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Pulsatile blood flow in the entire coronary arterial tree: theory and experiment

Department of Biomedical Engineering, University of California, Irvine, California

Submitted 23 February 2006 ; accepted in final form 7 April 2006

	ABSTRACT

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The pulsatility of coronary circulation can be accurately simulated on the basis of the measured branching pattern, vascular geometry, and material properties of the coronary vasculature. A Womersley-type mathematical model is developed to analyze pulsatile blood flow in diastole in the absence of vessel tone in the entire coronary arterial tree on the basis of previously measured morphometric data. The model incorporates a constitutive equation of pressure and cross-section area relation based on our previous experimental data. The formulation enables the prediction of the impedance, the pressure distribution, and the pulsatile flow distribution throughout the entire coronary arterial tree. The model is validated by experimental measurements in six diastolic arrested, vasodilated porcine hearts. The agreement between theory and experiment is excellent. Furthermore, the present pulse wave results at low frequency agree very well with previously published steady-state model. Finally, the phase angle of flow is seen to decrease along the trunk of the major coronary artery and primary branches toward the capillary vessels. This study represents the first, most extensive validated analysis of Womersley-type pulse wave transmission in the entire coronary arterial tree down to the first segment of capillaries. The present model will serve to quantitatively test various hypotheses in the coronary circulation under pulsatile flow conditions.

pulse wave transmission; Womersley's method; impedance; admittance; Fourier transform

The theory of pulsatile blood flow was initiated by Leonhard Euler in 1775 and Thomas Young in 1808. Later on, numerous attempts have been made at the study of pulsatile blood flow in the vascular system, e.g., Zamir's Womersley-type analysis (5, 6, 33, 34), and Hughes' one-dimensional theory (15). Basically, there are three major approaches to simulate the pulsatile blood flow and pressure waves in the vascular system: 1) the Womersley's solution of pulsatile blood flow (1, 6, 9–11, 14, 31, 32), 2) the one-dimensional wave propagation solution (7, 15, 24, 25, 27, 29), and 3) the three-dimensional solution of pulsatile blood flow (7). The latter two methods require appropriate outflow boundary conditions to simulate the pulsatile blood flow wave. Different outflow boundary conditions have been developed to satisfy the one- or three-dimensional equations, such as the pure resistive load boundary conditions (28), the windkessel model (16, 21), the nonreflecting outlet boundary conditions (7), or the impedance boundary conditions (24, 25, 29). All of the boundary conditions may be classified as lumped-parameter model boundary conditions. Despite the usefulness of lumped models, however, they are generally limited to the global aspects of coronary blood flow and fail to reveal the flow and pressure wave distribution through the vascular system. For example, lumped models cannot be used to predict the significant spatial distribution of coronary blood flow. Womersley's solution (1, 6, 9–11, 14, 31, 32) avoids those shortcomings because it assumes that pressure and flow are decoupled. Hence, this approximation can be applied throughout a complex vascular network such as the entire coronary arterial tree.

Here, we shall use the platform of the entire coronary arterial tree (22) to carry out a Womersley-type numerical analysis of pulse wave transmission. The constitutive equation that relates pressure to cross-sectional area for the several largest orders is based on experimental data (19). The Womersley model is modified to account for the measured constitutive relation of coronary arteries. Previous measurements of coronary wall thickness (13) are also incorporated into the numerical model. The predictions of the mathematical model are compared with the experimental measurements on diastolically arrested, vasodilated porcine hearts. The agreement between theory and experiment is excellent. The pulse wave transmission is analyzed along spatial pathways stemming from the trunk through various branches to the first vessel segments of capillaries. The limitations and implications of theory and experiment are contemplated.

	Glossary

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a,b

Arbitrary constants of integration in the equations for flow and pressure

A

Area of a vessel (cm²)

A(n)

Initial area of a vessel (cm²)

c,c₀

Wave velocity with/without the viscosity effect c = ·c₀ (c₀ = ) (cm/s)

D

Diameter of a vessel (cm)

D(n)

Initial diameter of a given vessel (cm)

E(E_stat)

Young's modulus (static) (dyn/cm²)

E_dyn

Young's modulus (dynamic) (dyn/cm²)

f

Wave frequency (Hz)

F₁₀

F₁₀() = [2J₁(i^3/2)]/[i^3/2J₀(i^3/2)], where J₀ and J₁ are Bessel function of the first kind, and order zero and one, respectively, and = R(n) is the Womersley number

h[h(n)]

Wall thickness of a vessel n (cm)

L(n)

Initial length of a vessel (cm)

n

Vessel segment number

p₀

External pressure, assumed zero in diastole (mmHg)

P(x,)

Pressure along the axial length x of the vessel (mmHg)

Q(x,)

Volumetric flow rate along the axial length of the vessel (ml/min)

R

Radius of a vessel (cm)

R(n)

R(n) = D(n)/2; Initial radius of a vessel (cm)

t

Time (s)

u_r

Radial velocity (cm/s)

u_x

Axial velocity (cm/s)

x

Axial position along the length of a vessel (cm)

Y₀

Y₀ = A(n)/c₀; Characteristic admittance (cm⁴·s/g)

Y(x,)

Y(x,) = 1/[Z(x,)]; Admittance along the length of a vessel (cm⁴·s/g)

Z₀

Z₀ = 1Y0 Characteristic impedance (g·s^–1·cm^–4)

Z(x,)

Z(x,) = [P(x,)]/[Q(x,)]; Impedance along the length of a vessel (g·s^–1·cm^–4)

= R(n); Womersley number

Blood density (g/cm³)

Mean circumferential stress in a vessel wall (dyn/cm²)

Angular frequency (rad/s)

µ

Blood viscosity (cp)

	MATHEMATICAL MODEL

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Governing equations. The details of mathematical derivations are shown in the APPENDIX. Briefly, the governing equations for flow and pressure may be written as

(1)

(2)

where a and b are arbitrary constants of integration, c = 1 – F₁₀()·c₀ (c₀ = ) is the wave velocity, Y₀ = A(n)/c₀ is the characteristic admittance, Z₀ = 1/Y₀ is the characteristic impedance, Y₁ = Y₀, and Z₁ = Z₀. Following the transmission line method (TLM) (3), we define the impedance and admittance as:

(3)

(4)

In a given vessel segment, at x = 0 and x = L, we have the following inlet and outlet impedance:

(5)

and

(6)

If we combine Eqs. 5 and 6, we obtain:

(7)

Equation 7 was used to calculate the impedance/admittance in the entire coronary tree from inlet to the capillary vessels.

Method of solution. The characteristic impedance, characteristic admittance, and velocity (including the viscous effect) were first calculated for every vessel segment in the entire coronary arterial tree. There are two or more vessels that emanate from the jth junction point (e.g., junction point B in Fig. 1) anywhere in the current tree structure. We assume that mass is conserved at each junction and pressure is continuous at the junction, which may be written as:

(8)

(9)

From Eqs. 8 and 9, we may write:

(10)

Once the terminal impedance/admittance of the first capillary is computed, we proceed backward to iteratively calculate the impedance/admittance in the entire coronary tree by using Eq. 7 (obtained from Eq. 3) and Eq. 10 as demonstrated in Fig. 1B. The sawtooth pulsatile pressure produced experimentally by the piston pump was discretized by a Fourier transformation to determine the constants a and b in Eqs. 1 and 2. The flow and pressure were then calculated by using Eqs. 1 and 2.

View larger version (34K): [in this window] [in a new window]   Fig. 1. A: schematic representation of a bifurcation. 1) Equations 1–4, respectively, represent the flow, pressure, impedance, and admittance, which are calculated in every vessel (e.g., segments AB, BC, BD). 2) Equations 8 and 9, respectively, represent the mass conversation and pressure continuity, which happen at junction point B. B: schematic representation of the governing equations associated with the anatomical models

	EXPERIMENTAL METHODS

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Animal preparation. Studies were performed on six 3- to 4-mo-old farm pigs weighing 28–38 kg. The experimental procedures of the animal preparation were similar to those described by Kassab et al. (20). All animal experiments were performed in accordance with national and local ethical guidelines, including the Institute of Laboratory Animal Research guide, Public Health Service policy, Animal Welfare Act, and an approved University of California, Irvine-Institutional Animal Care and Use Protocol.

Briefly, surgical anesthesia was induced with ketamine (33 mg/kg im) and atropine (0.05 mg/kg im) and was maintained with pentobarbital sodium (30 mg/kg iv in an ear vein). A midline sternotomy was performed, ventilation with room air was provided with a respiratory pump, and anticoagulation was provided with heparin (3 mg/kg). Arterial pressure was measured through a catheter inserted through the carotid artery into the ascending aorta and was monitored with a Biopac MP 100. An incision was made in the pericardium, and the heart was supported in a pericardial cradle. A 1-liter bottle containing a hypothermic (10°C), isotonic, cardioplegic rinsing solution (composed of 1 mM EGTA, 0.2 mg/l calcium-channel blocker nifedipine, and 80 mg/l adenosine) was hung above the heart. A 14-gauge needle was connected to the rinsing solution with Tygon tubing. The needle was inserted into the ascending aorta with no perfusion initially. The heart was then arrested with a saturated KCl solution given through a jugular vein, and the tubing was unclamped to allow immediate perfusion of the cardioplegic rinsing solution. The aorta was clamped proximal to the needle to allow perfusion of the coronaries through Valsalva's sinus. The superior and inferior venae cavae were cut to allow free drainage of blood. The heart was covered with crushed ice for several minutes during perfusion with 500 ml of cardioplegic solution. The heart was then excised with the ascending aorta still clamped to keep air bubbles out of the coronary arteries.

Measurement of pulsatile flow. After the heart was isolated, it was placed in a cold (0°C) saline bath as shown in Fig. 2. The major coronary arteries [right coronary artery (RCA), left anterior descending (LAD) artery, and left circumflex (LCX) artery] were cannulated under saline to avoid air bubbles. The coronary arteries were perfused with a piston pump that provides a continuous sawtooth pulsatile flow. The perivascular flow probe (Transonic Systems; relative error of ±2% at full scale) was mounted on the coronary arteries directly, and the pressure transducer (Summit Disposable Pressure Transducer, Baxter Healthcare; error of ±2% at full scale) was connected at the inlet of the coronary arteries through a Y tube. The flow probe was connected to the flow meter module by a flexible cable. Both flowmeter module and pressure transducer were controlled by Biopac MP 100 data-acquisition system using AcqKnowledge (ACQ). A hypothermic (10°C), isotonic, cardioplegic rinsing solution was contained in a 2-liter bottle and injected into the coronary arteries through the pulsatile flow produced by the piston pump. The rinsing solution was different from the one used previously in that it was composed of 1.5 g/l 2,3-butanedione monoxime, 0.1 g/l adenosine, and 10 g/l albumin, which was found to be effective in maintaining the heart in a relaxed state. The sawtooth pulsatile flow produced by the piston pump was similar to that measured in vivo at a heart rate of 90 beats/min, as shown in Fig. 3. Therefore, we used the piston pump to simulate the in vivo aortic pressure. In the present study, the wave amplitude of the pulsatile pressure was in the range of 50–100 mmHg with a mean pressure of 75 mmHg.

View larger version (39K): [in this window] [in a new window]   Fig. 2. Schematic of apparatus for pulsatile flow measurements.

View larger version (17K): [in this window] [in a new window]   Fig. 3. Comparison of the profile of pressure produced by the piston pump with measured in vivo aortic pressure.

	RESULTS

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The results are divided into three subsections. First, low-frequency flow ( 0) is compared with steady-state flow. Second, the pressure produced by the piston pump is applied as the inlet pressure boundary condition to calculate the pressure and flow distribution in the entire coronary arterial tree by using the current mathematical model in conjunction with a Fourier transform. The mathematical predictions are in turn compared with the experimental results. Finally, the effects of various parameters (e.g., frequency, viscosity, etc.) on pulsatile flow are examined for the entire coronary arterial tree.

Low-frequency flow ( 0) compared with previous steady-state flow model. It is desirable to verify the computer code for the values of the pressure and flow against available analytical or numerical solutions. In the present study, the low-frequency ( 0) pressure and flow were calculated by using the current mathematical model and compared with a previous steady-state model developed by our group (23). The two models were implemented with the same mean inlet pressure of 100 mmHg. The outlet pressures in the steady-state computation were set at 16 mmHg and the outlet impedances in the pulsatile computation were evaluated as 128 µ_capillaryL_capillary/D(g·s^–1·cm^–4) at the first capillary vessel segments (order 0). The wave frequency (f) of the pulsatile blood flow was assumed to be 0.001 Hz. The entire asymmetric LAD, LCX, and RCA trees were considered in the comparison of the two models. The direct relationship between mean segment flow |Q|_mean (average of flow rates at every order) and mean segment pressure |P|_mean {average of 0.5 x [|P(0,)| + |P(L,)|] at every order} in the coronary arteries is depicted in Fig. 4. The present pulsatile flow results at low frequency agree very well with previous steady-state model.

View larger version (17K): [in this window] [in a new window]   Fig. 4. Relationship between mean segment flow |Q|mean (average of flow rates at every order) and mean segment pressure |P|mean {average of 0.5 x[|P(0,)| + |P(L,)|] at every order} in the coronary arteries for left anterior descending coronary (LAD) artery (A), left circumflex (LCX) artery (B), right coronary artery (RCA) (C).

View larger version (44K): [in this window] [in a new window]   Fig. 5. Measured pressures (means ± SD; average and SD of the pressure in 6 pigs) and volumetric flow rates (means ± SD) at the inlet of left common coronary artery (LCCA) (A) and RCA (B).

View larger version (31K): [in this window] [in a new window]   Fig. 6. Comparison of flow rates between theoretical prediction and experimental measurement (means ± SD; average and SD of the flow rates in 6 hearts) at the inlet of LCCA (A) and RCA (B).

View larger version (21K): [in this window] [in a new window]   Fig. 7. Comparison between results predicted by the present mathematical model with those by other models (see Refs. 24 and 25 and Refs. 6, 33, and 34) for LCCA (A) and RCA (B).

View larger version (27K): [in this window] [in a new window]   Fig. 8. Pressure and flow wave form at the inlet of LAD artery with different heart rates (90, 120, and 180 beats/min) for experimental pressure (A), experimental flow (B), and numerical flow (C).

View larger version (13K): [in this window] [in a new window]   Fig. 9. A: impedance |Zinlet(0,)| at the inlet of LAD artery vs. angular frequency = 2f. B: phase of impedance Zinlet(0,) vs. angular frequency .

View larger version (21K): [in this window] [in a new window]   Fig. 10. A: schematic of the trunk of the LAD artery and several of the primary branches (branches A–C). B: relationship between the flow rate |Q(x,)| and the cumulative length of vessel segments from the root (order 11) to the first capillary vessel segment (order 0) along the trunk line (root-AP-capillary) with the harmonic wave frequency of 0.001 Hz (steady state) and 1.5 Hz (90 beats/min). C: relationship between the phase angle of Q(x,) and the cumulative length of vessel segments from the root (order 11) to the first capillary vessel segment (order 0) along the trunk line (root-AP-capillary) with the harmonic wave frequency of 0.001 Hz and 1.5 Hz.

View larger version (23K): [in this window] [in a new window]   Fig. 11. A: relationship between the flow rate |Q(x,)| and the cumulative length of vessel segments from the primary branches (branches A, B, and C) to the first capillary vessel segment (order 0) with the harmonic wave frequency of 0.001 Hz and 1.5 Hz. B: relationship between the phase angle of Q(x,) and the cumulative length of vessel segments from the primary branches (branches A, B, and C) to the first capillary vessel segment (order 0) with the harmonic wave frequency of 0.001 Hz and 1.5 Hz.

View larger version (24K): [in this window] [in a new window]   Fig. 12. A: flow waves sequentially at 2.0-cm intervals along the trunk starting from the inlet of LAD artery. B: pressure waves sequentially at different diameters along the trunk of LAD artery.

View larger version (17K): [in this window] [in a new window]   Fig. 13. Relationship between mean segment flow |Q|mean (average of the flow rate at every order) and mean segment pressure |P|mean {average of 0.5 x [|P(0,)| + |P(L,)|] at every order} at different viscosities [µ = 1.1 cp and µ using the viscosity model of Pries et al. (26)].

	DISCUSSION

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Low-frequency flow ( 0) compared with previous steady-state flow model.

(11)

where L(n) and D(n) are the length and diameter of the vessel segment, respectively; is the angular frequency; Z(0,0) and Z(L,0) are the impedance at the entrance and exit of the vessel, respectively; and µ is blood viscosity. It is known that the Womersley number, = (D/2), describes the ratio of transient inertia force to viscous force. When (1) << 1, the viscous force dominates and the flow can be assumed to be steady-state flow. In the present computation of the low-frequency flow ( 0), the Womersley number (1) 0.07, 0.055, and 0.07 for the inlet vessel segment of the LAD, LCX, and RCA trees, respectively. The Womersley number at the inlet vessel segment is greatest for the coronary arterial tree. Equation 11 and the Womersley's analysis suggest that the pulsatile flow should be steady-state flow when the frequency of the pulsatile flow approaches zero. Figure 4 shows that the pulsatile flow results agree reasonably well with the steady-state except that there is a small deviation below order 4 because of slightly different outlet boundary conditions between the steady-state and pulsatile models. This comparison suggests that the current mathematical model predicts the correct results for the special case of steady-state flow.

Experimental validation of mathematical model. To validate the present mathematical model, the numerical simulation must be compared with experimental data. Figure 5 shows the distribution of experimentally measured mean pressure and flow ± SD at the inlet of LCCA and RCA of six hearts. In Figure 5, it is observed that the mean flow rate at the inlet of LCCA is about twice as large as that of RCA, which is in agreement with previous studies (30). In Fig. 6, the predicted flow rate by the mathematical model agrees reasonably well with experimental results (within ± 1 SD). Therefore, the present mathematical model correctly simulates the pulsatile flow in diastole in the absence of vessel tone in the complex anatomical tree, which has more than 1 million vessels. It should be noted that the experimental preparation contains both the arterial and venous trees, whereas the current simulation only considers the arterial tree. It appears that the arterial tree dominates the wave form, however, given the present agreement. This may be due to the small diameter of the capillary network, which strongly retards the propagation of the flow wave. Furthermore, the resistance of the arterial tree is significantly larger than that of the venous counterpart.

Comparison with other studies. The present mathematical model is compared with Olufsen's model for the root impedance (24, 25). In the present model, the constitutive equation is derived from the experimental data for the coronary arteries (see details in APPENDIX), which may be written as follows:

(12)

where E(n)h(n)/R(n) is obtained directly from the linear curve fit of the experimental data. The dynamic elastic properties of the coronary artery are considered in the formulation (see details in APPENDIX). The constitutive equation in the Olufsen's model (24, 25) is written as:

(13)

where E(n)h(n)/R(n) = k₁exp(k₂r₀) + k₃ (see Refs. 24, 25). Figure 14 shows a comparison of the measured pressure-cross-sectional area (pressure-CSA) relationship between these two models. Figure 7 shows a comparison of the flow waves when these two models are implemented in the entire coronary arterial trees (22). Although there is a phase angle and amplitude difference between the results, it is found the mean flow rates are similar. The reason for the phase angle difference is due to the difference in E(n)h(n)/R(n).

View larger version (19K): [in this window] [in a new window]   Fig. 14. Relationship between pressure [P (dyn/cm2)] and normalized cross-sectional area [A(x,t)/A(n)] for the first generation of left coronary arteries (1.3 and 2.8 mm) and linear curve fits, which are compared with previous model (24, 25). The linear curve fits for 1.3- and 2.8-mm arterial vessels are, respectively, A(x,t)/A(n) = 3 x 10–6 x [P(x,t) – P0] + 1.09 (R2 = 0.919) and A(x,t)/A(n) = 2 x 10–6 x [P(x,t) – P0] + 1.06 (R2 = 0.919). The static Young's modulus E(n) is calculated as 8.0 x 106 (dyn/cm2) by using R(n)/E(n)h(n) = (2 x 10–6 + 3 x 10–6)/2 = 2.5 x 10–6 [R(n)/h(n) 20 determined from Ref. 13].

(14)

where Y(k,j) is the characteristic admittance of the mother vessel segment (k,j). Y_e(k + 1,2j – 1) and Y_e(k + 1,2j) are effective admittances at the entrance of the two daughter vessels [(k + 1,2j – 1) and (k + 1,2j)], respectively. The difference stems from the application of the characteristic admittance Y(k,j) rather than the effective admittance at the exit of the mother vessel. Although this error may be negligible for small networks (such as in Ref. 6), it is significantly propagated in a network of millions of vessels. Hence, the error may be accumulated when all effective admittances (see Ref. 6) are calculated by marching backward along the tree structure and hence the observed discrepancy. It should be noted that the Zamir (6) and Olufsen (24, 25) formulations were not intended for the present vascular tree structure or the present capillary boundary conditions, and hence a direct comparison is not entirely possible.

Effect of various parameters on pulsatile blood flow. In pulsatile flow, the wave frequency may have an effect. The pressure and flow waves at different heart rates produced by the piston pump are thus analyzed at the inlet of the LAD artery. In Fig. 8, the measured and computed flow wave show similar waveform at different heart rates. To ensure accurate comparison, the mean pressures (as shown in Table 1) are fixed at the same value. The computed mean flow rates are very similar to those of steady-state flow because the zero frequency in Fourier series has a dominant effect.

Because the study of harmonics is important, the dependence on frequency of flow wave is examined. In Table 2, the real part of the complex flow rate [Q_inlet(0,)] does not change with the wave frequency even if the flow wave is harmonic with the same inlet pressure [P_inlet(0,)]. The phase angle of the flow wave is increased with increase in wave frequency. The amplitude |Z_inlet(0,)| and the phase angle of the impedance Z_inlet(0,) depict similar tendency as those in Ref. 24, as shown in Fig. 9. Naturally, the amplitudes |Z_inlet(0,)| are different because our anatomical coronary arterial trees are different from those used in Ref. 24. On the basis of the above analysis, we conclude that the wave frequency has a small effect on the mean flow rate and has a significant effect on the phase angle and the flow waveform.

The significant asymmetry of coronary arteries is important to understand the spatial heterogeneity in coronary flow. At present, the flow wave along the trunk and major branches of the coronary arterial tree is analyzed with the harmonic wave frequencies of 0.001 Hz (steady state) and 1.5 Hz (90 beats/min). It is noted in Fig. 10, B and A, that the blood flow through the trunk shows a gradual drop along the path to the capillary vessels while the flow from the trunk to the primary branches has an abrupt drop because the trunk vessel segments have much larger diameter than the primary branches. The difference of the amplitudes |Q(x,)| is not apparent between the harmonic wave frequencies of 0.001 Hz and 1.5 Hz. However, the phase angle of Q(x,) at the harmonic wave frequency of 1.5 Hz reaches 36° at the inlet of the trunk and gradually decreases along the trunk. The phase angle of Q(x,) of the primary branches (i.e., branches A, B, and C) shows a nearly linear decrease toward the capillaries.

Most of the previous studies on pressure wave transmission have concentrated on changes along the aorta and into the lower limb arteries. This is the first investigation of the pressure and flow wave transmission throughout the entire coronary arterial tree. Because the patterns of waves in the LAD, LCX, and RCA trees are similar, the pressure and flow waves are only shown for the trunk of LAD artery from the inlet to the first capillary. As seen in Fig. 12A, changes in amplitude of the flow wave are apparent at 2.0-cm intervals along the trunk because the primary branches shunt the flow away from the trunk. A small phase angle shift of the flow waves is gradually developed along the trunk from the inlet to the first capillary. Figure 10B also predicts that the pressure only has an abrupt drop when the diameter is below 100 µm, which is in agreement with experimental measurements (2). Figure 12B depicts the relation between pressure waves and diameter along the trunk of LAD artery. It is noted that the attenuation of the amplitude of the pressure wave is rather small for larger vessels and becomes significant for vessels <100 µm. The phase angle shift of the pressure wave shows a similar tendency to that of the flow wave.

As shown in Fig. 13, the viscosity has an important effect on the blood flow in the entire coronary arterial trees, particularly in the lower-order vessels (smaller diameter). The viscosity retards the movement of blood flow. The Womersley number may be written as = (D/2)() = transient inertia force/viscous force. An increase in the viscosity of pulsatile blood flow will increase the effect of viscous force so that the flow rate will decrease as predicted by the model.

Critique of model. Although our model is based on detailed measured morphometric data of coronary vessels, there are still a number of assumptions made. For example, 1) the ad hoc resistance boundary conditions are imposed on capillary vessels to decouple the arterial from the venous system; 2) all vessels in the present model are assumed to be in vasodilated state in the absence of vessel tone where coronary flow reserve is substantially reduced; 3) in the absence of experimental data, we assume that the measured compliance of larger vessels (19) applies to the entire vasculature; and 4) the Womersley-type model is an approximate solution and does not incorporate nonlinear advective losses.

In future studies, the present arterial circuit will need to be extended to the entire coronary vasculature; i.e., the capillaries need to be connected to the entire venous system. Hence, the ad hoc resistance boundary condition on the capillary vessel will need to be replaced with a measured boundary condition at the outlet of the venous system. The present mathematical model must ultimately include the myocardial/vascular interaction during systole. Future studies are also needed to determine the mechanical properties of small coronary arterial vessels in situ. On the computational side, the one-dimensional/three-dimensional equations must be solved by using the impedance boundary condition based on the anatomical trees.

Significance of model. The present study yields a mathematical model of pulsatile blood flow that incorporates some of the detailed anatomical features that influence coronary blood flow. The accuracy of the model is determined through experimental validation. The validated model of normal hearts will serve as a physiological reference state. Pathological states such as hypertension and stenosis can then be studied in relation to changes in model parameters. The proposed study makes use of physical principles, with the help of anatomy and experimental measurement, to explain and predict some aspects of pulsatile nature of the coronary arterial circulation in quantitative terms. The present model will serve to quantitatively test various hypotheses in the coronary circulation under pulsatile flow conditions.

	APPENDIX

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In the present study, the blood flow is assumed to be impermeable, incompressible, Newtonian, and laminar. The blood flow in a vessel segment is taken as the flow in an axisymmetric cylinder. Therefore, the velocity of the fluid inside the vessel is denoted by u_r(r,x,t) and u_x(r,x,t), where r is the radial coordinate, x as the position along the length of vessel, t is time, u_r is the radial velocity, and u_x is the axial velocity. The cross-sectional area of the vessel A(n) is calculated from the diameter of the vessel D(n) (n is the vessel number) consistent with experimental measurements. The density and viscosity µ are considered to be constant.

Wave propagation in a tube (e.g., the wave in a vessel segment AB as shown in Fig. 1) is governed by wave equations for the pressure p(x,t) and volume rate of flow q(x,t). To derive the wave equations, the continuity (mass conversation) and momentum equations for tube flow may be written as follows:

(A1)

(A2)

It is noted that there are two equations and three independent variables [A(x,t), q(x,t) = 2₀^R(n)u_x(r,x,t)rdr, and p(x,t)]. Therefore, an additional equation in the form of constitutive relation is needed to solve for the three unknown variables.

The constitutive equation is based on a pressure-CSA relation for every vessel. From Laplace's law, we have:

(A3)

where is the mean circumferential stress that relates the transmural pressure p(x,t) – p₀ (intravascular pressure minus external pressure, which is assumed zero in diastole), the vessel radius R(n) = D(n)/2, and the wall thickness h(n) for each order of vessel n. If we assume that the stress-strain relation is linear, we may write:

(A4)

where E(n) is Young's modulus for each order vessel and [R² – R²(n)]/R²(n) is the circumferential strain. If we combine Eqs. A3 and A4, we obtain the following constitutive equation:

(A5)

where the radius is expressed in terms of cross-sectional area. At this point, a pressure-CSA relation is needed. Such relations were previously measured by our group for the coronary arteries proximal to 1.0 mm in diameter using digital subtraction angiography (19). It was found that the differences in the pressure-CSA relation of vessels proximal to 1.5 mm in diameter (orders 11, 10, 9) were relatively small (see Fig. 3 and Table 1 in Ref. 19). Therefore, the mean of the pressure-CSA data for the 1.3 mm and 2.8 mm diameter vessels (see Fig. 3 in Ref. 19) were adopted here. As determined by linear least-squares fits of data, the experimental data may be expressed as follows:

(A6)

which can be approximated as:

(A7)

If we combine Eqs. A5 and A7, we obtain E(n)h(n)/R(n) = 4.0 x 10⁵. The thickness-to-radius ratio for the 1.3- and 2.8-mm vessels is very similar and was used to calculate the Young's modulus E. When R(n) and h(n) are obtained from Guo and Kassab's study (13), the static Young's modulus E(n) is determined as 8.0 x 10⁶ (dyn/cm²). In the present simulations, it was assumed that this value is constant through the coronary arterial tree.

The dynamic Young's modulus must be considered because the current model calculates the impedance, pressure, and flow under various frequencies after a Fourier transform. Previous studies (4, 12) have investigated the dynamic elastic properties of canine coronary artery. In the current study, the ratio of E_dyn/E_stat (4) is used to adjust the dependence of Young's modulus on frequency.

By assuming the blood flow is harmonic and quasi-steady state, the variables can be written as:

(A8)

(A9)

(A10)

(A11)

where is the angular frequency. Also, P(x,), U_x(r,x,), Q_x(x,), and A(x,) may be simplified as P, U_x, Q, and A. If we substitute Eqs. A5–A11 into Eqs. A1 and A2, we obtain:

(A12)

and

(A13)

The velocity profile was first obtained by Womersley (31) for pulsatile flow in a rigid tube as a solution to Eq. A13 in the form:

(A14)

where = R(n) is the Womersley number and J₀ is the zero-order Bessel function. The volume flow rate is the same as that obtained by Duan and Zamir (5). Integrating the flow velocity over the cross-sectional area, we obtain the following wave equation:

(A15)

where F₁₀() = [2J₁(i^3/2)]/[i^3/2J₀(i^3/2)] and J₁ is the first-order Bessel function. Finally, by using Eqs. A12 and A15, the wave equations for the pressure P and volume rate of flow Q may be written as follows:

(A16)

(A17)

where c₀ = is the wave velocity without the viscous effect, and Y₀ = A(n)/c₀ is the characteristic admittance. Also we define Z₀ = 1/Y₀ as the characteristic impedance. If we combine Eqs. A16 and A17, we obtain:

(A18)

When the viscous effect is incorporated into the wave velocity, we may define the wave velocity c = ·c₀. Equation A18 can then be written as follows:

(A19)

Defining Y₁ = Y₀ and Z₁ = Z₀ and solving Eqs. A16 and A19 yield Eqs. 1 and 2 and subsequently Eqs. 3–7 as outlined in MATHEMATICAL MODEL.

	GRANTS

TOP ABSTRACT Glossary MATHEMATICAL MODEL EXPERIMENTAL METHODS RESULTS DISCUSSION APPENDIX GRANTS REFERENCES

 
This research is supported in part by National Heart, Lung, and Blood Institute Grant 2-R01-HL-055554–06.

	ACKNOWLEDGMENTS

 
We thank Dr. X. Lu and Dr. Y. Huang for assisting in the experimental preparation.

	FOOTNOTES

 

Address for reprint requests and other correspondence: G. S. Kassab, Dept. of Biomedical Engineering, Surgery, Cellular, and Integrative Physiology, SL-174, Indiana Univ. Purdue Univ. Indianapolis, 723 Michigan St., Indianapolis, IN 46202 (e-mail: gkassab{at}iupui.edu)

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

	REFERENCES

TOP ABSTRACT Glossary MATHEMATICAL MODEL EXPERIMENTAL METHODS RESULTS DISCUSSION APPENDIX GRANTS REFERENCES

 

1. Avolio AP. Multi-branched model of the human arterial system. Med Biol Eng Comput 18: 709–718, 1980.[CrossRef][ISI][Medline]
2. Chilian WM. Microvascular pressures, and resistances in the left ventricular subepicardium and subendocardium. Circ Res 69: 561–570, 1991.[Abstract/Free Full Text]
3. Christopoulos C. The Transmission-Line Modeling Method: TLM. Wiley-IEEE Press, 1995.
4. Douglas JE and Greenfield JC. Epicardial coronary artery compliance in the dog. Circ Res 27: 921–929, 1970.[Abstract/Free Full Text]
5. Duan B and Zamir M. Viscous damping in one-dimensional wave transmission. J Acoust Soc Am 92: 3358–3363, 1992.[CrossRef]
6. Duan B and Zamir M. Pressure peaking in pulsatile flow through arterial tree structures. Ann Biomed Eng 23: 794–803, 1995.[ISI][Medline]
7. Formaggia L, Gerbeau JF, Nobile F, and Quarteroni J. On the coupling of 3D and 1D. Navier-Stokers equations for flow problems in compliant vessels. Comput Methods Appl Mech Engrg 191: 561–582, 2001.[CrossRef]
8. Fung YC. Biodynamics: Circulation. New York; Springer-Verlag, 1984.
9. Gan RZ and Moodie TB. Broad-spectrum impedance calculation for arterial models. Int J Eng Sci 27: 63–76, 1989.[CrossRef]
10. Gan RZ and Yen RT. Vascular impedance analysis in dog lung with detailed morphometric and elasticity data. J Appl Physiol 77: 706–717, 1994.[Abstract/