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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. 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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.