mathematical model of interactive respiration/cardiovascular...

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.

Keywords—Cardiovascular System, Heart-Lung Interaction, Extracorporeal Membrane Oxygenation (ECMO)

I. INTRODUCTION 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.

II. DYNAMICS OF INTEGRATED RESPIRATORY AND CARDIOVASCULAR SYSTEMS

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

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] ).

model for the heart and systemic circulation mainly consists of six components: an AC power supply ( )(tPV

), a diode ( D ), two resistors ( 1R and 2R ), one capacitor ( C ) and one inductance ( L ). In physiological terms, they respectively represent the driving force by Left-Ventricle ( )(tPV ), the function of Aortic Semilunar Valve ( D ), the overall peripheral resistance ( 2R ), the overall compliance of the arterial system ( C ), the impedance of the aorta ( 1R ) and the overall inertia of blood ( L ). On the other hand, the variables, )(tPart and )(tPper ,

shown in Fig. 1, are the blood pressures of aorta and peripheral artery (arteriole) respectively. )(tQart is the overall blood (volume) flow rate in the blood circulation system and

)( ),( ),(1 tQtQtQ CLR and )(2 tQR are the blood (volume) flow rates through CLR , ,1 and 2R respectively.

)(Part t )(Pper t

)(Qart t L

1R

)(PV t

)(QL t

)(QR1 t

C 2R)(QC t )(QR2 t

Fig. 1 Equivalent Electrical Circuit for Heart and Blood Circulation

Since LRart QQQ += 1 and CRart QQQ += 2 both hold valid

under mass conservation law, the state equations of )(tPart and

)(tPper can be constructed as follows:

CRtP

CtQ

dttdP perartper

2

)()()(−= (1)

CRtP

CtQtPtP

LR

dttdQR

dttdP perart

perartartart

2

11

)()()]()([)()(−+−−= (2)

where

22

)()(

RtP

tQ perR = (3a)

dttdP

CtQ perC

)()( = (3b)

LtPtP

dttdQ perartL )()()( −

= (3c)

11

)()()(

RtPtP

tQ perartR

−= (3d)

Mathematical Model of Interactive Respiration/Cardiovascular Composite System

Rong-Mao Lee, Hsin-Lin Chiu, Nan-Chyuan Tsai#

T

International Conference on Trends in Mechanical and Industrial Engineering (ICTMIE'2011) Bangkok Dec., 2011

135

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,

),(tQart is hence a function of the driving force of Left-Ventricle, ),(tPV and the blood pressure of aorta, ),(tPart as follows:

)]()([)( tPtPaart

artVaectQ −= (4) where “ a ” is the valve coefficient which is a non-negative constant. As long as ),()( tPtP artV < 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 ( ,0,1R )” can be employed to expand the

exponential function in Eq. (4) as follows: AeA +≅ 1 (5)

where A is any real scalar. From Eq. (1), (2) the dynamics of blood pressures, )(tPart and

),(tPper can be rewritten respectively as follows:

)(

)]()(1[)(

2CRtP

taPtaPCac

dttdP per

artVper −−+= (6)

(7)

Eq. (6) and Eq. (7) respectively represent the dynamics of blood pressure at peripheral artery and blood pressure at aorta.

ii. Mathematic Model of Ventricular Dynamic Characteristics

For incompressible liquid, the flow rate is related to the applied pressure as follows:

dtdP

CQ fff = (8)

where ,fQ fP and fC are volume flow rate, liquid pressure

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:

dttdPCtQ V

Vart)()( = (9)

The compliance of left-ventricle is denoted by .VC Although the compliance, ,VC varies within the cardiac cycle [9], [10], yet it does not exert any effect on artQ 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, ,VC during the systolic half-cycle is pretty limited, the compliance of left-ventricle is simplified and assumed as a constant in this paper. Its actual value

depends on the exercise modes undertaken by the human body. Consequently, Eq. (7) can be further rewritten as follows: (10) The derivative term, ,/)( dttdPV in Eq. (7) is now completely

substituted by the driving force by Left-Ventricle, ).(tPV

B. Dynamics of Respiration The equivalent electric circuit for lung/air dynamics is

shown in Fig. 2 [11], [12]. The pressure at lung, ),(tPLg plays

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 )(tPLg

is positive. In addition, an inductance,

LgL a capacitor, LgC and a resistor, LgR are analogic to

represent the inertia of airflow, compliance of trachea and resistance of the respiratory tract respectively.

)(PLg t

)(QLg t

LgC

LgR)(QRLg t)(QCLg t

LgL

Fig. 2 Equivalent Electrical Circuit for Lung/Air Dynamics

The variable )(tQLg is the overall air volume flow rate

while )(tQCLg and )(tQRLg are its corresponding components

flowing through LgC and LgR 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, ,LgZ can be reduced to LgR such that

the dynamics of overall air volume flow rate, )(tQLg can be

described as follows:

Lg

LgLgLgLg tQRtPdt

tdQL

)()()( ×−= (11)

LgZ)(PLg t

LgL)(QLg t )(P1 t

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

)1]( )()1(

)()()()(

[)(

1-1

2

2

1

21

2

12

RacCactP

CRLR

tPCac

LRtP

Cac

dttdP

Racdt

tdP

per

artVVart

++−+

+−+=

11

2132

2

1

211

421

42

)1()]()()1(

)()()()[()(

−+×++−+

++−+=

RcaC

RacCcatP

CRLR

tPC

caLR

CRactP

Cca

CRac

dttdP

Vper

artV

VV

art


136

by HRf and BFf respectively. Assume the driving pressure for ventricular outlet, denoted by ),(tPV can be approximated by Eq. (12) [13], [14]. On the contrary, the driving force of respiration cycle, denoted by ),(tPLg can be approximated by

Eq. (15) [14]. 1)(sin)( 10 += tfAtP HRVV π (12)

tfAtP BFLgLg π2sin)( = (13)

where VA is the amplitude of the sinusoidal driving pressures.

It is noted that the pressure residue during diastole cycle always exists and therefore constitutes a bias for )(tPV [9]. Under 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, ,HRf can be assumed to be proportional to the breath frequency, ,BFf as follows: fBFHR Kff =/ (14)

where the value of constant, ,fK is determined by the mode of

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 )(tPV and )(tPLg can be described as follows:

)(),,()( tPtfKMtP LgBFfV = (15)

910

)~(sin~2sin

)(sin),,( tfKA

tfAtfKA

tfKM BFBFLg

BFfVBFf ==

ππ (16)

where the coefficients, ,~and~ KA in Eq. (16) are both constants. From Fig. 3, the pressure at lung and the air flow rate can be linked by: )()(~ tQRtP LgLgLg ×= (17)

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:

CRtP

taPtfKMRtaQCca

dttdP

per

artBFfLgLgper

2

)(

)](),,()(1 [)(

−

−××+= (18)

(19) Similarly, the driving pressure at lung, ),(tPLg can be

expressed in terms of the ventricular pressure, ),(tPV as follows: )(),,()( tPtfKNtP VHRfLg = (20)

1)(sin)/2sin(

),,( 10 +=

tfAKtfA

tfKNHRV

fHRLgHRf π

π (21)

In addition, suppose the artery blood pressure, ),(tPart 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: )()(

),,()(tQ

LR

tPL

tfKbNdt

tdQLg

Lg

Lgart

Lg

HRfLg −= (22)

Consequently, the major interaction of respiration and blood circulation can be represented by Eq. (18), (19), and (22).

III. INTERACTION BETWEEN CARDIAC AND RESPIRATORY SYSTEMS

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.

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, ),(tPV the proximal aorta artery, ),(tPart and the peripheral artery, ),(tPper are shown in Fig. 4 by assuming the

heart beat rate is 75 beats/minute. In comparison, they are pretty close to the empirical physiological reports [1], [9], [15].

Table 1 Physical Parameter Values for Blood Circulation and

Cardiac System

______Left Ventricular Pressure,

********

Aortic Pressure,

□ □ □ □ □

Arteriolar Pressure,

)(tPV

)(tPart

)(tPper

Fig. 4 Blood Circulation based on Proposed Dynamic Model (Heart

Beat Rate = 75 Beats/Min.)

B. Simulation of Blood Circulation Dynamics

11

2132

2

12

1142

142

)1()](

)()1()()(

),,()()[()(

−+×++

−+++−

××+=

RcaC

RacCca

tPCRL

RtPC

caLR

CRac

tfKMRtQCca

CRac

dttdP

V

perartV

BFfLgLgV

art


137

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

Vol

ume

with

in L

ung

(ml)

Air

Vol

ume

Flow

Rat

e at

Lun

g (m

l / se

c)

Time (sec) Fig. 5 Tadial Volume of Respiration and Flow Rate of Tadial

Volume

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, ),,(,, tfKMba BFf

and ),,( tfKN HRf . 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 3 Parameters and Eigenvalues under Rest Mode

3.3.1. Effect of Valvular Closure Incompetence

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, ),(tPV which becomes, in fact, unstable 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,

********

Aortic Pressure,

□□□□

Arteriolar Pressure,

)(tPV

)(tPart

)(tPper

Fig. 6 Blood Pressures under Valvular Closure Incompetenc

Table 4 Eigenvalues under Various Degrees of Valvular Closure

Incompetence

Table 5 Eigenvalues under Rest Mode and Valvular Closure Incompetence

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


138

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.

Table 6 Eigenvalues under Various Degrees of Pulmonary

Obstruction

Table 7 Eigenvalues under Both Pulmonary Obstruction

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] 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.

[2] Bjorn A. J. Angelsen and Stig A. Slordahl et. al., “Estimation of Regurgitant Volume and Orifice in Aortic Regurgitation Combining CW

Doppler and Parameter Estimation in a Windkessel-like Model,” IEEE Transactions on Biomedical Engineering, V. 37, N. 10, p. 930-936, 1990.

[3] Heldt T. and Shim E., “Computational Model of Cardiovascular Function During Orthostatic Stress,” Computers in Cardiology, V. 27, p. 777-780, 2000.

[4] 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.

[5] Schwarz and Oldham, “Electrical Engineering 2nd ,”Oxford University Press,1993.

[6] Pozzi A., “Application of Pade Approximation Theory in Fluid Dynamics, ” World Scientific Publishing Co Pte Ltd, 1993.

[7] George A. Baker, Jr., Peter Graves-Morris, “Pade Approximants,” Addison-Wesley, Advanced Book Program, 1981.

[8] 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.

[9] Palladino L., Mulier P. and Moordergraaf A., “Defining Ventricular Elastance,” Proc. 20th Int. Conf. IEEE Eng. Med. & Biol. Soc., 1998.

[10] 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.

[11] Weibel E. R., “Morphometry of the Human Lung,” Berlin: Springer-Verlag, 1963.

[12] 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.

[13] Dhanjoo N. Ghista, “Lung Ventilatory Performance Pressure-Volume Model and Parameteric Simulation for Disease Detection,” 19th International Conference – IEEE/EMBS, 1997.

[14] 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.

[15] 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.

[16] Kulish V., “Human Respiration,” WIT press, 2006.


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