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Identification of the Circulatory System
IDENTIFICATION OF THE CIRCULATORY SYSTEM
v.
I . Burakovsky, V. A. Lyshchuk and V. L. Stoljar
The Bakulev Institute for Cardio-vascular Surgery, AMS of the USSR, Moscow, USSR
Abstract. In this paper we discuss the problem of identification of the circulatory system in cardiosurgical clinic. Mathematical description of cardiovascular system oriented on cardiosurgical clinic is suggested. We present a method for parameter estimation of the circulatory system models with the use of second type sensibility functions and Kalman filter. Results of the identification of the circulatory system in clinic for evalu~ ation of the state of patients are presented. Kef words. Biomedical, models, cardiology, parameter estimation, Ka man filter.
INTRODUCTION
the operational models [3,11). The definition of coefficients [8,9] (for example coefficients of differential equations describing the cardio-vascular system) does not give any clinical effect.That is why it is necessary to estimate the set of model parameters in clinic Which are the constants of biophisical lows and physiological relations, forming the main point of the model.
The progress of cardiac surgery demanded the creations of methods for quick evaluation of the state of the cardio-vascular system of patients in the intensive care units, reanimation units and operating rooms. Tod~ this problem can be solved on the base of identification of the circulatory system in the real-time ~,11,16). Though the number of pa~ ers on this problem is increasingr~ the authors as a rule do not go further than a simple presentation of a formulation of problems and an identification method without giving any results of clinical application [10,14,15). One of the reasons for such matter of state is the untraditionality of used methods and the prejudice of physicians to the results obtained by the indirect way ~1], and besides the difficulties of the organisation of a technical assistance and a calculating process in clinical conditions [2].
It comes from the natural phySicians desire to get not only an adequate mathematical description of the circulatory system (that proves to be quite approximate) as a result of idem ification but also the possibility of analizing of a patient state with the use of model parameter estimates having clinical interpretation.
The problem of identification of the circulatory system includes the questions of controlability, observeability, and identmabil1ty uder the test actions present in the clinical conditions and a measurements [13], selection of an identification method and mainly the problem of valueability of identification results for a physician. The latter point which many authors do not take into consideration [7,10,14] puts a number of specific {clinical) requirements to 85 1
For a cardiosurgical clinic we suggested a mathematical descrition of the cardio-vascular system generalizing valuable physiological and clinical properties and relations which allows to construct concrete models in accordance with the reseaTCh task, measurements and test actions [4]. In this paper for such models we present a method for parameter estimation realized in the form of application medical program package "Kalman" by means of monitor-computer system of our Institute.
MDDEL PARAMETER ESTIMATION OF THE CIRCULAmJRY SYSTEM Problem formulation. The cardio-vas-
85 2
V. I. Burakov sky, V. A. Ly s hchuk a nd V. L. Stoljar
cular system is considered to be a totality of n subsystems (vascular sites, cardiac cavities, etc.) which are described in fixed parameters (4]. The description of the model of the circulatory system concerning mean pressures is like this
P(t) = ERTp(t)+ EIHt)
(1 )
Y(t) =CP(t)
where pet) - n-dimensioDal state vector (vector of pressures) pet) =[Pi(t»).Q(tl-n-dimensional vector of test actions (Ht)· [Gi (t)] (Qat) volume flow rate of blood infusion into the i-th element, e.g. from a syringe), yet) - m-dimentional vector of observations yet) : [Y,(t)] • E and R - nxn matrixes of properties of model elements E.dia9 [e.. ···e,,) ( et - rigidity of the ~-th element), R:[fjtl ( J'jt - conductivity for l;>lood flows between elements j and l ) , C - mxn matrix of observations with known elements C = [Cji] For an adequate description of a steady state we introduce a balance equation R
l e~ Pt(t) = VK + ~1
J'I tn
0 ~.
Qi
d.~
(2)
where ~k - initial blood volume (in a steaay state). While constructing the concrete model from Eq. (1)-(2) in accordance with the research task and its specific features it is required to show the number (n) and the type of mapping subsystems, the character of their relations (H), the measurements and test actions. It is necessary to estimate model parameter (elements of matrixes H and E) on the base of measurement results yet) and Q(t) at [to;to+T ) ( to -moment when the action was applied) •
sensibility functions and Kalman filter [5,6)on the base of measuring Yi and Qj at [io,to+TJ • At the second stage model parameters themselves ( fji, et ) are calculated as a solution of nonlinear algebraic equations .1\.:::. IP (eL' fKi) (4) where A -vector of coefficients of Eq. (3). If the number of unknown parameters (r) in Eq. (4) is greater than the number of independent equations (L), then for (r-L) parameters we take the mean statistical values for the given group of patients. Well make a detailed analyses of the two-stepped method for model parameter estimation at the example of a simple but physiologically valuable twoelemental model of the circulatory system. PARAMETER ESTIMATION OF THE TWOELEMENTAL IDDEL OF THE CIRCULATORY SYSTEM The cardio-vascular system is c .nsidered to be of th~ two elastic reservoirs - arterial ~index 1) and venous (index 2). Let's unvestigate a clinical situation when permanent measurement is possible only for arterial pressure (venous pressure is measured before the action). The test action (blood infusion) is aPElied to the arterial reservoir (Q2(t)=0). Then we suggest in Eq. (1) n=2 and specify matrixes R and E of subsystem properties and observation matrix in the following form E =- di.O-<} [e" ez.1 , R" [fjt]
~(t)+Q)f(t)=gHQ,(t)+g.OQ1(t)
(5)
where o/fe1+feZ+J e,z, !1;,11=e" r,10:(
SOlvi~
method. Parameter estimation is do~ed in two stages. At the first stage it is necessary to estimate the coefficients of model e~uations, which are transformed from (1) to
where
i: I , ••• , m·f
QIl'l -- ",
a· Oi = 0
The coefficients al(l and &jKL are defined by the identification method with the use of the second type
Model parameter definit ion'(f,ef ,e".:!..) is performed in two stages. At the first stage we define the coefficieDts of equation (5) after measuring P1 (t) and ~(t) at [O,T] , and at the second stage we calculate model parameters themselves. Let's introduce the basic e~ation with coefficieDts (11' gf~' "t(,aprior estimates of of' 6'1,6 10
x(t):-a~X(i) +E>~OQf(t)+ r,~ Q.{t)
(6)
Identificati on o f the c ir c ul a t or y sys t em
Difference in solutions of the equations (6) and(f5) with the same in:i.t• ial values Pi j) (OJ=X (n( 0 ) , l= 0,1 is referred to distinguishing of proper coefficients. Suppose further that there exist functions N1 (t), Mo (t), Mo, (t) which with any endillg deviations ~,b11' b10 from their basic values 0;, b~1' b~o lead to the satisfiability of equivalence for t~O
P'(t):X(t)+AQ,N 1 (t} +Llr,l1M,(t)+t!~,oMIJ(t)
where AQ1
(7)
=a,-a;,", A~11=g'l-t~,l!>~,o:~,o-~:
Now we find the terms by which it will be possible to define Mo(t), Mo,(t) and N1 (t). For this by substituting P1(t) and P1(t) from (7) into (5) we get X(t)+c.a.)jl(tl +dgUM,(t) +I1SlOMO(t) +(II.~+
* .
.
.
.
_(8)
where K~ t K , Z(K)::P,
H(K) = [N.CK),
1\
(9)
Because we supposed that Eq. (7) and consequently Eq. (9) are satisfied with any endings l! Cl" l!> r,11 , 6 1',10 , then all the terms in brackets must be equal zero for (9), that is
.. *. . NI(t) = -a, N,(t) - P1Ct)
V(K)
(11)
T
+
+W(K+1) R(KH) WT(KH)
Where I=d:ia g [1, ••. ,1). Initial value for covariance matrix S(O) = =52 o· Estimates of equation coefficients (5) are defined on the base of the known vector estimate A on the i-th step 1\.
~IO(t)
(10)
-I
H(KtO [~(K+f)S(K) ~(K+I) +R(K+tj
SCK+1)= [I -W(K+1) HCKH)] S(K)[I-W(KH) j.I(I
1\.
System (10) is the original system for definition Mo(t), ~(t), N1 (t), called the second type sensibility functions (6). Initial values for Mo(t), ~(t), N1 (t) are equal to zero that comes from (7) and the equalit~ of initial values for P1 (t) and X(t). To obtain unbiased coefficient estimates with minimal dispersion of the equation (5) on the base of measured with errors P1 (t) and Q1(-\:) we use Kalman filter. For this we rewrite (7) in the discrete form l(K):: H(K) ACK) -I-
+ W(K+1) [l(K+1)- HCK+f)A(K)] " value A (0)=0. Vector
with initial of gains is determined from relations
al(l); a
Mott):
1
S,ol,
1\
A(K+O:: A(K)
W(KH):: SCK)
Performing necessary transformations considering equations (6) and (5) we obtain Aa, (~
1
I>
T
* . (A~fO+ £"0) * Q,(t) =(A~If+~4!)~Ct)+
-afM o(t)+Q1 Ct) .. - "* . . M/t)- - Cl M Ct) + Q4(t)
t= [ao,JA&If'
M1(K), Mo(K)],
~ (k)-uncorrelated random sequence with zero mathematical expectation and dispersion R(k), characterizing measurement error. Due to the fact that model parameters are considered to be invariable durillg the interval of an order of time of transitional processes in the circulatory system we have A(K+1) = ). (K) (12) We may take Eq. (12) and Eq. (11) as equations of state and observation [8] of certain system with state vector A. Then we may use the algorithm of Kalman filter for estimation A
+a,) (l( ctJ+lIa,N i (t) +A~/fM,(t) +t,g10 M/t») -
- Q}t») +lI~-fo{Mo(t) +n;Mo(t) -Q(t)::: 0
853
* j
1\.
+6 0
4 (t)
" b (i): H
,
= b*"w + Ll ". f>,o(l}
,*"
~ +l!~ Cl) I1
11'
It is required to find four model parameters (P' d- • eo{ , e;z) but we have only three independent relations which involve them Q1' b11 , b10 · Because of this for unique parameter definition we take the relation of arteria 1 and venous pressures under steady state (as far as the model is concerned) as the fourth1 missing relation PI · P2 1= (d.,+.!) f- • Then for parameter calculation we get the following formulas:
J " (Cl
f
~f1 - ~ /0)
-2.
g11
'
ef
::
6
If'
cL" (Pf - P2.)P~ff' e 2= g~og~: p;lpZ j1. We obtained similar solut ions for other possible clinical versions of blood circulation measurement and test actions •
854
V. I. Burakovsky, V. A. Lyshchuk and V. L. Stoljar
We characterize an identification error (parameter estimation) by the criterion value
J (n =1. ~l [ ~ (1<) fL
P-r1 (K) PiCK)
]
2
I(of
where PT1 - is the curve of relaxation process of ~al pressure corresponding to parameter estimates defined on the base of [ descreete measurements Q., aDd P1 at [O,T]. Game example. As a subject of research we used three elemental model of circulatory system (4]. The action - infusion of 40 ml of blood in to the arterial reservoir with the rate of 10ml/sec. Disturbing was given by rounding of measuring data: arterial pressure, venous pressure, action. Measurement interval of 6 sec was sufficient for obtaining satisfying results (see Table 1). The relation of parameter estimates and observation int erval is shown in Fig. 1. Evaluation of the state of the circulatory system of a patient. As an example we take the estimates of the circulatory system of patient P. (see Table 2) who was operated on for aortic-coronary shunting. First experimental researches on the estimation of the state of a patient using this derived method were carried out in 1976 on the base of monitor-computer system of our Institute. Up to this day all technical means and neccesary software were developed and implemented at our Institute for constant use of the identification method in the treatment process [2,3]. The type of initial information for performing an identification and the individual therapy units are shown in Fig. 2 and 3. The clinical method for performing an identification of the circulatory system is presented in details in the paper [2]. CONCLUSION Mathematical models of the circulatory system combined with methods of their parameter estimation are the base of the Automated system for providing physician decisions that was worked out in our Institute. The identification method proved to be quite effective in our clinic because it allows a physician to detect quickly a number of quantitative changes and disorders of cardio-vascular system, for example medicamental heart overload on the back-
ground of pulmonary vasospasm, etc. The more exact qualitative analys of blood circulation using identification method requires the solving of some urgent problems. First of all - it is a reflection of test actions in the model doiIlg no barm to a patient. In this aspect the actions from the respiration to the circulatory system are most likely. The next group of problems is the analyses of an identification error in relation to the mpping accuratness of the actual sites of pressure measurements in the model, detail description of various elements of circulation, influence of central regulation. Experts in the field of modern control theory should render their assistance in solving the given problems. Our experience proves that the ue.in part of such help is their team-work with physicians on the problem of evaluation of the state of a patient in the clinic itself. REFERENCES tBurakovsky V.I., Lyshchuk V.A., Podgorny V.F., Sokolov M.V. (1974). PrinCiples of the individual therapy with the use of computers. Vestnik AMS of the USSR, §" 31-40. 2.Lyshchuk V.A., Stoljar V.L., Sadojan D.G., Podgorny V.F. (1978). Application the dynamic identification in the intensive care. Circulation, ~, 31. ~Lyshchuk V.A., Stoljar V.F. (1979) Identification of the circulatory system. Avtomatica, 1, 19-26. ~Lyshchuk V.A., Stoljar V.L. (1978). Generalized mathematical description of the cardiovascular system with the central nervous-reflex regulation. Reprints of the :td Union conference on biol0tiic and medical cibernetics,oscow,
v.1, 141-147. ~Pollak J.A. (1974). Parameter defini-
tion of the dynamic models of electropower system elements. In the col. Problems of researches andcontrol of autonomic power systems, cheljabinsk. &Serdjukov V.A., Serdjukova A.E. (1972). Sensibility functions of the second type. In the col. Present problems of the technical cibernetics. Moscow, 94=98. 7.Tettelbaum LM., Zjablova N.V. (1977). Identification of integral parameters of the cardio-vascular system by methods of electrical circuits theory. In the col.Frec. of MEI, 345, 94-101.
Identification of the ci rculatory system ~Aickoff
855
u.Cobelly C., Lepschy A., Pomanin-Jacur P. (1975). Bases of identiG. (1976). Identifiability Probfication of control systems, lems in Biological System. Proc. Moscow, "Mirtt. the 4-th IFAC Symp. "IdentificaqRyberman N.S., Chadejev V.M. (1975). tion and System Parameter", Construction of models Of~roduc Tbilisi, part 1, 390=400. tion processes, Moscow, If ergya". 14.Dennison J.C. and Cristian J.A. ta.Aaslid R. (1973). Estimation of car(1974). Estimation of Paramemedio-vascular parameters. Proc. ters of the Human Cardio-Vascuthe 3-rd IFAC Symp. "Identificalar System. Proc. the 7-th Hawai tion andA!;stem Parameter EstiInt. Cont. on Syst. Sci., Honomation'! terdam, part 1, 231lulu, 78-82. -234. I~Deswysen B.A. (1976). Optimum Choice of the Statistical Parameters of ltBekey G.A. and Beneken J .E.W. (1978). a Nonlinear Filter Applied to Identification of biological Cardio-vascular Parameter Estisystem: a survey. Automatica, mation. Proc. the 4-th IFAC Symp. v.14, 1, 41-7. "Identification and ~ystem para12.Chang P.P., Matson G.L., Kendrick meter Estimation", Tbilisi, part J.F. Rideout V.C. (1974). Para1, 424-434. meter estimation in the canine 16.Rideout V.C. and Beneken J.E.W. cardiovascular system. IEEE Trans. Autom . Control, ~, 6, (1975). Parameter estimation applied to physiological system. 927-31. Ann. ArCA, v.17, 1, 23-36.
856
V. I. Burakovsky. V. A. Lyshchuk and V. L. Stoljar
Table 1 Parameters
d-
ml
mm Itg sec 10.60
e2
~.
mm Hfi mI
mm Itg sec
9.44
Subject Model
~
1.00 0.99
mm Hfi
J(22)
iiiI
-
0.010 0.0099
0.50 0.507
0.003
Table 2
~et.rs
cl-
ml mm Itg sec
~ction
).07 -30 ml - - - -- - ~ -- - - -
o
i
f ml mm Itg sec 0.46
- - - - - -
3
e( mm~ 2.71
- -
--
ez
mm Hp: ml
J(25)
0.06
0.025
- - - - - - -
5
Fig. 1. Dependence of parameter estimates from observation
i(c)
interval ( ot -characteristic of heart function, f -common peripheral conductivity, e1-rigidity of arterial reservoir, e 2 -rigLdity of venous reservoir)
- --
Ide ntifica t io n o f the ci rc ul a t o r y sys t em
o
4
5
6
85 7
7 t(C)
Fig. 2. Response of the circulatory system to bloodletting of 30 ml of blood from arterial reservoir (P1 and P -arterial and venous pressures respectively, ~-~olume flow rate of blood loss)
Fig. 3. Individual therapy unit with monitor-computer system in the Bakulev Institute for cardio-vascular surgery of AMS of the USSR
v.
I . Burakovsky, V. A. Lyshchuk and V. L. Stoljar
The Bakulev Institute for Cardio-vascular Surgery, AMS of the USSR, Moscow, USSR
Abstract. In this paper we discuss the problem of identification of the circulatory system in cardiosurgical clinic. Mathematical description of cardiovascular system oriented on cardiosurgical clinic is suggested. We present a method for parameter estimation of the circulatory system models with the use of second type sensibility functions and Kalman filter. Results of the identification of the circulatory system in clinic for evalu~ ation of the state of patients are presented. Kef words. Biomedical, models, cardiology, parameter estimation, Ka man filter.
INTRODUCTION
the operational models [3,11). The definition of coefficients [8,9] (for example coefficients of differential equations describing the cardio-vascular system) does not give any clinical effect.That is why it is necessary to estimate the set of model parameters in clinic Which are the constants of biophisical lows and physiological relations, forming the main point of the model.
The progress of cardiac surgery demanded the creations of methods for quick evaluation of the state of the cardio-vascular system of patients in the intensive care units, reanimation units and operating rooms. Tod~ this problem can be solved on the base of identification of the circulatory system in the real-time ~,11,16). Though the number of pa~ ers on this problem is increasingr~ the authors as a rule do not go further than a simple presentation of a formulation of problems and an identification method without giving any results of clinical application [10,14,15). One of the reasons for such matter of state is the untraditionality of used methods and the prejudice of physicians to the results obtained by the indirect way ~1], and besides the difficulties of the organisation of a technical assistance and a calculating process in clinical conditions [2].
It comes from the natural phySicians desire to get not only an adequate mathematical description of the circulatory system (that proves to be quite approximate) as a result of idem ification but also the possibility of analizing of a patient state with the use of model parameter estimates having clinical interpretation.
The problem of identification of the circulatory system includes the questions of controlability, observeability, and identmabil1ty uder the test actions present in the clinical conditions and a measurements [13], selection of an identification method and mainly the problem of valueability of identification results for a physician. The latter point which many authors do not take into consideration [7,10,14] puts a number of specific {clinical) requirements to 85 1
For a cardiosurgical clinic we suggested a mathematical descrition of the cardio-vascular system generalizing valuable physiological and clinical properties and relations which allows to construct concrete models in accordance with the reseaTCh task, measurements and test actions [4]. In this paper for such models we present a method for parameter estimation realized in the form of application medical program package "Kalman" by means of monitor-computer system of our Institute.
MDDEL PARAMETER ESTIMATION OF THE CIRCULAmJRY SYSTEM Problem formulation. The cardio-vas-
85 2
V. I. Burakov sky, V. A. Ly s hchuk a nd V. L. Stoljar
cular system is considered to be a totality of n subsystems (vascular sites, cardiac cavities, etc.) which are described in fixed parameters (4]. The description of the model of the circulatory system concerning mean pressures is like this
P(t) = ERTp(t)+ EIHt)
(1 )
Y(t) =CP(t)
where pet) - n-dimensioDal state vector (vector of pressures) pet) =[Pi(t»).Q(tl-n-dimensional vector of test actions (Ht)· [Gi (t)] (Qat) volume flow rate of blood infusion into the i-th element, e.g. from a syringe), yet) - m-dimentional vector of observations yet) : [Y,(t)] • E and R - nxn matrixes of properties of model elements E.dia9 [e.. ···e,,) ( et - rigidity of the ~-th element), R:[fjtl ( J'jt - conductivity for l;>lood flows between elements j and l ) , C - mxn matrix of observations with known elements C = [Cji] For an adequate description of a steady state we introduce a balance equation R
l e~ Pt(t) = VK + ~1
J'I tn
0 ~.
Qi
d.~
(2)
where ~k - initial blood volume (in a steaay state). While constructing the concrete model from Eq. (1)-(2) in accordance with the research task and its specific features it is required to show the number (n) and the type of mapping subsystems, the character of their relations (H), the measurements and test actions. It is necessary to estimate model parameter (elements of matrixes H and E) on the base of measurement results yet) and Q(t) at [to;to+T ) ( to -moment when the action was applied) •
sensibility functions and Kalman filter [5,6)on the base of measuring Yi and Qj at [io,to+TJ • At the second stage model parameters themselves ( fji, et ) are calculated as a solution of nonlinear algebraic equations .1\.:::. IP (eL' fKi) (4) where A -vector of coefficients of Eq. (3). If the number of unknown parameters (r) in Eq. (4) is greater than the number of independent equations (L), then for (r-L) parameters we take the mean statistical values for the given group of patients. Well make a detailed analyses of the two-stepped method for model parameter estimation at the example of a simple but physiologically valuable twoelemental model of the circulatory system. PARAMETER ESTIMATION OF THE TWOELEMENTAL IDDEL OF THE CIRCULATORY SYSTEM The cardio-vascular system is c .nsidered to be of th~ two elastic reservoirs - arterial ~index 1) and venous (index 2). Let's unvestigate a clinical situation when permanent measurement is possible only for arterial pressure (venous pressure is measured before the action). The test action (blood infusion) is aPElied to the arterial reservoir (Q2(t)=0). Then we suggest in Eq. (1) n=2 and specify matrixes R and E of subsystem properties and observation matrix in the following form E =- di.O-<} [e" ez.1 , R" [fjt]
~(t)+Q)f(t)=gHQ,(t)+g.OQ1(t)
(5)
where o/fe1+feZ+J e,z, !1;,11=e" r,10:(
SOlvi~
method. Parameter estimation is do~ed in two stages. At the first stage it is necessary to estimate the coefficients of model e~uations, which are transformed from (1) to
where
i: I , ••• , m·f
QIl'l -- ",
a· Oi = 0
The coefficients al(l and &jKL are defined by the identification method with the use of the second type
Model parameter definit ion'(f,ef ,e".:!..) is performed in two stages. At the first stage we define the coefficieDts of equation (5) after measuring P1 (t) and ~(t) at [O,T] , and at the second stage we calculate model parameters themselves. Let's introduce the basic e~ation with coefficieDts (11' gf~' "t(,aprior estimates of of' 6'1,6 10
x(t):-a~X(i) +E>~OQf(t)+ r,~ Q.{t)
(6)
Identificati on o f the c ir c ul a t or y sys t em
Difference in solutions of the equations (6) and(f5) with the same in:i.t• ial values Pi j) (OJ=X (n( 0 ) , l= 0,1 is referred to distinguishing of proper coefficients. Suppose further that there exist functions N1 (t), Mo (t), Mo, (t) which with any endillg deviations ~,b11' b10 from their basic values 0;, b~1' b~o lead to the satisfiability of equivalence for t~O
P'(t):X(t)+AQ,N 1 (t} +Llr,l1M,(t)+t!~,oMIJ(t)
where AQ1
(7)
=a,-a;,", A~11=g'l-t~,l!>~,o:~,o-~:
Now we find the terms by which it will be possible to define Mo(t), Mo,(t) and N1 (t). For this by substituting P1(t) and P1(t) from (7) into (5) we get X(t)+c.a.)jl(tl +dgUM,(t) +I1SlOMO(t) +(II.~+
* .
.
.
.
_(8)
where K~ t K , Z(K)::P,
H(K) = [N.CK),
1\
(9)
Because we supposed that Eq. (7) and consequently Eq. (9) are satisfied with any endings l! Cl" l!> r,11 , 6 1',10 , then all the terms in brackets must be equal zero for (9), that is
.. *. . NI(t) = -a, N,(t) - P1Ct)
V(K)
(11)
T
+
+W(K+1) R(KH) WT(KH)
Where I=d:ia g [1, ••. ,1). Initial value for covariance matrix S(O) = =52 o· Estimates of equation coefficients (5) are defined on the base of the known vector estimate A on the i-th step 1\.
~IO(t)
(10)
-I
H(KtO [~(K+f)S(K) ~(K+I) +R(K+tj
SCK+1)= [I -W(K+1) HCKH)] S(K)[I-W(KH) j.I(I
1\.
System (10) is the original system for definition Mo(t), ~(t), N1 (t), called the second type sensibility functions (6). Initial values for Mo(t), ~(t), N1 (t) are equal to zero that comes from (7) and the equalit~ of initial values for P1 (t) and X(t). To obtain unbiased coefficient estimates with minimal dispersion of the equation (5) on the base of measured with errors P1 (t) and Q1(-\:) we use Kalman filter. For this we rewrite (7) in the discrete form l(K):: H(K) ACK) -I-
+ W(K+1) [l(K+1)- HCK+f)A(K)] " value A (0)=0. Vector
with initial of gains is determined from relations
al(l); a
Mott):
1
S,ol,
1\
A(K+O:: A(K)
W(KH):: SCK)
Performing necessary transformations considering equations (6) and (5) we obtain Aa, (~
1
I>
T
* . (A~fO+ £"0) * Q,(t) =(A~If+~4!)~Ct)+
-afM o(t)+Q1 Ct) .. - "* . . M/t)- - Cl M Ct) + Q4(t)
t= [ao,JA&If'
M1(K), Mo(K)],
~ (k)-uncorrelated random sequence with zero mathematical expectation and dispersion R(k), characterizing measurement error. Due to the fact that model parameters are considered to be invariable durillg the interval of an order of time of transitional processes in the circulatory system we have A(K+1) = ). (K) (12) We may take Eq. (12) and Eq. (11) as equations of state and observation [8] of certain system with state vector A. Then we may use the algorithm of Kalman filter for estimation A
+a,) (l( ctJ+lIa,N i (t) +A~/fM,(t) +t,g10 M/t») -
- Q}t») +lI~-fo{Mo(t) +n;Mo(t) -Q(t)::: 0
853
* j
1\.
+6 0
4 (t)
" b (i): H
,
= b*"w + Ll ". f>,o(l}
,*"
~ +l!~ Cl) I1
11'
It is required to find four model parameters (P' d- • eo{ , e;z) but we have only three independent relations which involve them Q1' b11 , b10 · Because of this for unique parameter definition we take the relation of arteria 1 and venous pressures under steady state (as far as the model is concerned) as the fourth1 missing relation PI · P2 1= (d.,+.!) f- • Then for parameter calculation we get the following formulas:
J " (Cl
f
~f1 - ~ /0)
-2.
g11
'
ef
::
6
If'
cL" (Pf - P2.)P~ff' e 2= g~og~: p;lpZ j1. We obtained similar solut ions for other possible clinical versions of blood circulation measurement and test actions •
854
V. I. Burakovsky, V. A. Lyshchuk and V. L. Stoljar
We characterize an identification error (parameter estimation) by the criterion value
J (n =1. ~l [ ~ (1<) fL
P-r1 (K) PiCK)
]
2
I(of
where PT1 - is the curve of relaxation process of ~al pressure corresponding to parameter estimates defined on the base of [ descreete measurements Q., aDd P1 at [O,T]. Game example. As a subject of research we used three elemental model of circulatory system (4]. The action - infusion of 40 ml of blood in to the arterial reservoir with the rate of 10ml/sec. Disturbing was given by rounding of measuring data: arterial pressure, venous pressure, action. Measurement interval of 6 sec was sufficient for obtaining satisfying results (see Table 1). The relation of parameter estimates and observation int erval is shown in Fig. 1. Evaluation of the state of the circulatory system of a patient. As an example we take the estimates of the circulatory system of patient P. (see Table 2) who was operated on for aortic-coronary shunting. First experimental researches on the estimation of the state of a patient using this derived method were carried out in 1976 on the base of monitor-computer system of our Institute. Up to this day all technical means and neccesary software were developed and implemented at our Institute for constant use of the identification method in the treatment process [2,3]. The type of initial information for performing an identification and the individual therapy units are shown in Fig. 2 and 3. The clinical method for performing an identification of the circulatory system is presented in details in the paper [2]. CONCLUSION Mathematical models of the circulatory system combined with methods of their parameter estimation are the base of the Automated system for providing physician decisions that was worked out in our Institute. The identification method proved to be quite effective in our clinic because it allows a physician to detect quickly a number of quantitative changes and disorders of cardio-vascular system, for example medicamental heart overload on the back-
ground of pulmonary vasospasm, etc. The more exact qualitative analys of blood circulation using identification method requires the solving of some urgent problems. First of all - it is a reflection of test actions in the model doiIlg no barm to a patient. In this aspect the actions from the respiration to the circulatory system are most likely. The next group of problems is the analyses of an identification error in relation to the mpping accuratness of the actual sites of pressure measurements in the model, detail description of various elements of circulation, influence of central regulation. Experts in the field of modern control theory should render their assistance in solving the given problems. Our experience proves that the ue.in part of such help is their team-work with physicians on the problem of evaluation of the state of a patient in the clinic itself. REFERENCES tBurakovsky V.I., Lyshchuk V.A., Podgorny V.F., Sokolov M.V. (1974). PrinCiples of the individual therapy with the use of computers. Vestnik AMS of the USSR, §" 31-40. 2.Lyshchuk V.A., Stoljar V.L., Sadojan D.G., Podgorny V.F. (1978). Application the dynamic identification in the intensive care. Circulation, ~, 31. ~Lyshchuk V.A., Stoljar V.F. (1979) Identification of the circulatory system. Avtomatica, 1, 19-26. ~Lyshchuk V.A., Stoljar V.L. (1978). Generalized mathematical description of the cardiovascular system with the central nervous-reflex regulation. Reprints of the :td Union conference on biol0tiic and medical cibernetics,oscow,
v.1, 141-147. ~Pollak J.A. (1974). Parameter defini-
tion of the dynamic models of electropower system elements. In the col. Problems of researches andcontrol of autonomic power systems, cheljabinsk. &Serdjukov V.A., Serdjukova A.E. (1972). Sensibility functions of the second type. In the col. Present problems of the technical cibernetics. Moscow, 94=98. 7.Tettelbaum LM., Zjablova N.V. (1977). Identification of integral parameters of the cardio-vascular system by methods of electrical circuits theory. In the col.Frec. of MEI, 345, 94-101.
Identification of the ci rculatory system ~Aickoff
855
u.Cobelly C., Lepschy A., Pomanin-Jacur P. (1975). Bases of identiG. (1976). Identifiability Probfication of control systems, lems in Biological System. Proc. Moscow, "Mirtt. the 4-th IFAC Symp. "IdentificaqRyberman N.S., Chadejev V.M. (1975). tion and System Parameter", Construction of models Of~roduc Tbilisi, part 1, 390=400. tion processes, Moscow, If ergya". 14.Dennison J.C. and Cristian J.A. ta.Aaslid R. (1973). Estimation of car(1974). Estimation of Paramemedio-vascular parameters. Proc. ters of the Human Cardio-Vascuthe 3-rd IFAC Symp. "Identificalar System. Proc. the 7-th Hawai tion andA!;stem Parameter EstiInt. Cont. on Syst. Sci., Honomation'! terdam, part 1, 231lulu, 78-82. -234. I~Deswysen B.A. (1976). Optimum Choice of the Statistical Parameters of ltBekey G.A. and Beneken J .E.W. (1978). a Nonlinear Filter Applied to Identification of biological Cardio-vascular Parameter Estisystem: a survey. Automatica, mation. Proc. the 4-th IFAC Symp. v.14, 1, 41-7. "Identification and ~ystem para12.Chang P.P., Matson G.L., Kendrick meter Estimation", Tbilisi, part J.F. Rideout V.C. (1974). Para1, 424-434. meter estimation in the canine 16.Rideout V.C. and Beneken J.E.W. cardiovascular system. IEEE Trans. Autom . Control, ~, 6, (1975). Parameter estimation applied to physiological system. 927-31. Ann. ArCA, v.17, 1, 23-36.
856
V. I. Burakovsky. V. A. Lyshchuk and V. L. Stoljar
Table 1 Parameters
d-
ml
mm Itg sec 10.60
e2
~.
mm Hfi mI
mm Itg sec
9.44
Subject Model
~
1.00 0.99
mm Hfi
J(22)
iiiI
-
0.010 0.0099
0.50 0.507
0.003
Table 2
~et.rs
cl-
ml mm Itg sec
~ction
).07 -30 ml - - - -- - ~ -- - - -
o
i
f ml mm Itg sec 0.46
- - - - - -
3
e( mm~ 2.71
- -
--
ez
mm Hp: ml
J(25)
0.06
0.025
- - - - - - -
5
Fig. 1. Dependence of parameter estimates from observation
i(c)
interval ( ot -characteristic of heart function, f -common peripheral conductivity, e1-rigidity of arterial reservoir, e 2 -rigLdity of venous reservoir)
- --
Ide ntifica t io n o f the ci rc ul a t o r y sys t em
o
4
5
6
85 7
7 t(C)
Fig. 2. Response of the circulatory system to bloodletting of 30 ml of blood from arterial reservoir (P1 and P -arterial and venous pressures respectively, ~-~olume flow rate of blood loss)
Fig. 3. Individual therapy unit with monitor-computer system in the Bakulev Institute for cardio-vascular surgery of AMS of the USSR