Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Cardiovascular disease wikipedia , lookup
Saturated fat and cardiovascular disease wikipedia , lookup
Aortic stenosis wikipedia , lookup
Antihypertensive drug wikipedia , lookup
Quantium Medical Cardiac Output wikipedia , lookup
Cardiac surgery wikipedia , lookup
Dextro-Transposition of the great arteries wikipedia , lookup
History of invasive and interventional cardiology wikipedia , lookup
Annals of Biomedical Engineering, Vol. 38, No. 10, October 2010 ( 2010) pp. 3195–3209 DOI: 10.1007/s10439-010-0083-6 Patient-Specific Modeling of Blood Flow and Pressure in Human Coronary Arteries H. J. KIM,1 I. E. VIGNON-CLEMENTEL,2 J. S. COOGAN,3 C. A. FIGUEROA,3 K. E. JANSEN,1 and C. A. TAYLOR3,4 1 Aerospace Engineering Sciences, University of Colorado at Boulder, Boulder, CO 80309, USA; 2INRIA, Paris-Rocquencourt BP 105, 78153 Le Chesnay Cedex, France; 3Department of Bioengineering, Stanford University, Stanford, CA 94305, USA; and 4 Department of Surgery, Stanford University, E350 Clark Center, 318 Campus Drive, Stanford, CA 94305, USA (Received 29 September 2009; accepted 20 May 2010; published online 18 June 2010) Associate Editor Peter E. McHugh oversaw the review of this article. and the prediction of the outcomes of interventions.16,23 As the computing capacity and numerical methods for simulation of blood flow advance, further applications are anticipated. However, computational simulations have been rarely used to predict pulsatile flow and pressure fields of three-dimensional coronary vascular beds, in part because the flow rate and pressure in the coronary vascular beds are highly related to the interactions between the heart and the arterial system. Unlike flow in other parts of the arterial system, coronary flow decreases when the ventricles contract and increase the intramyocardial pressure, which exerts an extravascular compressive force on the coronary vessels.17 Coronary flow increases when the ventricles relax, thereby, decreasing the intramyocardial pressure and extravascular compressive force. Therefore, to model coronary flow and pressure realistically, it is necessary to have a model of the heart and a model of the arterial system with consideration of the interactions between the two models. Because of this complexity in modeling coronary flow and pressure, most three-dimensional computational studies have been conducted with only the coronary arteries, ignoring the interactions between the heart and the arterial system and prescribing, not predicting, coronary flow. Further, these studies have not modeled realistic pressures and generally used tractionfree outlet boundary conditions whether the lateral surfaces of the coronary arteries are considered as rigid walls1,6 or compliant walls18,30 or the cardiac motions due to the contraction and relaxation of the heart are considered19,21 or not. Migliavacca and coworkers12,16 computed three-dimensional pulsatile coronary flow and pressure in a single coronary artery, but this study was performed with an idealized model and low mesh Abstract—Coronary flow is different from the flow in other parts of the arterial system because it is influenced by the contraction and relaxation of the heart. To model coronary flow realistically, the compressive force of the heart acting on the coronary vessels needs to be included. In this study, we developed a method that predicts coronary flow and pressure of three-dimensional epicardial coronary arteries by considering models of the heart and arterial system and the interactions between the two models. For each coronary outlet, a lumped parameter coronary vascular bed model was assigned to represent the impedance of the downstream coronary vascular networks absent in the computational domain. The intramyocardial pressure was represented with either the left or right ventricular pressure depending on the location of the coronary arteries. The left and right ventricular pressure were solved from the lumped parameter heart models coupled to a closed loop system comprising a threedimensional model of the aorta, three-element Windkessel models of the rest of the systemic circulation and the pulmonary circulation, and lumped parameter models for the left and right sides of the heart. The computed coronary flow and pressure and the aortic flow and pressure waveforms were realistic as compared to literature data. Keywords—Blood flow, Coronary flow, Coronary pressure, Coupled multidomain method. INTRODUCTION Computational simulations have been proven useful in studying blood flow in the cardiovascular system24: assessing hemodynamics of healthy and diseased blood vessels,4,6 helping in the design and evaluation of vascular medical devices,1,14 planning of vascular surgeries, Address correspondence to C. A. Taylor, Department of Surgery, Stanford University, E350 Clark Center, 318 Campus Drive, Stanford, CA 94305, USA. Electronic mail: [email protected] 3195 0090-6964/10/1000-3195/0 2010 Biomedical Engineering Society 3196 KIM et al. resolution. The analytic models used as boundary conditions were coupled explicitly, necessitating either subiterations within the same time step or a small time step size bounded by the stability of an explicit time integration scheme. To predict physiologically realistic flow rate and pressure in the coronary arterial trees of a patient, computational simulations should be robust and stable enough to handle complex flow characteristics and the coupling with different scales of computer models should be efficient and versatile.10 In previous studies, Vignon-Clementel et al.27,28 developed boundary conditions for three-dimensional models of blood flow that can represent physiological conditions of each subject. For outlet boundaries of the arterial system excluding the coronary vascular beds, simple analytic models such as resistance, impedance, and the three-element Windkessel model were assigned to model the arterial system absent in the computational domain. This method is fully implicit and has proven to be versatile and robust. In a related study, we implemented a lumped parameter heart model as an inflow boundary condition of a three-dimensional finite-element model of the aorta to consider the interactions between the ventricle and the arterial system.11 Again, a fully implicit method was used to couple the heart model to the aortic model. We showed physiologically realistic flow and pressure fields for a subject-specific three-dimensional model of the aorta with these boundary conditions.11 In addition, constraints on the shape of the velocity profile at the aortic inlet and select outlet boundaries were enforced using an augmented Lagrangian method to resolve numerical instabilities that result from reverse and complex flow structures at boundaries.10 In this paper, we describe a method to calculate the flow and pressure of three-dimensional coronary vascular beds by considering the models of the left and right sides of the heart, arterial system, and the interactions between them. For each coronary outlet, a lumped parameter coronary vascular bed model was assigned to represent the impedance of downstream coronary vascular networks absent in the computational domain. We solved for coronary flow and pressure as well as aortic flow and pressure in patientspecific models for rest and light and moderate exercise conditions. Additionally, to demonstrate that these methods can be utilized in clinical investigation, we studied blood flow and pressure with different degrees of stenosis. We created a stenosis in the left anterior descending coronary artery by reducing the local diameter of the geometric model by 40%, 50%, 60%, and 75% and simulated a baseline (resting) condition and a moderate exercise condition, where in the latter case we assumed that the downstream coronary vascular beds were maximally dilated. METHODS Three-Dimensional Finite Element Model of Blood Flow and Vessel Wall Dynamics Blood flow in the large vessels of the cardiovascular system can be approximated by the flow of a Newtonian fluid. Blood flow is modeled using the incompressible Navier–Stokes equations, and the motion of the vessel wall using the elastodynamics equations. In these equations, fluid–solid interface conditions, as well as the initial and boundary conditions, are required to solve for the blood flow and the vessel wall dynamics of the fluid and solid domains, respectively. In this study, we used a fixed fluid mesh assuming small displacements of the vessel wall.5 For a fluid domain X with its boundary C and a solid domain Xs with its boundary Csg ; the following equations are solved for velocities ~ v; pressure p, and wall displacement ~ u .5,27 Given ~ f: X ð0; TÞ ! R3 ; ~ fs : Xs ð0; TÞ ! R3 ; 3 s s ~ g: Cg ð0; TÞ ! R ; ~ g : Cg ð0; TÞ ! R3 ; ~ v0 : X ! R 3 ; s s 3 3 ~ u0 : X ! R ; and ~ u0;t : X ! R ; find ~ vð~ x; tÞ; pð~ x; tÞ; and ~ uð~ xs ; tÞ for 8~ x 2 X; 8~ xs 2 Xs ; and " t 2 (0, T), such that the following conditions are satisfied: q~ v;t þ q~ v r~ v ¼ rp þ divðs Þ þ ~ f for ð~ x;tÞ 2 X ð0;TÞ divð~ vÞ ¼ 0 for ð~ x;tÞ 2 X ð0;TÞ s s ~s x s ;tÞ 2 Xs ð0; TÞ ð1Þ q~ u;tt ¼ r r þ f for ð~ where s ¼ lðr~ v þ ðr~ vÞ T Þ 1 u þ ðr~ uÞT Þ and rs ¼ C : ðr~ 2 with the Dirichlet boundary conditions, ~ vð~ x; tÞ ¼ ~ gð~ x; tÞ for ð~ x; tÞ 2 Cg ð0; TÞ ~ uð~ x s ; tÞ ¼ ~ g s ð~ x s ; tÞ for ð~ x s ; tÞ 2 Csg ð0; TÞ ð2Þ the Neumann boundary conditions, ~ v; p; ~ ~ n ¼ hð~ x; tÞ for ~ x 2 Ch t~n ¼ ½p I þ s ~ ð3Þ and the initial conditions, ~ xÞ for ~ x2X vð~ x; 0Þ ¼ ~ v0 ð~ s s ~ uð~ x ; 0Þ ¼ ~ u0 ð~ x Þ for ~ x s 2 Xs ~ x s ; 0Þ ¼ ~ u0;t ð~ x s Þ for ~ x s 2 Xs u;t ð~ ð4Þ where the boundary C of the fluid domain is divided into a Dirichlet boundary portion Cg and a Neumann boundary portion Ch. Similarly, the boundary Cs of the solid domain is divided into a Dirichlet boundary portion Csg and a Neumann boundary portion Csh. They Patient-Specific Coronary Flow Modeling a model developed by Mantero et al.15 (Fig. 1). The coronary venous microcirculation compliance was eliminated from the original model in order to simplify the numerics. The coronary pressure and flow waveforms retained realistic waveforms without the coronary venous microcirculation compliance. The coronary vascular bed model consists of coronary arterial resistance Ra, coronary arterial compliance Ca, coronary arterial microcirculation resistance Ra-micro, myocardial compliance Cim, coronary venous microcirculation resistance Rv-micro, coronary venous resistance Rv, and intramyocardial pressure Pim(t). The modeling of the intramyocardial pressure is explained in the next subsection. For each coronary outlet surface Chcork we defined the operators M and H as follows by replacing the coronary outlet pressure P(t) with the ordinary differential equation obtained from the lumped parameter coronary vascular bed model: Z ~ v;pÞ þ H ¼ R vðtÞ ~ ndC M ð~ satisfy ðCh [ Cg Þ ¼ C; ðCsh [ Csg Þ ¼ Cs ; Ch \ Cg ¼ /; and Csh \ Csg ¼ /: The density q and the dynamic viscosity l of the fluid, and the density qs of the vessel walls are assumed to be constant. The external body force on the fluid domain is represented by ~ f: Similarly, ~ fs is the external body force on the solid domain, C is a fourth-order tensor of material constants, and rs is the vessel wall stress tensor. To assign coronary outlet boundary conditions, the coupled multidomain method27 is utilized with a lumped parameter coronary vascular bed model. Similarly to the treatment of other outflow boundary conditions, a lumped parameter model of coronary vascular ~c beds defines boundary operators M ¼ M ; M m Chcor ~c and H ¼ H ; H that represent the traction and Chcor flow at each coronary outlet surface Chcor . For the other boundaries, the same method is applied to assign an impedance using a three-element Windkessel model.28 The lateral surface of the fluid domain coincides with a membrane model of the vessel wall in the coupled momentum method for fluid–solid interaction.5 A stabilized semi-discrete finite element method25 is utilized to solve for Eqs. (1)–(4) in this study. m þ m C hcor Z t ek1 ðtsÞ Z1 Z 0 þ k ! ~ vðsÞ ~ ndCds I Chcor Z Chcor k t ek2 ðtsÞ Z2 Z 0 ! k Chcor ~ njChcor vðsÞ ~ ndCds ~ n s ~ þ s jChcor Aek1 t Bek2 t I k Z t Z t k1 ðtsÞ k2 ðtsÞ e Y1 Pim ðsÞds e Y2 Pim ðsÞds I 0 0 h i ~c ~c ð~ v;pÞ þ H ¼~ vjChcor M The Lumped Parameter Model of Downstream Coronary Vascular Beds The lumped parameter model of coronary vascular beds coupled to each coronary outlet surface is based on Chcor B C D EF k k A: Inlet - coupled to lumped parameter heart model G RLA-V L LA-V ELA ELA RLA-V LLA-V RLV-art LLV-art A V a i ELV ( t) k j Rpd Rpp LRV-art RRV-art LRA-V I ð5Þ k k A c b e h gd f RRA-V H ERA B-H: Outlets - coupled to three-element Windkessel model Rp Rd C RLV-art L LV-art ELV (t) V A -Inlet 2.1 Elastance function (mmHg/cm3) m 3197 0 V Time (s) 0 0.952 a-k: Coronary outlets - coupled to lumped parameter coronary vascular model Ra Ra-micro - Rv-mi v cro Cim Cp ERV (t ) Ca Pim V V Rv V FIGURE 1. Problem specification of the inlet, upper branch vessels, the descending thoracic aorta, and coronary outlets for simulations of blood flow in a normal thoracic aorta model with coronary outlets. Note that all the outlets of the three-dimensional computational model feed back in the lumped model at (V). 3198 KIM et al. where P(t) and Q(t) are the pressure and flow at the coronary outlet surface and I is an identity tensor. The coefficients k1, k2, A, B, R, Z1, Y1, Z2, and Y2 are defined using the lumped parameter model of downstream coronary vascular beds as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p1 þ p21 4p0 p2 k1 ¼ 2p2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p1 p21 4p0 p2 k2 ¼ 2p2 " 1 dQ A ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðq2 k1 þ q1 ÞQð0Þ þ q2 ð0Þ dt p21 4p0 p2 # dP þ b1 Pim ð0Þ þ p2 k2 Pð0Þ ð0Þ dt " 1 dQ B ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðq2 k2 þ q1 ÞQð0Þ þ q2 ð0Þ dt p21 4p0 p2 # dP þ b1 Pim ð0Þ þ p2 k1 Pð0Þ ð0Þ dt q2 R¼ p2 Z1 ¼ q2 k21 þ q1 k1 þ q0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p21 4p0 p2 b1 k1 þ b0 Y1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p21 4p0 p2 Z2 ¼ q2 k22 þ q1 k2 þ q0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p21 4p0 p2 b1 k2 þ b0 Y2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p21 4p0 p2 with the following coefficients: p0 ¼ 1 p1 ¼ Ra-micro Ca þ ðRv þ Rv-micro ÞðCa þ Cim Þ p2 ¼ Ca Cim Ra-micro ðRv þ Rv-micro Þ q0 ¼ Ra þ Ra-micro þ Rv þ Rv-micro q1 ¼ Ra Ca ðRa-micro þ Rv þ Rv-micro Þ þ Cim ðRa þ Ra-micro ÞðRv þ Rv-micro Þ q2 ¼ Ca Cim Ra Ra-micro ðRv þ Rv-micro Þ b0 ¼ 0 b1 ¼ Cim ðRv þ Rv-micro Þ Using these operators defined above, we couple the flow and pressure at each coronary outlet surface between the upstream three-dimensional finite element model and the downstream lumped parameter model. The velocity and pressure fields in the three-dimensional domain and its boundaries were solved implicitly. Computation of the Intramyocardial Pressure Depending on the location of the coronary vascular beds, the coronary beds experience different intramyocardial pressure Pim. Coronary flow in the left coronary vascular beds is decreased significantly in systole because the left ventricle operates in a high pressure range. However, the right coronary flow does not decrease in systole because the right ventricle operates in a lower pressure range than the left ventricle.17 Even the maximum right ventricular pressure is one-fifth of the average aortic pressure.17 To accommodate the change in the intramyorcardial pressure depending on the location of the coronary vascular beds, we implemented a closed loop system and computed the left and right ventricular pressure. Previously, we had developed a boundary condition coupling a lumped parameter model representing the left side of the heart to the inlet of the thoracic aorta to compute the left ventricular pressure.11 In this study, another lumped parameter right heart model was coupled to the pulmonary circulation to compute the right ventricular pressure. Depending on the location of the coronary outlet boundaries, we used the left ventricular pressure to represent the intramyocardial pressure of the left coronary arteries and the right ventricular pressure to represent the intramyocardial pressure of the right coronary arteries. The closed loop system implemented in these simulations is plotted in Fig. 1. Choice of the Parameter Values for the Coronary Model The flow distribution to each primary branch of the coronary arteries was calculated on the basis of morphology data and data from the literature.31 We assumed that the mean coronary flow is 4.0% of the cardiac output.17 For each coronary outlet surface, coronary venous resistance was calculated on the basis of the mean flow and assigned venous pressure according to literature data.17 We then computed coronary arterial resistance and coronary arterial microcirculation resistance on the basis of mean flow, mean arterial pressure, and the coronary impedance spectrum using literature data.3,26 The capacitance values were adjusted to give physiologically realistic coronary flow and pressure waveforms using literature data.3 During simulated exercise, the mean flow to the coronary vascular bed was increased to maintain the mean flow at 4.0% of the cardiac output. However, we set a threshold value below which the resistance cannot decrease. The coronary parameter values for each Patient-Specific Coronary Flow Modeling coronary outlet surface were modified by decreasing the total resistances and increasing the capacitances and the ratio of the coronary arterial resistance to the total coronary resistance based on literature data.3 The parameter values of the lumped parameter heart model and three-element Windkessel models were determined to match subject-specific cardiac output and pulse pressure.11 Exercise Simulations In this study, we simulated two different levels of exercise conditions. First, for the simulation with normal coronary arteries, we simulated rest and light exercise conditions. The light exercise condition is similar to a brisk walk. We doubled the cardiac output and heart rate, and the coronary flow was doubled to meet the increased metabolic demands accordingly. However, the downstream coronary vascular networks were not dilated maximally. Second, for the simulations with different degrees of stenosis, we simulated moderate exercise to approximate maximal dilation of downstream coronary vascular networks. This way, we can study how different degrees of stenosis reduce the coronary flow reserve affecting the coronary flow and pressure waveforms. To simulate moderate exercise, we tripled the cardiac output and heart rate and the mean coronary flow was increased by 350%. Simulation Details Blood was approximated as incompressible Newtonian fluid with a density of 1.06 g/cm3 and a dynamic viscosity of 0.04 dynes/cm2 s for all the simulations. The blood vessel walls were modeled with a linearly elastic material. Poisson’s ratio was 0.5 with a wall density of 1.0 g/cm3 and a uniform wall thickness of 0.1 cm. The inlet and the outlet rings were fixed in space and time.5 The mesh generated was anisotropic with local refinement on exterior surfaces and coronary vascular beds with five semi-structured layers within the boundary layer.20 The solutions were run until the relative pressure fields at the inlet and outlets did not change more than 1.0% compared to those in the solutions at the same phase in the previous cardiac cycle. We constructed the geometrical model used in these simulations from cardiac-gated computer tomography data of a 36-year-old healthy male subject. The model started from the root of the aorta, ended above the diaphragm, and included major coronary arteries (left anterior descending, left circumflex, and right coronary arteries) and major branches of them and main upper branch vessels (right subclavian, left subclavian, right vertebral, left vertebral, right carotid, and left carotid 3199 arteries). For the inlet, we coupled the lumped parameter heart model.11 For the coronary outlets, we assigned lumped parameter coronary vascular models.15 For the upper branch vessels and the descending thoracic aorta, we assigned three-element Windkessel models (Fig. 1). The flow distribution used to evaluate the resistance/impedance of each coronary, upper branch, and descending thoracic aortic outlet was set based on literature data.29 Boundary conditions were adjusted to match both the flow distribution and the measured brachial artery pulse pressure.13,22 Additionally, we adjusted the Young’s modulus of the blood vessel walls to match the measured wall deformations of the aorta on the basis of the cardiac-gated computer tomography data. In this study, we modeled the wall deformability with the uniform Young’s modulus of 6.26 9 106 dynes/cm2. We also found tethering areas using this patient’s cardiac-gated computer tomography data and fixed those areas in space and time in the computer model. For the coronary vessels, we identified thin strips approximating the intersection between the coronary arteries and epicardial surface of the heart and fixed the surface of the coronary arteries along the strips. To demonstrate that these methods can be utilized in clinical investigation, we studied blood flow and pressure with different degrees of stenosis. We created a stenosis in the left anterior descending coronary artery by reducing the local diameter of the geometric model by 40%, 50%, 60%, and 75% and simulated a rest condition and a moderate exercise condition, where in the latter case we assumed that the downstream coronary vascular beds were maximally dilated. RESULTS Simulations of Coronary Flow and Pressure of a Normal Thoracic Aortic Model with Epicardial Coronary Arteries at Rest and During Light Exercise In these simulations, we studied coronary flow and pressure of normal coronary arteries for rest and light exercise conditions. Solutions were obtained using a 1,768,953 element (339,664 node) finite element mesh with a time step size of 0.25 ms to simulate a resting condition and 0.125 ms to simulate a light exercise condition. The shape of the velocity profiles of the inlet and of the outlets of the upper branch vessels, and the descending thoracic aorta was constrained to an axisymmetric shape with a profile order of nine.10 To simulate light exercise, the resistance value of the descending thoracic aorta was decreased in order to increase flow to the lower extremities. The cardiac cycle was shortened to simulate increased heart rate until the systolic pressure of the thoracic aorta 3200 KIM et al. increased by 20% compared to the resting state.2 The boundary conditions of the upper branch vessels were unchanged. The parameter values of the closed loop system are shown in Table 1, and the parameter values of the Windkessel models are shown in Table 2. The contractility of this subject was not changed for the light exercise simulation. Table 3 shows the parameter values of the lumped parameter coronary vascular models assigned to each coronary outlet for rest and light exercise conditions. Figure 2 shows computed pressure and flow waveforms of the inlet and the outlets for rest and light exercise conditions of the normal thoracic aorta. The pressure–volume loops of the left and right ventricles for both conditions are also shown. The cardiac cycle decreased from 1.0 to 0.5 s, the cardiac output increased from 5 to 10.3 L/min, and systolic blood pressure increased from 124 to 149 mmHg. However, the stroke volume increased little from 83 to 86 cc signifying the increase in the cardiac output was mainly due to the shortening of the cardiac cycle, not due to the increase of the stroke volume. Figure 2 also shows that the upper branch vessels experience retrograde flow in diastole. Retrograde flow to the upper branch vessels is severe in the light exercise condition, even though the same boundary conditions were assigned to the upper branch vessels, likely due to the increased flow demand of the descending thoracic aorta. Figure 2 also shows the pressure waveforms of the upper branch vessels and the descending thoracic aorta. The pressure waveform of the descending thoracic aorta decays faster during exercise than in the resting condition. In Fig. 3, coronary flow and pressure waveforms of the left anterior descending, left circumflex, and right coronary arteries are plotted for rest and exercise TABLE 1. Parameter values of the closed loop system at rest (R), during light exercise (E1), and during moderate exercise (E2) for the simulations of thoracic aorta with coronary arteries. RLA-V (dynes s/cm5) LLA-V (dynes s2/cm5) RLV-art(dynes s/cm5) LLV-art (dynes s2/cm5) ELV,max (mmHg/cc) VLV, 0 (cc) VLA, 0 (cc) ELA (mmHg/cc) tmax (s) Rpp (dynes s/cm5) Rpd (dynes s/cm5) R E1 E2 5 5 10 0.69 2.0 0 260 270 0.33 16 144 5 5 10 0.69 2.0 0 260 350 0.25 16 144 5 5 10 0.69 2.2 0 260 360 0.15 2 38 RRA-V (dynes s/cm5) LRA-V (dynes s2/cm5) RRV-art (dynes s/cm5) LRV-art (dynes s2/cm5) ERV,max (mmHg/cc) VRV, 0 (cc) VRA, 0 (cc) ERA (mmHg/cc) Cardiac cycle (s) Cp (cm5/dynes) R E1 E2 5 1 10 0.55 0.5 0 260 60 1.0 0.022 5 1 10 0.55 0.5 0 260 80 0.5 0.022 5 1 10 0.55 0.5 0 260 80 0.3 0.02 TABLE 2. Parameter values of the three-element Windkessel models at rest (R), during light exercise (E1), and during moderate exercise (E2) for the simulations of thoracic aorta with coronary arteries. Rp (103 dynes s/cm5) C (1026cm5/dynes) Rd (103 s/cm5) Rp (103dynes s/cm5) C (1026 cm5/dynes) Rd (103 dynes s/cm5) 3 5 Rp (10 dynes s/cm ) C (1026 cm5/dynes) Rd (103 dynes s/cm5) B-R B-El B-E2 C-R C-El C-E2 D-R D-El D-E2 1.49 235 15.1 1.49 235 15.1 0.20 31.8 20.0 1.41 248 14.3 1.41 248 14.3 0.19 33.5 19.0 10.7 32.9 108 10.7 32.9 108 1.44 4.45 143 E-R E-El E-E2 F-R F-El F-E2 G-R G-El G-E2 1.75 201 17.6 1.75 201 17.6 0.24 27.1 23.4 7.96 44.0 80.5 7.96 44.0 80.5 1.08 5.95 107 1.80 195 18.2 1.08 195 18.2 0.24 26.3 24.1 H-R H-El H-E2 0.227 1540 2.29 0.225 1600 0.676 0.08 1200 0.45 B denotes right subclavian artery, C denotes right carotid artery, D denotes right vertebral artery, E denotes left carotid artery, F denotes left vertebral artery, G denotes left subclavian artery, and H denotes descending thoracic aorta. Note that the parameter values of the upper branch vessels are the same for the light exercise and resting conditions and change for the moderate exercise condition. Patient-Specific Coronary Flow Modeling 3201 TABLE 3. Parameter values of the lumped parameter models of the coronary vascular beds at rest (R), during light exercise (E1), and during moderate exercise (E2) for the simulations of thoracic aorta with coronary arteries. Ra a: LAD1 b: LAD2 c: LAD3 d: LAD4 e: LCX1 f: LCX2 g: LCX3 h: LCX4 i: RCA1 j: RCA2 k: RCA3 c*: LAD3 c**: LAD3 c***: LAD3 c****: LAD3 d*: LAD4 d**: LAD4 d***: LAD4 d****: LAD4 Ra-micro Rv + Rv-micro Ca cim R E1 E2 R E1 E2 R E1 E2 R E1 E2 R E1 E2 183 131 123 75 49 160 216 170 168 236 266 100 89 80 57 65 54 48 35 177 126 91 55 47 154 208 164 163 229 257 76 52 51 31 20 65 87 68 71 98 110 51 51 51 51 31 31 31 31 299 214 148 90 80 261 353 277 274 385 435 146 146 98 93 89 89 58 57 56 40 39 24 15 49 66 52 51 72 81 24 16 16 10 6.2 20 27 21 22 31 35 16 16 16 16 10 10 10 10 94 67 65 40 25 82 111 87 86 121 136 65 65 65 65 40 40 40 40 44 32 31 19 12 39 53 41 40 56 64 18 13 12 7 5 15 21 16 16 23 25 12 12 12 12 7 7 7 7 0.34 0.48 0.49 0.80 1.28 0.39 0.29 0.37 0.37 0.26 0.23 0.49 0.49 0.49 0.49 0.80 0.80 0.80 0.80 0.75 1.02 1.07 1.74 2.79 0.85 0.63 0.80 0.83 0.59 0.52 0.75 1.02 1.07 1.74 2.79 0.85 0.63 0.80 0.83 0.59 0.52 1.07 1.07 1.07 1.07 1.74 1.74 1.74 1.74 2.89 4.04 4.16 6.82 10.8 3.31 2.45 3.12 3.15 2.24 1.99 4.16 4.16 4.16 4.16 6.82 6.82 6.82 6.82 6.88 9.34 9.74 15.9 25.4 7.78 5.74 7.29 7.60 5.38 4.72 6.88 9.34 9.74 15.9 25.4 7.78 5.74 7.29 7.60 5.38 4.72 9.74 9.74 9.74 9.74 15.9 15.9 15.9 15.9 c* denotes LAD3 with 40% diameter reduction, c** denotes LAD3 with 50% diameter reduction, c*** denotes LAD3 with 60% diameter reduction, and c**** denotes LAD3 with 75% diameter reduction. Similarly, d* denotes LAD4 with 40% diameter reduction, d** denotes LAD4 with 50% diameter reduction, d*** denotes LAD4 with 60% diameter reduction, and d**** denotes LAD4 with 75% diameter reduction. Note that the unit of the resistance values are in 103 dynes s/cm5 and the unit of the capacitance values are in 1026 cm5/dynes. conditions. For the left anterior descending and circumflex coronary arteries, coronary flow is high in diastole and low in systole because the intramyocardial pressure approximated by the left ventricular pressure is elevated in systole. On the contrary, the right coronary artery has high flow in systole and low flow in diastole because the intramyocardial pressure caused by the right ventricular pressure operates in a low pressure range. By coupling the lumped parameter heart models and the lumped parameter coronary vascular models to a three-dimensional finite element model of the aorta, we were able to obtain realistic coronary flow and pressure waveforms17 because we considered the effects of the contraction and relaxation of the left and right ventricles. The coronary pressure waveforms are similar to the aortic pressure waveform unlike the coronary flow waveforms, which are dependent on the location of the myocardium the coronary arteries feed. During light exercise, the coronary flow doubled to meet the demands of the heart. For the resting condition, the mean coronary flow to the left anterior descending coronary artery was 1.3 cc/s, the mean flow to the left circumflex coronary artery was 1.5 cc/s, and the mean flow to the right coronary artery was 0.6 cc/s, to yield a total coronary flow of 3.4 cc/s. During light exercise, the coronary flow doubled achieving a mean flow of 2.8 cc/s to the left anterior descending coronary artery, 3.1 cc/s to the left circumflex coronary artery, and 1.1 cc/s to the right coronary artery, and a total coronary flow of 7.0 cc/s. Wall shear stress of the coronary arteries for the resting condition and the light exercise condition are also plotted for peak-systole, peak left coronary flow, and late diastole in Fig. 4. For the light exercise condition, the wall shear stress increased as higher volume of flow traveled to the coronary arteries from the left ventricle. The left coronary arteries experience higher wall shear stress fields in late systole and late diastole whereas the right coronary arteries experience higher wall shear stress fields in systole due to the asynchrony of the maximum flow in the left and right coronary arteries. Simulations of Coronary Flow and Pressure with Different Degrees of Stenosis at the Left Anterior Descending Coronary Artery In these simulations, we studied how the degree of a stenosis in the left anterior descending coronary artery affects coronary flow and pressure. For the normal case with no stenosis in the left anterior descending coronary artery, solutions were obtained using a 1,768,953 element and a 339,664 node mesh with a time step size of 0.25 ms to simulate a resting condition and 0.1 ms to simulate a moderate exercise condition. The shape of the velocity profiles at the inlet and at the 3202 KIM et al. Pressure-volume loops D-Left vertebral 150 E-Right carotid 138 Pressure (mmHg) Left ventricle (rest) Right ventricle (rest) Left ventricle (exercise) Right ventricle (exercise) Pressure (mmHg) Pressure (mmHg) 138 0 68 Time (s) 1 68 0 D-Left vertebral Flow rate (cc/s) C D E F G 0 1 39 B -1.5 Time (s) E-Right carotid Rest Exercise 7.5 0 150 Volume (cc) Flow rate (cc/s) 0 1 -7 0 Time (s) 1 Time (s) G-Right subclavian C-Left carotid 138 Pressure (mmHg) Pressure (mmHg) 138 68 68 0 Time (s) 0 1 A C-Left carotid Time (s) 1 G-Right subclavian 37 Flow rate (cc/s) Flow rate (cc/s) 31 0 1 -7 Time (s) H H-Descending thoracic aorta 138 138 Pressure (mmHg) Rest Exercise Pressure (mmHg) Pressure (mmHg) 138 68 68 Time (s) 68 0 1 B-Left subclavian 1 Time (s) 0 A-Aortic inlet 30 1 400 Rest Exercise Flow rate (cc/s) Flow rate (cc/s) 1 Time (s) Time (s) H-Descending thoracic aorta 580 -5 0 1 Time (s) A-Aortic inlet B-Left subclavian 0 0 Flow rate (cc/s) -6 -20 -40 0 Time (s) 1 0 Time (s) 1 FIGURE 2. Pressure–volume loops of the left and right ventricles and flow and pressure waveforms of the descending thoracic aorta and upper branch vessels for resting and light exercise cases with normal coronary anatomy. Patient-Specific Coronary Flow Modeling Pressure 3203 Flow Rest a-d: Left anterior descending coronary a-d: Left anterior descending coronary Exercise 4.5 Flow rate (cc/s) Pressure (mmHg) 138 68 0 0 Time (s) 0 1 Time (s) 1 e-h: Left circumflex coronary e-h: Left circumflex coronary 138 Flow rate (cc/s) Pressure (mmHg) 5 a i 68 0 Time (s) 1 0 0 c k i-k: Right coronary 138 b Time (s) 1 Right coronary j h gd 68 f e Flow rate (cc/s) Pressure (mmHg) 2 0 0 Time (s) 1 0 Time (s) 1 FIGURE 3. Flow and pressure waveforms of coronary arteries at rest and during light exercise case. outlets of the upper branch vessels and the descending thoracic aorta were constrained to an axisymmetric shape with a profile order of nine.10 For a stenosis case with 40% diameter reduction in the left anterior descending coronary artery, solutions were obtained using a 1,813,097 element and a 350,123 node mesh with the same time step size as in the normal case. For a stenosis case with 50% diameter reduction in the left anterior descending coronary artery, solutions were obtained using a 2,576,721 element and a 475,723 node mesh with the same time step size as in the normal case. For a stenosis case with 60% diameter reduction in the left anterior descending coronary artery, solutions were obtained using a 3,146,766 element and a 575,122 node mesh with the same time step size as in the normal case. Finally, for a stenosis case with 75% diameter reduction in the left anterior descending coronary artery, solutions were obtained using a 3,836,663 element and a 722,518 node mesh with the same time step size as in the normal case. The increase in the mesh size was due to the additional refinements in the area of stenosis. For resting conditions, we decreased the resistance of the coronary model downstream of the stenosed artery until we obtained the same mean flow to these outlets as with the normal case. We recorded the pressure loss through these stenosed regions in Table 4. To dilate the downstream coronary vascular networks to maximum, we simulated moderate exercise by decreasing the resistance value of the descending thoracic aorta and shortened the cardiac cycle to increase cardiac output threefold compared to the resting state. In the exercise simulations, we assumed that the downstream coronary vasculatures were maximally dilated and assigned the same boundary conditions with the normal case to all the stenosis cases. The flow difference downstream of the stenosed artery was recorded in Table 4. Table 3 shows the parameter values of the lumped parameter coronary vascular models assigned to each coronary outlet for rest and moderate exercise conditions for the normal case and all the stenosis cases. At rest, the downstream coronary boundary conditions were modified until the same mean flow was achieved through the stenosed artery. However, as the same mean flow traveled through the stenosed artery, a pressure loss occurred due to the viscous loss through the stenosis and complex flow structures distal to the 3204 KIM et al. Inlet flow rate (cc/s) 580 a Rest Exercise A -20 b c B C Time (s) 0 1 Rest A B C Exercise a b c Wall shear stress (dynes/cm2 ) 0 7.5 15 22.5 30 FIGURE 4. Wall shear stress of coronary arteries for peak systole, peak left coronary flow rate, and mid-diastole at rest and during light exercise. TABLE 4. Mean left anterior descending coronary artery flow and pressure at rest and during moderate exercise for the simulations of coronary flow and pressure with different degrees of stenosis in the left anterior descending coronary artery. Mean coronary flow (cc/s) Mean pressure (mmHg) Mean coronary flow and pressure at rest Normal 1.1 94 40% Diameter reduction 1.1 93 50% Diameter reduction 1.1 92 60% Diameter reduction 1.1 86 75% Diameter reduction 1.1 78 Mean coronary flow and pressure at moderate exercise Normal 3.7 85 40% Diameter reduction 3.4 77 50% Diameter reduction 3.2 73 60% Diameter reduction 3.0 67 75% Diameter reduction 1.8 36 stenosed region as shown in Table 4. During exercise, the same coronary boundary conditions were assigned to the coronary outlets. A noticeable pressure loss and flow rate decrease occurred in the case of the stenosis with 75% diameter reduction. The boundary conditions of the upper branch vessels were changed to obtain the same mean flow rate as in the resting state. The parameter values of the closed loop system is shown in Table 1 and the parameter values of the Windkessel models are shown in Table 2. The contractility of this subject was increased 10% for this moderate exercise simulation.2 Figure 5 depicts flow and pressure waveforms of the left anterior descending coronary artery for rest and moderate exercise conditions. Note that we plotted the flow and pressure waveforms of the 50% and 75% diameter reduction cases only. We see that the Patient-Specific Coronary Flow Modeling 50% diameter reduction Normal 75% diameter reduction Left anterior descending coronary flow at rest Left anterior descending coronary pressure at rest 3 Normal 50% diameter reduction 75% diameter reduction Pressure (mmHg) Flow rate (cc/s) 130 Normal 50% diameter reduction 75% diameter reduction 0 3205 60 0 0 1 Time (s) Left anterior descending coronary flow at exercise 11 Left anterior descending coronary pressure at exercise 130 Normal 50% diameter reduction 75% diameter reduction Pressure (mmHg) Normal 50% diameter reduction 75% diameter reduction Flow rate (cc/s) 1 Time (s) 0 0 0 0.3 Time (s) 0 Time (s) 0.3 FIGURE 5. Effect of stenosis on coronary artery flow and pressure at rest and during moderate exercise. The pressure was computed downstream of the stenosis. Note that only 50% and 75% diameter reduction cases are plotted. to the stenosed regions and they propagate longer for more severe stenosis. Note that we selected 50% and 75% diameter reduction cases only. Left anterior descending coronary Flow rate (cc/s) 5 4 3 2 DISCUSSION 1 0 0 20 40 60 80 100 Diameter reduction (%) Rest Exercise Rest (Gould et al) Hyperemia (Gould et al) FIGURE 6. Mean flow rate of the left anterior descending coronary artery vs. degree of stenosis for rest and moderate exercise. coronary flow tripled and the pressure pulse also became larger for the exercise condition. The pressure loss increased as more flow traveled through the stenosed regions in the cases of 50% and 75% diameter reduction. Figure 6 shows the mean flow in the stenosed artery in both resting and moderate exercise conditions. We also plotted the experimental results of Gould et al. for comparison.7 Figure 7 shows volume rendered velocity magnitudes of the coronary arteries during moderate exercise. Complex flow structures are observed distal We have successfully developed and implemented a coronary boundary condition that couples a lumped parameter coronary vascular model to each coronary outlet of a three-dimensional finite element model of the aorta and epicardial coronary arteries. We also used an inflow boundary condition coupling a lumped parameter heart model and a closed loop model to represent the intramyocardial pressure by considering the interactions between the heart and arterial system. Fluid– structure interaction simulations were performed to better represent flow and pressure waveforms. Additionally, we were able to obtain robust and stable solutions by constraining the shape of the velocity profiles of the boundaries that experienced retrograde flow. Using the lumped parameter coronary vascular model along with the inflow boundary condition that couples the lumped parameter heart model and the closed loop system, we studied how changes in cardiac and arterial properties affect coronary flow and 3206 KIM et al. Inflow waveform Flow rate (cc/s) 1000 Velocity magnitude (cm/s) A 0 15 30 45 60 B -50 0 0.3 Time (s) A B Normal 50% diameter reduction 75% diameter reduction FIGURE 7. Volume rendered velocity magnitudes of coronary arteries with different degrees of stenosis during moderate exercise. Note that only 50% and 75% diameter reduction cases are plotted. pressure. We examined how coronary flow and pressure change for resting and light exercise conditions for normal coronary anatomy. We then created stenoses in the left anterior descending coronary artery to investigate how the coronary flow and pressure change for different degrees of stenoses at rest and moderate exercise condition when the downstream coronary vascular beds are maximally dilated. For the simulations at rest and during light exercise, the computed coronary flow and pressure waveforms Patient-Specific Coronary Flow Modeling were realistic and the asynchrony of the left and right coronary arteries were represented as we approximated the intramyocardial pressure with the left and right ventricular pressure depending on the location of the coronary arteries. For the simulations with different degrees of stenosis in the left anterior descending coronary artery at rest and moderate exercise, the reduction in the coronary flow was more apparent for the moderate exercise case because more energy loss occurs as more coronary flow travels through the stenosed region. We also demonstrated a greater pressure loss across the stenosed region as the degree of stenosis increases. Especially, the pressure loss did not increase linearly resulting from the energy dissipated due to the viscous loss and the turbulence in the stenosed region. Figure 6 showed that the relationship between the mean coronary flow and the degree of stenosis is comparable to results obtained from experimental techniques.7 Gould et al.7 created temporary stenosis in the left circumflex artery of canines and injected Hypaque to simulate hyperemic conditions in canine coronary arteries. They observed that the mean flow did not decrease up to 85% diameter reduction at rest and 30–45% diameter reduction at hyperemia. Although our study focused on different physiologic conditions of moderate exercise rather than hyperemia, our results with resting and moderate exercise conditions yield similar behaviors because in both maximal hyperemia and simulated moderate exercise conditions, we assume that the downstream coronary vascular beds are maximally dilated to receive more flow from the aorta. We also observed that the flow remained the same up to 75% diameter reduction case at rest. In Gould et al.’s plot, the mean flow at rest slightly increased for mild stenosis cases but in our study, we aimed to have the same mean flow for mild stenosis cases, thus, we maintained the same mean flow at rest up to 75% diameter reduction case. The slope during moderate exercise is comparable to the slope in hyperemia although there is a difference in the magnitude of the mean flow. In Gould et al.’s paper, the mean flow at hyperemic state increased 409% compared to the baseline value whereas in our study, we designed that the mean flow increased 350% at moderate exercise compared to the baseline value. Our method has three primary limitations. First, we did not consider the motion of the heart during the cardiac cycle. We fixed the coronary arteries to the epicardial surface of the heart and fixed the surface in space and time. In reality, the heart moves significantly to contract and relax during the cardiac cycle. This movement is large and cannot be modeled using the coupled momentum method,5 which is a linearized approach using a fixed fluid mesh. A different approach, such as an arbitrary Lagrangian–Eulerian 3207 formulation, would be needed to represent the movement of the heart over the cardiac cycle. However, previous studies showed that the effects due to the movement of the heart were secondary and did not affect the pressure and flow fields as much as the geometry and the boundary conditions.18,19,21,30 Second, we assumed that the left coronary arteries transport blood to the left ventricle and the right coronary arteries transport blood to the right ventricle. In reality, however, the coronary vascular networks experience nonuniform intramyocardial pressure depending on the location of the coronary networks. To consider nonuniform intramyocardial pressure, a three-dimensional nonuniform model of the heart as well as the mapping of the coronary arteries to the location of the myocardium that the arteries perfuse should be considered. For example, a three-dimensional model of cardiac ventricular mechanics can be used to compute nonuniform intramyocardial pressure acting on the coronary vascular beds.8,9 Third, we assumed a uniform Young’s modulus for the whole computational model although we know that the vessel wall properties vary spatially. To consider nonuniform vessel wall properties, noninvasive methods of estimating wall thickness and elastic (viscoelastic) wall properties should be developed. In this study, rather than considering nonuniform wall properties, we adjusted the vessel wall properties so that the wall deformation of the descending thoracic aorta fit the experimental data reasonably well. CONCLUSIONS A coronary boundary condition that couples a lumped parameter coronary vascular model to each coronary outlet of a three-dimensional finite element model of the aorta and epicardial coronary arteries is developed. An inflow boundary condition coupling a lumped parameter heart model and a closed loop model is implemented to represent the intramyocardial pressure by considering the interactions between the heart and arterial system. We considered fluid– structure interaction to better represent flow and pressure waveforms. Using the lumped parameter coronary vascular model along with the inflow boundary condition that couples the lumped parameter heart model and the closed loop system, we can predict coronary flow and pressure realistically using anatomic data obtained from medical imaging techniques and study how changes in cardiac and arterial properties affect coronary flow and pressure and vice versa. Further, we can utilize these methods to investigate different vascular interventions of cardiovascular disease. KIM et al. 3208 ACKNOWLEDGMENTS Hyun Jin Kim was supported by a Stanford Graduate Fellowship. This material is based upon work supported by the National Science Foundation under Grant No. 0205741. The authors gratefully acknowledge the assistance of Dr. Nathan M. Wilson for assistance with software development. The authors gratefully acknowledge Dr. Farzin Shakib for the use of his linear algebra package AcuSolveTM (http://www. acusim.com) and the support of Simmetrix, Inc. for the use of the MeshSimTM (http://www.simmetrix.com) mesh generator. REFERENCES 1 Berry, J. L., A. Santamarina, J. E. Moore, S. Roychowdhury, and W. D. Routh. Experimental and computational flow evaluation of coronary stents. Ann. Biomed. Eng. 28(4):386–398, 2000. 2 Brooks, G. A., T. D. Fahey, T. P. White, and K. M. Baldwin. Exercise Physiology Human Bioenergetics and Its Applications. Berkshire, UK: McGraw-Hill Companies, 2004. 3 Burattini, R., P. Sipkema, G. van Huis, and N. Westerhof. Identification of canine coronary resistance and intramyocardial compliance on the basis of the waterfall model. Ann. Biomed. Eng. 13(5):385–404, 1985. 4 Cebral, J. R., M. A. Castro, J. E. Burgess, R. S. Pergolizzi, M. J. Sheridan, and C. M. Putman. Characterization of cerebral aneurysms for assessing risk of rupture by using patient-specific computational hemodynamics models. Am. J. Neuroradiol. 26(10):2550–2559, 2005. 5 Figueroa, C. A., I. E. Vignon-Clementel, K. E. Jansen, T. J. R. Hughes, and C. A. Taylor. A coupled momentum method for modeling blood flow in three-dimensional deformable arteries. Comput. Methods Appl. Mech. Eng. 195(41–43):5685–5706, 2006. 6 Gijsen, F. J. H., J. J. Wentzel, A. Thury, F. Mastik, J. A. Schaar, J. C. H. Schuurbiers, C. J. Slager, W. J. van der Giessen, P. J. de Feyter, A. F. W. van der Steen, and P. W. Serruys. Strain distribution over plaques in human coronary arteries relates to shear stress. Am. J. Physiol. Heart Circ. Physiol. 295(4):H1608–1614, 2008. 7 Gould, K. L., K. Lipscomb, and G. W. Hamilton. Physiologic basis for assessing critical coronary stenosis. Instantaneous flow response and regional distribution during coronary hyperemia as measures of coronary flow reserve. Am. J. Cardiol. 33(1):87–94, 1974. 8 Hunter, P. J., A. J. Pullan, and B. H. Smaill. Modeling total heart function. Annu. Rev. Biomed. Eng. 5(1):147–177, 2003. 9 Kerckhoffs, R. C. P., M. L. Neal, Q. Gu, J. B. Bassingthwaighte, J. H. Omens, and A. D. McCulloch. Coupling of a 3D finite element model of cardiac ventricular mechanics to lumped systems models of the systemic and pulmonic circulation. Ann. Biomed. Eng. 35(1):1–18, 2007. 10 Kim, H. J., C. A. Figueroa, T. J. R. Hughes, K. E. Jansen, and C. A. Taylor. Augmented Lagrangian method for constraining the shape of velocity profiles at outlet boundaries for three-dimensional finite element simulations of blood flow. Comput. Methods Appl. Mech. Eng. 198(45– 46):3551–3566, 2009. 11 Kim, H. J., I. E. Vignon-Clementel, C. A. Figueroa, J. F. LaDisa, K. E. Jansen, J. A. Feinstein, and C. A. Taylor. On coupling a lumped parameter heart model and a threedimensional finite element aorta model. Ann. Biomed. Eng. 37(11):2153–2169, 2009. 12 Lagana, K., R. Balossino, F. Migliavacca, G. Pennati, E. L. Bove, M. R. de Leval, and G. Dubini. Multiscale modeling of the cardiovascular system: application to the study of pulmonary and coronary perfusions in the univentricular circulation. J. Biomech. 38(5):1129–41, 2005. 13 Laskey, W. K., H. G. Parker, V. A. Ferrari, W. G. Kussmaul, and A. Noordergraaf. Estimation of total systemic arterial compliance in humans. J. Appl. Physiol. 69(1):112–119, 1990. 14 Li, Z., and C. Kleinstreuer. Blood flow and structure interactions in a stented abdominal aortic aneurysm model. Med. Eng. Phys. 27(5):369–382, 2005. 15 Mantero, S., R. Pietrabissa, and R. Fumero. The coronary bed and its role in the cardiovascular system: a review and an introductory single-branch model. J. Biomed. Eng. 14:109–115, 1992. 16 Migliavacca, F., R. Balossino, G. Pennati, G. Dubini, T. Y. Hsia, M. R. de Leval, and E. L. Bove. Multiscale modelling in biofluidynamics: application to reconstructive paediatric cardiac surgery. J. Biomech. 39(6):1010–1020, 2006. 17 Opie, L. H. Heart Physiology: From Cell to Circulation. Philadelphia, PA, USA: Lippincott Williams and Wilkins, 2003. 18 Qiu, Y., and J. M. Tarbell. Numerical simulation of pulsatile flow in a compliant curved tube model of a coronary artery. J. Biomech. Eng. 122(1):77–85, 2000. 19 Ramaswamy, S. D., S. C. Vigmostad, A. Wahle, Y. G. Lai, M. E. Olszewski, K. C. Braddy, T. M. H. Brennan, J. D. Rossen, M. Sonka, and K. B. Chandran. Fluid dynamic analysis in a human left anterior descending coronary artery with arterial motion. Ann. Biomed. Eng. 32(12):1628–1641, 2004. 20 Sahni, O., J. Muller, K. E. Jansen, M. S. Shephard, and C. A. Taylor. Efficient anisotropic adaptive discretization of the cardiovascular system. Comput. Methods Appl. Mech. Eng. 195(41–43):5634–5655, 2006. 21 Santamarina, A., E. Weydahl, Jr. J. M. Siegel, and J. E. Moore, Jr. Computational analysis of flow in a curved tube model of the coronary arteries: effects of time-varying curvature. Ann. Biomed. Eng. 26:944–954, 1998. 22 Stergiopulos, N., P. Segers, and N. Westerhof. Use of pulse pressure method for estimating total arterial compliance in vivo. Am. J. Physiol. Heart Circ. Physiol. 276(2):H424– H428, 1999. 23 Taylor, C. A., M. T. Draney, J. P. Ku, D. Parker, B. N. Steele, K. Wang, and C. K. Zarins. Predictive medicine: computational techniques in therapeutic decision-making. Comput. Aided Surg. 4(5):231–247, 1999. 24 Taylor, C. A., and C. A. Figueroa. Patient-specific model of cardiovascular mechanics. Annu. Rev. Biomed. Eng. 11: 109–134, 2009. 25 Taylor, C. A., T. J. R. Hughes, and C. K. Zarins. Finite element modeling of blood flow in arteries. Comput. Methods Appl. Mech. Eng. 158(1–2):155–196, 1998. 26 Van Huis, G. A., P. Sipkema, and N. Westerhof. Coronary input impedance during cardiac cycle as determined by impulse response method. Am. J. Physiol. Heart Circ. Physiol. 253(2):H317–H324, 1987. Patient-Specific Coronary Flow Modeling 27 Vignon-Clementel, I.E., C.A. Figueroa, K.E. Jansen, and C.A. Taylor. Outflow boundary conditions for threedimensional finite element modeling of blood flow and pressure in arteries. Comput. Methods Appl. Mech. Eng. 195(29–32):3776–3796, 2006. 28 Vignon-Clementel, I. E., C. A. Figueroa, K. E. Jansen, and C. A. Taylor. Outflow boundary conditions for three-dimensional simulations of non-periodic blood flow and pressure fields in deformable arteries. Comput. Methods Biomech. Biomed. Eng., 2008. doi: 10.1080/10255840903413565. 29 3209 Zamir, M., P. Sinclair, and T.H. Wonnacott. Relation between diameter and flow in major branches of the arch of the aorta. J. Biomech. 25(11):1303–1310, 1992. 30 Zeng, D., E. Boutsianis, M. Ammann, K. Boomsma, S. Wildermuth, and D. Poulikakos. A study on the compliance of a right coronary artery and its impact on wall shear stress. J. Biomech. Eng. 130(4):041014, 2008. 31 Zhou, Y., G. S. Kassab, and S. Molloi. On the design of the coronary arterial tree: a generalization of Murray’s law. Phys. Med. Biol. 44:2929–2945, 1999.