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Three-element model for total systemic circulation: Emphasis on the accuracy of parameter estimates
THREE-ELEMENT MODEL FOR TOTAL SYSTEMIC CIRCULATION: EMPHASIS ON THE ACCURACY OF PARAMETER ESTIMATES G. Cevenini, Received
P. Barbini,
October
A. Cappello’
and
1986, revised and accepted
G. Avanzolini*
May 1987
ABSTRACT In this study, the accuracy achievable in the parameter estimates o/a three-element linear model for the s_ystemic vascular bed is considered. The model neglects inertial efjects and includes only three elements representing arterial compliance, peripheral resistance and venous compliance, in agreement with recent sensitivity investigations. Parameter estimation startsfrom arterial and right atria1 pressure signals generated by a closed-loop simulator of the cardiovascular circulation and corrupted with normal noise to account for measurement errors. In thus way, the Keywords:
Circulatory
system, modelling,
systemic circulation,
INTRODUCTION Computer identification techniques have been applied in the past to the cardiovascular system for two main purposes: as a diagnostic procedure to quantify changes in internal (inaccessible) parameters of physiological or clinical interesP4 and as a tool for the development and evaluation of appropriate control strategies for circulatory assist device$‘. In both cases, major difficulties originate from complex non-linearities and the large number of unknown parameters present in cardiovascular modelss~g. Consequently, the challenge is to derive simple models which, while retaining some of the realistic behaviour of more complex models and their connection with the underlying physiological mechanisms, can be defined accurately from available measurements. Our aim is to validate the ability of a simple threeelement linear characterization of the systemic vascular bed to provide accurate estimates of important physiological parameters over a wide range of cardiocirculatory conditions. Data generated from a numerical simulator of the whole circulatory system were used to estimate arterial and venous compliances. This approach involves a model-to-model check and has been shown to be particularly convenient”*” because it enables the values of the simulator parameters to be changed easily in order to simulate different alterations both in the vasal properties and in the pumping cardiac function.
influences of a wide vanety of circulatory conditions were investigated. The results achieved give evidence that arterial compliance is generally well estimated, while venous compliance is more variable, particularly at high penpheral resistance where measured signals appear to be less sensitive to this parameter. However, presence of cardiac disease, such as heart failure and valvular stenosis has minimal influence on complzance estimates. These results suggest that this simple model can be conveniently applied even under noisy conditions. parameter
estimates,
estimation
accuracy
facilitate its implementation. The identification scheme employed for parameter estimation is then described and the results, obtained under a wide variety of circulatory conditions, are presented and discussed. SIMULATION CIRCULATORY
MODEL OF THE SYSTEM
The electric analogue of the model used for simulation is shown in Figure I. Ventricles are simulated by the classic left (EL) and right (&) variable elastances, while the diodes, S,, in series with a resistance, model the unidirectional and linearly resistive behaviour of inlet and outlet valves. Atria are considered purely passive and are included in the venous compliances (C, and c,). Two similar representations are used to characterize the systemic and pulmonary vascular bed; each vascular bed is represented by a twocompartment system consisting of arterial and venous compliances separated by a resistance due to the peripheral vessels. Thus, elastic and viscous effects are taken into account while inertia is neglected. This description of the circulation has been found to constitute a good approximation to a more complex experimental simulatoP. It is, therefore, possible to assign a precise physical meaning to
We outline the characteristics of the mathematical model adopted for the closed-loop cardiovascular system and the simulation language used to
POP zc3
S&one di Bioingegneria, Istituto di Chirurgia Toracica e Cardiovascolare, University of Siena, 53100 Siena. Italy. ‘Dipartimento di Elettronica. Informatica e Sistemistica, University Bologna, 40136 Bologna, Italy. Reprints from Professor P. Barbini 0 1987 Buttrtworth & Co (Publ~sttrrs) 0141-5425/87/040374-05 $03.00 374
J. Biomed.
Ltd
Eng. 1987. Vol. 9, October
of
Figure 1 system
,Electrical
analogue
of the closed-loop
circulatory
Model of the circulatory system: G. Ceuentni et al Table
1
Non-zero
elements of the system matrix A[i, .j] s
All, l]=-+’
A[l, 21 = + 1 ‘I
R,C,
A[l, 61 = 1 ,“(, W) I ‘1
2 I
A[% 1) =$
A[2, 21 = - &
- 1 2
1 ‘Z
s,
A[2, 31 = +-
RJ,
32
Wti
1 A(4. 31 = + 4
A[5, 41 = +-
3
A[4, 41 = - +
E&I
A[& 51 = - f,
5I
A[4, 51 = +
fW3
$4
A[$ 61 = --&
R,C,
A[6. 5]+
I 1 during
systole
5‘3
I 3
A[6. 61 = - +
6
6
E,N
E,(t) - +
1
EL(~)
1 during dim&
s,. s, =
s,, s, = 0 during
diastole
Nominal parameter
Resistive elements x IO” (mmHg cn-’ s) R, R, R, R, R, R,
- 5 1
A[6. 11=+
Table 2
4,3
- ~
= 3.3 = 664.7 = 12.6 = 2.0 = 51.7 = 55.6
0 during
systole
values
18
Capacitive elements (mmHg-’ cm’)
15 Ir) ‘E u
c,=
2 c, = 20 c,= 5 c, = 23
I” : 0.9 S g
06
-
s
Heart rate = 1.25 Hz
each parameter of this model and the simulation of different circulatory conditions may be obtained by varying its parameters. From an analysis of the electrical analogue shown in Figure I the state variables can be shown to be the ventricular volumes V, and vR and the pressures Pa,, P,,, Pap, Pvp. They can be combined in the vector X = [Pa,, P,,,, VR, Pap, Pvp, VL] and their dynamics are determined by a system of differential equations which may be obtained by applying network analysis theory to the electrical analogue. These equations can be expressed in the following compact matricial form:
k = A(t) X
IZ-
(I)
where the non-zero elements of the system matrix A(t) are shown in Table I. The presence is noted of non-linear elements such as the switches S,, whose state is determined by the pressure drop sign at their terminations. Therefore, equation (1) defines a non-linear, sixth order, time-varying system,
0
02
04
0.6
Time (s)
Figure 2 ventricular
Time courses elastance
of the left (EL) and
the right
(ER)
wherein the time variability is derived from periodical ventricular elastances. Table 2 gives the set of nominal parameter values assumed for simulation of the normal circulatorv condition in man, while the time-courses of veniricular elastances are shown in Figure 2. This model was implemented on a DEC computer VAX-1 l/750 using SIMNON”, an interactive simulation language developed by the Department of Automatic Control, Lund Institute of Technology, Sweden. This simulation language has advanced facilities for the integration of differential equations, the storage and retrieval of data, and displaying solutions as graphs. It also permits parameters and initial conditions to be changed.
,J_ Bicxricd. Erg. 1987, Vol. 9, OrtoLcr
375
Model of the circulatory system: G. Cevenini et al
PARAMETER
ESTIMATION
Table 3 simulated
PROCEDURE
Arterial pressure (Xi), right atria1 pressure (X,) (Figure 3) and cardiac output obtained by simulation, are used to determine the three parameters (one resistance and two compliances) the systemic circulation. Initially, peripheral resistance was directly calculated as the ratio between mean pressure gradient and cardiac output; subsequently, the two compliances were individually determined through an identification procedure which separately analyses systolic and diastolic phases.
RESULTS
AND
DISCUSSION
150
x 10’
(mmHg cm-’ s)
parameters
in different
C, (mmHg-’ cm’)
Heart rate HZ
E, (peak) CmmHg cm-l)
E, (peak)
R,
CmmHg cm -I)
x IO’ CmmHg cm-’ s)
1.25 1.25 1.25 1.25 1.25 2.50 ,833 1.25 1.25
1.6
0.4
3.3
1.6 1.6 1.6 1.6 1.6 1.6 0.8 1.6
0.4 0.4 0.4 0.4 0.4 0.4 0.2 0.4
3.3 3.3 3.3 3.3 3.3 3.3 3.3 167
2
664.7 1329.4
2 2
3 4 5 6 7 8 9
332.4 664.7 664.7 664.7 664.7 664.7 664.7
2 4 1 2 2 2 2
To allow for the inevitable presence of noise in real situations, a zero mean white noise with standard deviation approximately equal to 5% of the pressure peak in nominal conditions (SD = 7.5 mmHg for aortic pressure and 0.5 mmHg for atrial pressure) was superimposed on the output signal in all cases. Performing the ML identification shown in Table 4, and from these suggestions can be made.
gave the results a number of
Firstly, in calculating peripheral resistance, R2, we always obtained values very close to its nominal value. The result is not surprising since determination of this parameter does not require estimation procedures, but direct use of mean quantities (pressures and cardiac output), which are affected, to a minor extent, by noise. Secondly, arterial compliance is generally estimated with good accuracy, while the estimate of venous compliance is less precise, a finding which can be qualitatively explained by assuming the venous time constant, R,C2, to be much greater than the arterial time constant R2C,. In fact, their nominal values are 13.3 and 1.33 s, which corresponds to cutoff frequencies of 0.012 and 0.12 Hz. Consequently, we can expect venous compliance to be poorly estimated when we use physiological signals having harmonic frequencies higher than 1 Hz (neglecting the dc component). To obtain a better insight into this phenomenon, the sensitivities of the output signal with respect to the two compliances have been evaluated and their courses versus frequency are shown in Figure 4. We can see that the range of significant sensitivity is located at frequencies lower than 1 Hz, and that the phenomenon is more marked for venous compliance.
06
1.2
1.8
Time (s)
Figure 3 Arterial (a) and right atrial (b) pressures from the simulation model, in nominal condition. pressure is amplified by a factor of ten
376
R,
I
Nine different physiopathological conditions were simulated by changing the parameter values as illustrated in Table 3. Condition 1 refers to nominal values of parameters and conditions 2-5 correspond to changes in vascular parameters. To simulate tachycardia and bradycardia the heart rate was doubled (condition 6) and reduced by a third (condition 7) respectively. Condition 8 was obtained by a 50% reduction of both ventricular elastances throughout the cardiac cycle, to simulate reduced heart contractility. Condition 9 simulates a strong valvular stenosis localized on the aortic valve through a dramatic increase (C.50~) in resistance R, with respect to the basal condition. Similar pathologies for the other three valves were not considered since they have minimal influence on the estimation of systemic parameters.
0
Case number
of
In the systolic phase (atrium-ventricular valve closed) venous compliance was estimated by assuming atrial pressure as the output of a monocompartmental model driven by the arterial pressure. In the diastolic phase (aortic valve closed), arterial compliance was estimated by assuming arterial pressure as the output and atria1 pressure as the input of a new monocompartmental model. In this identification procedure, a maximum-likelihood (ML) criterion is used for parameter estimation, and its minimum is obtained by means of the Powell methodi3, which belongs to the class of conjugate directions algorithms.
Values of physiological circulatory conditions
J. Biomed. Eng. 1987, Vol. 9, October
obtained Atrial
Thirdly, accuracy of compliance estimates improves slightly when the corresponding time constant decreases (condition 3 for venous compliance, 3 and 5 for arterial compliance) - this is of course reversed on increasing the time constants (conditions 2 and 4). In particular, referring to arterial compliance, this phenomenon seems to be more marked when increase or decrease of the
Model o/ the cuculatory system: G. Ceuenim et al Table 4 Parameter and the simulated circulatory
related standard conditions
Case
C, zt SD% (mmHg_’ cm’)
C, + SD% jmmHg_’ cm’)
1 2 3 4 5 6 7 8 9
2.06 2.07 2.08 4.18 1.02 1.84 2.04 2.05 2.03
19.8 19.7 19.8 19.7 19.8 22.7 19.1 19.2 19.8
i 3.0 + 3.7 zk 2.8 k 6.0 t 1.6 + 7.9 + 1.8 i 3.3 + 3.5
* + + * k + + + +
deviation
R, x 10’ (mmHg
9.3 13.2 6.4 10.3 7.9 10.3 6.9 10.4 11.2
in
estimates
cm-) s)
664 1326 332 668 665 670 664 661 669
CONCLUSIONS In this paper the systemic vascular bed is characterized using a linear lumped-parameter model with three elements representing arterial compliance, total peripheral resistance and venous compliance. Inertial effects have been neglected, in agreement with recent sensitivity studies15. Robustness of model parameter estimates with respect to measurement noise has been examined over a range of simulated physiopathological conditions.
‘.“r
c
0.5
t
g
00
.Z$
5
cl7
-05
-1.0
-
0
I 0.01
I 0. I
I I
by increasing the resistance of the aortic valve, which simulates a valvular stenosis. In condition 9 a dramatic increase in the valvular resistance was required in order to obtain some change in estimation accuracy.
I IO
I 100
HZ Figure 4 Sensitivity of transfer function magnitude F in respect of C, (a) and C, (b). This sensitivity was calculated as S = C, dF/dC,, (; = 1, 2)
time constant is caused by the compliance, whereas it is modest when it follows on from a change in the peripheral resistance. This agrees with the findings of Burattini and Gnudi3 in their study of the arterial tree input impedance through a simulation and an experimental approach. For example, experimental results on dogs showed an increase in arterial compliance standard deviation by about seven times, corresponding to an increase of five times in the arterial compliance. Fourthly, a decrease in the heart rate (condition 7) provides a consistent increase in estimate accuracy, compared with what was seen previously; the ratio between the cutoff and fundamental frequency increases, as it did in condition 3. The opposite occurs in the presence of an increase in heart rate (condition 6). It can be seen that the results obtained for arterial compliance also agree with the findings obtained by Burattini and Gnudi through a simulation approach based on a complex model of the arterial tree14; their results also show decreasing accuracy in the arterial compliance estimate with increasing heart rate, and vice versa. Finally, changes in the waveforms of the simulated signals, consequent upon halving the cardiac elastances (cardiac failure was imitated in this way) do not significantly affect the accuracy of compliance estimates. Similar results were obtained
Peripheral resistance and total arterial compliance are estimated with good accuracy in all cases. The venous compliance, on the other hand, appears to be more critical since measured signals are not particularly sensitive to this parameter in the frequency range of haemodynamic variables. Nevertheless, if a 10% error can be tolerated the model shows promise as a research and clinical tool since it may provide a simple method to estimate haemodynamic parameters, even in the presence of heart disease.
ACKNOWLEDGEMENTS This work was supported by the Italian Ministry of Education (M.P.I.) and the Italian National Research Council (C.N.R.).
REFERENCES Deswysen, B.A. Parameter estimation of a simple model of the left ventricle and of the systemic vascular bed with particular attention to the physical meaning of the left ventricular parameters. IEEE Trans. 1977, BME-24: 29-38 Clark, J.W., Ling, R.Y.S., Scrinivasan, R., Cole, J.S. and Pruett, R.C. A two-stage identification scheme for the determination of parameters of a model of the left heart and systemic arterial circulation. IEEE Trans. 1980, BME-27: 20-29 Burattini, R. and Gnudi, G. Computer identification of models for the arterial tree input impedance: comparison between two new simple models and first experimental results. Med. Biol. Eng. Cornput. 1982, 20: 134-144 Avanzolini, G. and Cappello, A. Estimation ot time/ varying systolic properties of left ventricular mechanics. Med. Biol. Eng. Comput. 1986, 24:261-266 Kuklinski, W.S. Closed loop control of intraaortic balloon pumping: Studies using a computer simulation and animal experiments. Ph. D. dissertutioq Univ. Rhode Island, Kingston. 1979. Jaron, D., and Moore, T.W. Engineering techniques applied to the analysis and control of in-series cardiac assistance. IEEE Truns 1984, BME-31: 893-899 Mchnis, B.C., Guo, Z.W., Lu, P.C. and Wang, J.C. Adaptive control of left ventricular bypass assist devices. IEEE Trans 1985, AC-30: 322-329 Bekey, G.A. and Beneken, J.E.W. Identification of biological systems: a survey. Automatica 1978, 14: 41-47
J. Biomed.
Eng.
1987. Vol. 9, October
377
Model of the circulatory system: G. Cevenini et al 9
10
11
12
378
Moller, D.P.F., Popovic, D. and Thiele, G. Reliability of parameter estimation methods applied to the identification of biomedical multicompartment systems. hoc. 7th IFAC Symp. Identification and parameter estimation. Pergamon Press, 1985, 1385-1390 Sims, J.B. Estimation of arterial system parameters from dynamic records. Computers and Biomed Res 1972, 5: 131-147 Deswysen, B.A. Quantitative evaluation of the systemic arterial bed by parameter estimation of a simple model. Med. Biol. Eng. Comput. 1980, 18, 153-166 Astrom, K.J. and Wittenmark, B. Computer-Controlled Systems: Theory and Design Englewood Cliffs, New Jersey: Prentice-
J. Biomed. Eng. 1987, vol. 9, Octobe:
13
14
15
Hall, 1984, 408-420 Powell, M.J.D. An efficient method for finding the minimum of a function of several variables without calculating derivatives. Computer J 1964, 7: 155-162 Avanzolini, G., Belardinelli, E., Capitani, G. and Passigato, R. Steady-state numerical model of the human systemic arterial tree. F’roc IFAC Symposium on Automatic Control and Computers in the Medical Field, Brussells 197 1, 83-97 Barbini, P., Cappello, A., and Avanzolini G. Cadcs techniques in cardiovascular system simulation and identification. Proc. IEEE Frontiers of Eng. and Comput. in Health Care, Dallas-Forth Worth 1986; 51-54
P. Barbini,
October
A. Cappello’
and
1986, revised and accepted
G. Avanzolini*
May 1987
ABSTRACT In this study, the accuracy achievable in the parameter estimates o/a three-element linear model for the s_ystemic vascular bed is considered. The model neglects inertial efjects and includes only three elements representing arterial compliance, peripheral resistance and venous compliance, in agreement with recent sensitivity investigations. Parameter estimation startsfrom arterial and right atria1 pressure signals generated by a closed-loop simulator of the cardiovascular circulation and corrupted with normal noise to account for measurement errors. In thus way, the Keywords:
Circulatory
system, modelling,
systemic circulation,
INTRODUCTION Computer identification techniques have been applied in the past to the cardiovascular system for two main purposes: as a diagnostic procedure to quantify changes in internal (inaccessible) parameters of physiological or clinical interesP4 and as a tool for the development and evaluation of appropriate control strategies for circulatory assist device$‘. In both cases, major difficulties originate from complex non-linearities and the large number of unknown parameters present in cardiovascular modelss~g. Consequently, the challenge is to derive simple models which, while retaining some of the realistic behaviour of more complex models and their connection with the underlying physiological mechanisms, can be defined accurately from available measurements. Our aim is to validate the ability of a simple threeelement linear characterization of the systemic vascular bed to provide accurate estimates of important physiological parameters over a wide range of cardiocirculatory conditions. Data generated from a numerical simulator of the whole circulatory system were used to estimate arterial and venous compliances. This approach involves a model-to-model check and has been shown to be particularly convenient”*” because it enables the values of the simulator parameters to be changed easily in order to simulate different alterations both in the vasal properties and in the pumping cardiac function.
influences of a wide vanety of circulatory conditions were investigated. The results achieved give evidence that arterial compliance is generally well estimated, while venous compliance is more variable, particularly at high penpheral resistance where measured signals appear to be less sensitive to this parameter. However, presence of cardiac disease, such as heart failure and valvular stenosis has minimal influence on complzance estimates. These results suggest that this simple model can be conveniently applied even under noisy conditions. parameter
estimates,
estimation
accuracy
facilitate its implementation. The identification scheme employed for parameter estimation is then described and the results, obtained under a wide variety of circulatory conditions, are presented and discussed. SIMULATION CIRCULATORY
MODEL OF THE SYSTEM
The electric analogue of the model used for simulation is shown in Figure I. Ventricles are simulated by the classic left (EL) and right (&) variable elastances, while the diodes, S,, in series with a resistance, model the unidirectional and linearly resistive behaviour of inlet and outlet valves. Atria are considered purely passive and are included in the venous compliances (C, and c,). Two similar representations are used to characterize the systemic and pulmonary vascular bed; each vascular bed is represented by a twocompartment system consisting of arterial and venous compliances separated by a resistance due to the peripheral vessels. Thus, elastic and viscous effects are taken into account while inertia is neglected. This description of the circulation has been found to constitute a good approximation to a more complex experimental simulatoP. It is, therefore, possible to assign a precise physical meaning to
We outline the characteristics of the mathematical model adopted for the closed-loop cardiovascular system and the simulation language used to
POP zc3
S&one di Bioingegneria, Istituto di Chirurgia Toracica e Cardiovascolare, University of Siena, 53100 Siena. Italy. ‘Dipartimento di Elettronica. Informatica e Sistemistica, University Bologna, 40136 Bologna, Italy. Reprints from Professor P. Barbini 0 1987 Buttrtworth & Co (Publ~sttrrs) 0141-5425/87/040374-05 $03.00 374
J. Biomed.
Ltd
Eng. 1987. Vol. 9, October
of
Figure 1 system
,Electrical
analogue
of the closed-loop
circulatory
Model of the circulatory system: G. Ceuentni et al Table
1
Non-zero
elements of the system matrix A[i, .j] s
All, l]=-+’
A[l, 21 = + 1 ‘I
R,C,
A[l, 61 = 1 ,“(, W) I ‘1
2 I
A[% 1) =$
A[2, 21 = - &
- 1 2
1 ‘Z
s,
A[2, 31 = +-
RJ,
32
Wti
1 A(4. 31 = + 4
A[5, 41 = +-
3
A[4, 41 = - +
E&I
A[& 51 = - f,
5I
A[4, 51 = +
fW3
$4
A[$ 61 = --&
R,C,
A[6. 5]+
I 1 during
systole
5‘3
I 3
A[6. 61 = - +
6
6
E,N
E,(t) - +
1
EL(~)
1 during dim&
s,. s, =
s,, s, = 0 during
diastole
Nominal parameter
Resistive elements x IO” (mmHg cn-’ s) R, R, R, R, R, R,
- 5 1
A[6. 11=+
Table 2
4,3
- ~
= 3.3 = 664.7 = 12.6 = 2.0 = 51.7 = 55.6
0 during
systole
values
18
Capacitive elements (mmHg-’ cm’)
15 Ir) ‘E u
c,=
2 c, = 20 c,= 5 c, = 23
I” : 0.9 S g
06
-
s
Heart rate = 1.25 Hz
each parameter of this model and the simulation of different circulatory conditions may be obtained by varying its parameters. From an analysis of the electrical analogue shown in Figure I the state variables can be shown to be the ventricular volumes V, and vR and the pressures Pa,, P,,, Pap, Pvp. They can be combined in the vector X = [Pa,, P,,,, VR, Pap, Pvp, VL] and their dynamics are determined by a system of differential equations which may be obtained by applying network analysis theory to the electrical analogue. These equations can be expressed in the following compact matricial form:
k = A(t) X
IZ-
(I)
where the non-zero elements of the system matrix A(t) are shown in Table I. The presence is noted of non-linear elements such as the switches S,, whose state is determined by the pressure drop sign at their terminations. Therefore, equation (1) defines a non-linear, sixth order, time-varying system,
0
02
04
0.6
Time (s)
Figure 2 ventricular
Time courses elastance
of the left (EL) and
the right
(ER)
wherein the time variability is derived from periodical ventricular elastances. Table 2 gives the set of nominal parameter values assumed for simulation of the normal circulatorv condition in man, while the time-courses of veniricular elastances are shown in Figure 2. This model was implemented on a DEC computer VAX-1 l/750 using SIMNON”, an interactive simulation language developed by the Department of Automatic Control, Lund Institute of Technology, Sweden. This simulation language has advanced facilities for the integration of differential equations, the storage and retrieval of data, and displaying solutions as graphs. It also permits parameters and initial conditions to be changed.
,J_ Bicxricd. Erg. 1987, Vol. 9, OrtoLcr
375
Model of the circulatory system: G. Cevenini et al
PARAMETER
ESTIMATION
Table 3 simulated
PROCEDURE
Arterial pressure (Xi), right atria1 pressure (X,) (Figure 3) and cardiac output obtained by simulation, are used to determine the three parameters (one resistance and two compliances) the systemic circulation. Initially, peripheral resistance was directly calculated as the ratio between mean pressure gradient and cardiac output; subsequently, the two compliances were individually determined through an identification procedure which separately analyses systolic and diastolic phases.
RESULTS
AND
DISCUSSION
150
x 10’
(mmHg cm-’ s)
parameters
in different
C, (mmHg-’ cm’)
Heart rate HZ
E, (peak) CmmHg cm-l)
E, (peak)
R,
CmmHg cm -I)
x IO’ CmmHg cm-’ s)
1.25 1.25 1.25 1.25 1.25 2.50 ,833 1.25 1.25
1.6
0.4
3.3
1.6 1.6 1.6 1.6 1.6 1.6 0.8 1.6
0.4 0.4 0.4 0.4 0.4 0.4 0.2 0.4
3.3 3.3 3.3 3.3 3.3 3.3 3.3 167
2
664.7 1329.4
2 2
3 4 5 6 7 8 9
332.4 664.7 664.7 664.7 664.7 664.7 664.7
2 4 1 2 2 2 2
To allow for the inevitable presence of noise in real situations, a zero mean white noise with standard deviation approximately equal to 5% of the pressure peak in nominal conditions (SD = 7.5 mmHg for aortic pressure and 0.5 mmHg for atrial pressure) was superimposed on the output signal in all cases. Performing the ML identification shown in Table 4, and from these suggestions can be made.
gave the results a number of
Firstly, in calculating peripheral resistance, R2, we always obtained values very close to its nominal value. The result is not surprising since determination of this parameter does not require estimation procedures, but direct use of mean quantities (pressures and cardiac output), which are affected, to a minor extent, by noise. Secondly, arterial compliance is generally estimated with good accuracy, while the estimate of venous compliance is less precise, a finding which can be qualitatively explained by assuming the venous time constant, R,C2, to be much greater than the arterial time constant R2C,. In fact, their nominal values are 13.3 and 1.33 s, which corresponds to cutoff frequencies of 0.012 and 0.12 Hz. Consequently, we can expect venous compliance to be poorly estimated when we use physiological signals having harmonic frequencies higher than 1 Hz (neglecting the dc component). To obtain a better insight into this phenomenon, the sensitivities of the output signal with respect to the two compliances have been evaluated and their courses versus frequency are shown in Figure 4. We can see that the range of significant sensitivity is located at frequencies lower than 1 Hz, and that the phenomenon is more marked for venous compliance.
06
1.2
1.8
Time (s)
Figure 3 Arterial (a) and right atrial (b) pressures from the simulation model, in nominal condition. pressure is amplified by a factor of ten
376
R,
I
Nine different physiopathological conditions were simulated by changing the parameter values as illustrated in Table 3. Condition 1 refers to nominal values of parameters and conditions 2-5 correspond to changes in vascular parameters. To simulate tachycardia and bradycardia the heart rate was doubled (condition 6) and reduced by a third (condition 7) respectively. Condition 8 was obtained by a 50% reduction of both ventricular elastances throughout the cardiac cycle, to simulate reduced heart contractility. Condition 9 simulates a strong valvular stenosis localized on the aortic valve through a dramatic increase (C.50~) in resistance R, with respect to the basal condition. Similar pathologies for the other three valves were not considered since they have minimal influence on the estimation of systemic parameters.
0
Case number
of
In the systolic phase (atrium-ventricular valve closed) venous compliance was estimated by assuming atrial pressure as the output of a monocompartmental model driven by the arterial pressure. In the diastolic phase (aortic valve closed), arterial compliance was estimated by assuming arterial pressure as the output and atria1 pressure as the input of a new monocompartmental model. In this identification procedure, a maximum-likelihood (ML) criterion is used for parameter estimation, and its minimum is obtained by means of the Powell methodi3, which belongs to the class of conjugate directions algorithms.
Values of physiological circulatory conditions
J. Biomed. Eng. 1987, Vol. 9, October
obtained Atrial
Thirdly, accuracy of compliance estimates improves slightly when the corresponding time constant decreases (condition 3 for venous compliance, 3 and 5 for arterial compliance) - this is of course reversed on increasing the time constants (conditions 2 and 4). In particular, referring to arterial compliance, this phenomenon seems to be more marked when increase or decrease of the
Model o/ the cuculatory system: G. Ceuenim et al Table 4 Parameter and the simulated circulatory
related standard conditions
Case
C, zt SD% (mmHg_’ cm’)
C, + SD% jmmHg_’ cm’)
1 2 3 4 5 6 7 8 9
2.06 2.07 2.08 4.18 1.02 1.84 2.04 2.05 2.03
19.8 19.7 19.8 19.7 19.8 22.7 19.1 19.2 19.8
i 3.0 + 3.7 zk 2.8 k 6.0 t 1.6 + 7.9 + 1.8 i 3.3 + 3.5
* + + * k + + + +
deviation
R, x 10’ (mmHg
9.3 13.2 6.4 10.3 7.9 10.3 6.9 10.4 11.2
in
estimates
cm-) s)
664 1326 332 668 665 670 664 661 669
CONCLUSIONS In this paper the systemic vascular bed is characterized using a linear lumped-parameter model with three elements representing arterial compliance, total peripheral resistance and venous compliance. Inertial effects have been neglected, in agreement with recent sensitivity studies15. Robustness of model parameter estimates with respect to measurement noise has been examined over a range of simulated physiopathological conditions.
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by increasing the resistance of the aortic valve, which simulates a valvular stenosis. In condition 9 a dramatic increase in the valvular resistance was required in order to obtain some change in estimation accuracy.
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time constant is caused by the compliance, whereas it is modest when it follows on from a change in the peripheral resistance. This agrees with the findings of Burattini and Gnudi3 in their study of the arterial tree input impedance through a simulation and an experimental approach. For example, experimental results on dogs showed an increase in arterial compliance standard deviation by about seven times, corresponding to an increase of five times in the arterial compliance. Fourthly, a decrease in the heart rate (condition 7) provides a consistent increase in estimate accuracy, compared with what was seen previously; the ratio between the cutoff and fundamental frequency increases, as it did in condition 3. The opposite occurs in the presence of an increase in heart rate (condition 6). It can be seen that the results obtained for arterial compliance also agree with the findings obtained by Burattini and Gnudi through a simulation approach based on a complex model of the arterial tree14; their results also show decreasing accuracy in the arterial compliance estimate with increasing heart rate, and vice versa. Finally, changes in the waveforms of the simulated signals, consequent upon halving the cardiac elastances (cardiac failure was imitated in this way) do not significantly affect the accuracy of compliance estimates. Similar results were obtained
Peripheral resistance and total arterial compliance are estimated with good accuracy in all cases. The venous compliance, on the other hand, appears to be more critical since measured signals are not particularly sensitive to this parameter in the frequency range of haemodynamic variables. Nevertheless, if a 10% error can be tolerated the model shows promise as a research and clinical tool since it may provide a simple method to estimate haemodynamic parameters, even in the presence of heart disease.
ACKNOWLEDGEMENTS This work was supported by the Italian Ministry of Education (M.P.I.) and the Italian National Research Council (C.N.R.).
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