingthe influence of E on EBV as done by the Bourke andSmith formula leads to a further overestimation of‘real’ EBV.

The ratios obtained from applying equations 8 and9 are depicted in Fig. 4 for the experimental ANH andin Fig. 5 for the patients. Ratio1 differs significantly

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from ratio2 in the animals (0.85 ∫ 0.18 vs. 0.99 ∫ 0.15;P 0.05), as well as in the patients (0.8 ∫ 0.33 vs. 0.92∫ 0.33; P 0.05).

The dogs’ blood volume (Fig. 6) was not altered sig-nificantly by the hemodilution process (1330∫ 198 mlvs. 1348 ∫ 128 ml, NS).

The Bourke and Smith formula systematically over-estimates the exchangeable blood volume in animalsas well as in patients (ratio1 1.0). The application ofour new iterative model enabled a significantly betterprediction of EBV in both settings.

The new iterative model correctly predicts EBV if

Fig.5. Values of equation 7 and equation 8 obtained in the patients.BV, blood volume; ANH, acute normovolemic hemodilution.

Fig.6. The dogs’ blood volume before and at the end of the hemodilu-tion procedure. BV, blood volume.

ANH: prediction of exchangeable blood volume

blood volume is measured (ratio Ω 0.99). If blood vol-ume is calculated also, the new iterative model willoverestimate EBV (ratio Ω 0.92). Nevertheless, in thisclinically relevant setting calculation of EBV with theiterative model is significantly better than using theBourke and Smith formula (P 0.05). The overestima-tion of EBV by the iterative model can be neglected incomparison with that by the Bourke and Smith for-mula.

Discussion

The main findings of the present study are:

1. The formula of Bourke and Smith systematicallyand significantly overestimates EBV independentof the method for blood volume determination(measurement or estimation).

2. Using our new iterative model a more accurate pre-diction of EBV was achieved as compared in a stan-dardized laboratory situation as well as in a clinicalhemodilution procedure.

3. The accuracy of our new iterative model is suffi-ciently high to justify its preferred application inclinically performed ANH.

The efficacy of hemodilution in reducing perioperat-ive allogenic blood transfusions is mainly related tothe fact that in a case of surgical bleeding less RBCwill be lost per milliliter in a hemodiluted patient thanin a patient with normal hb. The lower the hb afterANH the higher becomes the probability to avoidallogenic transfusion (6). However the correct predic-tion of EBV, particularly in extreme hemodilution, isimportant because the margin of safety to guaranteeadequate tissue oxygenation narrows with higher de-grees of dilutional anemia. Exceeding a targeted lowhb due to overestimation of EBV might expose thepatient to the risk of tissue hypoxia. Therefore in ex-treme hemodilution the new iterative model is su-perior to the formula of Bourke and Smith concerninga patient’s safety.

Moreover in theoretical and mathematical studieson ANH, the use of the Bourke and Smith formulamay also result in an incorrect prediction of EBV. Mostof the studies investigating the clinical efficacy ofANH use one of two different approaches. Either thenumber of allogeneic transfused RBC units is com-pared between patient groups undergoing ANH ornot (4, 25), or mathematical models are developed topredict the amount of blood to be saved by ANH (3,5, 6, 26–28). In most of these studies the original for-mula of Bourke and Smith is applied, although bettermodels have already been found (10). As a conse-

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quence these results have to be interpreted carefully,as they usually do not account for the fact that theoriginal Bourke and Smith formula will systemicallyoverestimate the exchangeable blood volume.

EBViterative is approximately 10% lower thanEBVB π S if E is infinitely small (Fig. 2). Therefore anoverestimation of ‘true’ EBV is less likely to occurusing our new iterative model. This is confirmed bythe ratios obtained (equations 8 and 9) in the stan-dardized laboratory situation as well as in the clinicalsetting. Irrespective of the fact that blood volume ismeasured or estimated the new iterative model al-ways provides a more accurate description of the de-cline of hb during a hemodilution process than ob-tained by using the Bourke and Smith formula.

If blood volume is measured our new iterativemodel enables a very reliable calculation of EBV(ratio Ω 0.99). However, even if blood volume can onlybe estimated like in the usual clinical setting the newiterative model also provides a more accurate esti-mation of EBV than do the common methods (ratio Ω0.92). Nevertheless, optimal, e.g. correct results,would also be obtained if the patient’s blood volumeis measured.

The standard deviation of ratio 1 and ratio 2 in ouranimal model (equations 8 and 9) is as high as thestandard deviation of the respective ratios found inpatients (Figs 4 and 5). Therefore prediction of EBVusing our computer model supplies correct but nothighly precise values of EBV. Several reasons mightbe responsible for this. Volume shifts from or to theinterstitium might contribute to a fall or increase ofhemoglobin concentration in absence of actual bloodor fluid loss. Correct measurement or estimation ofBV is an indispensable precondition for correct andprecise calculation of EBV, but can usually not be per-formed in a clinical setting. Therefore any variationsof BV during the observation period will lead to anincorrect prediction of EBV. No changes of the bloodvolume of the dogs were observed before and afterthe hemodilution procedure. Therefore a systematicoverestimation of EBV by the Bourke and Smith for-mula can not be explained by systematic changes ofblood volume throughout the hemodilution processin this setting. As blood volume was not measured inpatients it can not be excluded that variations in bloodvolume can explain part of the systematic overestima-tion of EBV by the Bourke and Smith formula in pa-tients.

The ratio obtained by use of equation 5 might notbe precise enough to provide a correct relation of hbwb

and hblv. Because hblv is only one of the many factorscontributing to hbwb, it is not surprising that this cor-

J. Meier et al.

relation can not be complete. Obtaining a more preciseprediction of hbwb is difficult as many variables haveto be taken into account (species, body weight, height,sex, splenic contraction, blood volume, etc.).

In conclusion our results demonstrate that by usingour new iterative model, EBV can be predicted moreaccurately than by the Bourke and Smith formula be-cause physiological conditions are more adequatelytaken into account. Former formulas regarding di-lutional kinetics do not take into consideration thespecific problems arising with the use of ANH. Ournew iterative model is easy to apply, its use enhancespatient safety and it provides a correct physiologicalbasis for further studies addressing the efficacy ofANH.

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AddressProf. Dr h. c. mult. Konrad MessmerInstitute for Surgical ResearchLudwig-Maximilians-Universität MünchenKlinikum GroßhadernMarchioninistraße 1581366 MünchenGermanyE-mail: messmer/icf.med.uni-muenchen.de

Appendix AIf red blood cells (RBC) are distributed homogeneously withinthe vasculature (hb1), hb after one hemodilution step (hb2)reads:

hb2 Ωhb1 ¿ (BVªE)

BV[11]

And for any further hemodilution step:

hbnπ1 Ωhbn ¿ (BVªE)

BVΩhbn ¿1ª

EBV [12]

ANH: prediction of exchangeable blood volume

This is the recursive form of a geometric series if E and BVremain constant. If this can be ensured equation 12 can be trans-formed into a recursion-free presentation:

hbn Ωhb1 ¿1ªE

BVnª1[13]

Exchangeable blood volume can be calculated by multiplyingthe number of dilution steps with the volume replaced in onestep: EBV Ω (n-1)¿E. In order to obtain the resulting hb not fromthe number of dilution steps, but from the volume replaced inone step (E), one can substitute:

EΩEBVnª1

; nΩEBV

Eπ1 [14]

This expression has to be set in the recursion-free form of theequation. Several rearrangements give this result:

1ªE

BVEBV

E Ωhbn

hb1[15]

And further:

EBV ΩE¿ lnhbn

hb1

ln1ªE

BV [16]

Thus EBV can be calculated depending on the volume replacedby a single dilution step (E). During ANH, withdrawal of bloodand infusion of the diluents are performed simultaneously. Thisimplies that the single dilution step is nearly infinitely small.In other words: E»0. One can rearrange the last expression asfollows:

EBV ΩE

ln1ªE

BV¿ lnhbn

hb1 [17]

The limit for this expression can be calculated. Using a standard

45

algorithm (implemented in Maple V5, Brooks/Cole PublishingCompany, Pacific Grove, CA) the equation can be solved:

limE»0

(EBV) Ω ªBV¿ lnhbn

hb1ΩBV¿ lnhb1

hbn [18]

Hence, it is demonstrated that the formula of Bourke and Smithcan be derived exactly and that it is a special case of the iterativeprocedure described above. It therefore describes the process ofANH and its kinetics in a correct way presuming that three pre-conditions are met:

1. Homogeneous distribution of RBC within the vasculature.2. Constant and infinitely small E.3. Constant BV.

Appendix BThe error that is entailed by the fact that E is not infinitely smallcan be quantified by equation 17. It yields a term to calculateEBV depending on the amount of E. In the following EBV calcu-lated by equation 17 is called EBViterative. A ratio of the value forEBV calculated by the limitation process (EBVlim) and the orig-inal formula of the Bourke and Smith precise value for EBV(EBVprecise) can be derived and written as follows:

EBVBπS

EBViterativeΩ

BV¿ lnhct1

hctt¿ ln1ª

EBV

E¿ lnhctt

hct1 [19]

It is possible to substitute the ratio E/BV with¿ΩE/BV. Severaltransformations produce:

EBViterative

EBVBπSΩ ª

xln(1ªx)

[20]

This term is plotted in Fig.3. It depicts the relation of EBViterative

and EBVBπS, depending on the aliquot removed in a singlehemodilution step. The aliquot removed is given as a part of thetotal blood volume (E/BV).