Minimum realisations and modelling of biological systems R. N. M i s h r a
S.S.
Departmentof Electrical Engineering, University of Roorkee, Roorkee,
Department of Electrical Engineering, Guru Nanak Engng. College Ludhiana, Paniab, lndia
UP, India
Koonar
Abstract--A state variable model in the canonical form of Bucy is constructed from the given impulse response of a finite-dimensional, discrete-time, linear constant dgnamical biological (arterial circulatory) system. Keywords--Modelling, Arterial circulation
List of symbols
[H, F, G] = (1 • n), (n x n), (n x 1) matrices x = (n x 1) state vector u = single-dimensional input y = single-dimensional output Yk = matrix of impulse ~esponse N = total number of observations taken during diastole period Introduction
The arterial circulatory system is considered in this paper for minimum realisations and modelling. The arterial circulatory system can be represented by a finite-dimensional stable discrete-time linearconstant dynamical system x [ k + 1] = F x [ k ] + G u [ k ] yIk] = Hx[k]
.
.
.
.
.
.
.
(1)
with the following assumptions:
Yk --- y[k]
(a) there is no active vascular compression (b) during diastole there is neither flow into nor out of the heart thus dispensing with the necessity to consider ventricular effects. F o r this system, the input and output are single-dimensional. The model of the circulatory system considered consists of two elastic reservoirs connected by a long colunm of blood, with a provision for the acceleration of blood and with the presence of an additional compliance effect. It was assumed for mathematical simplicity that the blood flowing between the first and second reservoir was flowing in a rigid tube of uniform cross-sectional area and dissipation in the tube was neglected. It was also assumed that there was a linear relationship between the pressure in the second reservoir and flow into the vascular bed. " Fimt received 14th May and in final form 18th October 1973
56
Assumptions (a) and (b) are also included. During diastole it is also assumed that all hearts are identical, and hence the blood flow in is zero. In the cardiovascular system, the blood-pressure input to an artery is considered as a train of periodic pulses and the output as a periodic function of time. The model omits the distributed, nonlinear characteristics of the human vascular system but, on the other hand, enables one to perform an analysis of a pressure curve in terms of parameters that appear to be clinically important. One can obtain the passive network from the triple [H, F, G] obtained by minimum realisation. These model parameters can at least be useful in analysing the effects of drugs and disease on model parameters. The blood pressure is measured with respect to time during the diastole period after the system has come sufficiently close to the state x[0] = 0. Let the impulse response be denoted by where k = 1, 2, 3 . . . . N
(2)
The relation between the impulse response and the matrices H, F and G can be expressed as Yk = H F k - 1 G
.
.
.
.
.
.
.
(3)
The realisation problem is defined as follows: Given an impulse response Y k where k = 1, 2, . . , N, find a triple [H, F, G] which satisfies eqn. 3 for all k. We find only the minimal realisation, i.e. the triple [H, F, G], which is both observable and controllable, since the impulse response given by eqn. 3 depends only on the observable and controllable part of the system (KALMAN, 1963). Here, a new solution to the minimal realisation of biological systems is given. The method is based 9n the canonical form of Bucv (1968). The system (eqn. 1) can be described by, at most, 2n parameters. The realisation is performed only by computing these parameters.
Medical and Biological Engineering
January 1975
Realisation in the canonical form of Bucy H F " = fl .
In the system given by eqn. 1, let
.
.
.
.
.
.
(10)
n--1
The same linear equation must hold for rows on the right-hand side. Let d'(p) be the pth row of [Yp Yp+~ .. Yo+,-1];i.e.
/-/F
H1 =
.
o-1
d'=
rank HI = n
. . . . . . . .
(4)
[y(p) .. y ( n + p - 1 ) ]
. . . . .
(11)
. . . . .
(12)
Then, from eqns. 9 and 10,
The transformation
d'(1) d'(n+ 1) = / r
HF
(5)
z = Mx, M =
d'(2)
d'(n)
n--1
Procedure
takes the system into the form
Step 1
z[k + 1] = L z [ k ] + M G u [ k ] y[k] = [1
0..0]z[k]
. . . . .
(6)
The state dimension n of the minimal realisation may be determined as:
Then n • n matrix L has the form
n = max rank I1(1, i) i
[o ol o
Step 2
L=
"..
. . . . .
(7)
The biological data are arranged as f o l l o w s : d'(1) = [y(1) y(2) ... y(N)]
a'(2) = ry(2) yr Now the observability structure is exhibited in the following canonical form of the matrices F and H in eqn. 1 :
a'(,,+ 1) = B'
"~ F =
d
S.S.
Departmentof Electrical Engineering, University of Roorkee, Roorkee,
Department of Electrical Engineering, Guru Nanak Engng. College Ludhiana, Paniab, lndia
UP, India
Koonar
Abstract--A state variable model in the canonical form of Bucy is constructed from the given impulse response of a finite-dimensional, discrete-time, linear constant dgnamical biological (arterial circulatory) system. Keywords--Modelling, Arterial circulation
List of symbols
[H, F, G] = (1 • n), (n x n), (n x 1) matrices x = (n x 1) state vector u = single-dimensional input y = single-dimensional output Yk = matrix of impulse ~esponse N = total number of observations taken during diastole period Introduction
The arterial circulatory system is considered in this paper for minimum realisations and modelling. The arterial circulatory system can be represented by a finite-dimensional stable discrete-time linearconstant dynamical system x [ k + 1] = F x [ k ] + G u [ k ] yIk] = Hx[k]
.
.
.
.
.
.
.
(1)
with the following assumptions:
Yk --- y[k]
(a) there is no active vascular compression (b) during diastole there is neither flow into nor out of the heart thus dispensing with the necessity to consider ventricular effects. F o r this system, the input and output are single-dimensional. The model of the circulatory system considered consists of two elastic reservoirs connected by a long colunm of blood, with a provision for the acceleration of blood and with the presence of an additional compliance effect. It was assumed for mathematical simplicity that the blood flowing between the first and second reservoir was flowing in a rigid tube of uniform cross-sectional area and dissipation in the tube was neglected. It was also assumed that there was a linear relationship between the pressure in the second reservoir and flow into the vascular bed. " Fimt received 14th May and in final form 18th October 1973
56
Assumptions (a) and (b) are also included. During diastole it is also assumed that all hearts are identical, and hence the blood flow in is zero. In the cardiovascular system, the blood-pressure input to an artery is considered as a train of periodic pulses and the output as a periodic function of time. The model omits the distributed, nonlinear characteristics of the human vascular system but, on the other hand, enables one to perform an analysis of a pressure curve in terms of parameters that appear to be clinically important. One can obtain the passive network from the triple [H, F, G] obtained by minimum realisation. These model parameters can at least be useful in analysing the effects of drugs and disease on model parameters. The blood pressure is measured with respect to time during the diastole period after the system has come sufficiently close to the state x[0] = 0. Let the impulse response be denoted by where k = 1, 2, 3 . . . . N
(2)
The relation between the impulse response and the matrices H, F and G can be expressed as Yk = H F k - 1 G
.
.
.
.
.
.
.
(3)
The realisation problem is defined as follows: Given an impulse response Y k where k = 1, 2, . . , N, find a triple [H, F, G] which satisfies eqn. 3 for all k. We find only the minimal realisation, i.e. the triple [H, F, G], which is both observable and controllable, since the impulse response given by eqn. 3 depends only on the observable and controllable part of the system (KALMAN, 1963). Here, a new solution to the minimal realisation of biological systems is given. The method is based 9n the canonical form of Bucv (1968). The system (eqn. 1) can be described by, at most, 2n parameters. The realisation is performed only by computing these parameters.
Medical and Biological Engineering
January 1975
Realisation in the canonical form of Bucy H F " = fl .
In the system given by eqn. 1, let
.
.
.
.
.
.
(10)
n--1
The same linear equation must hold for rows on the right-hand side. Let d'(p) be the pth row of [Yp Yp+~ .. Yo+,-1];i.e.
/-/F
H1 =
.
o-1
d'=
rank HI = n
. . . . . . . .
(4)
[y(p) .. y ( n + p - 1 ) ]
. . . . .
(11)
. . . . .
(12)
Then, from eqns. 9 and 10,
The transformation
d'(1) d'(n+ 1) = / r
HF
(5)
z = Mx, M =
d'(2)
d'(n)
n--1
Procedure
takes the system into the form
Step 1
z[k + 1] = L z [ k ] + M G u [ k ] y[k] = [1
0..0]z[k]
. . . . .
(6)
The state dimension n of the minimal realisation may be determined as:
Then n • n matrix L has the form
n = max rank I1(1, i) i
[o ol o
Step 2
L=
"..
. . . . .
(7)
The biological data are arranged as f o l l o w s : d'(1) = [y(1) y(2) ... y(N)]
a'(2) = ry(2) yr Now the observability structure is exhibited in the following canonical form of the matrices F and H in eqn. 1 :
a'(,,+ 1) = B'
"~ F =
d
