MEDSOFT 2009 MEDSOFT 2009, sborník příspěvků

Transkript

1 MEDSOFT 09 Jiří Kofránek: Complex model of blood acid-base balance. In MEDSOFT 09, sborník příspěvků, (editor: Milena Ziethamlová Ed.) Praha: Agentura Action M, Praha, ISBN , str , originally published in Czech, English translation of the paper is available at cz/references/acidbasemedsoft09.pdf, model is available at physiome.cz/acidbase.

2

3 COMPLEX MODEL OF BLOOD ACID-BASE BALLANCE Ji í Kofránek Laboratory of Biocyberne cs, Departmernt of Pathological Physiology, First Faculty of Medicine, Charles University of Prague. Annotation Originally, the classic Siggaard-Andersen nomogram, widely used in clinical prac ce for the assessment of acid-base balance, experimentally obtained at 38 C with the precondi on of normal plasma protein concentra ons. However, a nomogram is used in clinical prac ce to calculate from the data measured in blood samples tempered at 37 C. We made a simula on recalcula on of the baseline experimental data to 37 C and set a new nomogram for 37 C. Compared with the original nomogram, there are no signi cant devia ons, if BE does not deviate by more than 10 mmol/l; the results are, however, di erent with the devia ons exceeding 15 mmol/l. We suggested an algorithm and a program, which enables calcula on of BE from and pco 2 according to the original as well as adjusted normograms. However, the data, having been a base of the normogram, count with normal plasma protein concentra ons. Furthermore, we combined Figge and Fencl plasma acid-base balance model with the data based on Siggaard-Andersen nomogram, adjusted to 37 C. Thus, BE was not only de ned in the dependence on haemoglobin concentra ons, but also on plasma protein and phosphate concentra ons. At these condi ons, BE corresponds to SID changes according to the modern concep on of acid-base balance by Stewart. Moreover, the model obviously suggests that the independence of SID and pco 2 is not applicable for the whole blood. The model is a core of a wider model of acid-base balance in the organism, enabling realisa on of the pathogenesis of acid-base balance disturbances, which is in accordance with our earlier publica on of the b lance approach to the interpreta on of acid-base balance. Originally published in Czech. Ji í Kofránek: Komplexní model acidobazické rovnováhy. In MEDSOFT 09. (Milena Ziethamlová Ed.) Prague: Crea ve Connec ons, Prague 08, pp ISBN Key words Acid-base balance, formalised descrip on, simula on model, blood gases, educa onal simulators Acid-base balance in the organism is controlled by two balances carbon dioxide ow balance (respira on control) and strong acid produc on/excre on balance (regula on of acidi ca on processes in the kidney). Both ows are connected via bu er systems. The balance disturbances result in changes in blood liquids. Dri s in the chemical balances of bu er systems, transport of substances between the bu er systems, H + /Na + H + /K + exchange between the cell and the inters al liquid (and, in a long-term scale, washing out NaHCO 3, KHCO 3 and, later, CaCO 3 and CaHPO 4 from the bone mineral mass in chronic acidemia) are only suppressive mechanisms in acid-base disturbances. The basic regula on organs, able to control acid-base balance (by their e ect on CO 2 and H + /HCO 3 - ows) include the respiratory system and kidney. From the clinical point of view, the arterial blood bu er system is an important indicator of the status of acid-base balance. CO 2 reten on or deple on during the change of carbon dioxide balance as well as H + /HCO 3 - reten on or deple on during the changes of strong acid produc on/excre on balance develop into the dri of the chemical balance in bicarbonate and nonbicarbonate bu er systems. Labelling the total concentra on of non-bicarbonate bases [Buf - ] which, in fact, are the bu er bases of plasma proteins and phosphates (and haemoglobin concentra ons in the whole blood) then the total concentra on of nonbicarbonate bu er bases forms the Bu er Base (BB) value: BB=[HCO 3- ]+[Buf - ] The varia ons in pco 2 result in changes; If the tra on curve of pco 2 and changes is plo ed in the semi-logarithmic scale, these tra on curves verge on lines in the range of lifecompa ble values. This precondi on was a base of blood acid-base balance tests introduced in the rst half of 1950s by Paul Astrup. At that me, there were no electrodes which enable direct measurement of plasma pco 2. There were, however, rela vely accurate methods of measurement. Astrup s method of pco 2 analysis (1956) was based on the following procedure: rst, blood was measured, then, the sample was automa cally equilibrated by O 2 /CO 2 mixture with accurately set pco 2. The blood sample was equilibrated with a high pco 2 gas mixture and the equilibra on was followed by measuring. Then, the blood was equilibrated with a mixture with low carbon dioxide par al pressure and the equilibra on was followed by another measurement. The points obtained were plo ed into a semi-logarithmic graph to create a line, used to read out pco 2 according to baseline (see Fig. 1). The Bu er Base concept made by Singer and Has ngs (1948) was further improved by Siggaard-Andersen in (1960,1962), who introduced the di erence of Bu er Base and its normal value - Normal Bu er Base (NBB) - as a clinically relevant factor: BE=BB-NBB At normal circumstances, BE values (for blood samples with any haemoglobin concentra ons) are zero. They are changed during a bu er reac on with strong acid or base added. 1

4 Logarithmic PCO 2 scale High pco 2 in O 2 /CO 2 mixture Derived pco 2 Value from titration line Low pco 2 in O 2 /CO 2 mixture Constructed log pco 2 titration line scale with the same BB crossed in the same points, too. Thus, a nomogram with BE and BB curves with semilogarithmic coordinates was obtained; the curves enabled the determina on of BE and BB in the samples having been tested. Siggaard-Andersen used this procedure to nd experimentally the dependence of hydrogen ion [H + ] concentra ons or on pco 2 and haemoglobin (Hb) concentra ons; the results obtained were used to create clinically applicable nomograms expressing the following dependence: [H + ]= func on (pco 2,BE,Hb) after equilibration with high pco 2 Measured before equilibration with pco 2 after equilibration with low pco 2 Fig. 1 The tra on curve of /PCO 2 varia ons a er blood equilibra on with carbon dioxide is prac cally a line. This therefore enabled pco 2 determina on in the tested blood sample according to the tra on curve plo ed a er blood equilibra on with low and high par al CO 2 pressure. Siggaard Andersen used the equilibrium tra on curves to determine BB and BE. He added de ned amounts of strong acids or bases to blood samples with various haematocrit concentra ons, changing their BE. Then, the samples were trated and the results were plo ed in log PCO 2 / coordinates. The tra on curves (being lines in the semilogarithmic coordinates) of the blood samples with various haematocrit and the same BE always crossed in the same points (see Fig. 2). Similarly, the tra on curves of the samples with various haematocrit concentra ons (and various BE), but BE= 15 meq/l BE=0 meq/l Fig. 2 Siggaard-Andersen nomogram. The tra on curves (lines in the semi-logarithmic scale) have di erent slopes a er blood tra on with carbon dioxide, depending on haemoglobin concentra ons. The curves with the same BE cut each other in one point. The intersec ons of these points were a base for experimental determina on of BE curve (Base Excess). Similarly, BB curve (Bu er Base) was experimentally determined as the intersec on of the points where the tra on curves of the blood samples with the same BB cut each other. The nomogram was experimentally created for 38 C. The tested blood sample is tempered to the standard temperature of 37 C in modern automats for the tests of acid-base balance. At present, the determina on of BE and BB, however, uses (digitalised) data based on the original Siggaard-Andersen nomogram. In the assessment of acid-base disturbances by BE and pco 2, it should be taken into considera on that the increase or the fall in CO 2 a ects neither the total concentra on of the bu er bases (BB) nor BE. The increase results in the increase in carbonic acid concentra on, dissocia ng into bicarbonate and hydrogen ions, which are, however, completely bound to non-carbonate bu er bases [Buf - ]; the increase in bicarbonate concentra ons therefore corresponds with the same fall in non-bicarbonate bu ers with the total [HCO3 - ]+[Buf - ] concentra ons and, thus, BB as well as BE remaining prac cally unchanged. BB and BE are therefore considered pco 2 independent. This applies for plasma exactly but not exactly for the whole blood pco 2 a ects haemoglobin oxygena on. However, as deoxygenated haemoglobin has higher a nity to protons than oxygenated haemoglobin (the oxygenated blood therefore contains virtually higher nonbicarbonate bu er concentra ons), the total concentra on of bu er bases BB also depends on haemoglobin oxygen satura on (suscep ble by pco 2 ). Hence, to make acid-base balance models, it is bene cial to de ne standardised Bu er Base oxy-value (BBox) as BB, poten ally found in the blood sample with full oxygen satura on of oxyhaemoglobin (i.e. full % oxygen satura on of haemoglobin). Similarly, the standardised Base Excess oxyvalue (BEox) is de ned as BE measured in the blood sample with full oxygen satura on of oxyhaemoglobin (Kofránek, 19). Thus, BEox is really pco 2 independent. It is necessary to say that the independence of pco 2 and BEox does not apply for in vivo whole blood completely, as the increase in pco 2 is connected with higher increase of bicarbonates in plasma compared with that in the inters um; thus, part of the bicarbonates is transported into the inters ary liquid during the increase in pco 2 (with a mild fall in BEox in acute pco 2 increase). BB and BE (or BBox and BEox) change a er addi on of a strong acid (or strong base) or bicarbonates to the blood sample. Addi on of one millimol of a strong acid to one litre of blood results in BE fall by one millimol; addi on of one millimol of bicarbonates (or withdrawal of one millimol of hydrogen ions by a reac on with a strong base) results in BB and BE (BBox and BEox) increase by one millimol. The varia ons in dissolved CO 2 plasma concentra ons (expressed as pco 2 ) and BE therefore characterise carbon dioxide ow balance and the varia on in strong acid produc on/excre on balance, respec vely. Thus, pco 2 and BE characterise the respiratory and metabolic parts of acidbase balance, respec vely. 2

5 To use, pco 2 and BE in clinical prac ce for the diagnosis of acid-base balance, so called compensa on diagrams were created, expressing the e ect of adapta on responses of the respiratory and renal systems to acid-base disturbances (Dell and Winters, 1970, Goldberg et al., 1973, Siggaard-Andersen, 1974, Grogono et al., 1976). Siggaard-Andersen nomogram (expressed in the form of approximate equa ons) became a base for algorithm assessment in a number of laboratory automats for the measurement of acid-base balance. A certain problem was that the experimental measurements for the construc on of Siggard-Andersen nomogram were carried out at 38 C. Modern devices for the measurement of acid-base balance (allowing direct measurement of pco 2, and po 2 ) usually give data for samples adjusted to 37 C. However, a more serious problem was that the tra on done to create an experimental nomogram was carried out with blood with normal plasma protein concentra ons (72 g/l). If the plasma protein concentra ons are lower (which is not rare in cri cally ill pa ents), the points on the nomogram are shi ed and all the clinical counts derived from this nomogram are incorrect. Later, Siggaard-Andersen published certain correc ons, considering various plasma protein concentra ons (Siggaard- Andersen, 1977, Siggaard-Andersen et al. 1985, Siggaard- Andersen, Fogh-Andersen, 1995); however, they were not included into clinical prac ce properly. The abovemen oned inaccuracies of the classical approach to the assessment of acid-base balance resulted in the a empt to nd new methods of the descrip on and assessment of blood acid-base balance in 19s. The most used method was Stewart s one (1983), improved later for clinical prac ce by Fencl et al. (1989, 1993, 00). Unlike Siggaard-Andersen s method, Stewart s descrip on is limited to plasma only; however, it enables accurate descrip on of hypo- and hyperalbuminaemia, dilute acidosis as well as concentra on alkalosis. Stewart s calcula ons are based on the combina on of physical-chemical equa ons. The original Stewart s calcula ons are based on simple precondi ons: 1. The equa on for water must apply: [H + ] [OH - ] = K w 2. The constancy of the sum of weak acid concentra ons (Buf - ), and their dissociated bu er bases (Buf - ): [Buf - ]+[HBuf] = [Buf TOT ] 3. Dissocia on balance of non-bicarbonate bu er system: [Buf - ] [H + ] = K BUF [HBuf] 4. Dissocia on balance of bicarbonate bu er: [H + ] [HCO 3- ] = M pco 2 5. Dissocia on balance between bicarbonate and carbonate: [H + ] [CO 3 2- ] = N [HCO 3- ] 6. Electroneutrality: SID + [H + ] [HCO 3- ] [Buf - ] [CO 3 2- ] [OH - ] = 0 with SID meaning the value of strong ion di erence (residual anion) de ned as the di erence between the concentra ons of fully dissociated anions and ca on (expressed in meq/l). Prac cally, the value can be found out by the following equa on: SID = [Na + ] + [K + ] + [Mg 2+ ] + [Ca 2+ ] - [Cl - ] Combining these two equa ons, the result is the fourth degree algebraic equa on, enabling calcula on of hydrogen ion concentra ons in dependence on SID, the total concentra on of weak acids and their bu er bases [Buf TOT ] and pco 2 (the dependent variable is underlined in the equa on, independent varia ons and constants are in bold and italic, respec vely): [H + ] 4 + (SID + K BUF ) [H + ] 3 + (K BUF (SID - [Buf TOT ]) - K w M pco 2 ) [H + ] 2 - (K BUF (K w 2 + M pco 2 ) - N M pco 2 ) [H + ] - K w N M pco 2 = 0 Solving of the equa on gives hydrogen ion concentra on, depending on the respiratory part of acid-base balance i.e. pco 2, and, moreover, on the respiratory part of SID independent metabolic parameters as well as on the total concentra on of non-bicarbonate bases and acids [Buf TOT ]: = func on ( pco 2, SID, [Buf TOT ] ) The total concentra on of non-bicarbonate bases [Buf TOT ] is related to the total plasma protein (albumin) concentra on. More detailed studies consider the total phosphate concentra ons, too. The results of these studies are rela onships enabling (by means of a computer programme) calcula on of (and other variables such as bicarbonate concentra ons etc.) from pco 2, SID, and total phosphate [Pi] and plasma albumin [Alb ] concentra ons (see, for example, TOT Watson, 1999): = func on ( pco 2, SID, [Alb TOT ], [Pi] ) One of the most detailed quan ta ve analyses of plasma acidbase balance (Figge, 09) improving Figge-Fencl s model (Figge et al. 1992) even corrects the e ect of externally added citrate [Cit] in the plasma sample used for the laboratory test. = func on ( pco 2, SID, [Alb TOT ], [Pi], [Cit] ) Mathema cal rela onships between the variables derived from the quan ta ve physical-chemical analysis enable calcula on of dependent variables, being a base for other dependent variables, i.e. bicarbonate concentra ons from independent variables (i.e. pco 2, SID, albumin and phosphate concentra ons or, as the case may be, concentra ons of the citrate added to the plasma sample). Stewart s approach enables more detailed descrip on of some of the pathophysiological condi ons (the e ect of hypo- and hyperalbuminaemia on acid-base balance, dilu on acidosis or concentra on alkalosis) and, at rst site, gives the clinicians the feeling of be er insight into the ethiology of acid base disturbances. To determine independent variables, used for the calcula on of other acid-base parameters, it is 3

6 necessary to do explicit measurements of phosphate, Na +, Cl -, HCO 3 - and other ion concentra ons, which clinicians work in their diagnos c forethought with. On the contrary, the drawbacks of Stewart s theory include the fact that he works with plasma only. Moreover, some Stewart s followers, fascinated by the possibility to calculate acid-base parameters - (and proper concentra ons of bicarbonates, carbonates and non-bicarbonate acids) from independent variables (pco 2, SID, [Alb TOT ], [Pi]), o en make objec vely incorrect conclusions in their interpreta on. In the calcula on, the independence of baseline variables, par cularly SID, is meant not in a causal but in a strictly mathema cal meaning. This is, however, o en forgo en in clinical-physiological prac ce, which o en results in incorrect interpreta on of the causality rela onship between the causes of acid-base disturbances. A number of Stewart s followers considered his mathema cal rela onships as oracle incorrect causal rela onships are deducted from substan ally correct mathema cal rela onships. The causality of mathema cal calcula ons (where independent variables are calculated from dependent ones) is confused with the causality of pathophysiological rela onships. For example, some authors deduct that one of the elementary causal rela onships of acid-base disturbances are changes in SID concentra ons. Sirker et al. (01) even states that the transport of hydrogen ions through membranes (via hydrogen channels) does not a ect their actual concentra on. Direct removal of H + from one compartment can alter neither the value of any independent variable nor [H + ] concentra on the equilibrium dissocia on of water balances any uctua ons in [H + ] concentra ons and serves as an inexhaus ble source or sink for H + ions. There is no ra onal explana on for the opinion that SID (as a mathema cal construct, not a physical-chemical characteris c) a ects [H + ] concentra ons in a certain mechanis c way to keep electroneutrality any bu er reac on is a shi ed chemical balance only; thus, there is no way how they could a ect the electroneutrality themselves (without membrane transport). Excited debates lead by supporters of both theories in interna onal journals (e.g. Dubin et al. 07, Dubin 07, Kaplan 07, Kurz et al., 08, Kelum 09) might suggest that both theories are completely di erent and their applicability will be proved during the me. In fact, both theories are complementary. If similar condi ons of their applicability are observed (i.e. they are used for plasma with normal albumin and phosphate concentra ons only), the results are, in fact, iden cal. It is obvious that if one of the theories is used out of the area which it was proposed for, it fails and the other theory seems to be more accurate. For example, reduced protein concentra ons do not correspond to the condi ons determined experimentally for Siggaard-Andersen nomogram; if this nomogram is used for BE assessment in pa ents with hyperalbuminaemia, incorrect values are obtained. In this case, the use of Stewart s method prevents incorrect diagnosis. On the other hand, Stewart does not 4 calculate with the e ect of such an important blood bu er - haemoglobin in erythrocytes. Stewart s approach is applicable neither for the calcula on of the amount of infusion solu ons for the correc on of the acid-base disturbance nor for the assessment of the grade of respiratory and renal compensa on of the acid-base disturbance. During the bedside diagnos cs it is advisable to consider both theories and to realise their bene ts and limits (Kelum, 05). The accordance and di erences of both approaches are as follows. Both Stewart and Siggaard-Andersen use pco 2 as a parameter describing the respiratory part of acid-base balance. According to the Danish School, the metabolic part is represented by BB or its devia on from the norm BE. According to Stewart, the metabolic part is represented by SID as the di erence of fully dissociated posi vely and nega vely charged anions and ca ons in the respect of keeping the principle of electroneutrality, it might seem at rst sight that, numerically, SID is iden cal with plasma BB (Fig. 3). SID = [HCO3 - ] + [Buf - ] = BB But is it true really? Siggaard-Andersen (06) states so. However, focused on the importance of non-bicarbonate bases, certain di erences can be seen. Plasma non-bicarbonate bases include phosphates and plasma proteins par cularly albumin (the e ect of globulins on acid-base balance is insigni cant). The albumin hydrogen ion can be bound to the following nega vely charged amino acids (Figge, 09): cysteine, glutamic and aspar c acid, tyrosine and carboxyl end of protein polymer. Labelling these binding sites as Alb -, the binding of hydrogen ions can neutralise the electric charge (as presumed in the classical Stewart s theory): Alb - + H + = HAlb Hydrogen ions van, however, be bound to imidazol cores of his dine as well as to arginine, lysine and NH 2 end of an albumin molecule. Labelling these binding sites as Alb, then the binding of hydrogen ions results in the crea on of posi ve charge: mmol/l Mg 2 Cations Anions Ca 2 SID K + Na + HCO 3 Pr Cl BB HPO H 2 PO 4 2 SO 4 Organic anions Fig. 3 SID and BB are nearly iden cal. The varia ons in SID and BB are completely iden cal: dsid=dbb.

7 Alb + H + = HAlb + Labelling the total concentra ons of non-bicarbonate bases by Stewart and Siggaard-Andersen as [Buf st- ] and [Buf sa- ], respec vely, a small di erence can be observed (the concentra ons are considered in miliequivalents): [Buf st- ] = [PO 4 3- ] + [HPO 4 2- ] + [H2PO 4- ] + [Alb - ] [HAlb + ] [Buf sa- ] = [PO 4 3- ] + [HPO 4 2- ] + [H2PO 4- ] + [Alb - ] + [Alb] The concentra on of non-bicarbonate bases is a bit higher by Siggaard-Andersen, as the rela onship [Alb]>[HAlb + ] applies at physiological condi ons. This obviously suggests the di erence between normal SID (around 38 mmol/l) and normal plasma BB (stated as 41.7 mmol/l). However, as it applies that the varia on in [Alb] concentra ons is related to the varia on in [HAlb + ] concentra ons: d[alb]=-d[halb + ] the varia on in the concentra ons of non-bicarbonate bases by Siggaard-Andersen will be iden cal with that of nonbicarbonate bases by Stewart: d[buf st- ] = d[buf sa- ] The varia on in BB or BE is therefore the same as that of SID: dbb=dsid Thus, it would meaningful for clinical purposes to calculate normal SID for various plasma protein and phosphate concentra ons: NSID=func on ([Alb TOT ], [Pi]), similarly as Siggard-Andersen calculates NBB as a variable dependent on haemoglobin concentra ons. It would not be complicated in any respect. However, the problem is that what circulates in the blood vessels is not plasma only, but plasma and erythrocytes. A more accurate quan ta ve analysis requires considering the whole blood and it is also necessary to re-evaluate and connect both the approaches. The outcome of the connec on will be the su ciently quan ed Figge-Fencl s model of plasma (Figge, 09) and experimental data for the whole blood, included in Siggaard- Andersen nomogram. The rst step necessary for the realisa on of this connec on is to formalise Siggaard-Andersen nomogram. The literature describes a number of equa ons which formalise Sigaard-Andersen nomogram with higher or lower accuracy (e.g. Siggaard-Andersen et al. 1988). Lang and Zander (02) compared the accuracy of BE calcula on in 7 approxima ons of various authors. The most accurate approxima on was that of Van Slyk equa on by Zander (1995). Surprisingly, it was, however, shown that the formalisa on of Siggaard-Andersen nomogram from 19, used in a lot of our models in the past, approximated Siggaard-Andersen nomogram with higher accuracy than the rela onships having been published later (Fig. 4) It is possible to try further speci ca on of our approxima on. Zander (1995): BE -6,1 1-0,0143cHb 0,0304 pco ,26 9,5 1,63 chb Kofránek (19): a1 996,35-10,35 chb a2 = 35, , chb a3 = -82, chb; a4 = -5, , chb a5 =121 - chb a6 = 2, ,025 chb a7 = -2,556-0,09 chb 2 a8 =13, , chb + 0, chb a9 = 0, , chb a10 = 0,274-0,0137 chb BE BEox a11 (1-sO2) However, the situa on in 19s was a bit di erent. At that me, the struggle was focused on the nding of such approxima ons which would not require a big memory (regarding the opportunity of their use in laboratory devices and the capacity of microprocessors at that me). At present, the approxima on of experimental curves is carried out by means of the approxima on of the original curve Siggaard- Andersen nomogram by splines. The aim is to create approxima on of the func on =BEINV(cHb,BEox,sO2,pCO2) Calculate d BE - BE from nomogram [m mmol/l] Y ( a7 a8 a9 BE )/a (a1 a2*y (a3 a4 Y) log (pco2))/(a5 a6 Y) (mmol/l) 10 BE Fig. 4 The comparison of the accuracy of BE curve approxima on by Siggard-Andersen nomogram for various haemoglobin concentra ons and BE. Approxima on by Kofránek (19): and by Zander (1995): +. where chb is haemoglobin concentra on (in g/ ml blood), BEox is BE (mmol/l) with % haemoglobin satura on (being therefore independent on haemoglobin oxygen satura on), so 2 is haemoglobin oxygen satura on and pco 2 is carbon dioxide par al pressure (torr). Hence, the spline approxima on of the coordinates of BE and BB curves on the curve Siggaard-Andersen nomogram (Fig 5. and 6) is created rst, being a base for the calcula on of according to BEox, haemoglobin concentra on chb, haemoglobin oxygen satura on so 2 and pco 2 (Fig. 5). The calcula on of takes advantage of the fact that the tra on curves are prac cally lines in the semi-logarithmic scale (log pco 2, ). Func on BEINV (Fig. 7) enables simula on of blood tra on with carbon dioxide at various haemoglobin concentra ons and haemoglobin oxygen satura on (at standard temperature 38 C and normal plasma protein concentra ons). The calcula on of BE and BEox from and pco 2, haemoglobin concentra on and haemoglobin oxygen satura on is based on the itera on calcula on using the abovemen oned equa ons. This calcula on is a base of ABEOX func on. Siggaard-Andersen nomogram was created at the standard temperature of 38 C. However, the standard temperature for measuring acid-base balance parameters is 37 C in modern diagnos c devices. Nevertheless, Siggard-Andersen 5

8 Coordinates of BB curve [log(pco2), ] = function (BB) log pco 2 BB LPCO 2 BB ( BB ) models have been iden ed for 37 C. Thus, it was necessary to correct Siggaard-Andersen nomogram from 38 C to the standard temperature of 37 C. In clinical prac ce, the temperature correc ons of and pco 2 from t to the standard temperature of 37 C are based on simple rela onships, e.g. (Ashwood et al. 1983): 37 C = t (37-t ) log 10 (pco 2 37 C ) = log 10 (pco2 37 C )(37-t ) Siggaard-Andersen curve nomogram BB [mmol/l] BB PHBB(BB) Fig. 5. Approxima on of BB curve from Siggaard-Andersen nomogram by means of splines. Coordinates of BE curve 1.3 [log(pco2), ]= function (BE) 1.2 Siggaard-Andersen curve nomogram BE PHBE(BE) Fig. 6 Approxima on of BE curve from Siggaard-Andersen nomogram by splines. lg PCO2 [to orr] BE [mmol/l] l pco 2 BE LPCO 2 BE BE log 10 function BEINV chb, BEox, so2, pco2 chb, Beox, so2, pco2 BE BEox 0,2(1 so2) * chb NBB , 42cHb BB NBB BE x1 log10 pco2bb LPCO2BB( BB) y1 y1 BB PHBB( BB) BB x2 x 2 log pco 2BE LPCO 2 BE BE 10 y2 y2 BE PHBE( BE) x log 10 pco 2 BE x1 BB BE pco X1,y1 chb so2 BEox NBB BB X2,y2 BE For proper temperature correc ons of Siggaard-Andersen nomogram it is advisable to use the more accurate rela onship by Ashwood et al. (1983): 37 C = ( t (37-t ) (7.4) (37 t ) (37 2 t 2 ))/( (37-t )) log 10 (pco 2 37 ) = log 10 (pco 2 t ) + ( ( C )) (37 - t ) (37 2 -t 2 ) However, to correct Siggaard-Andersen nomogram from 38 C to 37 C, it is insu cient to transfer simply log 10 pco 2 and, represen ng the coordinates of BE and BB curves in Siggaard- Andersen nomogram, from one temperature to another. The trouble is that, according to the de ni on, BE is calculated as a trable base in blood tra on to the standard values (pco 2 =40 torr and =7.4). BE is zero at these standard values. Thus, the zero point of the BE curve, where all tra on curves of blood with various haematocrit cut each other, lies in the coordinates of =7.4, and pco 2 =40 torr. Using a simple re-calcula on of the values from 38 C to 37 C, the zero point of the BE curve is transferred to pco 2 = torr and =7.421 then (Fig. 8). Our aim is, however, to achieve that pco 2 and corresponding to zero BE are 40 torr and 7.4 on the curve for 37 C. Thus, standard and pco 2 are re-calculated from 37 C to 38 C as follows: 37 C = C -> 38 C = C 37 C =7.4 pco2=40 torr Warming up 1 C 38 C = pco2= torr y ( x x 1)( y 1 y 2) /( x 1 x 2) y 1 Fig. 7 Algorithm of the calcula on of tra on curves by Siggaard-Andersen nomogram formalised by means of splines. nomogram is used for the assessment of measured nomograms without any correc on. Moreover, this nomogram is used for iden ca on of the models created for 37 C in a number of works (e.g. Reeves and Andreassen 05). On the contrary, models of plasma acid-base balance, e.g. Watson s (Watson, 1999) or Figge-Fencl s (Figge, 09) 6 Fig. 8 In the points 37 C = 7.4 and pco 2 37 C = 40 torr, there is an intersec on of plasma and erythrocyte tra on curves with various haematocrit and BE=0 mmol/l. A er the temperature increase by one degree cen grade, all lines are shi ed with the intersec on in the same point ( 38 C = and pco 2 38 C = torr); BEs are, however, non-zero and di er for each blood sample.

9 pco 2 37 C = 40 torr 38 C -> pco2 38 C = torr 38 C PCO O2 [torr] plasma, BE 38 C = mmol/l chb = 5 g/ml, BE 38 C = 0.54 mmol/l chb = 10 g/ml, BE 38 C = mmol/l chb = 15 g/ ml, BE 38 C = mmol/l chb = g/ ml, BE 38 C = mmol/l chb = 25 g/ ml, BE 38 C = mmol/l C pco 2 38 C =41,862 torr Vstup: chb SO 2 38 C =1 BEINV SO 2 38 C =1 ol/l] 38 C BE [mmol/l] 38 C Fig. 9 The algorithm for the calcula on of BE in the tra on curves with various haemoglobin concentra ons (chb) for 38 C, with the corresponding zero BE at 37 C (the curves cut each other in the point of =7.4 and pco 2 =40 torr at 37 C). = pco2= torr Ashwood(1983) pco 237 C =40 torr 37 C =7,4-0.1 BE 38 ABEOX pco 2 38 C =41,862 torr 38 C =7,3878 BE Ashwood(1983) [g/ml] chb [g/ ml] pco 2 37 C =40 torr 37 C =7,4 All tra on curves of fully oxygenated blood with various haematocrit will cut each other in these points (in fact, these curves will be lines in the semi-logarithmic scale). Their BE will be set to zero at 37 C. Their BE will be non-zero at 37 C, depending on haemoglobin concentra on (Fig 8). For the algorithm of the calcula on, see Fig. 9. If the tra on curves of the values calculated by this procedure are modelled, it is obvious that they cut each other in one point at 38 C (Fig. 10). The re-calcula on of the data of the tra on curves from 38 C to 37 C by the abovemen oned rela onships derived by Ashwood et al. (1983) enables to obtain a set of curves (or lines in the semi-logarithmic scale), cu ng each other at the standard values of =7.4 and pco 2 =40 torr (see Fig. 11). According to the de ni on, BE (at 37 C) will be therefore zero in all cases. At 38 C, their BE will be di erent, depending on haemoglobin concentra on (see Fig. 12). To obtain a set of the values characterising the BE curve for Siggaard-Andersen nomogram corrected to 37 C, it is advisable to carry out simula on experiments with carbon dioxide blood tra on in blood samples with various haemoglobin concentra ons for each BE 37 C, in the condi on of full oxygen satura on (see the calcula on algorithm scheme in Fig. 13). Correc on factor dbe 38 C (depending on haemoglobin concentra on and corresponding to BE zero value at 37 C) was always added to each BE 37 C. This correc on shi was a base for BE 38 C. BE 38 C = BE 37 C + dbe 38 C Fig. 10 The tra on curves at 38 C with various haemoglobin concentra ons and BE, calculated by the algorithm described in the previous picture, cut each other in the point whose and pco 2 coordinates correspond to =7.4 and pco 2 = 40 torr a er cooling the blood by one degree. A set of 38 C was calculated from a set of BE 38 C and pco 2 by Siggaard-Andersen nomogram (by means of BEINV 38 C algorithm see Fig. 7). pco 2 38 C and 38 C were then recalculated to the values corresponding to 37 C. This procedure enabled obtaining the tra on curves for 37 C. The intersec ons of the curves with the same BE 37 C and various haematocrit characterise the BE curve of Siggaard- Andersen nomogram corrected to 37 C (see Fig. 14) PCO O2 [torr] plasma, BE 37 C = 0 mmol/l chb = 5 g/ml, BE 37 C= 0 mmol/l chb = 10 g/ml, BE 37 C = 0 mmol/l chb = 15 g/ ml, BE 37 C = 0 mmol/l chb = g/ ml, BE 37 C = 0 mmol/l chb = 25 g/ ml, BE 37 C = 0 mmol/l =7.4 pco2=40 torr 37 C BE [m mmol/l] Fig. 11 The tra on curves at 37 C and pco 2 from par cular curves in the previous gure were re-calculated from 38 C to 37 C. The curves cut each other in the zero point of BE curve for 37 C, which lies on the coordinates pco 2 =40 torr and = chb [g/ml] Fig. 12. The dependence of BE on haemoglobin concentra ons at = and pco 2 = torr according to the data from Sigaard-Andersen nomogram at 38 C (in fully oxygensaturated blood). At 37 C, these values correspond to the standard values of =7.4 and pco 2 =40 torr, at which BE will be zero (at 37 C). 7

10 Ashwood(1983) BEINV pco2 [torr] I Input: t pco CO2 38 C Outputs: pco2 37 C 37 C 38 C C 38 C 40 PC CO237 C Input: chb SO2 38 C=1 Input: BE37 C BE38 C=BE37 C+dBE38 C BE37 C 30 BE38 C dbe38 ABEOX C 15 pco CO2 37 C=40 torr pco2 38 C=41,862 torr Ashwood(1983) C=7,4 38 C=7, BE [ l/l] BE mmol/l Fig. 13 The scheme of data calcula on of the tra on curves for various haemoglobin concentra ons and BE at 37 C. First, the re-calcula on of the normal values =7.4 and pco2=40 torr from 37 C to 38 C is done. These values and the given haemoglobin concentra on (supposed to be fully saturated by oxygen) is a base for the calcula on of the correc on shi of BE (dbe38 C) corresponding to zero BE value at 37 C. The given BE37 C at 37 C is re-calculated to BE38 C at 38 C; this value and the set of pco2 38 C values for the given haemoglobin concentra ons (supposing fully saturated blood with oxygen) are a base for the calcula on of 38 C. These values are then re-calculated to 37 C and PCO2 37 C, characterising the tra on curve for the given haemoglobin concentra on and selected BE at 37 C. Fig. 15 Results of BE curve correc on from 38 C to 37 C new coordinates in pco2 axis. SO2 38 C= C 38 C For new coordinates of BE curves, see Fig. 15 and The calcula on of new coordinates of BB curves (i.e. the coordinates where the curves or lines in the semi-logarithmic scale of blood samples with the same BB cut each other is simpler. In anaerobic hea ng (or cooling) must apply that: d[hco3-] = -d[buf-]+d[h+], as d[h+]<<d[hco3-], thus, it applies that d[hco3-]=-d[buf-], i.e. BB do not vary; thus: BB37 C=BB38 C BE mmol/l Fig. 16 Results of BE curve correc on from 38 C to 37 C new coordinates in. The 38 C and pco2 38 C on the tra on curve with a given BB are re-calculated from 38 C to 37 C to new 37 C a pco2 37 C by Ashwood et al. (1983) however, the tra on curve will correspond to the same BB (but to a di erent BE value). It therefore suggests that the coordinates of the points of the BB curve of Siggaard-Andersen nomogram for 37 C can be obtained by the transforma on of the coordinates of the points on the BB curve of the original Siggard-Andersen nomogram (represen ng the coordinates of the intersec ons of the tra on curves with the same BB vale at 38 C) into new values by Ashwood et al. (1983) BBs depend on BE normal BB (NBB). Although BB38 C and BB37 C are the same, it is possible to show that their normal values (NBB37 C and NBB37 C) are di erent for 37 C and 38 C: BE [mmol/l] 50 PCO O2 [torr] NBB37 C=BB37 C -BE37 C = BB38 C - BE37 C -15 As (see above): - BE37 C=BE38 C - dbe38 C then: Fig. 14 The tra on curves for haemoglobin concentra ons (0, 5, 10, 15,, 25 g/ ml) and various BE at 37 C cut each other in the points characterising BE curve on SiggaardAndersen nomogram corrected to 37 C. 8 NBB37 C=BB38 C - BE38 C + dbe38 C = NBB38 C + dbe38 C The value of dbe38 C shi is calculated by the algorithm stated in Fig. 13 and depends on haemoglobin concentra on. The

11 l] 8 [mmol/l dbe Fig. 17 Lineariza on of the dependence of BE shi on haemoglobin concentra on (chb) expressed in g/ ml during temperature change from 37 C to 38 C. consequent dependence can be linearised by the following rela onship (Fig. 17) dbe= chb where chb is haemoglobin concentra on in g/ml. NBB 38 C is calculated by the known, in clinical prac ce used, rela onship (Siggaard-Andersen, 1960): NBB 38 C = chb The subs tu on of NBB 37 C results in a slightly di erent rela onship: NBB 37 C = chb dbe38 = chb dbe38 = *cHb chb [g/ml] BB 37 C value will be calculated from BE 37 C and haemoglobin concentra on: BB 37 C = chb + BE 37 C For the comparison of the curve Siggaar-Andersen nomograms for 37 C and 38 C, see Fig. 18 and Table 1. In clinical laboratory prac ce, data ( and pco 2 ) are measured at the standard temperature of 37 C; however, PCO O2 [torr] BB 38 C 90 BE 38 C BB 37 C BE 37 C BB BE Fig. 18 Correc on of the values on BE and BB curves of Siggaard-Andersen nomogram (created originally for 38 C) to the standard temperature 37 C. BE 37 C 38 C [mmol/l] pco 2 [torr] pco 2 [torr] -22 7,226 11,6 7,221 12,4-21 7,229 13,5 7,225 14,2-7,233 15,3 7,230 16,0-19 7,237 17,1 7,235 17,8-18 7,242 18,9 7,241 19,5-17 7,247,6 7,246 21,2-16 7,253 22,3 7,252 22,9-15 7, ,258 24,6-14 7,266 25,7 7,265 26,3-13 7,273 27,3 7,272 27,9-12 7,281 28,9 7,2 29,4-11 7,289 30,4 7,289 30,8-10 7,297 31,7 7,297 32,1-9 7, ,306 33,3-8 7,315 34,1 7,315 34,4-7 7,324 35,2 7,324 35,4-6 7,334 36,1 7,334 36,3-5 7, ,344 37,2-4 7,354 37,8 7,354 37,9-3 7,365 38,5 7,365 38,7-2 7,377 39,1 7,377 39,2-1 7,388 39,6 7,388 39,6 0 7,4 40 7,400 40,0 1 7,412 40,3 7,412 40,3 2 7,425 40,5 7,424 40,5 3 7,438 40,6 7,438 40,5 4 7,451 40,6 7,450 40,6 5 7,465 40,6 7,463 40,7 6 7,479 40,5 7,477 40,5 7 7,494 40,3 7,492 40,3 8 7, ,507 40,0 9 7,525 39,6 7,523 39,6 10 7,541 39,1 7,539 39,1 11 7,558 38,5 7,555 38,6 12 7,576 37,9 7,572 38,0 13 7,594 37,2 7,590 37,3 14 7,613 36,4 7,608 36,5 15 7,633 35,5 7,628 35,6 16 7,654 34,5 7,648 34,7 17 7,676 33,5 7,669 33,7 18 7,699 32,3 7,691 32,6 19 7,724 31,1 7,714 31,6 7,75 29,8 7,740 30,2 21 7,777 28,4 7,767 28,8 22 7,6 26,9 7,795 27,3 9

12 BB 37 C 38 C [mmol/l] pco 2 [torr] pco 2 [torr] 14 6,903 4,0 6,887 4,2 15 6,904 9,6 6, ,904 14,9 6,889 15,6 17 6,905,1 6, ,906 25,1 6,891 26,2 19 6,908 29,8 6,893 31,1 6,911 34,3 6,896 35,8 21 6,916 38,6 6,901 40,3 22 6,923 42,7 6,908 44,6 23 6,932 46,6 6,917 48,7 24 6,940 50,3 6,925 52,6 25 6,949 53,9 6,934 56,3 26 6,958 57,2 6,943 59,8 27 6,967 60,4 6,952 63,1 28 6,976 63,3 6,961 66,2 29 6,985 66,1 6,97 69,1 30 6,994 68,7 6,979 71,8 31 7,004 71,1 6,989 74,3 32 7,013 73,4 6,998 76,7 33 7,022 75,6 7, ,032 77,6 7,017 81,1 35 7,042 79,4 7, ,051 81,0 7,036 84,7 37 7,060 82,5 7,046 86,3 38 7,070 83,9 7,056 87,7 39 7,0 85,1 7, ,090 86,3 7,076 90,2 41 7, 87,3 7,086 91,3 42 7,110 88,3 7,096 92,3 43 7,1 89,0 7,106 93,1 44 7,131 89,7 7,117 93,8 45 7,141 90,3 7,127 94,4 46 7,151 90,7 7,137 94,9 47 7,162 91,1 7,148 95,3 48 7,173 91,4 7,159 95,6 49 7,183 91,6 7,169 95,8 50 7,194 91,7 7,18 95,9 51 7,5 91,8 7, ,215 91,8 7, ,226 91,7 7,213 95,9 54 7,237 91,5 7,224 95,7 55 7,248 91,2 7,235 95,4 56 7,260 90,8 7, ,271 90,4 7,258 94,6 58 7,282 89,9 7,269 94,1 BB 37 C 38 C [mmol/l] pco 2 [torr] pco 2 [torr] 59 7,294 89,5 7,281 93,6 60 7,306 88,9 7, ,318 88,2 7,305 92,3 62 7,330 87,4 7,317 91,5 63 7,342 86,7 7,329 90,7 64 7,354 85,9 7,341 89,9 65 7,365 85,0 7, ,378 84,1 7, ,391 83,0 7,379 86,9 68 7,404 82,0 7,392 85,8 69 7,417,8 7,405 84,6 70 7,430 79,7 7,418 83,4 71 7,443 78,5 7,431 82,2 72 7,456 77,3 7,444,9 73 7,469 76,0 7,457 79,6 74 7,482 74,7 7,47 78,2 75 7,496 73,3 7,484 76,7 76 7,510 71,8 7,498 75,2 77 7,523 70,3 7,512 73,6 78 7,537 68,8 7, ,551 67,2 7,54 70,4 7,566 65,6 7,555 68,7 Table 1 Coordinates of BE and BB curves for original (37 C) and corrected (37 C) Siggaard-Andersen nomogram. they are assessed (BE calcula on) by means of Siggaard- Andersen nomogram, created originally for 38 C. Thus, the comparison of the course of the tra on curves according to the original and corrected Siggaard-Andersen nomogram (Fig. 19) is interes ng in the view of clinical outcomes. It is obvious that no ceable devia ons occur as late as with BE under 10 mmo/l and more signi cant ones at BE exceeding 15 mmol/l. pco 2 [torr] P C O 2 [torr] Without corrections (temperature 38 C) With corrections to temperature 37 C BE [mmol/l] BE [mmol/l] chb = 0,5,10,15, g/ml Fig. 19 Comparison of the tra on curves calculated according to original and corrected Siggaard-Andersen nomogram

13 Now, Siggaard-Andersen nomogram is formalised for the same temperature, which detailed models of plasma acidbase balance, created by Stewart s model, are iden ed for. These models (e.g. Figge 09), anyhow considering the details of the e ect of the dissocia on constants of par cular amino acids in an albumin molecule, en rely neglect the e ect of such a substan al bu er as haemoglobin in erythrocytes. On the other hand, the drawback of the models based on experimental data derived from Siggaard-Andersen nomogram, is a precondi on of normal plasma protein concentra on. The aim of this work is to connect both approaches into one model, poten ally usable as a subsystem of the complex model of homeostasis in the organism with the possibility to simulate complex osmo c, ion, volume and acid-base disturbances. First, using the experimental data from Siggaard-Andersen nomogram, the tra on curves of plasma and erythrocytes should be separated the result should be a model of the bu er behaviour of erythrocytes, connected with the detailed model of plasma acid-base balance, created by Stewart s approach, regarding various plasma protein and phosphate concentra ons. Siggaard-Andersen veri ed experimentally that the curves of plasma and blood samples with various haematocrit and the same BE cut each other in one point on the BE curve (see Fig. 2). Similarly, the curves of blood samples with the same BB cut each other in one point on the BB curve. It raises a ques on, why the BB and BE tra on curves cut each other in the same points on Siggaard-Andersen nomogram? To reply this ques on, it is necessary to realise that blood tra on with carbon dioxide results in the increase in bicarbonate concentra ons in plasma and erythrocytes during the increase in pco 2. Regarding the plasma itself by Stewart then, during plasma tra on with carbon dioxide, the sum of bicarbonates and all non-bicarbonate bu er bases, forming BB p and SID, are unchanged (Fig. ) SID and pco 2 are therefore mutually independent variables, which, together with another independent variable, plasma protein concentra on, determines the value of the dependent variable. This basic Stewart s canon does not apply in blood (see Fig. 21) in the tra on with carbon dioxide, plasma SID, corresponding (with the abovemen oned objec ons) with BB p, varies. The increase in pco 2 causes the increase in BB p (and SID), whereas the decrease in pco 2 causes the decrease in BB p. As the erythrocyte has more non-bicarbonate bases (par cularly due to haemoglobin) than plasma, and the dissocia on reac on of carbonic acid is more shi ed to the right, there is a higher increase in bicarbonate concentra ons in erythrocytes than in plasma. Bicarbonates are transported into plasma by the concentra on gradient (by exchange for chloride ions). Thus, the increase in CO 2 concentra ons is associated with the decrease or increase in BB concentra ons in erythrocytes or plasma, respec vely. Blood tra on with carbon dioxide helps achieve pco 2 at which BB concentra ons in erythrocytes and plasma equilibrate (BB e = BB p ). This value determines the place where the tra on curves with the same total BB and various haematocrit (Hk) will cut each other on Siggaard-Andersen nomogram. As: plasma H2O H2CO3 HCO3 - CO2 H + HBuf BB p =[HCO 3 - ] p +[Buf - ] p BE p =BB p -NBB p Buf - BB = BB p (1 - Hk) + BB e Hk = BB p + Hk (BB e BB p ) The second member of the sum is zero with BB e = BB p and the whole blood BB does not depend on haematocrit. With this pco 2 (and proper plasma ) when BB p =BB e, the blood exert any value of haematocrit; all tra on curves of blood samples with various haematocrit therefore cut each other in this point. Thus, the BB curve on Siggaard-Andersen nomogram is a geometric site of the points where plasma and erythrocytes have the same bu er base concentra ons, as at BB e =BB p the whole blood BB does not depend on haematocrit (Hk): A similar considera on applies for the BE curve, too. As: BE=BE p (1 Hk) + BE e Hk = BE p + Hk (BE e BE p ) Rise in range of milomols Rise in range of nanomols Drop in range of milimols d[hco - 3 ] p = -d[buf - ] p SID, BB p a BE p do not vary! Fig.. Plasma tra on with carbon dioxide BE p, BB p and SID do not vary. Thus, pco 2 and SID are mutually independent. the second member of the sum equals zero at BE e =BE p then and the whole blood BE does not depend on haematocrit (Hk) or haemoglobin concentra on. Thus, the BE curve on Siggaard-Andersen nomogram is a geometric site of the points with the same BE in the whole blood and plasma, as the whole blood BE does not depend on haematocrit at proper pco 2 and, when BE e =BE p. H2O CO2 H2CO3 BB e =[HCO - 3 ] e +[Buf - ] e BE e=bb e +NBB e H2O H2CO3 HCO3 - H + Buf - HBuf erythrocytes plasma HCO3 - CO2 H + HBuf BB =[HCO - - p 3 ] p +[Buf ] p Buf - BE p =BB p +NBB p Fig. 21 Blood tra on with carbon dioxide SID varies during pco 2 changes (thanks to the exchange of HCO 3 - for Cl - ). SID and pco2 in the whole blood are not mutually independent. Cl - Cl - 11

14 The BE curve can also be interpreted in other way. Regarding the fact that BE is the di erence between BB and normal proper NBB for the given haemoglobin concentra on, then the precondi on of the equality of BE in plasma and erythrocytes means: BB e NBB e = BB p - NBB p This can be speci ed: BB e BB p = NBB e NBB p = constant This means that the BE curve can be interpreted as the geometric site of the points (i.e. pco 2 and values) with a constant di erence between BB in erythrocytes and plasma, which equals the di erence between the proper values in erythrocytes and plasma (pco2=40 torr and plasma =7.4). If the equa on NBB 38 C = chb applies (Siggaard- Andersen, 1962), then haemoglobin concentra on in erythrocytes chb = g/ml is NBB e -NBB p = =14 mmol/l (according to our correc on of Siggaard-Andersen nomogram, this value was =13.4 mmol/l for 37 C). Siggaard-Andersen used the mixture of O 2 - CO 2 for blood tra on with fully oxygen-saturated blood in fact, the BE curves are those for fully oxygenated blood i.e. the abovemen oned standardised oxyvalues of Base Excess BEox (Kofránek, 19). BE or BB exert a linear increase in haemoglobin oxygen desatura on: BE = BEox chb (1-sO2) where chb is haemoglobin concentra on [g/ml] and so 2 is haemoglobin oxygen satura on (Siggaard-Andersen 1988). 10. Separation of plasma and erythrocyte titration curves on Siggaard-Andersen nomogram It is recommended to test if it is possible to make a model of blood acid-base balance from the experimental data 0<cHb<33.34 [g/ml] PCO2 [to rr] chb=0 [g/ ml] chb=33.34 [g/ml] if haematocrit (Hk) =1: chb= g/ ml BB = BBe, BE = BEe Transfer of bicarbonates during titration with CO 2 erythrocytes BE e BB e dbb e = dbe e = -mhco3ep/hk plasma - - HCO 3 mhco3ep HCO 3 BB p BE p dbb p = dbe p = mhco3ep / (1-Hk) Fig. 22 The transfer of bicarbonates and varia ons in plasma and erythrocyte BB and BE during the tra on with carbon dioxide. The tra on curve of blood (a line in the semi-logarithmic scale) is calculated from the combina on of plasma and erythrocyte tra on curves and from the transfer of bicarbonates between erythrocytes and plasma, which change proper BE and BB in plasma and erythrocytes (depending on haematocrit). on Siggaard-Andersen nomogram as a combina on of the models of plasma and erythrocyte tra on curves (Fig 22). The tra on curves (plo ed as lines in the semi-logarithmic scale) can be read out direct from the nomogram. The tra on curves of erythrocytes can be obtained from the nomogram as follows: chose the blood concentra on of haemoglobin g/ ml, which is the value with haematocrit having the value of one. The tra on curve of this virtual blood with carbon dioxide follows varia ons (measured on the outer side of the erythrocyte) during pco 2 changes. The tra on curve of the blood with a given haemoglobin and, thus, haematocrit concentra ons chb (in g/ml blood). Hk=cHb/33.34 (supposing the normal haemoglobin concentra on in erythrocytes g/ml) will lie between the tra on curves of plasma and erythrocytes in the semi-logarithmic coordinates log 10 (pco 2 ). It will cut the curves for plasma and erythrocytes in a point of the BE curve. As non-bicarbonate bu ers (haemoglobin and phosphates) have a higher bu er capacity in erythrocytes than those in plasma (plasma proteins and phosphates), and non-bicarbonate bases in erythrocytes bind more hydrogen ions than those in plasma during blood tra on with increasing concentra ons of carbon dioxide, the concentra on of bicarbonates increases more signi cantly in the erythrocyte than in plasma. The consequence is the transfer of bicarbonates between the erythrocyte and plasma (accompanied with a counter chloride transport). Labelling the amount of bicarbonates in 1 litre, transferred from erythrocytes into plasma during blood tra on with carbon dioxide: mhco3ep [mmol/l], then the varia ons in plasma BE and BB is: dbb p =dbe p =mhco3ep/(1-hk) The corresponding varia on of BE in erythrocytes is: dbb e =dbe e =-mhco3ep/hk Choosing, for example, haemoglobin concentra on 15 g/ ml (and haematocrit concentra on 15/33.34=0.4449) for the transfer of 1mmol of bicarbonate, there will be an increase and decrease in plasma and erythrocyte BE as well as BB concentra ons by 1/( )=1.15 mmol/l and by 1/0.4449= mmol/l, respec vely. There will be le and right shi s on plasma and erythrocyte tra on curves (see Fig. 23), respec vely their intersec on corresponds with the point on the tra on curve with haemoglobin concentra on 15 g/ ml, in which 1 ml of bicarbonates were transferred from erythrocytes into plasma during the increase of pco 2 from the baseline value of 40 tor. As seen in Figure 23, this intersec on lies on the tra on curve with haemoglobin concentra on 15 g/ ml, modelled according to the data in Siggaard-Andersen nomogram (by means of the abovemen oned func on BEINV). Similarly, this curve includes the intersec ons of the le and right s of plasma and erythrocyte curves a er the transfer of 2 and 1 mmol of bicarbonates from erythrocytes into plasma (during pco 2 increase) and from plasma into erythrocytes (during pco 2 increase), respec vely. Figures 24 and 25 show the results of the modelling of the tra on curves for blood tra on with carbon dioxide at BE -10 mmol/l and 10 mmol/l. Fig. 26 shows the results of the modelling of blood tra on with carbon dioxide in the range of BE - to mmol/l. 12

15 chb=15 g/ ml, BE=0 mmol/l chb=15 g/ ml, BE=-10 mmol/l 1 90 plasma chb=15 g/ ml Ery (chb = g/ ml) BE=0 mmol/l 1 90 plasma chb= 15 g/ml Ery (chb = g/ ml) ery -> 2 mmol HCO3 -> plasma BE=-10 mmol/l 70 ery -> 2mmol HCO3 ->plasma ery -> 1mmol HCO3 ->plasma 60 ery -> 1mmol HCO3 -> plasma PCO2 [torr] plasma ->1mmol HCO3 ->ery PCO2 [torr] plasma -> 1mmol HCO3 -> ery Fig. 23 Model of the tra on curves of plasma, erythrocytes and blood with haemoglobin concentra on 15 g/ ml with BE=0 mmol/l. The plasma and erythrocyte curves cut each other in point (1) and on Base Excess in point BE=0, respec vely. The transfer of bicarbonates from erythrocytes into plasma during blood tra on with carbon dioxide shi s the plasma and erythrocyte curves to the right and to the le (with the increase and decrease in plasma and erythrocyte BE and BB values), respec vely. The curves cut each other in points (2) and (3) on the tra on curve with haemoglobin concentra on 15g/ ml. The decrease in pco 2 causes the transfer of bicarbonates from plasma to erythrocytes with following decrease in plasma BE and BB, which results in the right shi of the tra on curve and increase in erythrocyte BB with the le shi of erythrocyte curve. The curves cut each other on the blood tra on curve (in point 4) with haemoglobin concentra on 15 g/ ml, modelled by the data in Siggaard-Andersen nomogram. This suggests that the tra on curves can be modelled by the intersec ons of the shi s on plasma and erythrocyte tra on curves. It has been shown that the tra on curves modelled by means of the intersec ons of the shi s of plasma and erythrocyte tra on curves (due to the transfer of bicarbonates between the erythrocyte and plasma) copy the tra on curves modelled direct by Siggaard-Andersen nomogram with a su cient accuracy. It therefore means that the modelling of blood tra on with carbon dioxide can be based on the combina on of plasma and erythrocyte tra on curves. The modelling of blood tra on with varied plasma protein concentra on can be based on the combina on of plasma tra on curve with various plasma protein concentra ons (for which, however, Siggaard- Andersen nomogram does not apply) for example by Figge- Fencl s model (Figge, 09), and erythrocyte tra on curve (obtained from the experimental data of Siggaard-Andersen nomogram, corrected to 37 C). Fig. 27 shows erythrocyte tra on curves with various BE by Siggaard-Andersen nomogram the erythrocytes are modelled as blood with haemoglobin concentra on g/ml (corresponding to the proper haematocrit value of 1). In the semi-logarithmic scale, these curves are lines with variable slopes (k) and o set (h), depending on BE concentra ons in erythrocytes (BE er ). The erythrocyte tra on curves will be approximated according to the following rela onships: log10(pco2) = k + h k=f(be er ) h=g (BE er ) Fig. 24 Model of the tra on curves of plasma, erythrocytes and blood with haemoglobin concentra on 15 g/ ml with BE=-10 mmol/l. If the blood tra on curve of the tra on with carbon dioxide is modelled by means of the intersec ons of the shi of blood and erythrocyte tra on curves caused by the transfer of bicarbonates between the erythrocyte and plasma, points of the tra on curve are obtained (similarly as in the previous gure), which cover the tra on curve of blood with haemoglobin concentra on 15 g/ ml, modelled by Siggaard-Andersen nomogram. PCO2 [torr] plasma ery->2 mmol HCO3 ->plasma ery->1 mmol HCO3 -> plasma chb=15 g/ ml, BE=10 mmol/l 3 chb=15g/ml Ery (chb = g/ml) BE = 10mmol/l plasma -> 1 mmol HCO3 -> ery Fig. 25 Model of the tra on curves of plasma, erythrocytes and blood with haemoglobin concentra on 15 g/ ml with BE=10 mmol/l. Similarly as in the previous gures, the intersec ons of the shi s of the plasma and erythrocyte tra on curves caused by the transfer of bicarbonates between the erythrocyte and plasma cover the tra on curve of blood with haemoglobin concentra on 15 g/ ml, modelled by Siggaard-Andersen nomogram. Func ons f and g are approximated by polynomic regression according to the data from Siggaard-Andersen nomogram, corrected to 37 C (see Fig. 28 and 29)

16 chb=15 g/ ml, -<BE< mmol/l Slopes (k) of erythocyte titration lines (log 10 (pco2) = k + h) BE [mmol/l] chb=15 g/ ml k -1.5 k k k = pk1 k=p1*be^6+p2*be^5+p3*be^4+p4*be^3+p5*be^2+p6*be+p7 BE 6 + pk2 BE BE^5+p3 5 + pk3 BE^4+p4 BE 4 + pk4 BE^3+p5 BE 3 + pk5 BE^2+p6 BE 2 + BE+p7 pk6 BE + pk PCO2 [torr] BE Fig. 26 Model of the tra on curves of plasma, erythrocytes and blood with haemoglobin concentra on with various BE concentra ons ranged from - to mmol/l by Siggaard- Andersen nomogram (con nuous lines). The crosses represent the tra on curves modelled as the intersec ons of the shi s of plasma and erythrocyte tra on curves caused by the transfer of bicarbonates between the erythrocyte and plasma. This means that the whole blood tra on curves on Siggaard- Andersen nomogram can be calculated from the plasma and erythrocyte tra on curves with su cient accuracy (modelled as blood with limit haematocrit 1 and haemoglobin concentra on g/ ml). Fig. 28 Polynomic regression of the variable slopes of erythrocyte tra on lines. h Offset (h) of erythocyte titration lines (log 10 (pco2) = k + h) h h h=p1*be^6+p2*be^5+p3*be^4+p4*be^3+p5*be^2+p6*be+p7= ph1 6 + ph2 5 + ph3 4 + ph4 3 + ph5 2 + ph6 BE + ph7 PCO O2 [torr] BE=- mmol/l BE=-10 mmol/l 60 BE= 0 mmol/l C 38 C BE= 10 mmol/l BE= mmol/l Fig. 29 Polynomic regression of the variable o set of erythrocyte tra on lines. BE er =erybeinv(pco 2,BE er ) BE Fig. 27 Erythrocyte tra on curves (lines in the semilogarithmic scale) by Siggaard-Andersen nomogram at 38 C and a er correc on to 37 C at various BE concentra ons. ( of the outer side of erythrocytes), depending on pco 2 and BE in erythrocytes (BE er ), is calculated by means of erybeinv func on; for its algorithm, see Fig. 30. =erybeinv(pco 2,BE er ) The erythrocyte model is connected with the plasma model. Figge-Fencl s model (Figge, 09), combined, in addi on, with the e ect of globulin concentra ons (calculated by means of their bu er value by Siggaard-Andersen, 1995), was selected as a plasma model. BEINV func on calculates blood in dependence on pco 2, total phosphate (Pitot), albumin (Alb), globulin (Glob) and haemoglobin concentra ons as well as on standardised oxyvalues BEox, (i.e. BE found in fully oxygenated blood), pco 2 and haemoglobin oxygen satura on: pk1 = e-009; pk2 = 1.328e-008; pk3 p = 2.228e-007; ; pk4 = 1.479e-005; pk5 = ; pk6 p = ; pk7 = ; ph1 = 8.229e-009; ph2 = e-008; ph3 p = -1.82e-006;; ph4 = ; ph5 = ; ph6 p = ; ph7 = ; k =pk1= BE 6 + +pk2 BE 5 + +pk3 BE 4 + +pk4 BE 3 + +pk5 BE 2 + +pk6 BE + pk7 h = =ph1 BE 6 + +ph2 BE 5 + +ph3 BE 4 + +ph4 BE 3 + +ph5 BE 2 + +ph6 BE + ph7 pco 2 lpco2=log 10 (pco 2 ) =(lpco2-h)/k; Fig. 30 Algorithm of the calcula on of erythrocyte tra on curves =bloodbeinv(pitot,alb,glob,chb,beox,pco 2,sO 2 ) For the principle of the calcula on and for the algorithm itself, see Fig. 31 and 32, respec vely. First, BE is calculated according to the grade of desatura on (from so 2 ) and BEox. This value is considered ini al for plasma and erythrocytes (BE). is calculated from pco 2. 14

17 Combination of blood and plasma acid-base models plazma BE blood BE log 10 (pco2) erythrocytes BE erythrocytes BE er p pco2 - mhco 3 BE p log 10(pCO2) pco2 (BEp) (BEp) (BEer) =bloodbeinv(pitot,alb,glob,chb,beox,pco 2,sO 2 ) (BEp) = (BEer) pco 2 er =erybeinv(pco 2,BE er ) Pitot, Alb,Glob SID BE - p =BE+mHCO 3 /(1-Hk) plasma BEp NSID BE p =SID-NSID BE er =BE-mHCO - 3 /Hk) BE er =BE-mHCO 3 - /Hk erythrocytes BEer BE ox BE=BEox+0,2(1-sO 2 ) chb mhco - 3 =(BEp-BE)(1-Hk) so 2 (BEp) (BEer) chb Hk=cHb/33.34 However, the plasma tra on curve has a smaller slope than that for erythrocytes (see Fig. 31) and plasma (H (BEp) ) is calculated according to plasma BE (BE p ); on the outer side of erythrocytes ( (BEer) ), calculated according to erythrocyte BE (BE er ), is di erent. Then, the transfer of bicarbonates between plasma and erythrocytes is calculated by itera on the transfer causes varia ons in plasma (BE p ) and erythrocyte (BE er ) BEs the ra o of BE varai ons in erythrocytes and plasma depend on haematocrit. The itera on converges to the nal value in plasma calculated according to both erythrocyte and plasma BEs ( = (BEp) = (BEer) ). The algorithm also calculates the normal SID (NSID) i.e. the SID, in which =7.4 with the given haemoglobin, albumin and phosphate concentra ons and pco 2 =40 torr. There is a wider de ni on of BE in this model compared with classical Siggaard-Andersen s nomogram interpreta on its normal value depends not only on haemoglobin concentra ons but also on albumin, globulin and phosphate concentra ons - like Siggaard-Andersen s van Slyke equa on (Siggaard-Andersen, 1977, 06). Unlike in classical plasma models by Stewart and his followers, this model enables to demonstrate that the rela onship between SID a pco 2 does not apply in the whole blood. The model (and the related formalised rela onships) can be used in a number of clinicalphysiological calcula ons. For the model, including its source text and the descrip on of all used mathema cal rela onships and algorithms, see = (BEer)? = (BE)(BEp) Fig. 31 Principle of the calcula on of the whole blood tra on curves. At given BE plasma and erythrocytes tra on curves (plasma BE and erythrocytes BE ) have a di erent slopes, hence at given pco 2 a di erent can be calculated. Searched blood tra on curve (blood BE ) lies between plasma BE and erythrocytes BE curves. In the blood at given haematocrit (Hk) plasma and erythrocyte BE (BE p and BE er ) is shi ed because of HCO 3- /Cl - erythrocyte-plasma exchanges. New tra on curves of plasma and erythrocytes (plasma BEp, erythrocytes BEer ) can be calculated. Algorithm seeks the intersec on of blood BE, plasma BEp, and erythrocytes BEer curves at given pco 2. Siggaard-Andersen nomogram was recalculated from original 38 C to standard 37 C. The experimental data of Fige and Fencl s model of plasma acid-base balance was combined with the data based on Siggaard-Andersen nomogram, Fig. 32 Algorithm of the calcula on of the whole blood tra on curves. corrected to 37 C. It was obtained a model of blood acid-base balance combining the plasma model with variable albumin, globulin and phosphate concentra ons and connected with the erythrocyte model. The model is a core of an extent model of acid-base balance which enables the realisa on of pathogenesis of acid-balance disturbances in compliance with the bilance approach to the interpreta on of ABB disturbances, published earlier (Kofránek et al., 07). The work was supported by the project of Na onal Programme of Research No. 2C06031, e-golem, the development project of Ministry of Educa on, Youth and Sports C/08 and by Crea ve Connec ons s.r.o. company. 1. Ashwood E.R., Kost G., Kenny M. (1983): Clinical Chemistry. 29: Astrup, P. (1956): A simple electrometric technique for the determina on of carbon dioxide tension in blood and plasma, total content of carbon dioxide in plasma, and bicarbonate content in separated plasma at a xed carbon dioxide tension (40 mm. Hg). Scand. J. clin. & Lab. Invest., 8: Dell R.D., Winters R.W. (1970) A model for the in vivo CO2 equilibra on curve. Am J Physiol. 219: Dubin A., Menises M.M., Masevicius F.D. (07): Comparison of three di erent methods of evalua on of metabolic acid-base disorders. Crit Care Med. 35: Dubin A. (07) Acid-base balance analysis: Misunderstanding the target Crit Care Med. 35: Fencl V., Rossing T.H. (1989): Acid-base disorders in cri cal care medicine. Ann Rev. Med. 40, 17-, Fencl V., Leith D.E. (1993): Stewart s quan ta ve acidbase chemistry: applica ons in biology and medicine. Respir. Physiol. 91: 1-16, Fencl J., Jabor A., Kazda A., Figge, J. (00): Diagnosis of metabolic acid-base disturbances in cri cally ill pa ents. Am. J. Respir. Crit. Care. 162:

18 9. Figge J., Mydosh T., Fencl V. (1992): Serum proteins and acid-base equilibria: a follow-up. The Journal of Laboratory and Clinical Medicine. 1992; 1: Figge J. (09): The Figge-Fencl quan ta ve physicochemical model of human acid-base physiology. Updtated version 15 January 09. Online Web site. Available at h p:// modelapplica on.html. Accessed Goldberg M., Green S.B, Moss M.L., Marbach C.B., Gar nkel D. (1973) Computerised instruc on and diagnosis of acid-base disorders. J. Am. Med. Assoc. 223: Grogono AW, Byles PH, Hawke W (1976): An in-vivo representa on of acid-base balance. Lancet, 1: , Kaplan L. (07): Acid-base balance analysis: A li le o target. Crit Care Med. 35: Kelum J.A. (05): Clinical review: Reuni ca on of acid-base physiology. Cri cal Care, 9: Kellum J.A. (Ed) (09): The Acid base orum. University of Pi sburgh School of Medicine, Department of Cri cal Care Medicine. Online Web site. Available at: h p:// educa on/ resources/phorum.html. 16. Kofránek, J. (19): Modelling of blood acid-base equilibium. Ph.D. Thesis. Charles University in Prague, Faculty of General Medicine, Prague, Kofránek J, Matoušek S, Andrlík M (07): Border ux ballance approach towards modelling acid-base chemistry and blood gases transport. In. Proceedings of the 6th EUROSIM Congress on Modeling and Simula on, Vol. 2. Full Papers (CD). (B. Zupanic, R. Karba, S. Blaži Eds.), University of Ljubljana, 1-9. Available at: h p://physiome.cz/publica ons/ Eurosim07ABB.pdf 18. Kurtz I. Kraut J, Ornekian V., Nguyen M. K. (08): Acid-base analysis: a cri que of the Stewart and bicarbonate-centered approaches. Am J Physiol Renal Physiol. 294: Lang W., Zander R (02): The accuracy of calculated Base Excess in blood. Clin Chem Lab Med. 40: Rees S.R., Andreasen S. (05): Mathema cal models of oxygen and carbon dioxide storage and transport: the acid-base chemistry of blood. Cri cal Reviews in Biomedical Engineering. 33: Siggaard-Andersen O, Engel K. (1960): A new acid-base nomogram. An improved method for the calcula on of the relevant blood acid-base data. Scand J Clin Lab Invest, 12: Siggaard-Andersen O. (1962): The, log pco2 blood acid-base nomogram revised. Scand J Clin Lab Invest. 14: Siggaard-Andersen O. (1974): An acid-base chart for arterial blood with normal and pathophysiological reference areas. Scan J Clin Lab Invest 27: Siggaard-Andersen O (1974) The acid-base status of the blood. Munksgaard, Copenhagen 25. Siggaard-Andersen O. (1977): The Van Slyke Equa on. Scand J Clin Lab Invest. Suppl 146: Siggaard-Andersen O., Wimberley P.D., Fogh- Andersen, Gøthgen I.H. (1988): Measured and derived quan es with modern and blood gas equipment: calcula on algorithms with 54 equa ons. Scand J Clin Lab Invest. 48, Suppl 189: Siggaard-Andersen O, Fogh-Andersen N. (1995): Base excess or bu er base (strong ion di erence) as measure of a non-respiratory acid-base disturbance. Acta Anaesth Scand. 39, Suppl. 107: Siggaard-Andersen O.(06): Acid-base balance. In: Laurent GJ, Shapiro SD (eds.). Encyclopedia of Respiratory Medicine. Elsevier Ltd. 06: Singer R.B. and Has ngs A.B. (1948): An umproved clinical method for the es ma on of disturbances of the acid-base balance of human blood. Medicine (Bal more) 27: Sirker, A. A., Rhodes, A., and Grounds, R. M. (01): Acid-base physiology: the tradi onal and modern approaches. Anesthesia 57: Schlich g R., Grogono A.W., Severinghaus J.W.: (1998) Human PaCO 2 and Standard Base Excess Compensa on for Acid-Base Imbalance. Cri cal Care Medicine. 26: Stewart PA. (1981): How to understand acid)base. A Quan ta ve Primer for Biology and Medicine. New York: Elsevier. 33. Stewart P.A. (1983): Modern quan ta ve acid-base chemistry. Can. J. Physiol. Pharmacol. 61, Watson, P.D. (1999): Modeling the e ects of proteins of in plasma. J. Appl Physiol. 86: Zander R. (1995): Die korrekte Bes mmung des Base- Excess (BE, mmol/l) im Blut. Anästhesiol Intensivmed No allmed Schmerzther. 30:Suppl 1: Corresponding author Ji í Kofránek, Laboratory of Biocyberne cs, Department of Pathophysiology, U nemocnice 5, Prague 2, Czech Republic, kofranek@gmail.com. 16

19 Physiological Research Jiří Kofránek, Jan Rusz: Restoration of Guyton Diagram for Regulation of the Circulation as a Base of Quantitative Physiological Model Development. Physiological Research. V recenzním řízení.

20

21 Restoration of Guyton Diagram for Regulation of the Circulation as a Base of Quantitative Physiological Model Development J. KOFRÁNEK 1, J. RUSZ 1,2 1 Charles University in Prague, 1 st Faculty of Medicine Department of Pathophysiology, Laboratory of Biocybernetics, Czech Republic 2 Czech Technical University in Prague, Faculty of Electrical Engineering, Department of Circuit Theory, Czech Republic Corresponding author Jan Rusz, Department of Circuit Theory, Czech Technical University, Technická 2, Prague 6, Czech Republic, ruszjan@fel.cvut.cz Summary We present the current state of complex circulatory dynamics model development based on famous Guyton diagram. The aim is to provide an open-source model that will allow the simulation of a number of pathological conditions on a virtual patient including cardiac, respiratory, and kidney failure. The model will also simulate the therapeutic influence of various drugs, infusions of electrolytes, blood transfusion etc. As a current result of implementation, we describe a core model of human physiology targeting the systemic circulation, arterial pressure and body fluid regulation, including short- and long-term regulations. The model can be used for educational purposes and general reflection on physiological regulation in pathogenesis of various diseases. Key words Body fluid homeostasis; Blood pressure regulation; Physiological modelling; Guyton diagram

22 Introduction The landmark achievement closely associated with integrative physiology development was the circulatory dynamics model published by prof. A. C. Guyton and his collaborators in 1972 (Guyton et al. 1972). Subsequently, its more detailed description was published in the monograph one year later (Guyton et al. 1973). This model represents the first large-scale mathematical description of the body s interconnected physiological subsystems. The model was described by a sophisticated graphic diagram with various computing blocks symbolizing quantitative physiological feedback connections. The diagram was published as a picture and the actual realization of the model was implemented in the FORTRAN language. Although the FORTRAN implementation worked correctly, the diagram contains a number of errors that cause wrong model behaviour. Moreover, FORTRAN implementation is not in correspondence with this famous graphic diagram, it is almost unavailable nowadays, and contains several programming and computation-related features that require special treatment (Thomas et al. 08). Despite the fact that the diagram was published over 30 years ago, it is currently used as a base for a number of research studies in the field of physiology (Montani and Van Vliet 09, Osborn et al. 09) and physiological modelling (Bassingthwaighte 00, Hunter et al. 02, Thomas et al. 08, Bassingthwaighte 09), including research on the physiological consequences of weightlessness in manned space flight (White et al. 1991, White et al. 03), or in a new approach to automation in medicine (Nguyen et al. 08). In addition, it is still reprinted with errors (Hall 04, Bruce and Montani 05). As the result of the only simple corrections of mathematical analyses, wrong interpretation of physiological relationships followed by incorrect model behaviour is occurred. The overall revision of the diagram requires exhaustive search for errors and sophisticated analyses of physiological regulations system. Here, we present prototype of core model of human physiology based on the original Guyton diagram targeting the short- and long-term regulation of blood pressure, body fluids and homeostasis of the major solutes. This model also includes the hormonal (antidiuretic hormone, aldosterone and angiotensine) and nervous regulators (autonomic control), and the main regulatory sensors (baro- and chemo-receptors). Our complex circulatory dynamics model corresponds to the same graphic notation of the original Guyton diagram and keeps an adherence of its basic physiological principles. While new models are continuously being developed (Srinivasan et al. 1996, Abram et al. 07, Hester et al. 08), our model finally brings fully functional modification of the original Guyton diagram, which is more suitable for the better and deeper understanding of the importance of physiological regulations and their use in development of many pathophysiological conditions by using simulation experiments. The resulting model can be used as a baseline for the quantitative physiological model development designated for physicians e-learning and acute care medicine simulators. Another use of the model consists in an effective learning aid for physiological regulation systems education, connected with biomedical engineering specialization. The model is provided as an open-source and it is downloadable at < Methods Mathematical model of global physiological regulation of blood pressure The model consists of 18 modules containing approximately 160 variables and including 36 state variables (see Table 1 for more details). Each module represents an interconnected physiological subsystem (kidney, tissue fluid, electrolytes, autonomous nervous regulation and hormonal control including antidiuretic hormone, angiotensine and aldosterone). The model is constructed around a central circulatory dynamics module in interaction with 17 peripheral modules corresponding to physiological functions (see figure 1) and complete model targeting the systemic circulation, arterial pressure and body fluid regulation, including short- and longterm regulations. Graphic presentation of the model allows a display of the connectivity among all physiological relationships. In essence, the model contains a total of approximately 500 numerical entities (model variables, parameters and constants). Members of the original Guyton laboratory have been continuously developing a more sophisticated version of the model, which is used for teaching (Abram et al. 07). Although it includes about 4000 variables, this more elaborate model is less well suited to our purposes then 1972 Guyton et al. model, because of its incomplete description and physiological relationships formulation. Physiological regulations system analyses

23 The original model represented as a sophisticated graphic diagram contains a number of errors which imply entirely incorrect physiological model behaviour. The correction of these errors demanded complicated physiological regulations system analyses. These include exhaustive revision of the complete model and its behaviour validation using several simulation experiments. In this stage, the original FORTRAN code of the Guyton et al. model was also used to compare the obtained simulation results. It is from reason that the original FORTRAN code run correctly; the errors were in the diagram only. Because it would be beyond the scope of this paper to discuss each error in the original Guyton diagram, as an example of the system analyses, we describe the five most significant errors which had the greatest role in creating the unpredictable model behaviour (see figure 2). The other errors are mostly caused by replaced mathematical operations, wrong set of normalization and damping constants, and replaced signs that determine the positive or negative feedback. The first error is the wrong flow direction marking of blood flow in the circulatory dynamics subsystem (see figure 2a). The rate of increase in systemic venous vascular blood volume (DVS) is the subtraction between all rates of inflows and rates of outflows. Blood flow from the systemic arterial system (QAO) means inflow and rate from veins into the right atrium (QVO) means outflow. Rate change of the vascular system filling as the blood volume changes (VBD) is calculated as the difference between the summation of vascular blood compartments and blood volume overall capacity, meaning that VBD is found in the outflow rate too. Equation (1) gives DVS: Correct eq.: DVS QAO VBD QVO, Erroneous eq.: DVS QAO VBD QVO. (1) The second error is an algebraic loop in the non-muscle oxygen delivery subsystem (see figure 2b). There is a wrong feedback connection in venous oxygen saturation (OSV), which would cause a constant rise of OSV and the model would rapidly became unstable. Equation (2) gives the OSV from the blood flow in non-renal, nonmuscle tissues (BFN), oxygen volume in aortic blood (OVA), rate of oxygen delivery to non-muscle cells (DOB) and hematocrit (HM), d( OSV ) BFN OVA DOB Correct eq.: OSV / Z 7, dt BFN HM 5 d( OSV ) BFN OVA DOB BFN OVA DOB Erroneous eq.:. (2) dt BFN HM 5 BFN HM 5 Z 7 Errors 3 and 4 involve simple subsystem red cells and viscosity. The third one is caused by positive feedback in the volume of red blood cells (VRC) computation (see figure 2c). Equation (3) gives the VRC from the red cell mass production rate (RC1) and rate factor for red cells destruction (RCK) where the product between VRC and RCK gives the red cell mass destruction rate, d( VRC) Correct eq.: RC1 VRC RCK, dt d( VRC) Erroneous eq.: RC1 VRC RCK. (3) dt The fourth error is caused by missing negative feedback in the portion of blood viscosity caused by red blood cells (VIE) computation (see figure 2d). VIE is computed from the output of integrator HM2 (HM after integration divided by the normalization parameter HKM). Without the negative feedback, HM2 would incessantly rise. Viscosity is proportionate to hematocrit and the integrator acts as a dampening element in the original Guyton et al. model. From experimental data it can be derived that dependence of blood viscosity on hematocrit is not linearly proportional (Guyton et al. 1973). In equation (4), we designed a negative feedback by adding HMK constant into the feedback and by changing the HKM normalization parameter, which caused stabilized behaviour of HM2, d( HM 2) HM 2 Correct eq.: HM, dt HMK d( HM 2) Erroneous eq.: HM. (4) dt

24 The fifth error is in the antidiuretic hormone control subsystem. The problem is in normalized antidiuretic hormone control computation (AHC) and normalized rate of antidiuretic hormone creation (AH ) computation (see figure 2e), when both values have a value of 1 under normal conditions. The solution emerges from the classic compartment approach. The hormone inflows into the whole-body compartment at the rate F I and outflows at the rate F O. Rate of its depletion is proportional to its concentration c, where F O = k c, and concentration depends on overall quantity of hormone M and on capacity of distribution area V. Equation (5) gives the quantity of hormone M in whole-body compartment, which depends on balance between hormone inflow and outflow, dm km F dt I V. (5) Provided that the capacity of distribution area V is constant, we will substitute the ratio k/v with constant k 1. Guyton calculated the concentration of hormone c 0 normalized as a ratio of current concentration c to its normal value c norm = c/c 0. At invariable distribution area V, ratio of concentrations is the same as a ratio of current hormone overall quantity M to overall hormone quantity under normal conditions M norm = M/c 0. When we formulate the rate of flow in a normalized way (as a ratio to normal rate), under normal conditions it holds that F I = 1, dm norm /dt = 0 and after substituting it into equation (5) we get the equation (6), 1 k 1M norm 0. (6) The relative concentration of hormone c 0 can be formulated as equation (7), M c0 k1m, (7) M norm and after final adjustments and inserting into a differential equation (7) we arrive at dc Correct eq.: 0 F c0 k I 1, (8) dt dc Erroneous eq.: 0 F c k I 0 1. (9) dt According to equation (8), the normalized concentration of hormone c 0 is calculated from normalized inflow of hormone F I. In the original Guyton diagram, the normalized concentration of aldosterone and angiotensine is calculated this way, which means that normalized rate of inflows is F I = AH and normalized concentration of hormone is c 0 = AHC. As a result, AHC is represented by equation (9) instead of equation (8) in the original Guyton diagram. Equation (10) gives the final relation of AHC represented in model: d( AHC) Correct eq.: ( AH AHC) 0.14, dt d( AHC) Erroneous eq.: AH AHC (10) dt Model under SIMULINK SIMULINK is a block-based language for describing dynamic systems, and also works as a modelling and simulation platform (we used version R07b, integrated with MATLAB, The MathWorks, Nattick, MA, USA). It is an interactive and graphic environment dedicated to the multi-domain simulation of hybrid continuous/discrete systems. During simulations, model and block parameters can be modified, and signals can be easily accessed and monitored. In the model, numerical integration was performed using ode13t (a MATLAB library) with a variable step size (maximum step size, auto; relative tolerance, 10K3). First, code operations and routines from the computer program were rendered into the SIMULINK graphical description, i.e. elementary blocks and subsystems were connected by appropriate signals and the graphic notation of the original Guyton diagram was kept as much as possible (Kofránek and Rusz 07). Second, subsystems are not treated as atomic subunits. This causes SIMULINK s solver to treat each subsystem as a complete functioning model. Technically, the model works in continuous time and performs all physiological regulations as a complete unit (as the original graphic diagram was designed the FORTRAN implementation of the model is characterized by a wide range of time scales in the different subsystems), which

25 provides an advantage when designing control systems using principles of complex physiological regulation. All calculations were performed using only the original damping constants obtained from Guyton diagram. Finally, to remove a lack of convergence due to oscillation and other run-time errors, the model has addressed the algebraic loops. Note that complex model behaviour depends also on correct communication between all subsystems. In this case, it was essential to normalize some of the experimental set and dumping constants and supervise model behaviour. The complete model is available as open-source on < Model validation In order to validate our corrected SIMULINK implementation of the Guyton diagram, we simulated four experiments described in the (Guyton et al. 1972) paper and compared the results with: 1) clinic data measurements in a series of six dogs, data adopted from (Chau et al. 1979); 2-4) the original Guyton et al. model implementation in the FORTRAN environment. The first experiment is the simulation of hypertension in a salt-loaded, renal-deficient patient by decreasing the functional renal mass to ~ 30% of normal and increasing the salt intake to about five times normal on day 0. This is very fundamental experiment revealing the importance of the kidneys in blood-pressure control and their influence in the development of essential hypertension (Langston et al. 1963, Douglas et al. 1964, Coleman and Guyton 1969, Cowley and Guyton 1975). The duration of the whole experiment is 12 days. The second benchmark experiment represents sudden severe muscle exercise and takes place over a much shorter time scale than other experiments (5 min). The exercise activity was increased to sixty times the normal resting level by setting the exercise activity-ratio with respect to activity at rest after 30 second, corresponding to an approximately 15-fold increase in the whole-body metabolic rate (in this case, the time constant for the local vascular response to metabolic activity was reduced by 1/40). The third benchmark experiment simulates the progress of nephrotic edema by increasing seven-fold the rate of plasma-protein loss on day 1. After seven days, the rate of plasma-protein loss is reduced to three-times above the norm. The duration of the whole experiment is 12 days. The fourth benchmark experiment simulates the atrioventricular fistula by opening the fistula on day 1 (the constant that represents fistula is set to 5%) and closing the fistula on day 5. The duration of the whole experiment is 9 days. The goodness-of-fit of model was also compared in terms of the chi-square (χ 2 ) test between observed simulation results and predicted clinical data. Results Figure 3 represents the results of the simulation of hypertension (1. experiment). The cardiac output rose at first to ~ 30% above normal but then was stabilized by the end of 12 days. The arterial pressure rises more slowly, requiring several days to reach high elevation. During the next days it remained at its new high level indefinitely, as long as the high salt intake was maintained. The simulation is quite sufficient to predict the available data with high statistical significances of χ 2 (11) = 1445; p < for simulation of the arterial pressure, χ 2 (10) = 939; p < for simulation of the heart rate, χ 2 (10) = 1388; p < for simulation of the stroke volume, χ 2 (10) = 1189; p < for simulation of the cardiac output, and χ 2 (10) = 1304; p < for simulation of the total peripheral resistance. Figure 4 presents the results of the muscle exercise simulation (2. benchmark experiment). At the onset of exercise, cardiac output and muscle blood flow increased considerably and within a second. Urinary output fell to its minimal level, while arterial pressure rose moderately. Muscle cell and venous PO 2 fell rapidly. Muscle metabolic activity showed an instantaneous increase but then decreased considerably because of the development of a metabolic deficit in the muscles. When exercise was stopped, muscle metabolic activity fell below normal, but cardiac output, muscle blood flow and arterial pressure remained elevated for a while as the person was repaying their oxygen dept. Figure 5 illustrates the results of the nephrosis simulation (3. benchmark experiment). The principal effect of nephrosis consists of urine protein excretion that may or may not be associated with any significant changes in other renal functions. A deficit of the total plasma protein reduces the oncotic pressure, resulting in a fluid redistribution from the blood to the interstitial compartment and an increase of the (mostly free-) interstitial-fluid volume. Another effect is mild decreases of cardiac output and arterial pressure. The initial hypoproteinemia

26 only slightly decreased both arterial pressure and cardiac output but induced a notable restriction of the urinary output. Thus, the fluid was being retained in the organism causing the interstitial swelling, although the volume of the free interstitial fluid remained relatively unchanged until the interstitial-fluid pressure stayed negative. After it reached positive values, an apparent edema occurred with a sharp drop in the arterial pressure. When the rate of renal loss of protein was increased to the point where the liver could increase the plasma protein level, the edema was relieved with high diuresis and increased cardiac output by the end of 12 days. In figure 6 are shown the results of atrioventricular fistula simulation (4. benchmark experiment). Opening the fistula caused an immediate dramatic change in cardiac output, total peripheral resistance and heart rate. Urinary output decreased to minimal threshold levels. As the body adapted, extracellular fluid volume and blood volume increased to compensate for the fistula with the result that after a few days arterial pressure, heart rate and urinary output were near normal levels, while cardiac output doubled and peripheral resistance halved. When the fistula was closed on, a dramatic effect occurred with a rapid decrease in cardiac output, rapid increase in peripheral resistance, moderate increase in arterial pressure and moderate decrease in heart rate. Marked diuresis reduced the extracellular fluid volume and blood volume to normal or slightly below. After 9 days, the patient was nearly normal. Discussion and Conclusion The main goal of this paper is the implementation of the core circulatory dynamics model based on Guyton s original diagram and its validation with real experimental data. It was shown how a model might furnish a physiological interpretation for the statistical results obtained on clinical data. We also used the output from Guyton experiments (Guyton et al. 1972) as a benchmark to validate our implementation. One such problem is the regulation of arterial blood pressure, as was well established by Guyton and his collaborators, since their quantitative systems models led them to a deep reorientation of the understanding of the causes of hypertension (Guyton et al. 1967, Guyton 19, Guyton 1990). This was our rationale for adopting Guyton diagram as the initial demonstrator of the core model. As an example of general reflection on physiological regulation, we further discuss significant differences between the output of the last two simulations including nephrosis and atrioventricular fistula. The both experiments are associated with significant changes in functions of kidneys; involve changes in urinary output, arterial pressure, cardiac output, and plasma or blood volume. In simulation of the circulatory changes in nephrosis, the seven-fold rate of plasma-protein loss caused fast decrease of proteins volume in plasma. Reduced oncotic pressure of proteins led to transfer of water from plasma into interstitium, and decrease of plasma volume which caused decrease of arterial pressure. Decreased volume of plasma led also to decrease of pressure in atriums followed by decrease of the cardiac output. As a result of decreased arterial pressure, vasoconstrictor effects of autonomic autoregulation caused rapid decrease of urinary output. Reduced volume of plasma proteins lowered intake of oncotic pressure of proteins in glomerular capillaries, and thus caused increase in glomerular filtration and sequential diuresis. Continuous transfer of water from plasma into interstitium and decrease of arterial pressure resulted in slow decrease of diuresis into minimal threshold levels. Considering that simulated patient could not loss more plasma proteins through the kidneys, the rate of plasma proteins was reduced to three-fold of norm after 7 days of experiment. This effect was sufficient to stop decrease and sequential increase of concentration of plasma proteins in consequence of proteins synthesis progress in liver. Considering water accumulation in interstitium, the interstitial fluid pressure increased, slight increase of proteins was sufficient to invert equilibrium on capillary membrane, and water began resorb from interstitium to plasma. This was associated with increased of plasma volume and sequential diuresis. The results from the simulation are almost identical with those that occur in patients with nephrosis (Guyton et al. 1972, Lewis et al. 1998). This includes the failure to develop sufficient amounts of edema until the protein concentration falls below a critically low level of about third of normal (Guyton et al. 1972). The simulation also shows the typical tendency for nephrotic patients to have a mild degree of circulatory collapse and slightly decreased plasma volumes (Guyton et al. 1972). Other important effect is the changing level of urinary output, a feature that also occurs in nephrotic patients, with urinary output falling very low during those periods where large amounts of edema are being actively formed and the urinary output becoming great during those periods when edema is being resorbed (Guyton et al. 1972).

27 Simultaneously as the simulation of nephrosis, simulation of the atrioventricular fistula was associated with inceptive rapid decrease of urinary output. Opening the fistula caused dramatic decrease of peripheral resistance and immediate increase of cardiac output. This resulted in acute reaction of autonomic system which rapidly decreased glomerular filtration using increase of resistance, and thus practically stopped the urinary output. In consequence of stopped urinary output, the blood volume was increased, vasoconstrictor reaction in kidneys was subsided, and diuresis was re-established. Circulatory system dynamics shifted to its new dynamic equilibrium with increased cardiac output and blood volume, and decreased peripheral resistance. After closure of the fistula, this whole process was reoriented. The kidneys rapidly urinated redundant blood volume and circulatory dynamics system was returned to normal levels. An important effect of fistula management can be listed in (Friesen et al. 00). This simulation among others shows the essential importance of renal blood volume control for maintenance of blood pressure. Our circulatory dynamics model can also be used to simulate other experiments including simulations of development of general heart failure, effects of removal of the sympathetic nervous system on circulatory function, effect of infusion of different types of substances, effects of vasoconstrictor agents acting on different parts of the circulation, effects of extreme reduction of renal function on circulatory function, and others. Created SIMULINK diagram involves tracking the values of physiological functions during simulation experiments and also disconnect the individual regulation circuits using switches. It allows tracking the importance of individual regulation circuits in progression of number various pathological conditions. As an example, in atrioventricular fistula experiment, when the AUM-parameter (sympathetic arterial effect on arteries) in kidneys is reconnected to norm value, the kidneys will not responded on increased autonomic system activity. Simultaneously in the nephrotic experiment, when the PPC-parameter (plasma colloid osmotic pressure) is reconnected to norm value, the kidneys will not increase diuresis in response of decrease of plasma proteins volume. The restored Gyuton diagram is became interactive educational aid that allows through model experiments better reflection on general physiological regulations in pathogenesis of various diseases. The result of this study is not only a complex functional model, but also a correction of the frequently published Guyton diagram, which still remains a landmark achievement. The model evolved over the years, but the core of the model and the basic concepts remained untouched and many of the principles contained in the original model have been incorporated by others into advanced models (Abram et al. 07, Hester et al. 08). The originality of our core model implementation is our commitment to providing documentation for each basic module and continuous interactive modification and development of any aspect of the model parameters or equation and its documentation. The complex medicine simulator based on the quantitative physiological model will make it possible to simulate a number of pathological conditions on a virtual patient and the effect of the artificial organ use on normal physiological function could have been simulated. These include artificial heart, artificial ventilator, dialysis, and others. Acknowledgement This research was supported by the research programs Studies at the molecular and cellular levels in normal and in selected clinically relevant pathologic states MSM , and Transdisciplinary Research in Biomedical Engineering MSM , and by the grants e-golem: medical learning simulator of human physiological functions as a background of e-learning teaching of critical care medicine MSM 2C06031, and Analysis and Modelling Biological and Speech Signals GAČR 102/08/H008, and by Creative Connection Ltd. We are obliged to R. J. White for provision of FORTRAN implementation of the original Guyton et al. model.

28 References ABRAM SR, HODNETT BL, SUMMERS RL, COLEMAN TG, HESTER RL: Quantitative circulatory physiology: an integrative mathematical model of human physiology for medical education. Adv. Physiol. Educ. 31: 2 210, 07. BASSINGTHWAIGHTE JB: Strategies for the Physiome Project. Ann. Biomed. Eng. 28: , 00. BASSINGTHWAIGHTE J, HUNTER P, NOBLE D: The Cardiac Physiome: perspectives for the future. Exp. Physiol. 94: , 09. BRUCE NVV, MONTANI J-P: Circulation and Fluid Volume Control. In: Integrative Physiology in the Proteomics and Post-Genomics Age. W. Walz (eds), Humana Press, Totowa, NJ, 05, pp CHAU NP, SAFAR ME, LONDON GM, WEISS YA: Essential Hypertension: An Approach to Clinical data by the Use of Models. Hypertension. 1: 86-97, COLEMAN TG, GUYTON AC: Hypertension caused by salt loading in the dog. III. Onset transients of cardiac output and other circulatory variables. Circ. Res. 25: , COWLEY AW, GUYTON AC: Baroreceptor reflex effects on transient and steady-state hemodynamics of saltloading hypertension in dogs. Circ. Res. 36: , DOUGLAS BH, GUYTON AC, LANGSTON JB., BISHOP VS: Hypertension caused by salt loading. II. Fluid volume and tissue pressure changes. Am. J. Physiol. 7: , FRIESEN CH, HOWLETT JG, ROSS DB: Traumatic coronary artery fistula management. Ann. Thorac. Surg. 69: , 00. GUYTON AC, COLEMAN TG: Long-term regulation of the circulation: interrelationships with body fluid volumes. In Physical bases of Circulatory Transport Regulation and Exchange, edited by E. B. Reeve and A. C. Guyton. Philadelphia, PA: Saunders, 1967, pp GUYTON AC, COLEMAN TG, GRANDER HJ: Circulation: Overall Regulation. Ann. Rev. Physiol. 41:13-41, GUYTON AC, JONES CE, COLEMAN TG: Circulatory Physiology: Cardiac Output and Its Regulation. WB Saunders Company, Philadelphia, 1973, p GUYTON AC: Arterial Pressure and Hypertension. Philadelphia, PA: Saunders, 19. GUYTON AC: The suprising kidney-fluid mechanism for pressure control its infinite gain! Hypertension. 16: , (1990). HALL JE: The pioneering use of system analysis to study cardiac output regulation. Am. J. Physiol. Regul. Integr. Comp. Physiol. 287: 9-1, 04. HESTER RL, COLEMAN T, SUMMERS R: A multilevel open source integrative model of human physiology. The FASEB Journal. 22: 756.8, 08. HUNTER PJ, ROBINS P, NOBLE D: The IUPS Physiome Project. Pflugers Archiv - European Journal of Physiology. 445: 1-9, 02. KOFRÁNEK J, RUSZ J: From graphic diagrams to educational models. Cesk. Fysiol. 56: 69 78, 07. LANGSTON JB, GUYTON AC, DOUGLAS BH, DORSETT PE: Effect of changes in salt intake on arterial pressure and renal function in nephrectomized dogs. Circ. Res. 12: , LEWIS DM, TOOKE JE, BEAMAN M, GAMBLE H, SHORE AC: Peripheral microvascular parameters in the nephrotic syndrome. Kidney Int. 54: , MANNING RD, COLEMAN TG, GUYTON AC, NORMAN RA, McCAA RE: Essential role of mean circulatory filling pressure in salt-induced hypertension. Am. J. Physiol. 236: 40-R7, MONTANI J-P, VAN VLIET BN: Understanding the contribution of Guyton s large circulatory model to longterm control of arterial pressure. Exp. Physiol. 94: , 09. NGUYEN CN, SIMANSKI O, KAHLER R, et al.: The benefits of using Guyton s model in a hypotensive control system. Comput. Meth. Prog. Bio. 89: , 08. OSBORN JW, AVERINA VA, FINK GD: Current computational models do not reveal the importance of the nervous system in long-term control of arterial pressure. Exp. Physiol. 94: , 09. SRINISAVAN RS, LEONARD JI, WHITE RJ: Mathematical modelling of physiological states. Space biology and medicine. 3: , 1996.

29 THOMAS SR, BACONNIER P, FONTECAVE J, et al.: SAPHIR: a physiome core model of body fluid homeostasis and blood pressure regulation. Phil. Trans. R. Soc. A. 366: , 08. WHITE RJ, LEONARD JI, SRINIVASAN RS, CHARLES JB: Mathematical modelling of acute and chronic cardiovascular changes during extended duration orbiter (EDO) flights. Acta Astronaut. 23: 41 51, WHITE RJ, BASSINGTHWAIGHTE JB, CHARLES JB, KUSHMERICK MJ, NEWMAN DJ: Issues of exploration: human health and wellbeing during a mission to Mars. Adv. Space Res. 31: 7 16, 03.

30 Table 1 List of state variables used in the original Guyton diagram with physiological significances, block numbers, and abbreviations. State variable in selected subsystem Block number Abbreviation Circulatory dynamics Venous vascular volume 6 VVS 02. Right atrial volume 13 VRA 03. Volume in pulmonary arteries 19 VPA 04. Volume in left atrium 25 VLA 05. Volume in systemic arteries 31 VAS Vascular stress relaxation Increased vascular volume caused by stress relaxation 65 VV7 Capillary membrane dynamics Plasma volume 71 VP 08. Total plasma protein PRP Tissue fluids, pressure and gel Total interstitial fluid volume 84 VTS 10. Volume of interstitial fluid gel 101 VG 11. Interstitial fluid protein 103 IFP 12. Total protein in gel 112 GPR Electrolytes and cell water Total extracellular sodium 118 NAE 14. Total extracellular fluid potassium 122 KE 15. Total intracellular potassium concentration 131 KI Pulmonary dynamics and fluids Pulmonary free fluid volume 142 VPF 17. Total protein in pulmonary fluids 149 PPR Angiotensin control Angiotensin concentration 159 ANC Aldosterone control Aldosterone concentration 170 AMC Antidiuretic hormone control Degree of adaption of the right atrial pressure 1 AHY 21. Antidiuretic hormone concentration 185 AHC Thirst and drinking Kidney dynamics and excretion Muscle blood flow control and PO Rate of increase in venous vascular volume 231 DVS 23. Total volume of oxygen in muscle cells 238 QOM 24. Muscle vascular constriction caused by local tissue control 254 AMM Non-muscle oxygen delivery Non-muscle venous oxygen saturation 260 OSV 26. Non-muscle total cellular oxygen 271 QO2 Non-muscle, non-renal local blood flow control Vasoconstrictor effects of rapid autoregulation 278 AR1 28. Vasoconstrictor effects of intermediate autoregulation 285 AR2 29. Vasoconstrictor effects of long-term autoregulation 289 AR3 Autonomic control Time delay for realization of autonomic drive 305 AU4 31. Overall activity of autonomic system 310 AUJ Heart rate and stroke volume Red cells and viscosity Volume of red blood cells 332 VRC 33. Hematocrit 336 HM2 Hearth hypertrophy or deterioration Hypertrophy effect on left ventricle 344 HPL 35. Hypertrophy effect on heart 349 HPR 36. Cardiac depressant effect of hypoxia 352 HMD

31 Figure 1 Block diagram of the original Guyton et al. model subscribed by subsystems

32 Figure 2 The most significant errors of the original diagram and their correction