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Lumped Parameter Model for Hemodynamic Simulation of Congenital Heart Diseases

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Lumped Parameter Model for Hemodynamic Simulation of Congenital Heart Diseases The Journal of Physiological Sciences (2018) 68:103–111 https://doi.org/10.1007/s12576-017-0585-1 REVIEW Lumped parameter model for hemodynamic simulation of congenital heart diseases Shuji Shimizu1 · Dai Une1 · Toru Kawada1 · Yohsuke Hayama1 · Atsunori Kamiya1 · Toshiaki Shishido2 · Masaru Sugimachi1 Received: 19 July 2017 / Accepted: 5 December 2017 / Published online: 21 December 2017 © The Physiological Society of Japan and Springer Japan KK, part of Springer Nature 2017 Abstract The recent development of computer technology has made it possible to simulate the hemodynamics of congenital heart dis- eases on a desktop computer. However, multi-scale modeling of the cardiovascular system based on computed tomographic and magnetic resonance images still requires long simulation times. The lumped parameter model is potentially benefcial for real-time bedside simulation of congenital heart diseases. In this review, we introduce the basics of the lumped param- eter model (time-varying elastance chamber model combined with modifed Windkessel vasculature model) and illustrate its usage in hemodynamic simulation of congenital heart diseases using examples such as hypoplastic left heart syndrome and Fontan circulation. We also discuss the advantages of the lumped parameter model and the problems for clinical use. Keywords Lumped parameter model · Time-varying elastance · Windkessel model · Congenital heart diseases · Hemodynamic simulation Introduction as three-dimensional computed tomography (CT) and mag- netic resonance imaging (MRI) provide specifc anatomical Hemodynamic management of patients with congenital heart information of complex heart anomalies [2]. Hemodynamic diseases remains a challenge for pediatric cardiologists and simulation based on these anatomical data will be helpful for pediatric cardiac surgeons, and the number of specialists surgical decision making [3]. However, because large-scale for pediatric hemodynamic management is limited. Recent computations are required to use these data, execution of the development of computer technology has allowed simu- simulation process is too long to meet the requirement of lations of hemodynamics of congenital heart diseases on real-time clinical applications [4]. Another approach is the desktop computers. The performance of a recent desktop simulation with a lumped parameter model, which can be computer has already reached a sufcient stage to perform performed even on a small bedside computer [5]. Although some kind of real-time simulation. this type of hemodynamic simulation does not include There are two diferent approaches to simulating car- specifc anatomical information, the greatest advantage of diovascular systems of congenital heart diseases. One this approach is that real-time simulation can be performed approach is multi-scale computational fluid dynamics while acquiring clinical data from patients. This approach (CFD) modeling, which is often combined with lumped is likely to be more helpful for perioperative hemodynamic parameter models [1]. Modern imaging technologies such management. In this review, we introduce computational modeling of congenital heart diseases using the lumped parameter model * Shuji Shimizu and discuss its usefulness and issues in clinical use. [email protected] 1 Department of Cardiovascular Dynamics, National Cerebral and Cardiovascular Center, 5‑7‑1 Fujishiro‑dai, Suita, Osaka 565‑5685, Japan 2 Department of Research Promotion, National Cerebral and Cardiovascular Center, 5‑7‑1 Fujishiro‑dai, Suita, Osaka 565‑8565, Japan Vol.:(0123456789)1 3 104 The Journal of Physiological Sciences (2018) 68:103–111 Lumped parameter model a maximal value (Ees), which is the slope of the ESPVR. Hence, ESPVR is described as: The lumped parameter model of the cardiovascular sys- Pes(t)=Ees[Ves(t)−V0], (2) tem consists of two components; a time-varying elastance P V chamber component and a (modifed) Windkessel vascular where es and es are end-systolic pressure and volume, component. The time-varying chamber component describes respectively. In contrast with ESPVR, end-diastolic pres- instantaneous ventricular and atrial pressure–volume rela- sure–volume relationship (EDPVR) is often represented by tions. The Windkessel vascular component describes hemo- an exponential curve: dynamics in arterial and venous trees. The combination of Ped(t)=A{exp[B(Ved(t)−V0)] − 1}, (3) these two components enables us to simulate the cardio- where Ped and Ved are end-diastolic pressure and volume, vascular system; the left and right heart, and systemic and respectively. A and B are constants. The parameter B is pulmonary circulations. called as a stifness constant. Instantaneous chamber pressure is described by the sum of Time‑varying elastance chamber component Ped and the developed pressure (diference between Pes and Ped) scaled by normalized elastance curve [e(t)]: In the lumped parameter model of cardiovascular system, each chamber (ventricle or atrium) is represented by a time- P(t)=[Pes(V(t)) − Ped(V(t))]e(t)+Ped(V(t)). (4) varying elastance model [6]. The basis of time-varying Parameters Ees, V0, A, and B are diferent among the four elastance chamber model has been reported by Suga et al. chambers (left and right ventricles and atria). [7]. The chamber pressure at time (t) after onset of contraction Elastance curve is approximated as: In the real world, elastance curve (E(t)) looks like a skewed P(t)≅E(t)[V(t)−V0], (1) sine curve [8]. However, in computational simulation, the t P t V t where is the time after onset of contraction, ( ) and ( ) skewed sine curve is often replaced by a simple sine (or are instantaneous chamber pressure and volume, respec- cosine) curve (Fig. 1a) [9, 10]: tively, and V is the volume axis intercept of the end-systolic 0 � ≤ < pressure–volume relationship (ESPVR). E(t) is an instan- e(t)=0.5[1 − cos( t∕Tes)] (0 t 2Tes) ≤ < (5) taneous elastance of the chamber, which is independent of e(t)=0 (2Tes t T), the loading conditions. At end-systole (tes), E(t) reaches Fig. 1 Simulated normalized ab elastance curves (Tes = 175 ms). 1 1 a A sine curve. b A sine curve combined with exponential curve. Time constant of relaxa- τ solid tion ( ) is 25 ms for the 0.8 0.8 line and 100 ms for the dotted line. An increase in τ prolongs the tail 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0200 400600 8000 200400 600 800 Time (ms) Time (ms) 1 3 The Journal of Physiological Sciences (2018) 68:103–111 105 where t is the time from the start of systole, Tes is the dura- range corresponding to the characteristic impedance. tion of systole, and T is the duration of cardiac cycle. Since aortic and/or pulmonary characteristic impedances A sine curve combined with an exponential curve is also in patients with congenital anomaly dramatically change used as a normalized elastance curve (Fig. 1b) [11]: between before and after the surgical treatment, three- or four-element Windkessel model may be suitable for hemo- e(t)=0.5[1 + sin(�t∕Tes − �∕2)] (0 < t < 3Tes∕2) dynamic simulation of congenital heart diseases. (6) e(t)=0.5 exp[(3Tes∕2 − t)∕�](3Tes∕2 ≤ t), At each resistance (R), a linear relation between pressure drop (ΔP) and fow (Q) applies, according to the Ohm’s law: τ τ where is the time constant of relaxation. An increase in ΔP = Q ⋅ R. prolongs the tail of the normalized elastance curve (Fig. 1b, (7) C dotted line). Because e(t) is discontinuous from the end of At each capacitance ( ), the relation between pressure P V V t t one cardiac cycle to the beginning of the next, this may ( ) and volume ( ), and the change in volume [d ( )/d ] Q cause an error in the simulation result. calculated by the diference between total infow (∑ infow) and total outfow (∑Qoutfow) are as follows: Windkessel vascular components V P = , C (8) In the lumped parameter model of cardiovascular system, vasculatures are simulated as Windkessel vasculature mod- dV(t) = Q (t)− Q (t). els. There are three major vasculature models: two-, three-, dt inflow outflow (9) and four-element Windkessel models [12]. The two-ele- ment Windkessel model consists of a resistor and a capaci- At each inductance (L), the relation between pressure ΔP tor (Fig. 2a). The resistor represents vascular resistance drop ( ) and change in fow [dQ(t)/dt] is as shown below: and the capacitor represents vascular volume capacitance. dQ(t) ΔP = L . In the three-element Windkessel model, another resistor dt (10) corresponding to characteristic impedance (Rc) is serially added to the two-element Windkessel resistor and capacitor Valves (Fig. 2b). In the four-element Windkessel model, an inductor is added to the three-element Windkessel model (Fig. 2c). There are several variations in the position of the inductor Each valve (aortic, pulmonary, mitral, or tricuspid valve) in the four-element Windkessel model [13]. is often represented as an ideal diode connected serially to When we consider the input impedance, the two-element a small resistor [14]. Several papers also used a diode con- Windkessel model is less accurate than the three- and four- nected with a non-linear resistance as a valve model [15], element Windkessel models especially in the high-frequency as described below: ⋅ 2 ΔP = k Q(t) . (11) a b Valve regurgitation is often simulated by introducing a second diode in the opposite direction with a small resistor R C [14]. RC RC Electrical analogs of cardiovascular systems Using the framework of the time-varying elastance chamber c model combined with the modifed three-element Windkes- RC sel vasculature model, cardiovascular systems are described in terms of electrical analogs. Figure 3 shows the electri- cal analogs of the cardiovascular systems of a normal heart L RC (Fig. 3a), Fontan circulation (total cavopulmonary connec- tion, Fig. 3b), atrial septal defect (ASD, Fig. 3c), and ven- tricular septal defect (VSD, Fig. 3d). In these electrical analogs, no non-linear components are used.
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