- Email: [email protected]
Securing Vital Organic Functions as a Basis of Automatic Control of Artificial Hearts
SECURING VITAL ORGANIC FUNCTIONS AS A BASIS OF AUTOMA.TIC CONTROL OF ARTIFICIAL HEARTS (Obespechenie zhiznenno-vazhnykh funktsii organizma kak osnova avtomaticheskogo upravleniya iskusstvennym serdtsem)
B. V. Petrovskii, V. I. Shumakov, V. N. Novosel'tsev, E. Sh. Shtengol'd, V. M. Baikovskii, and L. A. Dartau
Successes in modern cardiosurgery together with achievements in the physiology of extracorporal. circulation, chemistry, electronics, and technology, made possible the experimental creation of implantable artificial hearts, capable of sustaining organic circulation /1 - 3 / .
on the principle of securing vital functions of the organism. These functions are a unified association of physiological control systems, whose interrelation secures the organism's homeostatic properties and determines values of its essential variables
Later experience, however, revealed difficulties in controlling the artificial heart, both automatically and manually. Lack of unified and quantitatively correlated ideas about functional links between physiological systems in the organism hampers the selection of an algorithm of control capable of securing a prolonged and adequate perfusion.
The control problem arises with the need of prosthesis for some vital physiological function, the lack of which deprives the organism of its properties of physiological homeostasis. Successful control in such a case will depend on completing lost links, based on the data of work of the undamaged systems in the organism. Many results of physiological studies are, however, disconnected and merely qualitative. Their use as a basis for restoration of lost links is generally impossible. Therefore, an explanation of quantitative relations requires formulation of model concepts, which permit us to study the functions and correlations of vital systems in the organism. This does not require data on every organism variable. Thus, for example, prosthesis of heart function requires selection of such organic functions, whose condition provides information most necessary for control of this prosthesis. On the other hand, there are also systems whose condition during control need be taken into account to a much lesser degree.
/4-6 /.
The first unsuccessful attempts in automation of devices of artificial circulation and ventilation testify to the complexity of this problem. Adequate control of circulation and ventilation is currently based on the assumption that there are rigid settings concerning such parameters as blood pressure, pH, POz' Peoz' etc. In principle, this enables immediate creation of control systems, in order to do this, it would be sufficient to create monitors continuously measuring these parameters and to construct a system of control by error. Such an approach, however, is insufficiently substantiated due to variability of these parameters within the range of general organic activity. Besides, even in cases of inadequate circulation, most of the really important parameters can be sustained for a long time at a relatively constant level by the compensatory resources of the organism.
For example, let us look at the mechanism of blood pressure maintenance at a definite level. The value of this level is the result of a definite relationship between the volume of circulation and tissue requirements of this circulation (vascular tonus). Changes in such values occur far more easily and more often than in the system of circulation control. Hence, there are reasons to believe that the system of blood pressure control is hierarchically younger. Certain clinical observations also point to this conclusion / 7 / . If the main emphasis is laid on maintaining the blood pressure during artificial heart control, the heart work will be deranged by factors which have no effect on circulation under normal circumstances. In such a caSE:, the automatic system will be too sensitive to obstacles.
Finally, direct use of deviations of these parameters in order to control the artificial circulation device may be hampered by apparently contradictory situations where the value of one of the regulated parameters calls for increased circulation, while the value of another parameter requires decreased circulation. These difficulties do not arise when the solution of the problem of automatic control is based directly
564
On the other hand, when th e main emphasis is laid on systems hierarchi c ally older than the c ir c ulation system (for instance, the thermoregulati on system / 8 / or blood pH r e gulation / 9 / ), the control will obviously be ins e nsitive to obstac les too conservative and inert . From the point of view of automatic control theory, a problem is encountered which is somewhat similar to the problem of determining optimum correlation between the fluctuant and dynamic error in the common automatic systems during selection of the best intensification coefficient along the closed circuit / 10, 11 / . Hence during artificial heart control, the main emphasis should be on information obtained from systems which are hierarchically equal in age to the system regulating the circulation. Data on hierarchical dependence can be obtained by forming model concepts of links in the organism. Using the method of mathematical modeling, some artificial situations may be created in which various physiological control systems are reciprocally deranged and the younger systems yield to the older. Correlation of systems in the organism can also be judged by their sensitivity to various environmental parameters, by time constants of these systems (their mobility), and finally on grounds of the model concepts of processes in the entire organism. Let us examine a mathematical model of a complex of physiological systems linked with the circulation system. The principl e of adequate energy provision for the organism as a whole serves as the methodological precondition to selecting the structure of a model. The essence of controlling this process is the constant maintenance of its highenergy condition. Thus, metabolic tissue activity determines values of the circulation and ventilation systems. Among all the functions of energy metabolism, oxygen and carbon dioxide transportation are basically determlOlOg. If there was a possibility of continuously measuring the total oxygen requirements of an organism, the determination of adequate valu e s for cir c ulation and ventilation would present no difficulties. However, to obtain such information by a direct study is impossible at this stage. It is necessary to evaluate the general intensity of metabolism from data of methodically accessibl e measurements. Two generalized systems of tissues and organs can be singled out in connection with the organism's structural heterogeneity and functional tissue specifi c ity. The peripheral system consists of striated muscles, connective tissue and skin. Its energy specificity is determined by
anaer obi c fun c ti oni :.g with subsequent elimination of the oxygen debt . The system of internal organs includes the brain, liver, kidneys, and heart, the function of whi ch is realized exclusiv ely under c onditions of aerobic energy resynthesis. The general circulation may thus be divided into circulation through peripheral tissues and through internal organs. Metabolites of energy metabolism penetrate into venous blood, where buffering processes take place together with formation of an acid-base complex determining the hydrogen ion's tension. Alveolar ventilation and pulmonary gas metabolism close the circle of physiological systems sustaining energy balance of the organism. Mathematical description of such a complex gives an autonomous system of equations with a single entrance signal represented by intensity of energy metabolism in the organism (evaluated in terms of oxygen). The behavior of the system is totally determined by its task and the gas composition of the inspired air / 12 / . The complete system of equations describing the complex of physiological system is listed in supplement No. 1. A complex mathematical model permits us to effect complex processes, embracing all the systems, for example, to determine the future condition of all systems by data of current observations, reproduce on the model the results of a number of physiological experiments (reactive hyperemia, perfusion, isolated carotid sinus, etc.). The model enables us to investigate processes in systems of the organism while artificially securing its vital activity (artificial ventilation, artificial circulation, perfusion of indi vidual organs). Without changing the mathematical model of the systems complex, we may eliminate the equation describing the natural heart and its regulation and replace it by equations describing an artificial heart and the algorithm of its control, therebyobtaining a model of an organism with an artificial heart. Such preliminary modeling was performed in several of the simplest cases. The following 4 va riables were selected, according to which the artificial heart was controlled : arterial and tissue c o ncentration of incompletely oxidized metabolites (l a ctic acid), magnitude of alveolar ventilation, and mean blood pressure. A linear algorithm was selected:
where Z is one of the aforementioned parameters and Q[ is the complete left heart circulation.
565
V a lues of coefficients which accompany the mod eling of control pro ce sses a re listed in Table 1 . T .-\ SLE 1.
:-<0.
1:
I
Con trol algo rithm
I
l.
°r=
~Ct ~ ~ Ot
0t
= 0. 157 !.\n in
2.
Or
=
o aCa~ +B a
a
=
3.
Or
= ct:V V A + Bv
mg"lo
0.1 83 l /m in m g"l< l/mi n a v = 0.1 63 l/min a
\sI
B = 0.1 83 l;tnin
I Ba = 2.67 l/min
i dv
= 2. 37 I/ min
result of this work is an approach to elaborati on o f such algorithms. This approach, connected with s e curing vital fun c tions of the organism, pres c ribes that artifici a l heart control should not be carried out with th e narrow aim of sustaining some parameters of the organism on a constant and predetermined level, but should be directed to the restoration of damag e d links in the united complex of the organism system and to secure its stability.
I
4.
Or =
() pPA + Bp
l 'm in a p = 0. 05 0 mm. Hg
IBp =
0
Algorithms 1 - 3 were stable within the entire range of the functional activity of organism (oxygen consumption 200-2,500 ml O:!/ min, circulation 4.525 l/min). As might ha ve been expected, algorithm 4 secured stability of the complex only within a limited range (circulation 4.5 - 8 l / min) and an increase in circulation was accompanied by a drop in the blood pressure (from 100 mm Hg to 50 mm Hg). Algorithms 1 - 2 secured levels in the processes of the complex which were closest to physiological levels. With the use of algorithm 3, some fluctuations appeared in the course of the control process connected with changes in the metabolic level. When information on concentration of incompletely oxidized metabolites in tissues and arterial blood is used as the control signal, the relative adequacy of circulation differs in both established and transitional processes. At present, however, any direct measurements of this concentration meet with difficulties. Therefore, the further search for the control algorithms may result in detection of magnitudes accessible to measurement and linked indirectly with the concentration of incompletely oxidized metabolites. When the value of alveolar ventilation VA is used as the control signal, the control system secures adequate circulation in all established conditions. In this case, however, the time of circulation establishment will be determined by the time constant of the respiratory system, which is 3 - 4 times greater than in the circulation system of an intact organism 113 I . Consequently, this way of regula ting Q is insufficient, especially since patients in need of heart prosthesis usually undergo changes in the external respiration level as well. Only the simplest ! algorithms of artificial heart control are discussed in this paper . The preliminary results cannot serve as a basis for reaching practical conclusions about the expediency of elaborate algorithms of any definite type. The main
In the elaboration of control algorithms on the basis of this suggested approac h, the first stage involves the inclusion of the organic physiological systems complex in the model. This enables us to obtain preliminary data on the stability and quality of algorithms of artificial heart control, to recommend elaboration of practical control algorithms, and to chec k them under experimental and clinical conditions.
BIBLIOGRAPHY 1. Petrovskii,B.V. and V.I.Shumakov. Implantiruemoe v organizm iskusstvennoe serdtse (Artificial Heart Implanted into the Organism).Kardiologiya, No.8. 1967.
2. De Ba key, M. E. and C. W. Hall. Towards the Artificial Heart. - New Scientist, Vol. 22, No.393. 1964. 3. Ko 1 f f, W. J. Les coeurs artificiels intrathoraciques. - Das Medizinische Prisma, Vol. 66, No. 4. 1966. 4 . Be r n a r d, C. Le
The Wisdom of the Body. -
6. Ash by, U. R. Design for a Brain. - Chapman and Hall, London. 1954 . 7. S hi k, L. L. Kislorodnyi zapros i krovoobrashchenie (Oxygen Demand and Circulation). - In: Kislorodnyi rezhim organizma i ego regulirovanie, Naukova Dumka, Kiev. 1966. 8. Barton,A. andO. Edholm. Chelovekv usloviyakh kholoda (Man in Cold Conditions). - IIL. 1957. [Russian Translation.] 9. V 1 a dim i r 0 v a, G. E. and N. S. Pan t e 1 e e v a . Funktsional' naya biokhimiya (Functional Biochemistry). - Izd. LGU. 1965 .
566
10. Newton,J.C., L.A.Gould a ndJ.F. Ka i z er. Teoriya lineinykh sledy ashchikh sistem (Theory of Linear Tra c ing Systems). - Fizmatgiz. 1961.
The internal organs :.ltiliz e incompletely ox idiz ed metabolites in prop o rti on to their concentr ation:
11. S h ch u kin, A. N. Dinamicheskie i flyuktuatsionnye oshibki upravlyaemykh ob"ektov (Dynamic and Fluctuation Errors of the Controlled Processes). - Sovetskoe radio. 1961.
Equations describing dynamics of 02 and CO2 in t h e int er nal organs system are similar to equations (3)- (6).
12. Petrovskii, B. V., V. I. Shumakov, V. N. No v 0 se l' t s e v, E. Sh. S h ten go l' d and L. A. Da r tau. Problema avtomaticheskogo upravleniya iSkusstvennym serdtsem i matematicheskoe modelirovanie (Automatic Control of an Artificial Heart and Mathematical Modeling). - Khirurgiya, No.4. 1968.
13. N a v rat i 1, M., K. K a dIe t sand S. D a u m . Patofiziologiya dykhaniya (Patophysiology of Respiration). - Izd. Meditsina. 1967.
(9)
The heart system provides th e blood flow in the systemic and pulmonary c ir c ulation under condition that: (10)
and systemic circulation is d ete rmined by neurohumoral factors:
The mean blood pressure depends on cardiac output and vascular system resistance: (12 )
Supplement No. 1 The model of physiological homeostasis includes models of the following systems: metabolism of muscle tissue, of internal organs : of circulation control and its distribution, and of external respiration.
Dynamics of vascular tonus are determined according to the metabolism theory of work hyperemia. (13 ) (14)
(15 )
Energy losses in the peripheral system compensated by aerobic and anaerobic means:
Circulations Qiand Qp are determined by equations: (1 )
(16 ) and under established conditions: w ~naer
=Q
Wp , a
= const.
(2)
The pulmonary system secures the oxygen entry into the organism from the atmospheric air and removal of carbon dioxide:
Oxygen and carbon dioxide diffusion in peripheral organs is determined by the equations: (3)
(4)
(5)
The alveolar ventilation under established conditions is determined from the ratio: (19 )
(6)
The acid-alkaline equilibrium in the organism is determined by the Henderson-Hasselbach formula
Dynamics of incompletely oxidized metabolites (lactic acid) is described by the equation:
i
pH = 6 1 + log [HC03 1
(7)
(8)
.
[CO:!l
(20)
In equations (1) - (20), d is the respiration coefficient; is the coefficient which takes into account the
p
567
increase in the arteriovenous disparity due to intensity of metabolism, despite changes in the gradient; M i is the quantity of incompletely oxidiz e d metabolites utilized by the internal organ's system in a unit of time; Qv is the coefficient of oxygen utilization. Subscripts mean : t - tissue, a - arterial, v - venous, A - alveolar, p - peripheral, i-related to the internal organs, ~ - related to the organism as a whole, and J.I. related to incompletely oxidized metabolites. Coefficients Cl - C21 are positive numbers.
568
B. V. Petrovskii, V. I. Shumakov, V. N. Novosel'tsev, E. Sh. Shtengol'd, V. M. Baikovskii, and L. A. Dartau
Successes in modern cardiosurgery together with achievements in the physiology of extracorporal. circulation, chemistry, electronics, and technology, made possible the experimental creation of implantable artificial hearts, capable of sustaining organic circulation /1 - 3 / .
on the principle of securing vital functions of the organism. These functions are a unified association of physiological control systems, whose interrelation secures the organism's homeostatic properties and determines values of its essential variables
Later experience, however, revealed difficulties in controlling the artificial heart, both automatically and manually. Lack of unified and quantitatively correlated ideas about functional links between physiological systems in the organism hampers the selection of an algorithm of control capable of securing a prolonged and adequate perfusion.
The control problem arises with the need of prosthesis for some vital physiological function, the lack of which deprives the organism of its properties of physiological homeostasis. Successful control in such a case will depend on completing lost links, based on the data of work of the undamaged systems in the organism. Many results of physiological studies are, however, disconnected and merely qualitative. Their use as a basis for restoration of lost links is generally impossible. Therefore, an explanation of quantitative relations requires formulation of model concepts, which permit us to study the functions and correlations of vital systems in the organism. This does not require data on every organism variable. Thus, for example, prosthesis of heart function requires selection of such organic functions, whose condition provides information most necessary for control of this prosthesis. On the other hand, there are also systems whose condition during control need be taken into account to a much lesser degree.
/4-6 /.
The first unsuccessful attempts in automation of devices of artificial circulation and ventilation testify to the complexity of this problem. Adequate control of circulation and ventilation is currently based on the assumption that there are rigid settings concerning such parameters as blood pressure, pH, POz' Peoz' etc. In principle, this enables immediate creation of control systems, in order to do this, it would be sufficient to create monitors continuously measuring these parameters and to construct a system of control by error. Such an approach, however, is insufficiently substantiated due to variability of these parameters within the range of general organic activity. Besides, even in cases of inadequate circulation, most of the really important parameters can be sustained for a long time at a relatively constant level by the compensatory resources of the organism.
For example, let us look at the mechanism of blood pressure maintenance at a definite level. The value of this level is the result of a definite relationship between the volume of circulation and tissue requirements of this circulation (vascular tonus). Changes in such values occur far more easily and more often than in the system of circulation control. Hence, there are reasons to believe that the system of blood pressure control is hierarchically younger. Certain clinical observations also point to this conclusion / 7 / . If the main emphasis is laid on maintaining the blood pressure during artificial heart control, the heart work will be deranged by factors which have no effect on circulation under normal circumstances. In such a caSE:, the automatic system will be too sensitive to obstacles.
Finally, direct use of deviations of these parameters in order to control the artificial circulation device may be hampered by apparently contradictory situations where the value of one of the regulated parameters calls for increased circulation, while the value of another parameter requires decreased circulation. These difficulties do not arise when the solution of the problem of automatic control is based directly
564
On the other hand, when th e main emphasis is laid on systems hierarchi c ally older than the c ir c ulation system (for instance, the thermoregulati on system / 8 / or blood pH r e gulation / 9 / ), the control will obviously be ins e nsitive to obstac les too conservative and inert . From the point of view of automatic control theory, a problem is encountered which is somewhat similar to the problem of determining optimum correlation between the fluctuant and dynamic error in the common automatic systems during selection of the best intensification coefficient along the closed circuit / 10, 11 / . Hence during artificial heart control, the main emphasis should be on information obtained from systems which are hierarchically equal in age to the system regulating the circulation. Data on hierarchical dependence can be obtained by forming model concepts of links in the organism. Using the method of mathematical modeling, some artificial situations may be created in which various physiological control systems are reciprocally deranged and the younger systems yield to the older. Correlation of systems in the organism can also be judged by their sensitivity to various environmental parameters, by time constants of these systems (their mobility), and finally on grounds of the model concepts of processes in the entire organism. Let us examine a mathematical model of a complex of physiological systems linked with the circulation system. The principl e of adequate energy provision for the organism as a whole serves as the methodological precondition to selecting the structure of a model. The essence of controlling this process is the constant maintenance of its highenergy condition. Thus, metabolic tissue activity determines values of the circulation and ventilation systems. Among all the functions of energy metabolism, oxygen and carbon dioxide transportation are basically determlOlOg. If there was a possibility of continuously measuring the total oxygen requirements of an organism, the determination of adequate valu e s for cir c ulation and ventilation would present no difficulties. However, to obtain such information by a direct study is impossible at this stage. It is necessary to evaluate the general intensity of metabolism from data of methodically accessibl e measurements. Two generalized systems of tissues and organs can be singled out in connection with the organism's structural heterogeneity and functional tissue specifi c ity. The peripheral system consists of striated muscles, connective tissue and skin. Its energy specificity is determined by
anaer obi c fun c ti oni :.g with subsequent elimination of the oxygen debt . The system of internal organs includes the brain, liver, kidneys, and heart, the function of whi ch is realized exclusiv ely under c onditions of aerobic energy resynthesis. The general circulation may thus be divided into circulation through peripheral tissues and through internal organs. Metabolites of energy metabolism penetrate into venous blood, where buffering processes take place together with formation of an acid-base complex determining the hydrogen ion's tension. Alveolar ventilation and pulmonary gas metabolism close the circle of physiological systems sustaining energy balance of the organism. Mathematical description of such a complex gives an autonomous system of equations with a single entrance signal represented by intensity of energy metabolism in the organism (evaluated in terms of oxygen). The behavior of the system is totally determined by its task and the gas composition of the inspired air / 12 / . The complete system of equations describing the complex of physiological system is listed in supplement No. 1. A complex mathematical model permits us to effect complex processes, embracing all the systems, for example, to determine the future condition of all systems by data of current observations, reproduce on the model the results of a number of physiological experiments (reactive hyperemia, perfusion, isolated carotid sinus, etc.). The model enables us to investigate processes in systems of the organism while artificially securing its vital activity (artificial ventilation, artificial circulation, perfusion of indi vidual organs). Without changing the mathematical model of the systems complex, we may eliminate the equation describing the natural heart and its regulation and replace it by equations describing an artificial heart and the algorithm of its control, therebyobtaining a model of an organism with an artificial heart. Such preliminary modeling was performed in several of the simplest cases. The following 4 va riables were selected, according to which the artificial heart was controlled : arterial and tissue c o ncentration of incompletely oxidized metabolites (l a ctic acid), magnitude of alveolar ventilation, and mean blood pressure. A linear algorithm was selected:
where Z is one of the aforementioned parameters and Q[ is the complete left heart circulation.
565
V a lues of coefficients which accompany the mod eling of control pro ce sses a re listed in Table 1 . T .-\ SLE 1.
:-<0.
1:
I
Con trol algo rithm
I
l.
°r=
~Ct ~ ~ Ot
0t
= 0. 157 !.\n in
2.
Or
=
o aCa~ +B a
a
=
3.
Or
= ct:V V A + Bv
mg"lo
0.1 83 l /m in m g"l< l/mi n a v = 0.1 63 l/min a
\sI
B = 0.1 83 l;tnin
I Ba = 2.67 l/min
i dv
= 2. 37 I/ min
result of this work is an approach to elaborati on o f such algorithms. This approach, connected with s e curing vital fun c tions of the organism, pres c ribes that artifici a l heart control should not be carried out with th e narrow aim of sustaining some parameters of the organism on a constant and predetermined level, but should be directed to the restoration of damag e d links in the united complex of the organism system and to secure its stability.
I
4.
Or =
() pPA + Bp
l 'm in a p = 0. 05 0 mm. Hg
IBp =
0
Algorithms 1 - 3 were stable within the entire range of the functional activity of organism (oxygen consumption 200-2,500 ml O:!/ min, circulation 4.525 l/min). As might ha ve been expected, algorithm 4 secured stability of the complex only within a limited range (circulation 4.5 - 8 l / min) and an increase in circulation was accompanied by a drop in the blood pressure (from 100 mm Hg to 50 mm Hg). Algorithms 1 - 2 secured levels in the processes of the complex which were closest to physiological levels. With the use of algorithm 3, some fluctuations appeared in the course of the control process connected with changes in the metabolic level. When information on concentration of incompletely oxidized metabolites in tissues and arterial blood is used as the control signal, the relative adequacy of circulation differs in both established and transitional processes. At present, however, any direct measurements of this concentration meet with difficulties. Therefore, the further search for the control algorithms may result in detection of magnitudes accessible to measurement and linked indirectly with the concentration of incompletely oxidized metabolites. When the value of alveolar ventilation VA is used as the control signal, the control system secures adequate circulation in all established conditions. In this case, however, the time of circulation establishment will be determined by the time constant of the respiratory system, which is 3 - 4 times greater than in the circulation system of an intact organism 113 I . Consequently, this way of regula ting Q is insufficient, especially since patients in need of heart prosthesis usually undergo changes in the external respiration level as well. Only the simplest ! algorithms of artificial heart control are discussed in this paper . The preliminary results cannot serve as a basis for reaching practical conclusions about the expediency of elaborate algorithms of any definite type. The main
In the elaboration of control algorithms on the basis of this suggested approac h, the first stage involves the inclusion of the organic physiological systems complex in the model. This enables us to obtain preliminary data on the stability and quality of algorithms of artificial heart control, to recommend elaboration of practical control algorithms, and to chec k them under experimental and clinical conditions.
BIBLIOGRAPHY 1. Petrovskii,B.V. and V.I.Shumakov. Implantiruemoe v organizm iskusstvennoe serdtse (Artificial Heart Implanted into the Organism).Kardiologiya, No.8. 1967.
2. De Ba key, M. E. and C. W. Hall. Towards the Artificial Heart. - New Scientist, Vol. 22, No.393. 1964. 3. Ko 1 f f, W. J. Les coeurs artificiels intrathoraciques. - Das Medizinische Prisma, Vol. 66, No. 4. 1966. 4 . Be r n a r d, C. Le
The Wisdom of the Body. -
6. Ash by, U. R. Design for a Brain. - Chapman and Hall, London. 1954 . 7. S hi k, L. L. Kislorodnyi zapros i krovoobrashchenie (Oxygen Demand and Circulation). - In: Kislorodnyi rezhim organizma i ego regulirovanie, Naukova Dumka, Kiev. 1966. 8. Barton,A. andO. Edholm. Chelovekv usloviyakh kholoda (Man in Cold Conditions). - IIL. 1957. [Russian Translation.] 9. V 1 a dim i r 0 v a, G. E. and N. S. Pan t e 1 e e v a . Funktsional' naya biokhimiya (Functional Biochemistry). - Izd. LGU. 1965 .
566
10. Newton,J.C., L.A.Gould a ndJ.F. Ka i z er. Teoriya lineinykh sledy ashchikh sistem (Theory of Linear Tra c ing Systems). - Fizmatgiz. 1961.
The internal organs :.ltiliz e incompletely ox idiz ed metabolites in prop o rti on to their concentr ation:
11. S h ch u kin, A. N. Dinamicheskie i flyuktuatsionnye oshibki upravlyaemykh ob"ektov (Dynamic and Fluctuation Errors of the Controlled Processes). - Sovetskoe radio. 1961.
Equations describing dynamics of 02 and CO2 in t h e int er nal organs system are similar to equations (3)- (6).
12. Petrovskii, B. V., V. I. Shumakov, V. N. No v 0 se l' t s e v, E. Sh. S h ten go l' d and L. A. Da r tau. Problema avtomaticheskogo upravleniya iSkusstvennym serdtsem i matematicheskoe modelirovanie (Automatic Control of an Artificial Heart and Mathematical Modeling). - Khirurgiya, No.4. 1968.
13. N a v rat i 1, M., K. K a dIe t sand S. D a u m . Patofiziologiya dykhaniya (Patophysiology of Respiration). - Izd. Meditsina. 1967.
(9)
The heart system provides th e blood flow in the systemic and pulmonary c ir c ulation under condition that: (10)
and systemic circulation is d ete rmined by neurohumoral factors:
The mean blood pressure depends on cardiac output and vascular system resistance: (12 )
Supplement No. 1 The model of physiological homeostasis includes models of the following systems: metabolism of muscle tissue, of internal organs : of circulation control and its distribution, and of external respiration.
Dynamics of vascular tonus are determined according to the metabolism theory of work hyperemia. (13 ) (14)
(15 )
Energy losses in the peripheral system compensated by aerobic and anaerobic means:
Circulations Qiand Qp are determined by equations: (1 )
(16 ) and under established conditions: w ~naer
=Q
Wp , a
= const.
(2)
The pulmonary system secures the oxygen entry into the organism from the atmospheric air and removal of carbon dioxide:
Oxygen and carbon dioxide diffusion in peripheral organs is determined by the equations: (3)
(4)
(5)
The alveolar ventilation under established conditions is determined from the ratio: (19 )
(6)
The acid-alkaline equilibrium in the organism is determined by the Henderson-Hasselbach formula
Dynamics of incompletely oxidized metabolites (lactic acid) is described by the equation:
i
pH = 6 1 + log [HC03 1
(7)
(8)
.
[CO:!l
(20)
In equations (1) - (20), d is the respiration coefficient; is the coefficient which takes into account the
p
567
increase in the arteriovenous disparity due to intensity of metabolism, despite changes in the gradient; M i is the quantity of incompletely oxidiz e d metabolites utilized by the internal organ's system in a unit of time; Qv is the coefficient of oxygen utilization. Subscripts mean : t - tissue, a - arterial, v - venous, A - alveolar, p - peripheral, i-related to the internal organs, ~ - related to the organism as a whole, and J.I. related to incompletely oxidized metabolites. Coefficients Cl - C21 are positive numbers.
568