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RESEARCH Open Access

Modern and traditional approachescombined into an effective gray-boxmathematical model of full-blood acid-baseFilip Ježek1,2* and Jiří Kofránek2

Abstract

Background: The acidity of human body fluids, expressed by the pH, is physiologically regulated in a narrow range,which is required for the proper function of cellular metabolism. Acid-base disorders are common especially inintensive care, and the acid-base status is one of the vital clinical signs for the patient management. Because acid-base balance is connected to many bodily processes and regulations, complex mathematical models are needed toget insight into the mixed disorders and to act accordingly. The goal of this study is to develop a full-blood acid-base model, designed to be further integrated into more complex human physiology models.

Results: We have developed computationally simple and robust full-blood model, yet thorough enough to covermost of the common pathologies. Thanks to its simplicity and usage of Modelica language, it is suitable to beembedded within more elaborate systems. We achieved the simplification by a combination of behavioralSiggaard-Andersen’s traditional approach for erythrocyte modeling and the mechanistic Stewart’s physicochemicalapproach for plasma modeling. The resulting model is capable of providing variations in arterial pCO2, base excess,strong ion difference, hematocrit, plasma protein, phosphates and hemodilution/hemoconcentration, but insensitiveto DPG and CO concentrations.

Conclusions: This study presents a straightforward unification of Siggaard-Andersen’s and Stewart’s acid-basemodels. The resulting full-blood acid-base model is designed to be a core part of a complex dynamic whole-bodyacid-base and gas transfer model.

Keywords: Acid-base modeling, Physicochemical acid-base, Behavioral acid-base, Siggaard-Andersen, Modelica,Physiology, Physiolibrary

BackgroundAcid-base disturbances are associated with a number offluid, electrolyte, metabolic and respiratory disorders. Un-derstanding the pathogenesis and the underlying patho-physiological processes is crucial for proper diagnostics andtreatment, especially in acute medicine, anaesthesiologyand during artificial respiration. Complex mathematicalmodels help to uncover the pathogenesis of complex bodilydisorders.Two major approaches for mathematical modeling of

the acid-base status of blood are widely employed. The

traditional model by Siggaard-Andersen [1] (SA) is abehavioral model of full blood acid-base (i.e. includingerythrocytes), but it is originally defined for standard albu-min and phosphates only (though added later [2]) and, onits own, the model is unable to assess hemodilution andhemoconcentration (e.g. during fluid replacement), andthe individual levels of ions are not considered. Thesecond model, Stewart’s physicochemical [3] model, or theso-called modern approach, is a structural model ofplasma only, but it is essential for assessing hemodilution,ion and protein imbalances, which are common in critic-ally ill patients. However neither of these is satisfactory asa complete model.The Stewart’s physicochemical approach has been

further extended to overcome some of its disadvantages.Rees and Andreasen shown the extension to full blood

* Correspondence: [email protected] of Cybernetics, Faculty of Electrical Engineering, CzechTechnical University in Prague, Prague, Czech Republic2Institute of Pathological Physiology, First Faculty of Medicine, CharlesUniversity, U nemocnice 5, 128 00 Prague 2, Czech Republic

© The Author(s). 2019 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, andreproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link tothe Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver(http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.

Ježek and Kofránek Theoretical Biology and Medical Modelling (2018) 15:14 https://doi.org/10.1186/s12976-018-0086-9

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and enhanced it by circulation, blood gases, interstitiumand cellular compartments [4]. Wooten later presented anextension to the extracellular compartment [5] and mostrecently, Wolf [6] proposed a profound steady-state physi-cochemical model of erythrocyte-plasma-interstitium-cellcompartments, including detailed ion and water balance.It has been built with the purpose to mechanistically de-scribe complex physicochemical processes, which is, how-ever, computationally demanding and hard to solve due toa large system of non-linear equations. This prevents usfrom integrating the detailed models into more complexmodels, or from usage for patient-specific identification.Described approaches [1–6] are, however, designed for

steady-state situations. To extend the whole-bodyacid-base assessment by bodily regulatory loops and toshow pathogenesis of developing disorders in time, oneneeds to construct even larger integrative model, includ-ing important bodily compartments (interstitial fluid,cells) and regulations (kidney, liver, respiratory regula-tion) interconnected by a circulation of full-blood, thecore component of our approach [7]. From the authors’experience, these subcomponents however need to bebased on computationally robust submodels, which areyet precise enough to describe the physiology.The aim of this study is to develop a detailed, yet

computationally effective, full blood model by com-bining the two predominant approaches to bloodacid-base balance. The resulting combined model willbecome an essential part of a future complex dynamicwhole-body acid-base and blood gas transfer modelfollowing the border flux theory [7] to assess dynamicbodily compensations.

MethodsIf the detailed processes in erythrocyte are not the object-ive, an empirical description of their contribution is oftensatisfactory, which is substantially lowering the modelcomplexity. Therefore, instead of modeling erythrocytemechanistically, we propose to substitute it with behav-ioral description, whereas the plasma should follow thephysicochemical mechanistic design to include ion trans-fer to adjacent body compartments (Fig. 1). This approachwould simplify complex physicochemical computations ofmembrane balance.

The Siggaard-Andersen’s (SA) approach (also knownas the traditional approach), built around SA nomogram[8], uses measured values of pH and arterial pCO2 tocompute buffer base (BB), a concentration of buffer an-ions and cations, which can take a buffering action. Adifference between BB and normal buffer base (NBB) iscalled base excess (BE), which expresses how many mili-moles of strong acid must be added to 1 l of blood to re-gain normal pH (at normal pCO2 = 40 mmHg). Morerecently, the term BE has been substituted by the Con-centration of Titratable Hydrogen Ion (ctH+), whichhowever equals to the negative of BE [9]. The BE provedto be a handy indicator for the clinicians to quickly as-sess the level of metabolic acid-base disturbance. In thefollowing years, the importance of the printed acid-basenomogram declined in favor of its numericalformalization, e.g. the Van Slyke eq. [9], so that we canform a function for Siggaard-Andersen’s pH as:

pHSA ¼ fSA BE; pCO2;Hbð Þ ð1Þ

The physicochemical, (also referred to as a modern orStewart’s approach), does not rely on measured nomo-gram. Instead, a so-called strong ion difference (SID) iscalculated from the difference of strong (i.e. mostly orfully dissociated at physiologic ranges) cations andanions concentrations (which equals charge forfully-dissociated substances) in plasma. Because of theelectroneutrality principle, the SID must equal the sumof charges of HCO− 3, phosphates and albumin. SID canbe calculated from the difference of ion concentrations,i.e. [Na+] + [K+] + 2[Mg2+] + 2[Ca2+] - Cl− - lactate− (butsometimes simplified to [Na+] + [K+] + 2[Mg2+] + 2[Ca2+]- Cl− for clinical use) and is then known as an apparentSID (SIDa) for omitting the role of weak acids. Or onthe contrary, the SID is calculated from approximationof HCO− 3, phosphates and albumin charges, called thenthe effective SID (SIDe) [10]. The difference betweenSIDa and SIDe comprises the unmeasured anions (sul-fate, keto-acids, citrate, pyruvate, acetate and gluconate).For computer modeling, we consider that SIDe = SID.The pH is then believed to be specified physicochemi-cally by a function:

Full blood

HTC1 HTC0

Other body compartments

Other body compartments

Behavioral Mechanistic

Cl-

HCO3-

Fig. 1 Diagram of described full blood model, connected to other body compartments. The HCT1 compartment contains erythrocytes (bloodwith hematocrit reaching 1) and is modeled behaviorally, whereas the plasma (HCT0) is described mechanistically. Other body compartments areconnected directly to plasma

Ježek and Kofránek Theoretical Biology and Medical Modelling (2018) 15:14 Page 2 of 10

pHPC ¼ fPC SID; pCO2;Alb; Pið Þ ð2Þwhere the Alb and Pi are the plasma concentrations ofthe albumin and phosphates, respectively. The main ideaof our approach is to divide the blood into two theoret-ical limit compartments. The first compartment is con-taining only erythrocytes (HCT1, i.e. with hematocrit 1)and the other, only plasma (HCT0). Some ions maymove between the two compartments.The HCT1 compartment is described by the behavioral

Siggaard-Andersen model [1], extrapolated to fullhematocrit (of fully oxygenated blood). This is satisfactory,as we are not interested in the inner state of the erythro-cyte. On the contrary, the HCT0 compartment is gov-erned by the mechanistic physicochemical model,following the Stewart’s approach. When the two compart-ments are mixed together into full-blood, their pH mustbe equal:

pHHCT1 ¼ pHHCT0 ¼ pHBLOOD ð3Þwhere pHHCT1 = pHSA and pHHCT0 = pHPC. Note that

pHHCT1 does not represent the pH inside an erythrocyte,as modeled by Raftos et al., Rees et al. or Wolf and De-Land [11–13]. In contrast, it is a limit of the pH outsideerythrocytes, as the hematocrit of this theoretical com-partment (the ratio of erythrocytes) approaches 1. Forcomputation of pHHCT1, we employ our formalization ofSiggaard-Andersen’s nomogram, using a set of threesixth-order and a fourth order polynomials, whereaspHHCT0 (effectively plasma only) is given by Fencl’s sim-ple plasma description [14]. However, the particularmodels used can be replaced by more complex descrip-tions, with some trade-off between level of detail andnumerical complexity. Please refer to the online supple-ment for the details regarding the calculation of pHHCT1by own formalization of Siggaard-Andersen’s nomogram(eqs. 1–11 in the Additional file 1) and on calculation ofpHHCT0 from the physicochemical model (eqs. 12–16 inthe Additional file 1).Interconnection of these two compartments is made

possible by the definition of BEPC (base excess calculatedfor physico-chemical domain). Similar to the idea ofSIDEx (or SID excess) [15] and analogically to the defin-ition of normal buffer base (NBB) [8], let the normalSID (NSID) be the SID under standard conditions, i.e.pH 7.4 and pCO2 5.32 kPa (40 mmHg) at actual levelsof total phosphates (Pi) and albumin (Alb) concentra-tions, so that we can assemble a function fNSID:

NSID ¼ fNSID Alb;Pið Þ; at pH ¼ 7:4; pCO2¼ 5:32 kPa ð4Þ

For computation of fNSID we employ the physicochem-ical model with Alb, Pi, pH and pCO2 as inputs and SID

as an output. For derivation of such function, pleaserefer to the eqs. 16–18 of the online supplement. Then,in addition to Wooten’s ΔBE =ΔSID [16] and again, ana-logically to definition of Siggaard-Andersen’s BESA = BB- NBB [17], we define the BEPC as:

BEPC ¼ SID‐NSID ¼ BEHCT0 ð5ÞBEHCT0 (Base excess of the zero-hematocrit compart-

ment) is formed by the physicochemical model ofplasma, so in this case it equals BEPC.Now, when we mix the virtual limit HCT0 and HCT1

compartments, the mixture must establish a new BE(that is the BE of the whole blood, further referred to asBE unless stated otherwise) by a well-known Cl− forHCO3

− passive exchange between the HCT0 and HCT1compartments (that is shift of HCO3

− ions in theSiggaard-Andersen approach, or Cl− in Stewart’s terms,for further considerations see the Discussion section).Based on this assumption, we have developed the follow-ing equations to quantify the total charge of transferredions ZTI (meq/l) ion transport between compartments:

BEHCT0 ¼ BE‐ZTI � 1‐Hctð Þ ð6ÞBEHCT1 ¼ BEþ ZTI �Hct ð7Þ

where the BEHCT1 is the BE of the HCT1 compartment(all erythrocytes) and hematocrit (Hct, unitless) repre-sents the size of the HCT1 compartment and scales theion transfer accordingly.The set of eqs. (1–7) leads to an iterative numerical

solution, but we take advantage of the Modelica lan-guage and let the Modelica tool (Dymola 2016, DassaultSystémes) find the solution of the BE – pH relationship.Due to the possibility of Modelica to interchange the in-put with output, we can form a function for BE:

BE ¼ fCombinedModelBE pH; pCO2;Alb; Pið Þ ð8Þwhere pH is the independent variable (previously knownor varied) or a function for pH:

pH ¼ fCombinedModelpH BE; pCO2;Alb;Pið Þ ð9Þwhere the BE is the independent property (known orvaried). The illustrative simplest complete Modelicasource code for this case is listed in Appendix.A Modelica tool (tested in Dymola 2016 by Dassault

Systémes and OpenModelica 1.11 by OpenModelicaConsortium) automatically solves the set of coupledequations, employs necessary numerical methods andfinds the steady-state solution.The model is then validated by a visual comparison

with the contemporary models in use.We executed a steady-state sensitivity analysis of the

buffer capacity for albumin and phosphate concentration


levels on the combined model in comparison with con-temporary models. The initial plasma concentrationswere varied from 50 to 200% of the nominal value(4.4 g/dl for albumin and 1.15 g/dl for phosphate). TheBE was held constant during the changes; therefore, ac-cording to eq. (5), the plasma SID was also varied.Dilution by saline solution is simulated by the multi-

plication of SID by a given dilution factor. We thencompute BE as

BE ¼ Nominal SID� DilutionFactor�NSID ð10Þ

where the NSID is, according to the above definition, anormal SID for the actual (i.e. here diluted) Alb and Pi,and the Nominal_SID is the SID preceding dilution. Thehemoglobin is also diluted by the same factor.For the lack of established metric to compare the com-

putational complexity and solvability of equation-basedmodels, the former is demonstrated by a sum ofnon-trivial equations and the latter by the initializationtime (through the initial value of CPU time variable, pro-vided by the Modelica tool). We show the values of ourCombined model compared to our Modelica implemen-tation of the Wolf model, as a representative of acomplete physicochemical approach. The models werecompared in Dymola 2016, on a reference computerwith Windows 10 64b and i7-3667 U processor. To cor-rectly count small time spans, each model was run 1000times in parallel and then the CPU time was divided bythe same factor.The model source code implemented in the Mode-

lica language, including our implementation of theWolf ’s model and source codes for the figures, isaccessible at [18].

ResultsThe main result of the present study is the combinationof the Siggaard-Andersen and Stewart’s physicochemicalmodels into a single model, so that we can perform cal-culations for dilution, albumin, phosphate and the buffercapacity of erythrocytes within a joint computationallyeffective combined model. The secondary result is thedefinition of NSID, an indicator showing the relation ofBE and SID, each a flagship of its own approach. Thanksto NSID, we can quantify shifts in BE due to the dilutionand/or changes in albumin levels or differences in SIDdue to varying pCO2 in full blood.We validate the combined model by comparison

with the Siggaard-Andersen’s nomogram [8], later Sig-gaard-Andersen’s updated albumin-sensitive Van Slykeformalization [19], Figge-Fencl’s physicochemical model (FF)[14], updated to version 3.0 [20]) and the recent full bloodmodel by Wolf [6] (reduced to the plasma-erythrocyte com-partments). A graphic comparison (Fig. 2a) shows thatthe Combined model fits the Siggaard-Andersen’smodel, as well as the more recent model of Wolf, whereasthe Figge-Fencl physicochemical model of plasma alonesubstantially differs from full blood models.The results of sensitivity analysis at various plasma albu-

min concentrations at constant BE are outlined in Fig. 2b(compared with the Figge-Fencl model and the lateralbumin-sensitive Van-Slyke equation). Due to the dilutionof plasma volume by erythrocytes, the sensitivity to albuminconcentration in full blood is lower than the sensitivity inplasma. The results of sensitivity analysis of phosphate con-centration were negligible at this scale and are not shown.We can also use the Combined model to show the com-

position of the plasma SID. The differences of SID compos-ition for pCO2 titration and for change in BE in the fullblood and in the plasma is shown in Fig. 3. Note the

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BE curveCombined modelPlasma by FFVan Slyke

BE curveCombined modelPlasma by FFSAWolf

Fig. 2 Results of Combined model in the pH-pCO2 diagram. a Comparison of our Combined model with the formalised SA nomogram reflecting fullblood, with the Figge-Fencl (FF) model of plasma and with the recent full blood model by Wolf [6] reduced to the erythrocyte-plasma compartments.b Sensitivity analysis for albumin of our combined model, the updated Van Slyke Eq. [18] and the Figge-Fencl (FF) model of plasma


https://paperpile.com/c/NJAIJc/d430

different profiles for plasma and for full blood, especiallyduring the change in pCO2 – the SID independence onpCO2 no longer holds in full blood (as confirmed in otherliterature, e.g. [21]). In Fig. 2 we use semi-logarithmic axisscale visualisation (pH vs log(pCO2)) to be consistent withthe SA nomogram and the SID composition in mEq/l forvariable pCO2 and BE (added or removed H

+ ions) in Fig. 3to compare with the Figge-Fencl model.The model can predict the effect of dilution with

saline solution. Figure 4a shows the comparison ofthe reaction of full blood versus sole plasma to dilu-tion in the closed system (i.e. the total CO2 is alsodiluted) and the open system (constant pCO2, as invivo) during dilution, represented as a percentage ofthe original hemoglobin content. Here, the pH of theclosed system is no longer insensitive to the dilutionand the the open system is additionally buffered bythe erythrocytes. Plotting the dependency of theHCO3

− for the open system together with comparisonto Figge-Fencl and recent Wolf [13] (reduced toplasma and erythrocyte compartments) models re-veals, that the Combined model shows a perfect fit tomeasured values, as reproduced in [22] (Fig. 4b).The resulting Combined model is numerically stable

within BE range of − 25 to 25 and, in the described com-bination, requires only 25 non-trivial (i.e. not direct valueassignment) equations. In contrast, our implementation of

the plasma-erythrocyte model by Wolf (version 3.51) has77 non-trivial equations and its initialization problem con-sists of set of 64 non-linear equations (in contrast to only13 of the Combined model). Our Combined model alsoneeds significantly lower time for computation of theinitialization problem (Fig. 5).

DiscussionUnification of modern and traditional approachesThe preferred bed-side approach to acid-base evaluationstill remains under debate, for further information on thisissue see [23–25]. A number of authors have strived tocompare the traditional approach to the physicochemical,to be used at the bedside (e.g. [26, 27]) or even make useof both at once [28]. However, although different math-ematical formalism, these two major approaches give simi-lar information [29] and some studies conclude that, inprinciple, neither of the two methods offer a notable ad-vantage [29–31].Some previous works already attempted to find a rela-

tionship, e.g. Schlichtig [15] addressed the question of howbase excess (BE) could have been increased, even thoughthe strong ion difference (SID) had remained unchangedamong hypoproteinaemia patients, by proposing SIDExbased on the SA approach to the Van Slyke eq. [9]. Wootenlater demonstrated mathematically, that “BE and thechange in SID are numerically the same for plasma,

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Fig. 3 Composition of the SID. For various BE (a), i.e. addition of strong acid or bases, and during pCO2 titration (b) for the Figge-Fencl model ofplasma (FF, black dashed) and our Combined model (red solid). The SID is no longer the independent variable in full blood

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WolfCombined modelPlasma by FFReported measurements

Combined model, closed systemPlasma by FF, closed system

Combined model, open systemPlasma by FF, open system

Fig. 4 Prediction of dilution. a dilution with saline in the closed system, i.e. the pCO2 is also diluted (dashed) and in the open system, where thepCO2 is regulated to a constant level as in vivo (full line). b dilution predicted for the closed system – comparison with other models and withmeasured data, as reported in [21]


provided that the concentrations of plasma noncarbonatebuffers remain constant” [16], Matousek continued in themathematical comparison [29, 32]. We extend the compari-son, which was made mostly for plasma, to full blood andvariable albumin and phosphate levels and present amethod of combining both traditional and physicochemicalapproaches. NSID also indicates the desired value of SIDduring albumin and phosphate disorders.

The combined modelThe resulting Combined model fits the full-bloodSiggaard-Andersen model (by definition) and also the con-temporary physicochemical model by Wolf [13] reduced tothe plasma-erythrocyte compartment. Our contribution liesin the extending the classical physicochemical Stewart ap-proach with the buffering effect of erythrocytes and their ef-fects on SID during variable pCO2, while employing alsothe BE metric for quantization of H+/HCO3

− flow balances.The resulting model is computationally effective and thesolution does not exhibit numerical problems within nor-mal input ranges.During the pCO2 titration (Fig. 3b), it can be seen that

the SID of plasma in whole blood is not independent ofpCO2 and therefore could not serve as an independentstate property. On the contrary, the BEOX (BE correctionfor virtual fully oxygenated blood, which could be calcu-lated e.g. as in [33]), together with total O2 and totalCO2 blood concentrations, are then a truly independentstate properties of the full-blood component (as imple-mented in accompanied model [18]).The concept of NSID, redefined for standard intersti-

tium conditions, is then together with the eq. (5) usablealso for calculations of BE in interstitial fluid to inter-connect the blood with the interstitial compartment.The proposed model does not depend on any new

parameter or set any limitation to the contemporaryapproaches; rather, it solely joins them using the add-itional assumption of passive ion exchange only.

Combined model assumptions and limitationsThe Combined model relies on fundamental assump-tions of each of the combined approaches. The

erythrocytes in the HCT1 compartment provide an add-itional buffer, mostly due to the binding of H+ tohemoglobin. This is associated with interchange of Cl−

for HCO− 3, also during the change in pCO2, which hasbeen described in numerous textbooks. The H+ /HCO3

− flows are not directly conserved, as it might bebuffered by a number of mechanisms, but persists in theform of BE metric. In Stewart’s terms, the H+/HCO3

−

flows are expressed in the form of SID change, i.e. hereas exchange for Cl−.To maintain the electroneutrality, we assume 1:1 transfer.

As a complement of the reduced Cl− in the plasma, theHCO3

− in Fig. 3b is rising and so is the total SID. Again,this could be viewed from two standpoints: inSiggaard-Andersen’s traditional approach it is the change ofBE, e.g. + 1 M of HCO3

− equals the change of BE by plusone. In Stewart’s terms, this equals to the change in SID by+ 1 (as shown by [5]). In the current case of HCO3

− to Cl−

exchange, it is decrease of Cl− by exactly 1 M.Our approach is generally limited by the measured data

for behavioral model - the plasma is mechanistically ex-tendable and replaceable by more complex models (seethe next section), but the reactions in erythrocytes aremeasured for standard conditions only - that is normalconcentration of DPG (5,0 mmol/l), fetal hemoglobin(0 mmol/l) and CO (0 mmol/l). To take these into ac-count, the behavioral description would need to be radic-ally extended by a number of dimensions, impairing thecomputational effectivity.The slight inaccuracy of the fit to Siggaard-Andersen’s

model for very low BE values is caused by the inconsistencybetween the SA and the Figge-Fencl (FF) models (Fig. 6).The SA model, extrapolated to zero hematocrit, i.e. plasma,does not exactly fit the FF model in limit BE ranges. Thiserror could possibly be caused by using the SA model out-side the measured boundaries, while extrapolatinghematocrit to zero or close to one (an unphysical state), al-though the linearity is one of the assumptions of SA. Yet,based on the literature data, we are unable to distinguishwhether the error is caused by the extrapolation error,measurement inaccuracy of SA, or incompleteness of theFF model. Otherwise, for standard conditions (i.e. normal

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Fig. 5 Computational complexity. CPU time needed for model initialization. The Combined model requires significantly lower CPU time forinitialization (a) due to better numerical robustness than the latest version of the Wolf model (v3.51) reduced to P-E compartments and also has lowernumber of non-trivial equations (b)


hematocrit, albumin, phosphates, and dilution) the Com-bined model performs according to Siggaard-Andersen.

The various-complexity implementationOur implementation in Modelica language enables switch-ing complexity of a particular compartment. That is, wecan use three different models for formulation of the physi-cochemical plasma compartment with various complexity:the simplest model of Fencl [34], the Figge-Fencl model[14], updated to version 3.0 [20] to quantify albumin in de-tail, or Wolf ’s plasma compartment from [35] in currentversion v3.51 to consider also effects of Mg2+ and Ca2+

binding on albumin). The trade-off is computational com-plexity. For description of the Siggaard-Andersen erythro-cytes compartment, we include the original formulation ofthe Van-Slyke eq. [9], our exact formalization using the setof four sixth-order polynomials and the simplest model ofZander [36]. Optionally, the computation can be enhancedwith dilution. This allows to choose a simple tomedium-complex description, based on current modelingneeds. The full-blood component is supposed to be used inmultiple arterial and venous parts of a complex model, thusthe low complexity and robustness are vital.

Computation complexityBecause the acid-base computations are highlynon-linear, iterative algorithms which converge to the

solution are often used (as employed by e. g.Siggaard-Andersen’s OSA [37], Figge-Fencl calculations[20] or even by the newest Wolf model [13] in the form ofa running constraint-unknown optimization problem).From the authors observation, the algorithm may divergeand fail to find any solution or there might be more pos-sible solutions to the formulation of the problem,therefore these models are often sensitive to the ini-tial guess of unknown variables and could have prob-lems for states far from the physiological norm(usually being the initial guess). This makes acid-basemodeling particularly challenging, as small changes toguesses of state variables may lead to potentiallyinvalid solution.The numerical solvability of the equation-based

models depends especially on the non-linearity of theproblem, particularly on the Jacobian matrix conditionnumber of the equation system. This is, however, veryspecific to the chosen tool and usage of the certainmodel and, regrettably, no such formal metric is cur-rently established (F. Casella, personal communication,November 2017). For model comparison, we use a num-ber of non-trivial equations, which might be a good indi-cator for overall model complexity. To assess thesolvability, we propose to use the time needed to con-sistently initialize the equation set, that is to find exactlyone solution. For harder tasks, the solver employs sev-eral iterative methods one after another to overcome theconvergence problems caused by non-linearities,which will affect the initialization time. Sometimes,the solution may not be found. From our experience,these two metrics does not necessarily correspond toeach other.Wolf model [35] is the most complete mechanistic

description of whole body acidbase. The model isavailable as a VisSim simulator, which is howeverunsuitable for integration into other models. Conver-gence to a solution of its original implementation takesapproximately 5 - 20s on the reference computer. Thecomputation time of our reimplementation in modernequation-based Modelica language is negligible (almost in-stant - see Fig. 5), however it uncovers numerical instabil-ities and thus this model, as a whole, is unsuitable to be apart of larger integrative model. Our Modelica implemen-tation of Wolf ’s E-P (erythrocyte - plasma) has 77non-trivial (other than direct assignment) equations and isnumerically hard to solve - some input settings (e.g. lowpCO2 and high BE) lead to invalid solutions and someModelica tools are even unable to compute theerythrocyte-plasma model correctly at all (yet does nothave any problems with the proposed Combined model).Although we admire the level of detail of the Wolf ’smechanistic model, for our needs it is unnecessarily com-plex, even when reduced to E-P compartments.

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mH

g]

Fig. 6 Magnification of Fig. 2a at BE = − 15: Disparity between thebase models. The difference between Figge-Fencl (FF, black) andSiggaard-Andersen’s (SA, black dashed) models of plasma (blood withHCT0) is causing the Combined model (red bold) not being exactlyaligned with the SA model (green bold dashed) even for otherwisestandard conditions (hematocrit 0.45, normal levels of Alb and Pi)


AlbuminThe Combined model has been originally designed withfocus on precise albumin computation (when theFigge-Fencl plasma model is employed) in combinationwith hemoglobine buffering. Although the albumin con-centration is considered important, Figs. 2b and 3 suggestthat the plasma protein buffer capacity is significantlylower than the buffer capacity of hemoglobin.However, the reaction to the albumin depletion may be

different in vivo than what is presented during constant BEin Fig. 2b. In acute hypoalbuminemia, we can consider twocases of albumin depletion or addition, based on the elec-troneutrality assumption. The first is that an albumin isadded or removed together with a cation (Na+, K+, etc.) ina neutral pH solution. This would change the SID (bythe albumin charge), but keep the BE (and thus, thepH) constant (Fig. 7a). The second case is that albu-min is bound with H+ only, in an acidic solution. Asvirtually each H+ recombines to form HCO3

−, the lossof the albumin charge is compensated by the HCO3

−

charge. Therefore, the SID is constant, but the BE(and thus, the pH) changes (Fig. 7b). The predictionof the BE change is made possible by the eq. (5).Clinical observations favor the second case, where the

SID is reported normal during hypoproteinemic alkalosis[38]. Some later studies [27, 39] challenge the existence ofhypoproteinemic alkalosis in their study dataset, but wetheorize that the effect (elevated BE) of albumin depletionin these studies has already been compensated. The exactexplanation of this phenomenon is yet to be addressed bythe full-body model, which would extend the currently pre-sented model.

ConclusionWhen the inner details of any component are not the ob-jective in complex integrative modeling, we can

significantly reduce the complexity by substituting it withthe behavioral description, yet still retain mechanisticproperties of other components and its interactions.We present a method to quantify the interconnection of

two generally used and well-known approaches toacid-base balance, using no additional parameters or as-sumptions other than passive ion exchange.The resulting Combined model of full-blood acid-base

balance unites the advantages of each approach: it cansimulate variations in albumin level, buffer the effect oferythrocytes and predict a reaction to hemodilution andhemoconcentration, yet remains computationally simple.On the other hand, the proposed approach is insensitiveto non-normal DPG, HbF and CO concentrations.The combination gives an additional insight to the

acid-base balance by establishing the relationship betweenthe SID and the BE (using the defined NSID in the eq. (5)).The model is designed to have a variable computational

complexity and to be effectively extended by other bodilycompartments (interstitial fluid, intracellular fluid, metab-olism) and regulations (respiratory and renal) to assess thewhole-body dynamic acid-base status.

AppendixListing of the simplest flat implementation of Combinedmodel in Modelica. To keep the implementation lucid,the dilution is not incorporated here for the sake ofsimplicity of this example. In this example, the Fencl’s[14] approximation is used for plasma, whereas forerythrocytes here we use a simple formalization ofSiggaard-Andersen nomogram by Zander and Lang [36].Use any Modelica tool (e.g., OpenModelica) to run thismodel. For the complete set of used models, includingsources for presented figures, implementation of theWolf model and the complete full-blood component,please refer to [18].

A B

2 3 4 5 6 70

10

20

30

40

7.2

7.6

Cha

rge

[mE

q/l]

pH

Albumin concentration [g/dl]

HCO3-

pH

P-Alb-

2 3 4 5 6 7Albumin concentration [g/dl]

0

10

20

30

40

7.2

7.6

Cha

rge

[mE

q/l]

pH

BE

BE

HCO3-

pH

P-Alb-

- 8

0

8

- 8

0

8

Total SIDTotal SID

Fig. 7 The effect of changing the albumin level in plasma. a Adding (or removing) the albumin together with its anti-charged cation equivalent (e.g.,Na+) varies the the SID accordingly (bold red) and the BE and pH remain constant, which contrasts with clinical observations. b Adding (or removing)the albumin in electroneutral solution with corresponding amount of H+ maintains the SID unchanged (bold black), but the pH rises to alkalosis forhypoalbuminemia and falls to acidosis during hyperalbuminemia, which corresponds with Stewart’s theory of acute hypoalbuminemic alkalosis


Additional file

Additional file 1: Formalization of Siggaard-Andersen nomogram and der-ivation of the base excess for the physicochemical domain. (PDF 806 kb)

AcknowledgementsWe would like to thank to Karel Roubík and Arnošt Mládek for a carefulreview, to S.E. Rees for a great insight and invaluable comments and toFrancesco Casella for discussion on numerical solvability of the equation-based models.

FundingThis study has been supported by the TRIO MPO FV20628 grant.

Availability of data and materialsThe models analysed within the current study are available in the referencedGithub repository. No additional datasets have been employed.

� Project name: Full blood acid-base� Project home page: https://github.com/filip-jezek/full-blood-

acidbase/� Archived version: DOI https://doi.org/10.5281/zenodo.1134853

� Operating system(s): Platform independent� Programming language: Modelica 3.2.1� Other requirements: For some parts, Physiolibrary is required

(available at physiolibrary.org or bundled with OpenModelica)� License: GNU GPLv3� Any restrictions to use by non-academics: additional licence needed

Authors’ contributionsJK initiated the study, FJ implemented the models and performed thecomparative analysis. FJ and JK wrote the manuscript. All authors read andapproved the final manuscript.

Ethics approval and consent to participateNot applicable.

Consent for publicationNot applicable.

Competing interestsThe authors declare that they have no competing interests.


https://doi.org/10.1186/s12976-018-0086-9https://github.com/filip-jezek/full-blood-acidbase/https://github.com/filip-jezek/full-blood-acidbase/https://doi.org/10.5281/zenodo.1134853http://physiolibrary.org

Publisher’s NoteSpringer Nature remains neutral with regard to jurisdictional claims inpublished maps and institutional affiliations.

Received: 3 January 2018 Accepted: 24 July 2018

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https://doi.org/10.5281/zenodo.1134853https://doi.org/10.5281/zenodo.1134853http://www.siggaard-andersen.dk/OsaTextbook.htmhttp://www.siggaard-andersen.dk/OsaTextbook.htmhttp://www.figge-fencl.org/model.htmlhttps://is.cuni.cz/webapps/zzp/download/140030054/?lang=cshttps://is.cuni.cz/webapps/zzp/download/140030054/?lang=cs

AbstractBackgroundResultsConclusions

BackgroundMethodsResultsDiscussionUnification of modern and traditional approachesThe combined modelCombined model assumptions and limitationsThe various-complexity implementationComputation complexityAlbumin

ConclusionAppendixAdditional fileAcknowledgementsFundingAvailability of data and materialsAuthors’ contributionsEthics approval and consent to participateConsent for publicationCompeting interestsPublisher’s NoteReferences

modern and traditional approaches combined into an ...€¦ · erythrocytes), but it is originally...

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