Download Mathematical Model of Interactive Respiration/Cardiovascular

International Conference on Trends in Mechanical and Industrial Engineering (ICTMIE'2011) Bangkok Dec., 2011
Mathematical Model of Interactive
Respiration/Cardiovascular Composite System
Rong-Mao Lee, Hsin-Lin Chiu, Nan-Chyuan Tsai#
model for the heart and systemic circulation mainly consists of
six components: an AC power supply ( PV (t ) ), a diode ( D ), two
Abstract—A set of nonlinear dynamic models for the interactive
respiration/cardiovascular mechanism is constructed and analyzed.
By employing equivalent electric circuits for heart/blood and lung/air
systems, the dynamics of cardiovascular system and respiration cycle
are established. In order to verify the validity of the dynamic models,
numerical simulations and analysis on heart-lung interactions,
including the valvular closure incompetence and pulmonary
obstruction, are presented and compared with the empirical reports in
literature. The derived dynamics of heart-lung interactions can be
realized and examined in the biomechanical and medical engineering
fields. In addition, the dynamic models can also be employed for the
model-based controller synthesis in medical instrumentations, e.g., the
Extracorporeal Membrane Oxygenation (ECMO), to retain the
function of blood circulation and/or respiration by artificial
intelligence.
resistors ( R1 and R2 ), one capacitor ( C ) and one inductance
( L ). In physiological terms, they respectively represent the
driving force by Left-Ventricle ( PV (t ) ), the function of Aortic
Semilunar Valve ( D ), the overall peripheral resistance ( R2 ),
the overall compliance of the arterial system ( C ), the
impedance of the aorta ( R1 ) and the overall inertia of blood
( L ). On the other hand, the variables, Part (t ) and Pper (t ) ,
shown in Fig. 1, are the blood pressures of aorta and peripheral
artery (arteriole) respectively. Qart (t ) is the overall blood
(volume) flow rate in the blood circulation system and
QR1 (t ), QL (t ), QC (t ) and QR 2 (t ) are the blood (volume) flow
rates through R1 , L, C and R2 respectively.
Keywords—Cardiovascular System, Heart-Lung Interaction,
Extracorporeal Membrane Oxygenation (ECMO)
Q R1 (t )
I. INTRODUCTION
Part (t )
T
he major goal of this research is to establish a mathematic
model to describe the dynamics of heart–lung interaction.
By employing equivalent electric circuits for heart/blood and
lung/air systems, the dynamics of cardiovascular system and
respiration cycle can be represented by electric elements and
circuits. In addition, numerical simulations and analysis of
heart–lung interactions are presented and compared with the
available and well-known reported literature to verify the
validity of the proposed dynamic models. At last, both of
semilunar valvular closure incompetence and resulted
pulmonary obstruction are investigated as well.
R1
Pper (t )
Q L (t )
Q art (t )
PV (t )
L
Q C (t )
C R2
Q R2 (t )
Fig. 1 Equivalent Electrical Circuit for Heart and Blood Circulation
Since Qart = QR1 + QL and Qart = QR 2 + QC both hold valid
under mass conservation law, the state equations of Part (t ) and
II. DYNAMICS OF INTEGRATED RESPIRATORY AND
Pper (t ) can be constructed as follows:
CARDIOVASCULAR SYSTEMS
dPper (t )
A. Dynamics of Heart/Blood Pressure
The dynamics of heart/blood pressure can be schematically
described by an equivalent electric circuit [1]-[4], as shown in
Fig. 1. It is noticed that since the blood pressure and impedance
for systemic circulation are much higher and more influential
than the ones for pulmonary circulation, merely the systemic
circulation is investigated in our work. The lumped-parameter
dt
=
Qart (t ) Pper (t )
−
C
R2C
dPart (t )
dQ (t ) R
Q (t ) Pper (t )
= R1 art − 1 [ Part (t ) − Pper (t )] + art −
dt
dt
L
C
R2C
(1)
(2)
where
QR 2 (t ) =
Pper (t )
QC (t ) = C
#
Nan-Chyuan Tsai , he is now with the Department of Mechanical
Engineering, National Cheng Kung University, Tainan City, Taiwan.
(corresponding author to provide phone: +886-6-2757575 ext. 62137; e-mail:
[email protected] ).
dPper (t )
(3b)
dt
Part (t ) − Pper (t )
(3c)
dQL (t )
=
dt
L
Part (t ) − Pper (t )
QR1 (t ) =
R1
135
(3a)
R2
(3d)
International Conference on Trends in Mechanical and Industrial Engineering (ICTMIE'2011) Bangkok Dec., 2011
i. Mathematic Model of Semilunar Valve
The function of Semilunar Valve (SLV) can be basically
realized by a diode (see Fig. 1). Based on the characteristic
equation of diode [5], the blood volume flow at the aorta,
Qart (t ), is hence a function of the driving force of
depends on the exercise modes undertaken by the human body.
Consequently, Eq. (7) can be further rewritten as follows:
Left-Ventricle, PV (t ), and the blood pressure of aorta, Part (t ),
as follows:
(4)
Qart (t ) = c ae a[ P (t )− P (t )]
The derivative term, dPV (t ) / dt , in Eq. (7) is now completely
where “ a ” is the valve coefficient which is a non-negative
constant. As long as PV (t ) < Part (t ), the value of “ a ” is set to
be zero. Otherwise, “ a ” is retained to be positive. The
variable “ c ” in Eq. (4) is a constant and is to appropriately
scale-down the value of “ a ” so that “ ca ” is constrained within
a limited range. For the sake of simplifying Eq. (4), “Pade
Approximation” is employed in our work. Pade approximation
is a rational function and can be represented by the ratio of two
polynomials [6-8]. Since the difference is bounded, the
“Maclaurin series ( R1, 0 , )” can be employed to expand the
B. Dynamics of Respiration
The equivalent electric circuit for lung/air dynamics is
shown in Fig. 2 [11], [12]. The pressure at lung, PLg (t ), plays
V
dPart (t )
c 2 a 4 R1 ca
c 2 a 4 R1 R1 ca 2
) Part (t )
= [(
+ ) PV (t ) − (
+
+
L
C
CV
C
CV
dt
+(
R1
1
ca c 2 a 3 R1
) Pper (t ) + ( +
)] × (1 + ca 2 R1 ) −1
−
CV
L R2 C
C
substituted by the driving force by Left-Ventricle, PV (t ).
art
the role of driving force for the respiration system. For
example, as the air is inhaled to lung, the physical value of the
relative pressure PLg (t ) is positive. In addition, an inductance,
LLg a capacitor, C Lg and a resistor, RLg are analogic to
represent the inertia of airflow, compliance of trachea and
resistance of the respiratory tract respectively.
Q Lg (t )
exponential function in Eq. (4) as follows:
eA ≅ 1+ A
dt
=
Pper (t )
ca
[1 + aPV (t ) − aPart (t )] −
C
R2 C
2
PLg (t )
The variable QLg (t ) is the overall air volume flow rate
while QCLg (t ) and QRLg (t ) are its corresponding components
(7)
flowing through C Lg and RLg respectively.
It has been
well-known that the respiration is passively triggered by the
“Transpulmonary Pressure”, which is regulated by the muscles
against the chest wall [9]. In other words, the compliance does
not play the key role of triggering the respiration cycle so that
the circuit shown in Fig. 2 can be further simplified to Fig. 3.
That is, the impedance, Z Lg , can be reduced to RLg such that
(8)
the dynamics of overall air volume flow rate, QLg (t ) can be
dt
described as follows:
where Q f , Pf and C f are volume flow rate, liquid pressure
dQLg (t )
and hydraulic capacitance respectively. Assume blood is
incompressible and thus Eq. (8) can be also employed to
characterize the function of left-ventricle as follows:
dPV (t )
(9)
Q (t ) = C
V
Q RLg (t )
Fig. 2 Equivalent Electrical Circuit for Lung/Air Dynamics
ii. Mathematic Model of Ventricular Dynamic
Characteristics
For incompressible liquid, the flow rate is related to the
applied pressure as follows:
art
C Lg
2
Eq. (6) and Eq. (7) respectively represent the dynamics of
blood pressure at peripheral artery and blood pressure at aorta.
dPf
R Lg
Q CLg (t )
(6)
dPart (t )
dP (t ) c a
R ca
= [ c a 2 R1 V
+
PV (t ) − ( 1 +
) Part (t )
dt
dt
C
L
C
R
ca
1
+( 1 −
) Pper (t ) +
](1 + c a 2 R1 ) -1
L R2 C
C
Qf = C f
L Lg
(5)
where A is any real scalar.
From Eq. (1), (2) the dynamics of blood pressures, Part (t ) and
Pper (t ), can be rewritten respectively as follows:
dPper (t )
(10)
dt
=
PLg (t ) − RLg × QLg (t )
Q Lg (t )
dt
(11)
L Lg
L Lg
P1 (t )
PLg (t )
The compliance of left-ventricle is denoted by CV .
Z Lg
Although the compliance, CV , varies within the cardiac cycle
[9], [10], yet it does not exert any effect on Qart during the
diastole half-cycle due to the realistic function and definition of
valve coefficient, a, in last section. In addition, since the
variation of the compliance, CV , during the systolic half-cycle
Fig. 3 Modified Electrical Circuit for Lung/Air Dynamics
C. Interaction between Respiration and Blood Circulation
Let the heart beat rate and breath frequency be represented
is pretty limited, the compliance of left-ventricle is simplified
and assumed as a constant in this paper. Its actual value
136
International Conference on Trends in Mechanical and Industrial Engineering (ICTMIE'2011) Bangkok Dec., 2011
In addition, suppose the artery blood pressure, Part (t ), is
proportional to the left ventricle. In other words, the dynamics
of respiration, in fact, can be described in terms of artery
pressure and heart beat frequency:
dQLg (t ) bN ( K f , f HR , t )
RLg
(22)
=
P (t ) −
Q (t )
by f HR and f BF respectively. Assume the driving pressure
for ventricular outlet, denoted by PV (t ), can be approximated
by Eq. (12) [13], [14]. On the contrary, the driving force of
respiration cycle, denoted by PLg (t ), can be approximated by
Eq. (15) [14].
PV (t ) = AV (sin π f HR t )10 + 1
(12)
PLg (t ) = ALg sin 2π f BF t
(13)
dt
A. Simulation of Blood Circulation Dynamics
The dynamics of heart beat and blood circulation are referred
to Eq. (9) and (13). The exact physical values of parameters are
given in Table 1. The simulation results for the pressures of the
left-ventricle, PV (t ), the proximal aorta artery, Part (t ), and the
where the value of constant, K f , is determined by the mode of
peripheral artery, Pper (t ), are shown in Fig. 4 by assuming the
human body activity. Assume the residue pressure in Eq. (12)
is fairly limited so that it can be dropped off, in comparison
with the other sinusoidal term, then the relation between PV (t )
heart beat rate is 75 beats/minute. In comparison, they are
pretty close to the empirical physiological reports [1], [9], [15].
and PLg (t ) can be described as follows:
Table 1 Physical Parameter Values for Blood Circulation and
Cardiac System
(15)
AV (sin π K f f BF t )
ALg sin 2π f BF t
~
~
= A(sin K f BF t ) 9
Lg
In order to verify the validity of the mathematic model for
the integrated respiration/blood circulation system, a few
numerical simulations are undertaken and compared with the
reported works which have been well-known.
stationary conditions, i.e., for a specific activity mode in terms
of human body, the cycle-by-cycle variation, for either blood
flow through left-ventricular accommodation or the air volume
flow through the lung capacitance, can be neglected. In other
words, for a specified activity mode (e.g., sleeping, walking or
resting), the heart beat rate, f HR , can be assumed to be
proportional to the breath frequency, f BF , as follows:
(14)
f HR / f BF = K f
M ( K f , f BF , t ) =
LLg
III. INTERACTION BETWEEN CARDIAC AND RESPIRATORY
SYSTEMS
It is noted that the pressure residue during diastole cycle always
exists and therefore constitutes a bias for PV (t ) [9]. Under
10
art
Consequently, the major interaction of respiration and blood
circulation can be represented by Eq. (18), (19), and (22).
where AV is the amplitude of the sinusoidal driving pressures.
PV (t ) = M ( K f , f BF , t ) PLg (t )
LLg
(16)
~
~
where the coefficients, A and K , in Eq. (16) are both
constants. From Fig. 3, the pressure at lung and the air flow
rate can be linked by:
~
(17)
PLg (t ) = RLg × QLg (t )
By inclusion of Eq. (15)-(17), the dynamics of blood
circulation can be in terms of the variables and parameters of
respiration system as follows:
dPper (t )
dt
=
−
ca
[ 1 + aQLg (t ) × R Lg × M ( K f , f BF , t ) − aPart (t )]
C
Pper (t )
(18)
______
R2 C
Left Ventricular Pressure, PV (t )
dPart (t )
c 2 a 4 R1 ca
= [(
+ )QLg (t ) × RLg × M ( K f , f BF , t )
dt
CV
C
********
Aortic Pressure, Part (t )
c 2 a 4 R1 R1 ca 2
R
1
−(
+
+
) Part (t ) + ( 1 −
) Pper (t )
CV
L
C
L R2 C
ca c 2 a 3 R1
+( +
)] × (1 + ca 2 R1 ) −1
C
CV
□□□□□
Arteriolar Pressure, Pper (t )
(19)
Similarly, the driving pressure at lung, PLg (t ), can be
expressed in terms of the ventricular pressure, PV (t ), as
follows:
PLg (t ) = N ( K f , f HR , t ) PV (t )
N ( K f , f HR , t ) =
ALg sin( 2π f HR t / K f )
AV (sin π f HR t )10 + 1
Fig. 4 Blood Circulation based on Proposed Dynamic Model (Heart
Beat Rate = 75 Beats/Min.)
(20)
(21)
B. Simulation of Blood Circulation Dynamics
137
International Conference on Trends in Mechanical and Industrial Engineering (ICTMIE'2011) Bangkok Dec., 2011
As aforesaid, the valve coefficient, a, plays the role to
control the blood output from the left-ventricle. Ideally, the
valve has to be closed during the diastole cycle. However, for
some reasons, known or unknown, the valve is not completely
closed for some patients.
Numerical simulations on blood pressures for the case of
mild valvular closure incompetence are shown in Fig. 6. It is
noticed that though the blood pressure during systolic cycle is
basically normal, yet the blood pressure during the diastolic
cycle is not retained any more (by comparison between Fig. 4
and Fig. 6). The eigenvalues of the respiration/blood
circulation system under various degrees of valvular closure
incompetence are listed in Table 4. As long as the valvular
closure incompetence occurred, the left-ventricle would have to
provide stronger systolic momentum to retain the necessary
driving pressure, PV (t ), which becomes, in fact, unstable
The physical values of parameters for respiration dynamics
in Fig. 2 are listed in Table 2. The tadial volume of respiration
and the flow rate of tadial volume at lung versus time are shown
in Fig. 5. In comparison, the behavior of the air flow at the
lung, based on the proposed dynamic model, is fairly identical
to the physiological experiments [9], [16].
Table 2 Physical Parameter Values for Respiration System
Air Volume within Lung (ml)
Air Volume Flow Rate at Lung (ml / sec)
during the systole cycle. The eigenvalues under rest mode are
listed in Table 5. In comparison with the ones in Table 3
(normal condition), the dynamics of the interactive
respiration/blood circulation becomes unstable and needs to be
controlled by medicine or external devices to prevent from
heart failure or other side-effects.
______
Left Ventricular Pressure, PV (t )
********
Aortic Pressure, Part (t )
□□□□
Arteriolar Pressure, Pper (t )
Time (sec)
Fig. 5 Tadial Volume of Respiration and Flow Rate of Tadial
Volume
Fig. 6 Blood Pressures under Valvular Closure Incompetenc
C. Numerical Analysis of Respiration Dynamics and Blood
Circulation System
The dynamic interaction model between the respiration and
blood circulation system is described by Eq. (23). It is
observed that there are four parameters, a, b, M ( K f , f BF , t )
Table 4 Eigenvalues under Various Degrees of Valvular Closure
Incompetence
and N ( K f , f HR , t ) . The system parameters and eigenvalues
under rest mode are listed in Table 3. Since the semilunar valve
is fairly flexible and deformable, for some patients it cannot be
completely closed during the diastole cycle [9]. To some
extent, the impacts by the valvular closure incompetence and
the resulted pulmonary obstruction will be addressed in the
next section.
Table 5 Eigenvalues under Rest Mode and Valvular Closure
Incompetence
Table 3 Parameters and Eigenvalues under Rest Mode
3.3.2. On Pulmonary Obstruction
There are many kinds of pulmonary diseases which can
affect the respiration behavior. However, most cases can be
simplified to be in terms of the severe increase of pulmonary
3.3.1. Effect of Valvular Closure Incompetence
138
International Conference on Trends in Mechanical and Industrial Engineering (ICTMIE'2011) Bangkok Dec., 2011
flow resistance. The influences regarding the abnormal
variations of pulmonary flow resistance under rest state are
listed in Table 6. Two abnormal modes, mild and severe
pulmonary obstructions, are numerically studied by setting the
pulmonary flow resistance to be 100 and 1000 times to the
normal mode. By comparison between Table 3 and Table 6,
though the interaction of respiration and blood circulation can
be still stable at the beginning, yet the function of the integrated
interaction system will be ruined eventually, due to the
autonomous regulation of heart beat rate by brain. The
eigenvalues of the interaction system under rest mode for the
case in which the pulmonary obstruction and valvular closure
incompetence concurrently occur, are listed in Table 7. It can
be noticed that a few eigenvalues are unstable.
[3]
[4]
[5]
[6]
[7]
[8]
[9]
Table 6 Eigenvalues under Various Degrees of Pulmonary
Obstruction
[10]
[11]
[12]
[13]
Table 7 Eigenvalues under Both Pulmonary Obstruction
[14]
[15]
[16]
IV. CONCLUSIONS
The dynamics of cardiovascular system, respiration and
heart-lung interaction is constructed and studied. By setting a
set of parameters to link the heart beat cycle and respiration
mechanism, the models of cardiovascular system and
respiration can be simply integrated. The numerical simulation
results and theoretical analysis on dynamics of cardiovascular
system, respiration system and the abnormal cases are
compared with the empirical reports to verify the validity of the
proposed dynamic models. It is noticed that the heart-lung
interaction is inherently unstable, especially if certain
heart/lung disease or injuries are present. For realistic
contribution, the proposed models can be employed for
controller synthesis for medical equipments, e.g.,
Extracorporeal Membrane Oxygenation (ECMO) system, to
regulate the blood circulation and respiration for severely
injured patients.
REFERENCES
[1]
[2]
Segers P. and Stergiopulos N., “Systemic and Pulmonary Hemodynamics
Assessed with A Lumped-Parameter Heart-Arterial Interaction Model,”
Journal of Engineering Mathematics, V.47, p. 185-199, 2003.
Bjorn A. J. Angelsen and Stig A. Slordahl et. al., “Estimation of
Regurgitant Volume and Orifice in Aortic Regurgitation Combining CW
139
Doppler and Parameter Estimation in a Windkessel-like Model,” IEEE
Transactions on Biomedical Engineering, V. 37, N. 10, p. 930-936, 1990.
Heldt T. and Shim E., “Computational Model of Cardiovascular Function
During Orthostatic Stress,” Computers in Cardiology, V. 27, p. 777-780,
2000.
Watanabe H. and Hisada T., “Computer Simulation of Blood Flow Left
Ventricular Wall Motion and Their Interrelationship by Fluid-Structure
Interaction Finite Element Method,” JSME, Series C, V. 45, N. 4, p.
1003-1012, 2002.
Schwarz and Oldham, “Electrical Engineering 2nd ,”Oxford University
Press,1993.
Pozzi A., “Application of Pade Approximation Theory in Fluid
Dynamics, ” World Scientific Publishing Co Pte Ltd, 1993.
George A. Baker, Jr., Peter Graves-Morris, “Pade Approximants,”
Addison-Wesley, Advanced Book Program, 1981.
Gragg W. B., “The Pade Table and Its Relation to Certain Algorithms of
Numerical Analysis,” Society for Industrial and Applied Mathematics,
Vol.14 No.1 pp.1-62, 1972.
Palladino L., Mulier P. and Moordergraaf A., “Defining Ventricular
Elastance,” Proc. 20th Int. Conf. IEEE Eng. Med. & Biol. Soc., 1998.
Zhong L., Ghista D. N., Ng E. Y. and Lim S. T., “Passive and Active
Ventricular Elastances of the Left Ventricle,” BioMedical Engineering
Online, V.4, 2005.
Weibel E. R., “Morphometry of the Human Lung,” Berlin:
Springer-Verlag, 1963.
Nucci G., Tessarin S. and Cobelli C., “A Morphometric Model of Lung
Mechanics for Time-Domain Analysis of Alveolar Pressures during
Mechanical Ventilation,” Annals of Biomedical Engineering, V. 30, p.
537-545, 2002.
Dhanjoo N. Ghista, “Lung Ventilatory Performance Pressure-Volume
Model and Parameteric Simulation for Disease Detection,” 19th
International Conference – IEEE/EMBS, 1997.
Meste O., Khaddoumi B., Blain G. and Bermon S., “Time-Varying
Analysis Methods and Models for the Respiratory and Cardiac System
Coupling in Graded Exercise,” IEEE Transactions on Biomedical
Engineering, V. 52, N. 11, .p.1921-1930, 2005.
Segers P., Stergiopulos N., Westerhof N., Wouters P., Kolh P. and
Verdonck P., “ Systemic and Pulmonary Hemodynamics Assessed with A
Lumped-Parameter Heart-Arterial Interaction Model,” Journal of
Engineering Mathematics, V. 47, p. 85-199, 2003.
Kulish V., “Human Respiration,” WIT press, 2006.