 Research
 Open Access
 Published:
Modern and traditional approaches combined into an effective graybox mathematical model of fullblood acidbase
Theoretical Biology and Medical Modelling volume 15, Article number: 14 (2018)
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. Acidbase disorders are common especially in intensive care, and the acidbase status is one of the vital clinical signs for the patient management. Because acidbase balance is connected to many bodily processes and regulations, complex mathematical models are needed to get insight into the mixed disorders and to act accordingly. The goal of this study is to develop a fullblood acidbase model, designed to be further integrated into more complex human physiology models.
Results
We have developed computationally simple and robust fullblood model, yet thorough enough to cover most of the common pathologies. Thanks to its simplicity and usage of Modelica language, it is suitable to be embedded within more elaborate systems. We achieved the simplification by a combination of behavioral SiggaardAndersen’s traditional approach for erythrocyte modeling and the mechanistic Stewart’s physicochemical approach 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 insensitive to DPG and CO concentrations.
Conclusions
This study presents a straightforward unification of SiggaardAndersen’s and Stewart’s acidbase models. The resulting fullblood acidbase model is designed to be a core part of a complex dynamic wholebody acidbase and gas transfer model.
Background
Acidbase disturbances are associated with a number of fluid, electrolyte, metabolic and respiratory disorders. Understanding the pathogenesis and the underlying pathophysiological processes is crucial for proper diagnostics and treatment, especially in acute medicine, anaesthesiology and during artificial respiration. Complex mathematical models help to uncover the pathogenesis of complex bodily disorders.
Two major approaches for mathematical modeling of the acidbase status of blood are widely employed. The traditional model by SiggaardAndersen [1] (SA) is a behavioral model of full blood acidbase (i.e. including erythrocytes), but it is originally defined for standard albumin and phosphates only (though added later [2]) and, on its own, the model is unable to assess hemodilution and hemoconcentration (e.g. during fluid replacement), and the individual levels of ions are not considered. The second model, Stewart’s physicochemical [3] model, or the socalled modern approach, is a structural model of plasma only, but it is essential for assessing hemodilution, ion and protein imbalances, which are common in critically ill patients. However neither of these is satisfactory as a 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 and enhanced it by circulation, blood gases, interstitium and cellular compartments [4]. Wooten later presented an extension to the extracellular compartment [5] and most recently, Wolf [6] proposed a profound steadystate physicochemical model of erythrocyteplasmainterstitiumcell compartments, including detailed ion and water balance. It has been built with the purpose to mechanistically describe complex physicochemical processes, which is, however, computationally demanding and hard to solve due to a large system of nonlinear equations. This prevents us from integrating the detailed models into more complex models, or from usage for patientspecific identification.
Described approaches [1,2,3,4,5,6] are, however, designed for steadystate situations. To extend the wholebody acidbase assessment by bodily regulatory loops and to show pathogenesis of developing disorders in time, one needs to construct even larger integrative model, including important bodily compartments (interstitial fluid, cells) and regulations (kidney, liver, respiratory regulation) interconnected by a circulation of fullblood, the core component of our approach [7]. From the authors’ experience, these subcomponents however need to be based on computationally robust submodels, which are yet precise enough to describe the physiology.
The aim of this study is to develop a detailed, yet computationally effective, full blood model by combining the two predominant approaches to blood acidbase balance. The resulting combined model will become an essential part of a future complex dynamic wholebody acidbase and blood gas transfer model following the border flux theory [7] to assess dynamic bodily compensations.
Methods
If the detailed processes in erythrocyte are not the objective, an empirical description of their contribution is often satisfactory, which is substantially lowering the model complexity. Therefore, instead of modeling erythrocyte mechanistically, we propose to substitute it with behavioral description, whereas the plasma should follow the physicochemical mechanistic design to include ion transfer to adjacent body compartments (Fig. 1). This approach would simplify complex physicochemical computations of membrane balance.
The SiggaardAndersen’s (SA) approach (also known as the traditional approach), built around SA nomogram [8], uses measured values of pH and arterial pCO_{2} to compute buffer base (BB), a concentration of buffer anions and cations, which can take a buffering action. A difference between BB and normal buffer base (NBB) is called base excess (BE), which expresses how many milimoles of strong acid must be added to 1 l of blood to regain normal pH (at normal pCO_{2} = 40 mmHg). More recently, the term BE has been substituted by the Concentration of Titratable Hydrogen Ion (ctH^{+}), which however equals to the negative of BE [9]. The BE proved to be a handy indicator for the clinicians to quickly assess the level of metabolic acidbase disturbance. In the following years, the importance of the printed acidbase nomogram declined in favor of its numerical formalization, e.g. the Van Slyke eq. [9], so that we can form a function for SiggaardAndersen’s pH as:
The physicochemical, (also referred to as a modern or Stewart’s approach), does not rely on measured nomogram. Instead, a socalled strong ion difference (SID) is calculated from the difference of strong (i.e. mostly or fully dissociated at physiologic ranges) cations and anions concentrations (which equals charge for fullydissociated substances) in plasma. Because of the electroneutrality principle, the SID must equal the sum of charges of HCO^{−} _{3}, phosphates and albumin. SID can be calculated from the difference of ion concentrations, i.e. [Na^{+}] + [K^{+}] + 2[Mg^{2+}] + 2[Ca^{2+}]  Cl^{−}  lactate^{−} (but sometimes simplified to [Na^{+}] + [K^{+}] + 2[Mg^{2+}] + 2[Ca^{2+}]  Cl^{−} for clinical use) and is then known as an apparent SID (SIDa) for omitting the role of weak acids. Or on the contrary, the SID is calculated from approximation of HCO^{−} _{3}, phosphates and albumin charges, called then the effective SID (SIDe) [10]. The difference between SIDa and SIDe comprises the unmeasured anions (sulfate, ketoacids, citrate, pyruvate, acetate and gluconate). For computer modeling, we consider that SIDe = SID. The pH is then believed to be specified physicochemically by a function:
where the Alb and Pi are the plasma concentrations of the albumin and phosphates, respectively. The main idea of our approach is to divide the blood into two theoretical limit compartments. The first compartment is containing only erythrocytes (HCT1, i.e. with hematocrit 1) and the other, only plasma (HCT0). Some ions may move between the two compartments.
The HCT1 compartment is described by the behavioral SiggaardAndersen model [1], extrapolated to full hematocrit (of fully oxygenated blood). This is satisfactory, as we are not interested in the inner state of the erythrocyte. On the contrary, the HCT0 compartment is governed by the mechanistic physicochemical model, following the Stewart’s approach. When the two compartments are mixed together into fullblood, their pH must be equal:
where pH_{HCT1} = pH_{SA} and pH_{HCT0} = pH_{PC}. Note that pH_{HCT1} does not represent the pH inside an erythrocyte, as modeled by Raftos et al., Rees et al. or Wolf and DeLand [11,12,13]. In contrast, it is a limit of the pH outside erythrocytes, as the hematocrit of this theoretical compartment (the ratio of erythrocytes) approaches 1. For computation of pH_{HCT1}, we employ our formalization of SiggaardAndersen’s nomogram, using a set of three sixthorder and a fourth order polynomials, whereas pH_{HCT0} (effectively plasma only) is given by Fencl’s simple plasma description [14]. However, the particular models used can be replaced by more complex descriptions, with some tradeoff between level of detail and numerical complexity. Please refer to the online supplement for the details regarding the calculation of pH_{HCT1} by own formalization of SiggaardAndersen’s nomogram (eqs. 1–11 in the Additional file 1) and on calculation of pH_{HCT0} from the physicochemical model (eqs. 12–16 in the Additional file 1).
Interconnection of these two compartments is made possible by the definition of BE_{PC} (base excess calculated for physicochemical domain). Similar to the idea of SIDEx (or SID excess) [15] and analogically to the definition of normal buffer base (NBB) [8], let the normal SID (NSID) be the SID under standard conditions, i.e. pH 7.4 and pCO_{2} 5.32 kPa (40 mmHg) at actual levels of total phosphates (Pi) and albumin (Alb) concentrations, so that we can assemble a function f_{NSID}:
For computation of f_{NSID} we employ the physicochemical model with Alb, Pi, pH and pCO_{2} as inputs and SID as an output. For derivation of such function, please refer to the eqs. 16–18 of the online supplement. Then, in addition to Wooten’s ΔBE = ΔSID [16] and again, analogically to definition of SiggaardAndersen’s BE_{SA} = BB  NBB [17], we define the BE_{PC} as:
BE_{HCT0} (Base excess of the zerohematocrit compartment) is formed by the physicochemical model of plasma, so in this case it equals BE_{PC}.
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 as BE unless stated otherwise) by a wellknown Cl^{−} for HCO_{3}^{−} passive exchange between the HCT0 and HCT1 compartments (that is shift of HCO_{3}^{−} ions in the SiggaardAndersen approach, or Cl^{−} in Stewart’s terms, for further considerations see the Discussion section). Based on this assumption, we have developed the following equations to quantify the total charge of transferred ions Z_{TI} (meq/l) ion transport between compartments:
where the BE_{HCT1} is the BE of the HCT1 compartment (all erythrocytes) and hematocrit (Hct, unitless) represents the size of the HCT1 compartment and scales the ion transfer accordingly.
The set of eqs. (1–7) leads to an iterative numerical solution, but we take advantage of the Modelica language and let the Modelica tool (Dymola 2016, Dassault Systémes) find the solution of the BE – pH relationship. Due to the possibility of Modelica to interchange the input with output, we can form a function for BE:
where pH is the independent variable (previously known or varied) or a function for pH:
where the BE is the independent property (known or varied). The illustrative simplest complete Modelica source code for this case is listed in Appendix.
A Modelica tool (tested in Dymola 2016 by Dassault Systémes and OpenModelica 1.11 by OpenModelica Consortium) automatically solves the set of coupled equations, employs necessary numerical methods and finds the steadystate solution.
The model is then validated by a visual comparison with the contemporary models in use.
We executed a steadystate sensitivity analysis of the buffer capacity for albumin and phosphate concentration levels on the combined model in comparison with contemporary models. The initial plasma concentrations were varied from 50 to 200% of the nominal value (4.4 g/dl for albumin and 1.15 g/dl for phosphate). The BE was held constant during the changes; therefore, according to eq. (5), the plasma SID was also varied.
Dilution by saline solution is simulated by the multiplication of SID by a given dilution factor. We then compute BE as
where the NSID is, according to the above definition, a normal SID for the actual (i.e. here diluted) Alb and Pi, and the Nominal_SID is the SID preceding dilution. The hemoglobin is also diluted by the same factor.
For the lack of established metric to compare the computational complexity and solvability of equationbased models, the former is demonstrated by a sum of nontrivial equations and the latter by the initialization time (through the initial value of CPU time variable, provided by the Modelica tool). We show the values of our Combined model compared to our Modelica implementation of the Wolf model, as a representative of a complete physicochemical approach. The models were compared in Dymola 2016, on a reference computer with Windows 10 64b and i73667 U processor. To correctly count small time spans, each model was run 1000 times in parallel and then the CPU time was divided by the same factor.
The model source code implemented in the Modelica language, including our implementation of the Wolf’s model and source codes for the figures, is accessible at [18].
Results
The main result of the present study is the combination of the SiggaardAndersen and Stewart’s physicochemical models into a single model, so that we can perform calculations for dilution, albumin, phosphate and the buffer capacity of erythrocytes within a joint computationally effective combined model. The secondary result is the definition of NSID, an indicator showing the relation of BE and SID, each a flagship of its own approach. Thanks to NSID, we can quantify shifts in BE due to the dilution and/or changes in albumin levels or differences in SID due to varying pCO_{2} in full blood.
We validate the combined model by comparison with the SiggaardAndersen’s nomogram [8], later SiggaardAndersen’s updated albuminsensitive Van Slyke formalization [19], FiggeFencl’s physicochemical model (FF) [14], updated to version 3.0 [20]) and the recent full blood model by Wolf [6] (reduced to the plasmaerythrocyte compartments). A graphic comparison (Fig. 2a) shows that the Combined model fits the SiggaardAndersen’s model, as well as the more recent model of Wolf, whereas the FiggeFencl physicochemical model of plasma alone substantially differs from full blood models.
The results of sensitivity analysis at various plasma albumin concentrations at constant BE are outlined in Fig. 2b (compared with the FiggeFencl model and the later albuminsensitive VanSlyke equation). Due to the dilution of plasma volume by erythrocytes, the sensitivity to albumin concentration in full blood is lower than the sensitivity in plasma. The results of sensitivity analysis of phosphate concentration were negligible at this scale and are not shown.
We can also use the Combined model to show the composition of the plasma SID. The differences of SID composition for pCO_{2} titration and for change in BE in the full blood and in the plasma is shown in Fig. 3. Note the different profiles for plasma and for full blood, especially during the change in pCO_{2} – the SID independence on pCO_{2} no longer holds in full blood (as confirmed in other literature, e.g. [21]). In Fig. 2 we use semilogarithmic axis scale visualisation (pH vs log(pCO_{2})) to be consistent with the SA nomogram and the SID composition in mEq/l for variable pCO_{2} and BE (added or removed H^{+} ions) in Fig. 3 to compare with the FiggeFencl model.
The model can predict the effect of dilution with saline solution. Figure 4a shows the comparison of the reaction of full blood versus sole plasma to dilution in the closed system (i.e. the total CO_{2} is also diluted) and the open system (constant pCO_{2}, as in vivo) during dilution, represented as a percentage of the original hemoglobin content. Here, the pH of the closed system is no longer insensitive to the dilution and the the open system is additionally buffered by the erythrocytes. Plotting the dependency of the HCO_{3}^{−} for the open system together with comparison to FiggeFencl and recent Wolf [13] (reduced to plasma and erythrocyte compartments) models reveals, that the Combined model shows a perfect fit to measured 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 combination, requires only 25 nontrivial (i.e. not direct value assignment) equations. In contrast, our implementation of the plasmaerythrocyte model by Wolf (version 3.51) has 77 nontrivial equations and its initialization problem consists of set of 64 nonlinear equations (in contrast to only 13 of the Combined model). Our Combined model also needs significantly lower time for computation of the initialization problem (Fig. 5).
Discussion
Unification of modern and traditional approaches
The preferred bedside approach to acidbase evaluation still remains under debate, for further information on this issue see [23,24,25]. A number of authors have strived to compare the traditional approach to the physicochemical, to be used at the bedside (e.g. [26, 27]) or even make use of both at once [28]. However, although different mathematical formalism, these two major approaches give similar information [29] and some studies conclude that, in principle, neither of the two methods offer a notable advantage [29,30,31].
Some previous works already attempted to find a relationship, e.g. Schlichtig [15] addressed the question of how base excess (BE) could have been increased, even though the strong ion difference (SID) had remained unchanged among hypoproteinaemia patients, by proposing SIDEx based on the SA approach to the Van Slyke eq. [9]. Wooten later demonstrated mathematically, that “BE and the change in SID are numerically the same for plasma, provided that the concentrations of plasma noncarbonate buffers remain constant” [16], Matousek continued in the mathematical comparison [29, 32]. We extend the comparison, which was made mostly for plasma, to full blood and variable albumin and phosphate levels and present a method of combining both traditional and physicochemical approaches. NSID also indicates the desired value of SID during albumin and phosphate disorders.
The combined model
The resulting Combined model fits the fullblood SiggaardAndersen model (by definition) and also the contemporary physicochemical model by Wolf [13] reduced to the plasmaerythrocyte compartment. Our contribution lies in the extending the classical physicochemical Stewart approach with the buffering effect of erythrocytes and their effects on SID during variable pCO_{2}, while employing also the BE metric for quantization of H^{+}/HCO_{3}^{−} flow balances. The resulting model is computationally effective and the solution does not exhibit numerical problems within normal input ranges.
During the pCO_{2} titration (Fig. 3b), it can be seen that the SID of plasma in whole blood is not independent of pCO_{2} and therefore could not serve as an independent state property. On the contrary, the BE_{OX} (BE correction for virtual fully oxygenated blood, which could be calculated e.g. as in [33]), together with total O_{2} and total CO_{2} blood concentrations, are then a truly independent state properties of the fullblood component (as implemented in accompanied model [18]).
The concept of NSID, redefined for standard interstitium conditions, is then together with the eq. (5) usable also for calculations of BE in interstitial fluid to interconnect the blood with the interstitial compartment.
The proposed model does not depend on any new parameter or set any limitation to the contemporary approaches; rather, it solely joins them using the additional assumption of passive ion exchange only.
Combined model assumptions and limitations
The Combined model relies on fundamental assumptions of each of the combined approaches. The erythrocytes in the HCT1 compartment provide an additional buffer, mostly due to the binding of H^{+} to hemoglobin. This is associated with interchange of Cl^{−} for HCO^{−} _{3}, also during the change in pCO_{2}, which has been described in numerous textbooks. The H^{+} / HCO_{3}^{−} flows are not directly conserved, as it might be buffered by a number of mechanisms, but persists in the form of BE metric. In Stewart’s terms, the H^{+}/HCO_{3}^{−} flows are expressed in the form of SID change, i.e. here as exchange for Cl^{−}.
To maintain the electroneutrality, we assume 1:1 transfer. As a complement of the reduced Cl^{−} in the plasma, the HCO_{3}^{−} in Fig. 3b is rising and so is the total SID. Again, this could be viewed from two standpoints: in SiggaardAndersen’s traditional approach it is the change of BE, e.g. + 1 M of HCO_{3}^{−} equals the change of BE by plus one. In Stewart’s terms, this equals to the change in SID by + 1 (as shown by [5]). In the current case of HCO_{3}^{−} 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 extendable and replaceable by more complex models (see the next section), but the reactions in erythrocytes are measured for standard conditions only  that is normal concentration of DPG (5,0 mmol/l), fetal hemoglobin (0 mmol/l) and CO (0 mmol/l). To take these into account, the behavioral description would need to be radically extended by a number of dimensions, impairing the computational effectivity.
The slight inaccuracy of the fit to SiggaardAndersen’s model for very low BE values is caused by the inconsistency between the SA and the FiggeFencl (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. This error could possibly be caused by using the SA model outside the measured boundaries, while extrapolating hematocrit to zero or close to one (an unphysical state), although the linearity is one of the assumptions of SA. Yet, based on the literature data, we are unable to distinguish whether the error is caused by the extrapolation error, measurement inaccuracy of SA, or incompleteness of the FF model. Otherwise, for standard conditions (i.e. normal hematocrit, albumin, phosphates, and dilution) the Combined model performs according to SiggaardAndersen.
The variouscomplexity implementation
Our implementation in Modelica language enables switching complexity of a particular compartment. That is, we can use three different models for formulation of the physicochemical plasma compartment with various complexity: the simplest model of Fencl [34], the FiggeFencl model [14], updated to version 3.0 [20] to quantify albumin in detail, or Wolf’s plasma compartment from [35] in current version v3.51 to consider also effects of Mg^{2+} and Ca^{2+} binding on albumin). The tradeoff is computational complexity. For description of the SiggaardAndersen erythrocytes compartment, we include the original formulation of the VanSlyke eq. [9], our exact formalization using the set of four sixthorder polynomials and the simplest model of Zander [36]. Optionally, the computation can be enhanced with dilution. This allows to choose a simple to mediumcomplex description, based on current modeling needs. The fullblood component is supposed to be used in multiple arterial and venous parts of a complex model, thus the low complexity and robustness are vital.
Computation complexity
Because the acidbase computations are highly nonlinear, iterative algorithms which converge to the solution are often used (as employed by e. g. SiggaardAndersen’s OSA [37], FiggeFencl calculations [20] or even by the newest Wolf model [13] in the form of a running constraintunknown optimization problem). From the authors observation, the algorithm may diverge and fail to find any solution or there might be more possible solutions to the formulation of the problem, therefore these models are often sensitive to the initial guess of unknown variables and could have problems for states far from the physiological norm (usually being the initial guess). This makes acidbase modeling particularly challenging, as small changes to guesses of state variables may lead to potentially invalid solution.
The numerical solvability of the equationbased models depends especially on the nonlinearity of the problem, particularly on the Jacobian matrix condition number of the equation system. This is, however, very specific to the chosen tool and usage of the certain model and, regrettably, no such formal metric is currently established (F. Casella, personal communication, November 2017). For model comparison, we use a number of nontrivial equations, which might be a good indicator for overall model complexity. To assess the solvability, we propose to use the time needed to consistently initialize the equation set, that is to find exactly one solution. For harder tasks, the solver employs several iterative methods one after another to overcome the convergence problems caused by nonlinearities, which will affect the initialization time. Sometimes, the solution may not be found. From our experience, these two metrics does not necessarily correspond to each other.
Wolf model [35] is the most complete mechanistic description of whole body acidbase. The model is available as a VisSim simulator, which is however unsuitable for integration into other models. Convergence to a solution of its original implementation takes approximately 5  20s on the reference computer. The computation time of our reimplementation in modern equationbased Modelica language is negligible (almost instant  see Fig. 5), however it uncovers numerical instabilities and thus this model, as a whole, is unsuitable to be a part of larger integrative model. Our Modelica implementation of Wolf’s EP (erythrocyte  plasma) has 77 nontrivial (other than direct assignment) equations and is numerically hard to solve  some input settings (e.g. low pCO2 and high BE) lead to invalid solutions and some Modelica tools are even unable to compute the erythrocyteplasma model correctly at all (yet does not have any problems with the proposed Combined model). Although we admire the level of detail of the Wolf’s mechanistic model, for our needs it is unnecessarily complex, even when reduced to EP compartments.
Albumin
The Combined model has been originally designed with focus on precise albumin computation (when the FiggeFencl plasma model is employed) in combination with hemoglobine buffering. Although the albumin concentration is considered important, Figs. 2b and 3 suggest that the plasma protein buffer capacity is significantly lower than the buffer capacity of hemoglobin.
However, the reaction to the albumin depletion may be different in vivo than what is presented during constant BE in Fig. 2b. In acute hypoalbuminemia, we can consider two cases of albumin depletion or addition, based on the electroneutrality assumption. The first is that an albumin is added or removed together with a cation (Na^{+}, K^{+}, etc.) in a neutral pH solution. This would change the SID (by the albumin charge), but keep the BE (and thus, the pH) constant (Fig. 7a). The second case is that albumin is bound with H+ only, in an acidic solution. As virtually each H^{+} recombines to form HCO_{3}^{−}, the loss of the albumin charge is compensated by the HCO_{3}^{−} charge. Therefore, the SID is constant, but the BE (and thus, the pH) changes (Fig. 7b). The prediction of 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 of hypoproteinemic alkalosis in their study dataset, but we theorize that the effect (elevated BE) of albumin depletion in these studies has already been compensated. The exact explanation of this phenomenon is yet to be addressed by the fullbody model, which would extend the currently presented model.
Conclusion
When the inner details of any component are not the objective in complex integrative modeling, we can significantly reduce the complexity by substituting it with the behavioral description, yet still retain mechanistic properties of other components and its interactions.
We present a method to quantify the interconnection of two generally used and wellknown approaches to acidbase balance, using no additional parameters or assumptions other than passive ion exchange.
The resulting Combined model of fullblood acidbase balance unites the advantages of each approach: it can simulate variations in albumin level, buffer the effect of erythrocytes and predict a reaction to hemodilution and hemoconcentration, yet remains computationally simple. On the other hand, the proposed approach is insensitive to nonnormal DPG, HbF and CO concentrations.
The combination gives an additional insight to the acidbase balance by establishing the relationship between the 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 bodily compartments (interstitial fluid, intracellular fluid, metabolism) and regulations (respiratory and renal) to assess the wholebody dynamic acidbase status.
References
SiggaardAndersen O, Others. The acidbase status of the blood. Munksgaard.; 1974.
SiggaardAndersen O, FoghAndersen N. Base excess or buffer base (strong ion difference) as measure of a nonrespiratory acidbase disturbance. Acta Anaesthesiol Scand Suppl. 1995;107:123–8.
Stewart PA. How to understand acidbase: A quantitative acid base primer for biology and medicine: Elsevier; 1981.
Andreassen S, Rees SE. Mathematical models of oxygen and carbon dioxide storage and transport: interstitial fluid and tissue stores and wholebody transport. Crit Rev Biomed Eng. 2005;33:265–98.
Wooten EW. The standard strong ion difference, standard total titratable base, and their relationship to the Boston compensation rules and the Van Slyke equation for extracellular fluid. J Clin Monit Comput. 2010;24:177–88.
Wolf MB. Whole body acidbase and fluidelectrolyte balance: a mathematical model. Am J Physiol Renal Physiol. 2013;305:F1118–31.
Kofranek J, Matousek S, Andrlik M. Border flux balance approach towards modelling acidbase chemistry and blood gases transport. In: In: Proceedings of the 6th EUROSIM congress on modelling and simulation. Ljubljana: University of Ljubljana; 2007. p. 1–9.
Andersen OS. The pHlog pCO2 blood acidbase nomogram revised. Scand J Clin Lab Invest. 1962;14:598–604.
SiggaardAndersen O. The van Slyke equation. Scand J Clin Lab Invest Suppl. 1977;146:15–20.
Figge J, Rossing TH, Fencl V. The role of serum proteins in acidbase equilibria. J Lab Clin Med. 1991;117:453–67.
Raftos JE, Bulliman BT, Kuchel PW. Evaluation of an electrochemical model of erythrocyte pH buffering using 31P nuclear magnetic resonance data. J Gen Physiol. 1990;95:1183–204.
Rees SE, Klæstrup E, Handy J, Andreassen S, Kristensen SR. Mathematical modelling of the acid–base chemistry and oxygenation of blood: a mass balance, mass action approach including plasma and red blood cells. Eur J Appl Physiol. 2010;108:483–94.
Wolf MB, Deland EC. A comprehensive, computermodelbased approach for diagnosis and treatment of complex acidbase disorders in criticallyill patients. J Clin Monit Comput. 2011;25:353–64.
Fencl V, Jabor A, Kazda A, Figge J. Diagnosis of Metabolic Acid–Base Disturbances in Critically Ill Patients. Am J Respir Crit Care Med. 2000;162:2246–51.
Schlichtig R. [Base Excess] vs [Strong ION Difference]. In: Nemoto EM, JC LM, Cooper C, Delpy D, Groebe K, Hunt TK, et al., editors. Oxygen Transport to Tissue XVIII. US: Springer; 1997. p. 91–5.
Wooten EW. Calculation of physiological acidbase parameters in multicompartment systems with application to human blood. J Appl Physiol. 2003;95:2333–44.
Singer RB, Hastings AB. An improved clinical method for the estimation of disturbances of the acidbase balance of human blood. Medicine. 1948;27:223–42.
Ježek F. fullbloodacidbase. Github. https://doi.org/10.5281/zenodo.1134853.
SiggaardAndersen O. Textbook on the AcidBase and Oxygen Status of the Blood. AcidBase and Oxygen Status of the Blood. 2010; http://www.siggaardandersen.dk/OsaTextbook.htm. Accessed 7 Jul 2016
Figge J. The FiggeFencl Quantitative Physicochemical Model of Human AcidBase Physiology (Version 3.0). FiggeFencl.org  FiggeFencl Quantitative Physicochemical Model of Human AcidBase Physiology. 27 October, 2013. http://www.figgefencl.org/model.html. Accessed 22 Jun 2016.
Morgan TJ. The Stewart approachone clinician’s perspective. Clin Biochem Rev. 2009;30:41–54.
Lang W, Zander R. Prediction of dilutional acidosis based on the revised classical dilution concept for bicarbonate. J Appl Physiol. 2005;98:62–71.
Kurtz I, Kraut J, Ornekian V, Nguyen MK. Acidbase analysis: a critique of the Stewart and bicarbonatecentered approaches. Am J Physiol Renal Physiol. 2008;294:F1009–31.
Severinghaus JW. SiggaardAndersen and the “Great TransAtlantic AcidBase Debate.”. Scand J Clin Lab Invest. 1993;53:99–104.
Berend K. Acidbase pathophysiology after 130 years: confusing, irrational and controversial. J Nephrol. 2013;26:254–65.
Kellum JA. Clinical review: reunification of acidbase physiology. Crit Care. 2005;9:500–7.
Dubin A, Menises MM, Masevicius FD, Moseinco MC, Kutscherauer DO, Ventrice E, et al. Comparison of three different methods of evaluation of metabolic acidbase disorders*. Crit Care Med. 2007;35:1264.
Kishen R, Honoré PM, Jacobs R, JoannesBoyau O, De Waele E, De Regt J, et al. Facing acidbase disorders in the third millennium  the Stewart approach revisited. Int J Nephrol Renovasc Dis. 2014;7:209–17.
Matousek S, Handy J, Rees SE. Acidbase chemistry of plasma: consolidation of the traditional and modern approaches from a mathematical and clinical perspective. J Clin Monit Comput. 2011;25:57–70.
Masevicius FD, Dubin A. Has Stewart approach improved our ability to diagnose acidbase disorders in critically ill patients? Pediatr Crit Care Med. 2015;4:62–70.
Rastegar A. Clinical utility of Stewart’s method in diagnosis and management of acidbase disorders. Clin J Am Soc Nephrol. 2009;4:1267–74.
Matoušek S. Reunified description of acidbase physiology and chemistry of blood plasma: PhD. Charles University in Prague; 2013. https://is.cuni.cz/webapps/zzp/download/140030054/?lang=cs
Lutz J, Schulze HG, Michael UF. Calculation of O2 saturation and of the oxyhemoglobin dissociation curve for different species, using a new programmable pocket calculator. Pflugers Arch. 1975;359:285–95.
Fencl V, Jabor A, Kazda A, Figge J. Appendix of Diagnosis of Metabolic Acid–Base Disturbances in Critically Ill Patients. Am J Respir Crit Care Med. 2000;162:2246–51.
Wolf MB. Comprehensive diagnosis of wholebody acidbase and fluidelectrolyte disorders using a mathematical model and wholebody base excess. J Clin Monit Comput. 2015;29:475–90.
Zander R. Die korrekte Bestimmung des Base Excess (BE, mmol/l) im Blut. AINS  Anästhesiologie · Intensivmedizin · Notfallmedizin. Schmerztherapie. 1995;30(S 1):S36–8.
SiggaardAndersen M, SiggaardAndersen O. Oxygen status algorithm, version 3, with some applications. Acta Anaesthesiol Scand Suppl. 1995;107:13–20.
McAuliffe JJ, Lind LJ, Leith DE, Fencl V. Hypoproteinemic alkalosis. Am J Med. 1986;81:86–90.
Tuhay G, Pein MC, Masevicius FD, Kutscherauer DO, Dubin A. Severe hyperlactatemia with normal base excess: a quantitative analysis using conventional and Stewart approaches. Crit Care. 2008;12:R66.
Acknowledgements
We would like to thank to Karel Roubík and Arnošt Mládek for a careful review, to S.E. Rees for a great insight and invaluable comments and to Francesco Casella for discussion on numerical solvability of the equationbased models.
Funding
This study has been supported by the TRIO MPO FV20628 grant.
Availability of data and materials
The models analysed within the current study are available in the referenced Github repository. No additional datasets have been employed.

Project name: Full blood acidbase

Project home page: https://github.com/filipjezek/fullbloodacidbase/

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 nonacademics: additional licence needed
Author information
Authors and Affiliations
Contributions
JK initiated the study, FJ implemented the models and performed the comparative analysis. FJ and JK wrote the manuscript. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Ethics approval and consent to participate
Not applicable.
Consent for publication
Not applicable.
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Additional file
Additional file 1:
Formalization of SiggaardAndersen nomogram and derivation of the base excess for the physicochemical domain. (PDF 806 kb)
Appendix
Appendix
Listing of the simplest flat implementation of Combined model in Modelica. To keep the implementation lucid, the dilution is not incorporated here for the sake of simplicity of this example. In this example, the Fencl’s [14] approximation is used for plasma, whereas for erythrocytes here we use a simple formalization of SiggaardAndersen nomogram by Zander and Lang [36]. Use any Modelica tool (e.g., OpenModelica) to run this model. For the complete set of used models, including sources for presented figures, implementation of the Wolf model and the complete fullblood component, please refer to [18].
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the 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.
About this article
Cite this article
Ježek, F., Kofránek, J. Modern and traditional approaches combined into an effective graybox mathematical model of fullblood acidbase. Theor Biol Med Model 15, 14 (2018). https://doi.org/10.1186/s1297601800869
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s1297601800869
Keywords
 Acidbase modeling
 Physicochemical acidbase
 Behavioral acidbase
 SiggaardAndersen
 Modelica
 Physiology
 Physiolibrary