MEDSOFT 2009

JiříKofránek:Complexmodelofbloodacid-basebalance.InMEDSOFT 2009, sborník příspěvků,(editor:MilenaZiethamlováEd.)Praha:AgenturaActionM, Praha, ISBN 978-80-904326-0-4, str. 23-60, originally published inCzech,Englishtranslationofthepaperisavailableathttp://www.physiome.cz/references/AcidBaseMedsoft2009.pdf, model is available at http://physiome.cz/AcidBase.

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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 pH 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 NaHCO3, KHCO3 and, later, CaCO3 and CaHPO4 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 CO2 and H+/HCO3

- � 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. CO2 reten� on or deple� on during the change of carbon dioxide balance as well as H+/HCO3

- 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 non-bicarbonate 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 non-bicarbonate bu� er bases forms the Bu� er Base (BB) value:

BB=[HCO3-]+[Buf-]

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The varia� ons in pCO2 result in pH changes; If the � tra� on curve of pCO2 and pH changes is plo� ed in the semi-logarithmic scale, these � tra� on curves verge on lines in the range of life-compa� ble pH 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 pCO2. There were, however, rela� vely accurate methods of pH measurement. Astrup’s method of pCO2 analysis (1956) was based on the following procedure: � rst, blood pH was measured, then, the sample was automa� cally equilibrated by O2/CO2 mixture with accurately set pCO2. The blood sample was equilibrated with a high pCO2 gas mixture and the equilibra� on was followed by measuring pH. Then, the blood was equilibrated with a mixture with low carbon dioxide par� al pressure and the equilibra� on was followed by another pH measurement. The points obtained were plo� ed into a semi-logarithmic graph to create a line, used to read out pCO2 according to baseline pH (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.

COMPLEX MODEL OF BLOOD ACID-BASE BALLANCEJi�í Kofránek

Laboratory of Biocyberne� cs, Departmernt of Pathological Physiology, First Faculty of Medicine, Charles University of Prague.e-mail: kofranek@email.com

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 pH and pCO2 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 pCO2 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 2009. (Milena Ziethamlová Ed.) Prague: Crea� ve Connec� ons, Prague 2008, pp. 23-60. ISBN 978-80-904326-0-4

Key words

Acid-base balance, formalised descrip� on, simula� on model, blood gases, educa� onal simulators

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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 PCO2/pH coordinates. The � tra� on curves (being lines in the semi-logarithmic 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

with the same BB crossed in the same points, too.

Thus, a nomogram with BE and BB curves with semi-logarithmic 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 pH on pCO2 and haemoglobin (Hb) concentra� ons; the results obtained were used to create clinically applicable nomograms expressing the following dependence:

[H+]= func� on (pCO2,BE,Hb)

In the assessment of acid-base disturbances by BE and pCO2, it should be taken into considera� on that the increase or the fall in CO2 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 pCO2 independent. This applies for plasma exactly but not exactly for the whole blood – pCO2 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 non-bicarbonate bu� er concentra� ons), the total concentra� on of bu� er bases BB also depends on haemoglobin oxygen satura� on (suscep� ble by pCO2).

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 100% oxygen satura� on of haemoglobin). Similarly, the standardised Base Excess oxy-value (BEox) is de� ned as BE measured in the blood sample with full oxygen satura� on of oxyhaemoglobin (Kofránek, 1980). Thus, BEox is really pCO2 – independent.

It is necessary to say that the independence of pCO2 and BEox does not apply for “in vivo” whole blood completely, as the increase in pCO2 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 pCO2 (with a mild fall in BEox in acute pCO2 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 CO2 plasma concentra� ons (expressed as pCO2) and BE therefore characterise carbon dioxide � ow balance and the varia� on in strong acid produc� on/excre� on balance, respec� vely. Thus, pCO2 and BE characterise the respiratory and metabolic parts of acid-base balance, respec� vely.

Logarithmic�PCO2 scale

Constructed logpCO2 pHtitration line

High pCO2 inO2/CO2 mixture

Derived pCO2Value fromValue fromtitration line

Low pCO2 inO /CO mixture

pH scaleO2/CO2 mixture

pHafter equilibrationwith low pCO2

Measured pHbeforeequilibration

with pCO2

pHafter equilibrationwith high pCO2

Fig. 1 The � tra� on curve of pH/PCO2 varia� ons a� er blood equilibra� on with carbon dioxide is prac� cally a line. This therefore enabled pCO2 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 CO2 pressure.

BE=0�mEq/l

BE=�15�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.

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To use pH, pCO2 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 pCO2, pH and pO2) 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.

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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 1980s. The most used method was Stewart´s one (1983), improved later for clinical prac� ce by Fencl et al. (1989, 1993, 2000).

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] = [BufTOT]

3. Dissocia� on balance of non-bicarbonate bu� er system:

[Buf-] [H+] = KBUF × [HBuf]

4. Dissocia� on balance of bicarbonate bu� er:

[H+] [HCO3-] = M × pCO2

5. Dissocia� on balance between bicarbonate and carbonate:

[H+] [CO32-] = N × [HCO3

-]

6. Electroneutrality:

SID + [H+] – [HCO3-] – [Buf-] – [CO3

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+] + [Mg2+] + [Ca2+] - [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 [BufTOT] and pCO2 (the dependent variable is underlined in the equa� on, independent varia� ons and constants are in bold and italic, respec� vely):

[H+]4 + (SID + KBUF) × [H+]3 + (KBUF × (SID - [BufTOT]) - K’w – M × pCO2) × [H+]2 - (KBUF × (K’w2 + M × pCO2) - N × M × pCO2) × [H+] - K’w × N × M × pCO2 = 0

Solving of the equa� on gives hydrogen ion concentra� on, depending on the respiratory part of acid-base balance – i.e. pCO2, 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 [BufTOT]:

pH = func� on ( pCO2, SID, [BufTOT] )

The total concentra� on of non-bicarbonate bases [BufTOT] 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 pH (and other variables such as bicarbonate concentra� ons etc.) from pCO2, SID, and total phosphate [Pi] and plasma albumin [Alb TOT] concentra� ons (see, for example, Watson, 1999):

pH = func� on ( pCO2, SID, [AlbTOT], [Pi] )

One of the most detailed quan� ta� ve analyses of plasma acid-base balance (Figge, 2009) 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.

pH = func� on ( pCO2, SID, [AlbTOT], [Pi], [Cit] )

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Mathema� cal rela� onships between the variables derived from the quan� ta� ve physical-chemical analysis enable calcula� on of dependent variables – pH, being a base for other dependent variables, i.e. bicarbonate concentra� ons – from independent variables (i.e. pCO2, 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

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necessary to do explicit measurements of phosphate, Na+, Cl-, HCO3

- 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 - pH (and proper concentra� ons of bicarbonates, carbonates and non-bicarbonate acids) – from independent variables (pCO2, SID, [AlbTOT], [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.

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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. (2001) 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).

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Excited debates lead by supporters of both theories in interna� onal journals (e.g. Dubin et al. 2007, Dubin 2007, Kaplan 2007, Kurz et al., 2008, Kelum 2009) 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

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, 2005).

The accordance and di� erences of both approaches are as follows.

Both Stewart and Siggaard-Andersen use pCO2 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 (2006) 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, 2009): 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 NH2–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:

Ca2�Mg2� Cations Anions150

mmol/L

BBSID

Ca K+

HCO3�

Organic anionsSO4

2�HPO4

2� +�H2PO4�Pr�

Organic anions100

Na+ Cl�Na+ Cl

50

Fig. 3 SID and BB are nearly iden� cal. The varia� ons in SID and BB are completely iden� cal: dSID=dBB.

5

Alb + H+ = HAlb+

Labelling the total concentra� ons of non-bicarbonate bases by Stewart and Siggaard-Andersen as [Bufst

-] and [Bufsa-],

respec� vely, a small di� erence can be observed (the concentra� ons are considered in miliequivalents):

[Bufst-] = [PO4

3-] + [HPO42-] + [H2PO4

-] + [Alb-] – [HAlb+]

[Bufsa-] = [PO4

3-] + [HPO42-] + [H2PO4

-] + [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 non-bicarbonate bases by Stewart:

d[Bufst-] = d[Bufsa

-]

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 ([AlbTOT], [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, 2009) and experimental data for the whole blood, included in Siggaard-Andersen nomogram.

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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 (2002) 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 1980, 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.

However, the situa� on in 1980s 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

pH=BEINV(cHb,BEox,sO2,pCO2)

where cHb is haemoglobin concentra� on (in g/100 ml blood), BEox is BE (mmol/l) with 100% haemoglobin satura� on (being therefore independent on haemoglobin oxygen satura� on), sO2 is haemoglobin oxygen satura� on and pCO2 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 pH according to BEox, haemoglobin concentra� on cHB, haemoglobin oxygen satura� on sO2 and pCO2 (Fig. 5). The calcula� on of pH takes advantage of the fact that the � tra� on curves are prac� cally lines in the semi-logarithmic scale (log pCO2, pH).

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 pH and pCO2, 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.

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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

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���

BE (mmol/l)Y)a65(pCO2))/(alogY)a4(a3Y*a2(a1pH 10 �������

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 (1980): ‘×’ and by Zander (1995): ‘+’.

6

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 2005).

On the contrary, models of plasma acid-base balance, e.g. Watson´s (Watson, 1999) or Figge-Fencl´s (Figge, 2009)

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 pH and pCO2 from t° to the standard temperature of 37°C are based on simple rela� onships, e.g. (Ashwood et al. 1983):

pH37°C = pHt° - 0.0147 (37-t°)

log10(pCO2 37°C) = 0.019 log10(pCO2 37°C)(37-t°)

For proper temperature correc� ons of Siggaard-Andersen nomogram it is advisable to use the more accurate rela� onship by Ashwood et al. (1983):

pH37°C = (pHt° - 0.0276(37-t°) - 0.0065 (7.4) (37 – t°) + 0.000205 (372 – t°2))/(1-0.0065(37-t°))

log10(pCO2 37) = log10(pCO2 t°) + (0.02273 – 0.00126 (7.4 - pH37°C))(37 - t°) – 0.0000396(372-t2)

However, to correct Siggaard-Andersen nomogram from 38°C to 37°C, it is insu� cient to transfer simply log10pCO2 and pH, 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 (pCO2=40 torr and pH=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 pH=7.4, and pCO2=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 pCO2=38.2195 torr and pH=7.421 then (Fig. 8). Our aim is, however, to achieve that pCO2 and pH corresponding to zero BE are 40 torr and 7.4 on the curve for 37°C.

Thus, standard pH and pCO2 are re-calculated from 37°C to 38°C as follows:

pH37°C = 7.4 37°C -> pH38°C = 7.3878 38°C

2

1.4

1.6

1.8

Coordinates of BB curve

1

1.2[log(pCO2), pH ] = function (BB)

� � )(22log BBBBLPCOpCO �

10 20 30 40 50 60 70 80

0.8

� � )(22log10 BBBBLPCOpCO BB �

7.5

7.6

7.2

7.3

7.4

pH

6.9

7

7.1

10 20 30 40 50 60 70 806.8

BB [mmol/l]

)(BBPHBBpHBB �Siggaard-Andersen curve nomogram

Fig. 5. Approxima� on of BB curve from Siggaard-Andersen nomogram by means of splines.

Fig. 6 Approxima� on of BE curve from Siggaard-Andersen nomogram by splines.

1.8

1.5

1.6

1.7

orr]

Coordinates of BE curve[log(pCO2), pH ]= function (BE) 1.2

1.3

1.4

lg P

CO

2 [to

-25 -20 -15 -10 -5 0 5 10 15 20 251

1.1

BE [mmol/l]

� � � �BEBELPCOCO 22l � � � �BEBELPCOpCO BE 22log10 �

7.9

8

7.6

7.7

7.8

7.3

7.4

7.5

)(BEPHBEpHBE �

-25 -20 -15 -10 -5 0 5 10 15 20 257.2

Siggaard-Andersen curve nomogram

Fig. 7 Algorithm of the calcula� on of � tra� on curves by Siggaard-Andersen nomogram formalised by means of splines.

� �2,2,, pCOsOBEoxcHbBEINVfunctionpH �

cHbsOBEoxBE *)21(20 ��

cHb sO2 BEoxcHb,Beox,sO2,pCO2

cHbsOBEoxBE )21(2,0��

cHbNBB 42,07.41 �� BBNBBX1,y1

BENBBBB ��

� � )(22log1 10 BBBBLPCOpCOx BB �� pCO2BB

x1

)(1 BBPHBBpHy BB ��

� � � �BEBELPCOCO 22l2 BE

BB

x2

y1

BB

� � � �BEBELPCOpCOx BE 22log2 10 ��

)(2 BEPHBEpHy BE ��BE

y2

X2,y2

1)21/()21)(1( yxxyyxxypH ���

2log 10 pCOx �BE

1)21/()21)(1( yxxyyxxypH �pH

pH

Fig. 8 In the points pH37°C = 7.4 and pCO2 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 (pH38°C = 7.3878 and pCO2 38°C = 41.862 torr); BEs are, however, non-zero and di� er for each blood sample.

Warming up 1°C

38°CH 7 3878

37°CH 7 4 pH=7.3878

pCO2=41.862 torrpH=7.4

pCO2=40 torr

7

pCO2 37°C = 40 torr38°C -> pCO2 38 °C = 41.862 torr 38°C

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 pH=7.4 and pCO2=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 BE37°C, in the condi� on of full oxygen satura� on (see the calcula� on algorithm scheme in Fig. 13). Correc� on factor dBE38°C (depending on haemoglobin concentra� on and corresponding to BE zero value at 37°C) was always added to each BE37°C. This correc� on shi� was a base for BE38°C.

BE38°C = BE37°C + dBE38°C

A set of pH38°C was calculated from a set of BE38°C and pCO2

38°C by Siggaard-Andersen nomogram (by means of BEINV algorithm – see Fig. 7). pCO2 38°C and pH38°C were then re-calculated 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 BE37°C and various haematocrit characterise the BE curve of Siggaard-Andersen nomogram corrected to 37°C (see Fig. 14).

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 pH and pCO2 coordinates correspond to pH=7.4 and pCO2 = 40 torr a� er cooling the blood by one degree.

90

100

120

plasma,BE38°C =0.3036mmol/lcHb =5g/100ml,BE38°C=0.2054mmol/lcHb =10g/100ml,BE38°C=0.1166mmol/l

70

80

90cHb =15g/100ml,BE38°C=0.0252mmol/lcHb =20g/100ml,BE38°C=0.0592mmol/lcHb =25g/100ml,BE38°C=0.1469mmol/l

50

60 38°C38°C

40

O2

[torr

]

pH=7.3878pCO2=41.862 torr

30PC

O

20

6.8 6.9 7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.810

pH

Fig. 11 The � tra� on curves at 37°C – pH and pCO2 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 pCO2=40 torr and pH=7.4.

90

100

120

plasma,BE37°C =0mmol/lcHb =5g/100ml,BE37°C=0mmol/lcHb =10g/100ml,BE37°C=0mmol/l

70

80

90cHb =15g/100ml,BE37°C=0mmol/lcHb =20g/100ml,BE37°C=0mmol/lcHb =25g/100ml,BE37°C=0mmol/l

50

60 37°C37°C

40

O2

[torr

]

pH=7.4pCO2=40 torr

30PC

O

20

6.8 6.9 7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.810

pH

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 pH=7.4 and pCO2=40 torr at 37°C).

pCO2 37°C=40 torr

pH38°CBEINV

Ashwood(1983)pCO2 38°C=41,862 torrpH37°C=7,4

p 2 37 C 4

SO2 38°C=1

Vstup: cHb

0 3

0.35

0 15

0.2

0.25

0.3

ol/l

]

0

0.05

0.1

0.15

BE

[mm

ol/l]

38°C

[mm

o

BE38°ABEOX-0.1

-0.05

0

BE

38

CO

ABEOX0 5 10 15 20 25

-0.15

cHb [g/100ml]cHb [g/100ml]

pH37°C=7,4

pCO2 37°C=40 torr

pH38°C=7,3878

pCO2 38°C=41,862 torrSO2 38°C=1

pH38°C=7,3878

pCO2 38°C=41,862 torrAshwood(1983)

Fig. 12. The dependence of BE on haemoglobin concentra� ons at pH=7.3878 and pCO2 =41.862 torr according to the data from Sigaard-Andersen nomogram at 38°C (in fully oxygen-saturated blood). At 37°C, these values correspond to the standard values of pH=7.4 and pCO2=40 torr, at which BE will be zero (at 37°C).

0.35

0.25

0.3

0.2

0.1

0.15

mm

ol/l]

0.05BE

[m

-0.05

0

-0.1

0 5 10 15 20 25-0.15

cHb [g/100ml]

8

For new coordinates of BE curves, see Fig. 15 and 16.

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

The pH38°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 pH37°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:

NBB37°C=BB37°C -BE37°C = BB38°C - BE37°C

As (see above):

BE37°C=BE38°C - dBE38°C

then:

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

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 pH=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 pH38°C. These values are then re-calculated to pH37°C and PCO2 37°C, characterising the � tra� on curve for the given haemoglobin concentra� on and selected BE at 37°C.

Outputs: pCOI t CO

pH38 CBEINV

Outputs: pCO2 37 C

pH 37 CAshwood(1983)Input: pCO2 38 C

SO2 38 C=1

Input: cHb

8090

100120

CO

2 37

C

Input: BE37 C

40506070P

CBE

BE38 CBE38 C=BE37 C+dBE38 C

20

30BE37 C

dBEABEOX 6.8 6.9 7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.810dBE38

CO

ABEOX 6.8 6.9 7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8pH37 C

pH37 C=7,4

pCO2 37 C=40 torrSO2 38 C=1

pH38 C=7,3878

pCO2 38 C=41,862 torr Ashwood(1983)

Fig. 14 The � tra� on curves for haemoglobin concentra� ons (0, 5, 10, 15, 20, 25 g/100 ml) and various BE at 37°C cut each other in the points characterising BE curve on Siggaard-Andersen nomogram corrected to 37°C.

100

120

70

80

90100

50

60

70

BE [mmol/l]

40

O2

[torr

] 010

15

20

5

-10

-5

30PCO 20

-15

20-20

6.8 6.9 7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.810

pH

Fig. 15 Results of BE curve correc� on from 38°C to 37°C – new coordinates in pCO2 axis.

4537°C38°C

2[to

rr]

40pCO

2

35

30

25

20

15

-25 -20 -15 -10 -5 0 5 10 15 20 2510

BE [ l/l]

BE mmol/l

Fig. 16 Results of BE curve correc� on from 38°C to 37°C – new coordinates in pH.

837°C38°CpH

7.9

7.8

7 6

7.7

7.5

7.6

7.4

7.3

-25 -20 -15 -10 -5 0 5 10 15 20 257.2

BE mmol/l

9

consequent dependence can be linearised by the following rela� onship (Fig. 17)

dBE=0.3 – 0.018 cHb

where cHb is haemoglobin concentra� on in g/100ml.

NBB38°C is calculated by the known, in clinical prac� ce used, rela� onship (Siggaard-Andersen, 1960):

NBB38°C = 41.7 + 0.42 cHb

The subs� tu� on of NBB37°C results in a slightly di� erent rela� onship:

NBB37°C = 42.0 + 0.402 cHb

BB37°C value will be calculated from BE37°C and haemoglobin concentra� on:

BB37°C = 42.0 + 0.402 cHB + BE37°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 (pH and pCO2) are measured at the standard temperature of 37°C; however,

Fig. 17 Lineariza� on of the dependence of BE shi� on haemoglobin concentra� on (cHb) expressed in g/100 ml during temperature change from 37°C to 38°C.

0.4

dBE38 = 0 018*cHb + 0 3dBE38= 0.018cHb +0.3

0 2

0.3 dBE38 = - 0.018 cHb + 0.3

0.1

0.2

l]

0

8 [m

mol

/l

-0.1dBE

38

-0.2

-0.3

0 5 10 15 20 25 30 35-0.4

cHb [g/100ml]

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.

80

90100 BB 38°C

BE 38°CBB 37°CBE 37°C

50

60

70BE 37°C

40

50

BB

20

30

O2

[torr

] BE

20

PCO

10

6.8 6.9 7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8pH

BE [mmol/l]

37°C 38°CpH pCO2 [torr] pH pCO2 [torr]

-22 7,226 11,6 7,221 12,4

-21 7,229 13,5 7,225 14,2

-20 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 20,6 7,246 21,2

-16 7,253 22,3 7,252 22,9

-15 7,259 24 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 7,280 29,4

-11 7,289 30,4 7,289 30,8

-10 7,297 31,7 7,297 32,1

-9 7,306 33 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 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 7,509 40 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

20 7,75 29,8 7,740 30,2

21 7,777 28,4 7,767 28,8

22 7,806 26,9 7,795 27,3

10

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.

0��1�'����'��������������

Fig. 19 Comparison of the � tra� on curves calculated according to original and corrected Siggaard-Andersen nomogram.

90

100

110

cHB =0,5,10,15,20g/100ml

pCO

2[t

orr]

60

70

80

50

0 50 5

40

CO

2 [to

rr]

-515

20

100

-10

-515

20

100

-1010

5

0 510

15

2030P

C

-15-15

Without corrections(temperature 38°C)

15

20

-20-20

(temperature 38 C)

20

BE[mmol/l]BE[mmol/l]

With correctionstotemperature

37°CBE

[mmol/l]

6.8 6.9 7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.810

pHpH

BB [mmol/l]

37°C 38°CpH pCO2 [torr] pH pCO2 [torr]

14 6,903 4,0 6,887 4,2

15 6,904 9,6 6,888 10

16 6,904 14,9 6,889 15,6

17 6,905 20,1 6,89 21

18 6,906 25,1 6,891 26,2

19 6,908 29,8 6,893 31,1

20 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 7,007 79

34 7,032 77,6 7,017 81,1

35 7,042 79,4 7,027 83

36 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 7,080 85,1 7,066 89

40 7,090 86,3 7,076 90,2

41 7,100 87,3 7,086 91,3

42 7,110 88,3 7,096 92,3

43 7,120 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 7,205 91,8 7,191 96

52 7,215 91,8 7,202 96

53 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 7,247 95

57 7,271 90,4 7,258 94,6

58 7,282 89,9 7,269 94,1

BB [mmol/l]

37°C 38°CpH pCO2 [torr] pH pCO2 [torr]

59 7,294 89,5 7,281 93,6

60 7,306 88,9 7,293 93

61 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 7,353 89

66 7,378 84,1 7,366 88

67 7,391 83,0 7,379 86,9

68 7,404 82,0 7,392 85,8

69 7,417 80,8 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 80,9

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 7,526 72

79 7,551 67,2 7,54 70,4

80 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 .

11

Now, Siggaard-Andersen nomogram is formalised for the same temperature, which detailed models of plasma acid-base balance, created by Stewart´s model, are iden� � ed for. These models (e.g. Figge 2009), 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 pCO2.

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 BBp and SID, are unchanged (Fig. 20) – SID and pCO2 are therefore mutually independent variables, which, together with another independent variable, plasma protein concentra� on, determines the value of the dependent variable – pH.

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 BBp, varies. The increase in pCO2 causes the increase in BBp (and SID), whereas the decrease in pCO2 causes the decrease in BBp. 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 CO2 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 pCO2 at which BB concentra� ons in erythrocytes and plasma equilibrate (BBe = BBp). 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:

BB = BBp (1 - Hk) + BBe Hk = BBp + Hk (BBe – BBp)

The second member of the sum is zero with BBe = BBp and the whole blood BB does not depend on haematocrit. With this pCO2 (and proper plasma pH) when BBp=BBe, 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 BBe=BBp the whole blood BB does not depend on haematocrit (Hk):

A similar considera� on applies for the BE curve, too. As:

BE=BEp (1 – Hk) + BEe Hk = BEp + Hk (BEe – BEp)

the second member of the sum equals zero at BEe=BEp 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 pCO2 and pH, when BEe=BEp.

Rise in rangeplasma

i i

Rise in rangeof milomols

H2O

H2CO3

HCO3 - Rise in rangeof nanomols

CO2 H +

HBuf

Buf -

i

BBp=[HCO3-]p+[Buf-]p

Drop in rangeof milimols

BEp=BBp-NBBp

SID BB a BE do not vary!

d[HCO3-]p = -d[Buf-]p

SID, BBp a BEp do not vary!

Fig. 20. Plasma � tra� on with carbon dioxide – BEp, BBp and SID do not vary. Thus, pCO2 and SID are mutually independent.

H2O HCO3 -

CO2

H2CO3

H +CO2 H

HBufBBe=[HCO3

-]e+[Buf-]e

Cl -

Buf -BEe=BBe+NBBe

erythrocytes

H2O HCO3 -

Cl -erythrocytes

plasmaH2O

H2CO3

H +CO2 H +

HBufBBp=[HCO3

-]p+[Buf-]pBuf -

BBp [HCO3 ]p+[Buf ]p

BEp=BBp+NBBp

Fig. 21 Blood � tra� on with carbon dioxide – SID varies during pCO2 changes (thanks to the exchange of HCO3

- for Cl-). SID and pCO2 in the whole blood are not mutually independent.

12

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:

BBe – NBBe = BBp - NBBp

This can be speci� ed:

BBe – BBp = NBBe – NBBp = constant

This means that the BE curve can be interpreted as the geometric site of the points (i.e. pCO2 and pH 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 pH=7.4).

If the equa� on NBB38°C = 41.7 + 0.42 cHb applies (Siggaard-Andersen, 1962), then haemoglobin concentra� on in erythrocytes cHb = 33.34 g/100ml is NBBe-NBBp=0.42×33.34 =14 mmol/l (according to our correc� on of Siggaard-Andersen nomogram, this value was 0.402×33.34 =13.4 mmol/l for 37°C).

Siggaard-Andersen used the mixture of O2 - CO2 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, 1980). BE or BB exert a linear increase in haemoglobin oxygen desatura� on:

BE = BEox + 0.2 cHB (1-sO2)

where cHb is haemoglobin concentra� on [g/100ml] and sO2 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

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 33.34 g/100 ml, which is the value with haematocrit having the value of one. The � tra� on curve of this “virtual blood” with carbon dioxide follows pH varia� ons (measured on the outer side of the erythrocyte) during pCO2 changes. The � tra� on curve of the blood with a given haemoglobin and, thus, haematocrit concentra� ons cHb (in g/100ml blood).

Hk=cHb/33.34

(supposing the normal haemoglobin concentra� on in erythrocytes 33.34 g/100ml) will lie between the � tra� on curves of plasma and erythrocytes in the semi-logarithmic coordinates log10(pCO2) – pH. 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:

dBBp=dBEp=mHCO3ep/(1-Hk)

The corresponding varia� on of BE in erythrocytes is:

dBBe=dBEe=-mHCO3ep/Hk

Choosing, for example, haemoglobin concentra� on 15 g/100 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/(1-0.4449)=1.8015 mmol/l and by 1/0.4449=2.2477 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/100 ml, in which 1 ml of bicarbonates were transferred from erythrocytes into plasma during the increase of pCO2 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/100 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 pCO2 increase) and from plasma into erythrocytes (during pCO2 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 -20 to 20 mmol/l.

120

0<cHb<33.34[g/100ml]

if haematocrit (Hk) =1:Hb 33 34 /100 l

60

70

80

90100

rr]

cHb=33.34 [g/100ml]

• cHb=33.34 g/100 ml• BB = BBe, BE = BEe

30

40

50

PC

O2

[to

cHb=0[ /100 l]

6.8 6.9 7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.820

pH

[g/100 ml]

Transfer of bicarbonates during titration with CO2erythrocytes plasma

BB

BEe

BBp

BEp

erythrocytes plasma

HCO3- HCO3

-

BBe

mHCO3ep

dBBp= dBEp= mHCO3ep / (1-Hk)

dBBe= dBEe= -mHCO3ep/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).

13

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, 2009), and erythrocyte � tra� on curve (obtained from the experimental data of Siggaard-Andersen nomogram, corrected to 37°C).

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Fig. 27 shows erythrocyte � tra� on curves with various BE by Siggaard-Andersen nomogram – the erythrocytes are modelled as blood with haemoglobin concentra� on 33.34 g/100ml (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 (BEer).

The erythrocyte � tra� on curves will be approximated according to the following rela� onships:

log10(pCO2) = k pH + h

k=f(BEer)

h=g (BEer)

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).

cHB=15 g/100 ml, BE=0 mmol/l

100

120

cHb=15 g/100 ml

E ( Hb 33 34 /100 l)l

BE=0 mmol/l

80

90Ery (cHb = 33.34 g/100 ml)plasma

60

70

1

2

3

ery -> 2mmol HCO3 ->plasma

ery -> 1mmol HCO3 ->plasma

50

PC

O2

[torr] 13

40plasma ->1mmol HCO3 ->ery

30

6.8 6.9 7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.820

pH

4

Fig. 23 Model of the � tra� on curves of plasma, erythrocytes and blood with haemoglobin concentra� on 15 g/100 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/100 ml. The decrease in pCO2 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/100 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.

cHB=15 g/100 ml, BE=-10 mmol/l

100

120

BE=-10 mmol/l cHb= 15 g/100ml

E ( HB 33 34 /100 l)

80

90Ery (cHB = 33.34 g/100 ml)

plasmaery -> 2 mmol HCO3 -> plasma

60

70

2

ery -> 1mmol HCO3 -> plasma

50

PC

O2

[torr]

1

2

3

40

plasma -> 1mmol HCO3 -> ery30 4

6.8 6.9 7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.820

pH

cHB=15 g/100 ml, BE=10 mmol/l

100

120

BE = 10mmol/l cHb=15g/100mlplasma

80

90 Ery (cHb = 33.34 g/100ml)ery->2 mmol HCO3 ->plasma

60

70

2ery->1 mmol HCO3 -> plasma

50

PC

O2

[torr]

1

3

40

4

30

6.8 6.9 7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.820

pH

plasma -> 1 mmol HCO3 -> ery

Fig. 24 Model of the � tra� on curves of plasma, erythrocytes and blood with haemoglobin concentra� on 15 g/100 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/100 ml, modelled by Siggaard-Andersen nomogram.

Fig. 25 Model of the � tra� on curves of plasma, erythrocytes and blood with haemoglobin concentra� on 15 g/100 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/100 ml, modelled by Siggaard-Andersen nomogram.

14

pH (pH of the outer side of erythrocytes), depending on pCO2 and BE in erythrocytes (BEer), is calculated by means of eryBEINV func� on; for its algorithm, see Fig. 30.

pH=eryBEINV(pCO2,BEer)

The erythrocyte model is connected with the plasma model . Figge-Fencl´s model (Figge, 2009), 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 pH in dependence on pCO2, 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), pCO2 and haemoglobin oxygen satura� on:

pH=bloodBEINV(Pitot,Alb,Glob,cHb,BEox,pCO2,sO2)

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 sO2) and BEox. This value is considered ini� al for plasma and erythrocytes (BE). pH is calculated from pCO2.

cHB=15 g/100 ml, -20<BE<20 mmol/l

70

80-20BE [mmol/l] -18BE [mmol/l] -16BE [mmol/l] -14BE [mmol/l] -12BE [mmol/l] -10BE [mmol/l] -8BE [mmol/l] -6BE [mmol/l] -4BE [mmol/l] -2BE [mmol/l] 0BE [mmol/l] 2BE [mmol/l] 4BE [mmol/l] 6BE [mmol/l] 8BE [mmol/l] 10BE [mmol/l] 12BE [mmol/l] 14BE [mmol/l] 16BE [mmol/l] 18BE [mmol/l] 20BE [mmol/l]

cHb=15 g/100 ml

60

70

50

40

PC

O2

[torr]

30

6.8 6.9 7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.820

pH

Fig. 26 Model of the � tra� on curves of plasma, erythrocytes and blood with haemoglobin concentra� on with various BE concentra� ons ranged from -20 to 20 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 33.34 g/100 ml).

100

708090

BE=-20 mmol/l

BE=-10 mmol/l

50

60 BE= 0 mmol/l

BE= 10 mmol/l37 C

40

O2

[torr]

BE= 20 mmol/l38 C

30

PC

O

20

6.8 6.9 7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.910

pH

Fig. 27 Erythrocyte � tra� on curves (lines in the semi-logarithmic scale) by Siggaard-Andersen nomogram at 38°C and a� er correc� on to 37°C at various BE concentra� ons.

Slopes (k) of erythocyte titration lines (log10(pCO2) = k pH + h)

k -1.5

kk=p1*BE 6̂+p2*BE 5̂+p3*BE 4̂+p4*BE 3̂+p5*BE 2̂+p6*BE+p7

kk = pk1 BE6 + pk2 BE5 + pk3 BE4 + pk4 BE3 + pk5 BE2 + pk6 BE + pk7

-2

k=p1 BE 6̂+p2 BE 5̂+p3 BE 4̂+p4 BE 3̂+p5 BE 2̂+p6 BE+p7p p p p p p p

-2.5

-3

-3.5

-4

BE-20 -15 -10 -5 0 5 10 15 20

Fig. 28 Polynomic regression of the variable slopes of erythrocyte � tra� on lines.

Offset (h) of erythocyte titration lines (log10(pCO2) = k pH + h)

h 30hh=p1*BE^6+p2*BE^5+p3*BE^4+p4*BE^3+p5*BE^2+p6*BE+p7

hh = ph1 BE6 + ph2 BE5 + ph3 BE4 + ph4 BE3 + ph5 BE2 + ph6 BE + ph7

26

28

24

20

22

18

14

16

BE-20 -15 -10 -5 0 5 10 15 20

Fig. 29 Polynomic regression of the variable o� set of erythrocyte � tra� on lines.

Fig. 30 Algorithm of the calcula� on of erythrocyte � tra� on curves

pH=eryBEINV(pCO2,BEer)BEer

pk1 = -1.159e-009;pk2 = 1.328e-008;pk3 = 2.228e-007;

pk1 = -1.159e-009;pk2 = 1.328e-008;pk3 = 2.228e-007;

ph1 = 8.229e-009;ph2 = -8.913e-008;ph3 = -1.82e-006;

ph1 = 8.229e-009;ph2 = -8.913e-008;ph3 = -1.82e-006;p ;

pk4 = 1.479e-005;pk5 = -0.0005606;pk6 = 0.04644;

pk4 = 1.479e-005;pk5 = -0.0005606;pk6 = 0.04644;

p ;ph4 = -0.0001034;ph5 = 0.003499;ph6 = -0.3109;

ph4 = -0.0001034;ph5 = 0.003499;ph6 = -0.3109;p

pk7 = -2.431;pk7 = -2.431;

k k1 BE6 + k2 BE5 + k3 BE4 + k4 BE3 + k5 BE2 + k6 BE + k7k = pk1 BE6 + pk2 BE5 + pk3 BE4 + pk4 BE3 + pk5 BE2 + pk6 BE + pk7

pph7 = 19.5915;ph7 = 19.5915;

k = pk1 BE6 + pk2 BE5 + pk3 BE4 + pk4 BE3 + pk5 BE2 + pk6 BE + pk7k = pk1 BE6 + pk2 BE5 + pk3 BE4 + pk4 BE3 + pk5 BE2 + pk6 BE + pk7

h h1 BE6 h2 BE5 h3 BE4 h4 BE3 h5 BE2 h6 BE h7h = ph1 BE6 + ph2 BE5 + ph3 BE4 + ph4 BE3 + ph5 BE2 + ph6 BE + ph7h = ph1 BE6 + ph2 BE5 + ph3 BE4 + ph4 BE3 + ph5 BE2 + ph6 BE + ph7h = ph1 BE6 + ph2 BE5 + ph3 BE4 + ph4 BE3 + ph5 BE2 + ph6 BE + ph7

pH=(lpCO2-h)/k;pH=(lpCO2-h)/k;pCO2 lpCO2=log10(pCO2)lpCO2=log10(pCO2) pH

15

However, the plasma � tra� on curve has a smaller slope than that for erythrocytes (see Fig. 31) and plasma pH (H(BEp)) is calculated according to plasma BE (BEp); pH on the outer side of erythrocytes (pH(BEer)), calculated according to erythrocyte BE (BEer), is di� erent. Then, the transfer of bicarbonates between plasma and erythrocytes is calculated by itera� on – the transfer causes varia� ons in plasma (BEp) and erythrocyte (BEer) 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 (pH = pH(BEp) = pH(BEer)).

The algorithm also calculates the normal SID (NSID) – i.e. the SID, in which pH=7.4 with the given haemoglobin, albumin and phosphate concentra� ons and pCO2=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, 2006). Unlike in classical plasma models by Stewart and his followers, this model enables to demonstrate that the rela� onship between SID a pCO2 does not apply in the whole blood. The model (and the related formalised rela� onships) can be used in a number of clinical-physiological calcula� ons.

For the model, including its source text and the descrip� on of all used mathema� cal rela� onships and algorithms, see www.physiome.cz/acidbase.

���������� �

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,

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., 2007).

)�#����*������

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 C20/2008 and by Crea� ve Connec� ons s.r.o. company.

�4���������

1. Ashwood E.R., Kost G., Kenny M. (1983): Clinical Chemistry. 29:1877-1885.

2. 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., 8:33-43.

3. Dell R.D., Winters R.W. (1970) A model for the in vivo CO2 equilibra� on curve. Am J Physiol. 219:37–44

4. Dubin A., Menises M.M., Masevicius F.D. (2007): Comparison of three di� erent methods of evalua� on of metabolic acid-base disorders. Crit Care Med. 35:1264–1270

5. Dubin A. (2007) Acid-base balance analysis: Misunderstanding the target Crit Care Med. 35:1472–1473.

6. Fencl V., Rossing T.H. (1989): Acid-base disorders in cri� cal care medicine. Ann Rev. Med. 40, 17-20, 1989

7. Fencl V., Leith D.E. (1993): Stewart‘s quan� ta� ve acid-base chemistry: applica� ons in biology and medicine. Respir. Physiol. 91: 1-16, 1993

8. Fencl J., Jabor A., Kazda A., Figge, J. (2000): Diagnosis of metabolic acid-base disturbances in cri� cally ill pa� ents. Am. J. Respir. Crit. Care. 162:2246-2251.

Combination of blood and plasma acid-base modelsplazmaBE bloodBE

log10(pCO2)plazmaBE

erythrocytesBE

BE

BEerythrocytes

ppCO2

BEer

BEpmHCO3

-

BE =BE+mHCO - /(1 Hk)

BEp

plasmaBEp

BEp=BE+mHCO3 /(1-Hk)

erythrocytesBEer

BEer=BE-mHCO3- /Hk

H pH

?pH(BE)pH=pH(BEer)=pH(BEp)

pHpH(BEp) pH(BEer)

Fig. 31 Principle of the calcula� on of the whole blood � tra� on curves. At given BE plasma and erythrocytes � tra� on curves (plasmaBE and erythrocytesBE) have a di� erent slopes, hence at given pCO2 a di� erent pH can be calculated. Searched blood � tra� on curve (bloodBE) lies between plasmaBE and erythrocytesBE curves. In the blood at given haematocrit (Hk) plasma and erythrocyte BE (BEp and BEer) is shi� ed because of HCO3

-/Cl- erythrocyte-plasma exchanges. New � tra� on curves of plasma and erythrocytes (plasmaBEp, erythrocytesBEer) can be calculated. Algorithm seeks the intersec� on of bloodBE, plasmaBEp, and erythrocytesBEer curves at given pCO2.

Fig. 32 Algorithm of the calcula� on of the whole blood � tra� on curves.

pH=bloodBEINV(Pitot,Alb,Glob,cHb,BEox,pCO2,sO2)

log10(pCO2)

pH(BEp)

pH(BEp)pH(BEp) pH(BEer)

pH(BEer)== pHpCO2

pH(BEer)

pCO2 pHer=eryBEINV(pCO2,BEer)pHer=eryBEINV(pCO2,BEer)pHpH(BEp)

Pitot, Alb,Glob SIDSID

NSIDNSID BEp=SID-NSIDBEp=SID-NSIDBEer=BE-mHCO3

-/Hk)BEer=BE-mHCO3-/Hk)

BEox BE=BEox+0,2(1-sO2) cHbBE=BEox+0,2(1-sO2) cHb mHCO3-=(BEp-BE)(1-Hk)mHCO3-=(BEp-BE)(1-Hk)

sO2

cHb Hk=cHb/33.34Hk=cHb/33.34

16

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Corresponding author

Ji�í Kofránek, Laboratory of Biocyberne� cs, Department of Pathophysiology, U nemocnice 5, 128 53 Prague 2, Czech Republic, e-mail: kofranek@gmail.com.

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.

Restoration of Guyton Diagram for Regulation of the Circulation as a Base of Quantitative Physiological Model Development

J. KOFRÁNEK1, J. RUSZ1,2 1Charles University in Prague, 1st Faculty of Medicine Department of Pathophysiology, Laboratory of Biocybernetics, Czech Republic

2Czech 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, 166 27 Prague 6, Czech Republic, E-mail: 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

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. 2008). 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 2009, Osborn et al. 2009) and physiological modelling (Bassingthwaighte 2000, Hunter et al. 2002, Thomas et al. 2008, Bassingthwaighte 2009), including research on the physiological consequences of weightlessness in manned space flight (White et al. 1991, White et al. 2003), or in a new approach to automation in medicine (Nguyen et al. 2008). In addition, it is still reprinted with errors (Hall 2004, Bruce and Montani 2005). 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. 2007, Hester et al. 2008), 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 <http://physiome.cz/guyton/>. 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 long-term 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. 2007). 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

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.: ,QVOVBDQAODVS Erroneous eq.: .QVOVBDQAODVS (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, non-muscle tissues (BFN), oxygen volume in aortic blood (OVA), rate of oxygen delivery to non-muscle cells (DOB) and hematocrit (HM),

Correct eq.:

,7/5

)( ZOSVHMBFN

DOBOVABFNdt

OSVd

Erroneous eq.:

.755

)(ZHMBFN

DOBOVABFNHMBFN

DOBOVABFNdt

OSVd

(2)

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,

Correct eq.:

,1)( RCKVRCRCdt

VRCd

Erroneous eq.:

.1)( RCKVRCRCdt

VRCd

(3)

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,

Correct eq.: HMK

HMHMdt

HMd 2)2( ,

Erroneous eq.: .)2( HM

dtHMd

(4)

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 0.3333) 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 FI and outflows at the rate FO. Rate of its depletion is proportional to its concentration c, where FO = 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,

VkMF

dtdM

I . (5)

Provided that the capacity of distribution area V is constant, we will substitute the ratio k/V with constant k1. Guyton calculated the concentration of hormone c0 normalized as a ratio of current concentration c to its normal value cnorm = c/c0. 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 Mnorm = M/c0. When we formulate the rate of flow in a normalized way (as a ratio to normal rate), under normal conditions it holds that FI = 1, dMnorm/dt = 0 and after substituting it into equation (5) we get the equation (6),

01 1 normMk . (6)

The relative concentration of hormone c0 can be formulated as equation (7),

MkM

Mcnorm

10 , (7)

and after final adjustments and inserting into a differential equation (7) we arrive at

Correct eq.:

,100 kcF

dtdc

I

(8)

Erroneous eq.: .100 kcF

dtdc

I (9)

According to equation (8), the normalized concentration of hormone c0 is calculated from normalized inflow of hormone FI. In the original Guyton diagram, the normalized concentration of aldosterone and angiotensine is calculated this way, which means that normalized rate of inflows is FI = AH 0.3333 and normalized concentration of hormone is c0 = 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:

Correct eq.:

,14.0)3333.0()( AHCAH

dtAHCd

Erroneous eq.: .14.03333.0)( AHCAH

dtAHCd

(10)

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 7.5.0.342 - R2007b, 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 2007). 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

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 <http://physiome.cz/guyton/>. 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 (χ2) 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 χ2 (11) = 1445; p < 0.001 for simulation of the arterial pressure, χ2 (10) = 939; p < 0.001 for simulation of the heart rate, χ2 (10) = 1388; p < 0.001 for simulation of the stroke volume, χ2 (10) = 1189; p < 0.001 for simulation of the cardiac output, and χ2 (10) = 1304; p < 0.001 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 PO2 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

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 1980, 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).

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. 2000). 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. 2007, Hester et al. 2008). 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 0021620806, and “Transdisciplinary Research in Biomedical Engineering” MSM 6840770012, 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.

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69: 1973-1982, 2000. 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. 179-201.

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GUYTON AC, JONES CE, COLEMAN TG: Circulatory Physiology: Cardiac Output and Its Regulation. WB Saunders Company, Philadelphia, 1973, p. 486.

GUYTON AC: Arterial Pressure and Hypertension. Philadelphia, PA: Saunders, 1980. GUYTON AC: The suprising kidney-fluid mechanism for pressure control—its infinite gain! Hypertension. 16:

725-730, (1990). HALL JE: The pioneering use of system analysis to study cardiac output regulation. Am. J. Physiol. Regul.

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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 1 - 60 - 01. 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 61 - 65 - 06. Increased vascular volume caused by stress relaxation 65 VV7 Capillary membrane dynamics 66 - 82 - 07. Plasma volume 71 VP 08. Total plasma protein 80 PRP Tissue fluids, pressure and gel 83 - 113 - 09. 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 114 - 135 - 13. Total extracellular sodium 118 NAE 14. Total extracellular fluid potassium 122 KE 15. Total intracellular potassium concentration 131 KI Pulmonary dynamics and fluids 136 - 152 - 16. Pulmonary free fluid volume 142 VPF 17. Total protein in pulmonary fluids 149 PPR Angiotensin control 153 - 163 - 18. Angiotensin concentration 159 ANC Aldosterone control 164 - 174 - 19. Aldosterone concentration 170 AMC Antidiuretic hormone control 175 - 189 - 20. Degree of adaption of the right atrial pressure 180 AHY 21. Antidiuretic hormone concentration 185 AHC Thirst and drinking 190 - 194 - Kidney dynamics and excretion 195 - 222 - Muscle blood flow control and PO2 223 - 254 - 22. 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 255 - 272 - 25. Non-muscle venous oxygen saturation 260 OSV 26. Non-muscle total cellular oxygen 271 QO2 Non-muscle, non-renal local blood flow control 273 - 290 - 27. 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 291 - 320 - 30. Time delay for realization of autonomic drive 305 AU4 31. Overall activity of autonomic system 310 AUJ Heart rate and stroke volume 321 - 328 - Red cells and viscosity 329 - 339 - 32. Volume of red blood cells 332 VRC 33. Hematocrit 336 HM2 Hearth hypertrophy or deterioration 329 - 352 - 34. Hypertrophy effect on left ventricle 344 HPL 35. Hypertrophy effect on heart 349 HPR 36. Cardiac depressant effect of hypoxia 352 HMD

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

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

Figure 3 Simulation of changes in circulatory function at the onset of hypertension caused by reduction of renal mass and increase in salt intake. Left: Transient changes of different variables in a series of six dogs in which 70% of the renal mass had been removed and intravenous infusion of saline at a rate of 2 to 3 liters per day was given for 12 days (data adopted from Chau et al. 1979). Right: Model simulation of the same experiment as that seen on the left in dogs; performed by our corrected implementation in SIMULINK (solid lines) in comparison with original 1972 Guyton et al. implementation in FORTRAN (dashdot lines). Changes in all variables are essentially similar to those found in the animals.

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Figure 5 Benchmark experiment 3: simulation of circulatory dynamics in nephrosis. At the day 1, the kidneys began to excrete large amount of plasma protein. As a consequence, the fall of total circulating plasma protein is occurred. When the plasma total protein fell below a critical level, it becomes the enormous increase in interstitial free fluid. At the end of simulation, increase in total plasma protein caused marked diuresis and beginning resorption of the edema. Total experiment time (x -axis) was 288 hours (12 days). Comparison of simulation results of our SIMULINK model (solid lines) with the original Guyton et al. model implementation in FORTRAN (dashdot lines).

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Figure 6 Benchmark experiment 4: simulation of atrioventricular fistula. At the day 1, the opening of the fistula caused an extremely increase in cardiac output, and decrease in total peripheral resistance. It remains until 5 day where the fistula was closed. At the end of the record, patient was nearly normal. Total experiment time (x -axis) was 9 days. Comparison of simulation results of our SIMULINK model (solid lines) with the original Guyton et al. model implementation in FORTRAN (dashdot lines).

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