## Abstract

A hemodynamic analysis is implemented in the entire coronary arterial tree of diastolically arrested, vasodilated pig heart that takes into account vessel compliance and blood viscosity in each vessel of a large-scale simulation involving millions of vessels. The feed hematocrit (Hct) is varied at the inlet of the coronary arterial tree, and the Fahraeus-Lindqvist effect and phase separation are considered throughout the vasculature. The major findings are as follows: *1*) vessel compliance is the major determinant of nonlinearity of the pressure-flow relation, and *2*) changes in Hct influence wall shear stress (WSS) in epicardial coronary arteries more significantly than in transmural and perfusion arterioles because of the Fahraeus-Lindqvist effect. The present study predicts the flow rate as a second-order polynomial function of inlet pressure due to vessel compliance. WSS in epicardial coronary arteries increases >15% with an increase of feed Hct from 45% to 60% and decreases >15% with a decrease of feed Hct from 45% to 30%, whereas WSS in small arterioles is not affected as feed Hct changes in this range. These findings have important implications for acute Hct changes under vasodilated conditions.

- feed hematocrit
- vessel compliance
- hemodynamic analysis

vessel compliance is an important predictor of cardiovascular disease. A reduction of vessel compliance correlates with the onset of such diseases as atherosclerosis, hypertension, and diabetes mellitus (4). Vessel compliance is also an important determinant of the pressure-flow relation of an organ. Indeed, the pressure-flow relation of a circulatory network is expected to be linear if the vessels are rigid and the fluid is Newtonian according to Poiseuille's equation (12). Since neither is strictly true, we intend to evaluate the effect of compliance and blood rheology [hematocrit (Hct)] on the nonlinearity of the pressure-flow relation in the entire coronary arterial tree, for which anatomic and rheological data exist (10, 13).

The relation between the apparent viscosity and Hct for different-sized vessels is highly nonlinear, as described by the Fahraeus-Lindqvist effect (2). The effect of this nonlinearity on the pressure-flow relation, as well as on different-sized vessels, remains unknown in the context of a full vascular network. Our hypothesis is that the effect of vessel compliance is a stronger determinant of the nonlinearity of the pressure-flow relation than the blood rheology, since the change in diameter is amplified by the fourth power, as predicted by Poiseuille's equation. Blood rheology, on the other hand, plays a larger role in wall shear stress (WSS). Specifically, we hypothesize that changes in Hct have a greater effect on WSS of larger vessels than smaller vessels because of the Fahraeus-Lindqvist effect. Accordingly, we consider the effects of changes in inlet feed Hct on WSS throughout the entire coronary arterial tree.

A model of the entire coronary arterial tree (15) is used to evaluate the effects of acute changes of Hct on steady-state hemodynamic parameters. The compliance of arteries (3, 5, 13) and an in vivo viscosity model (20) are considered in each vessel. The cascading effect of numerous bifurcations on microvascular Hct (17, 18) is included in the model. The predictions of the mathematical model are compared with the experimentally measured pressure-flow relation of the right coronary arterial (RCA) tree perfused by cardioplegic solution (11). The agreement between theory and experiment is good. WSS is calculated on the basis of the pressure and flow results and analyzed in the left circumflex arterial (LCx) tree. Vessel compliance mainly contributes to the nonlinearity of the pressure-flow relation and has a small effect on WSS. Conversely, Hct has a small effect on the nonlinearity of the pressure-flow relation but causes a significant redistribution of WSS in the large epicardial arteries compared with the small transmural and perfusion arterioles.

## METHODS

#### Anatomic model.

The anatomic model is that of coronary arterial trees recently reconstructed by Mittal et al. (15), based on measured morphometric data (10). Briefly, ≥40-μm-diameter vessels were reconstructed from cast data, and <40-μm-diameter vessels were reconstructed from histological data. The RCA and LCx trees were reconstructed with 1,676,925 and 1,151,729 vessel segments, respectively, down to the first capillary segments. Similar to the previous study (9), the coronary arterial trees can be classified as follows: *1*) epicardial (orders ≥8; see Ref. 10 for the definition of diameter-defined Strahler order), *2*) transmural (5 ≤ orders < 8), and *3*) perfusion subnetworks (1 ≤ orders < 5).

#### Vessel compliance.

Using a video-densitometric technique, we previously determined the compliance of the first several generations (orders 9–11) of pig coronary arteries (5, 13). The diameter-compliance (Δ*D*/ΔP, where *D* is diameter and P is pressure) is 2.42 ± 2.3 μm/mmHg in ∼1,000-μm-diameter vessels in the arrested heart, so that the normalized diameter-compliance [diameter-compliance normalized by diameter, represented as (Δ*D*/ΔP)(1/*D*)] is ∼2.4 × 10^{−3} mmHg^{−1} in orders 9–11. The diameter-compliance is 1.7 ± 0.91 × 10^{−2} μm/mmHg in <8-μm-diameter pig capillary segments (12); on the basis of this value, the normalized diameter-compliance is estimated as 2.1 × 10^{−3} mmHg^{−1} in order 0 (capillary). We assume that normalized diameter-compliance is the same for perfusion vessels (orders 1–4) and capillaries, because the entire vasculature is deeply embedded in the myocardium. The normalized cross-sectional area-compliance (3) is ∼3.8 × 10^{−3} mmHg^{−1} in pig coronary arterioles (∼200 μm); this value can be converted to a normalized diameter-compliance of 2.0 × 10^{−3} mmHg^{−1} for orders 5–8. Table 1 summarizes the normalized diameter-compliances in different orders used in the simulations. Spaan (23) estimated the normalized volume-compliance of capillaries, venules, smaller arterioles, and larger arteries, obtained from organs other than the heart. The normalized volume-compliance (equal to the normalized cross-sectional area-compliance for a unit length) was ∼4.0 × 10^{−3} mmHg^{−1} when pressure was >60 mmHg. When the normalized volume-compliance is converted to the normalized diameter-compliance, it is ∼2.0 × 10^{−3} mmHg^{−1}, which is similar to the present data for pig heart in Table 1.

#### Mathematical model.

The basic equations are similar to those published previously (8, 16). Compared with the steady-state flow (8, 16), an iterative hemodynamic analysis is implemented by a variation of the inlet pressure from 60 to 140 mmHg. An additional iterative procedure is carried out to satisfy the criteria of convergence in each step of inlet pressure change. The distensibility and blood viscosity in each vessel are incorporated at every step. Different feed Hcts, in the range of 0.25–0.65, are considered at the inlet of the coronary arterial tree. The details of the mathematical method are described in the appendix.

## RESULTS

The pressure-flow relation was experimentally measured at the inlet of the entire RCA tree of arrested and vasodilated porcine heart (11). A hypothermic (10°C), isosmotic, cardioplegic rinsing solution was selected for perfusion of the RCA tree. Figure 1*A* shows the computed pressure-flow relation, with inlet pressure changing from 60 to 100 mmHg. The model of constant viscosity (1.3 cP) agrees very well with the experimental measurements (11). As determined by least-squares fits of the data, the pressure-flow relation that accounts for compliance of vessels depicts a second-order polynomial function (*R*^{2} = 1) compared with the linear pressure-flow relation in the rigid tree model subject to Poiseuille's law. Figure 1*B* shows that the total arterial volume increases linearly with inlet pressure, with a slope of 2.3 × 10^{−3} ml/mmHg (volume-compliance), which is consistent with the previous measurements (2.6 ± 1.8 × 10^{−3} ml/mmHg) (5).

Figure 2*A* shows the pressure-flow relations at the inlet of the LCx tree with feed Hcts of 0.25–0.65 as inlet pressure is varied from 60 to 140 mmHg. Figure 2*B* shows the flow rates as a second-order polynomial function of feed Hct at the inlet of the LCx tree when inlet pressure is 100 mmHg. The low feed Hct results in high coronary inlet flow. The slope of the pressure-flow relation is related to the conductance of flow, which is the inverse of flow resistance. Clearly, total resistance to flow decreases with a decrease in inlet Hct.

The relation between WSS and vessel diameter in the vessels of the LCx tree with inlet feed Hct of 0.45 is shown in Fig. 3. A least-squares fit shows a power-law relation (WSS = 3,716 × *D*^{−0.957}, *R*^{2} = 0.989) between WSS (dyn/cm^{2}) and vessel diameter (*D*, μm), in the diameter range of 30–1,000, μm. This is consistent with the experimental measurements in >50-μm-diameter vessels of dog hearts (25). There is a relatively uniform WSS in arterioles however, with diameters between 10 and 30 μm (orders 1–3), which agrees reasonably well with experimental measurements (21).

Figure 4 shows the Hct-induced relative difference of WSS in each order with inlet feed Hcts of 0.6 and 0.3 where the inlet pressure was fixed at 100 mmHg. The change of inlet feed Hct has a much larger effect on WSS in the epicardial coronary arteries (orders 8–10) than in the transmural and perfusion arterioles (orders <8). WSS in orders 8–10 increases by 12–20% with the increase of feed Hct from 0.45 to 0.6 and decreases by 13–21% with the decrease of feed Hct from 0.45 to 0.3. WSS in orders <7 is not significantly affected by the change of Hct, and WSS in order 7 (a transition from larger arteries to smaller arterioles) changes by ∼8%. A two-way ANOVA shows no significant difference of WSS in the arterioles (orders <7) as Hct changes.

The relative differences of flow rate and effective blood viscosity in each order are shown in Fig. 5. The relative change of flow rate is uniform in each order of vessels. The mean flow rate (averaged over all vessel segments in each order) decreases by ∼22% and increases by 16% with the change of feed Hct from 0.45 to 0.6 and 0.3, respectively. The mean viscosity (averaged over all vessel segments in each order) changes more significantly in the larger arteries; in orders 0–6, an approximately uniform increase of 30% and decrease of 22% were observed with the change of feed Hct from 0.45 to 0.6 and 0.3, respectively.

## DISCUSSION

The present study predicts the effects of compliance and Hcts on coronary arterial flows in the entire coronary arterial tree. The major effect of vessel compliance is the nonlinearity of the passive pressure-flow relation compared with other parameters of rheology. The change of inlet feed Hct can affect WSS (particularly in larger arteries) and the resistance of the entire coronary arterial tree, which may be associated with atherosclerosis and other biological responses. In future studies, the present model may be extended to diseased coronary arterial trees when the anatomic and rheological data become available.

#### Effect of compliance.

The compliance of coronary vessels is important, because it affects the pressure-flow relation and, hence, the resistance to flow (6). A number of researchers have investigated the compliance of coronary arteries under in vitro, in situ, and in vivo conditions (5, 13, 22). Table 1 summarizes the in vitro diameter-compliance and the value normalized by diameter in different orders of vessels of the coronary arterial tree in pig heart. The present study (Figs. 1 and 2) shows that vessel compliance mainly affects the nonlinearity of the pressure-flow relation. Vessel compliance has a very small effect on WSS in the entire coronary arterial tree, which is consistent with the finding in the epicardial coronary arterial tree (7).

#### Hct and pressure-flow relation.

Fahraeus and Lindqvist (2) found a reduction of blood viscosity in small tubes, which is due to a decrease of Hct as the vessel diameter decreases (1). Pries and colleagues showed that the in vitro (19) and in vivo (20) relations predict the relative apparent blood viscosity from vessel diameter and Hct. The in vivo flow resistances of <40-μm-vessels are markedly higher and show a stronger dependence on Hct than in vitro flow resistances. From in vivo viscosity law (20), viscosity decreases in the epicardial and transmural subnetworks (orders >4) and increases in the perfusion subnetwork (orders ≤4) as the vessel diameter decreases, and the minimum occurs at order 4 with vessel diameter of ∼30 μm. This result is due to the Fahraeus-Lindqvist effect.

The model of phase separation shows a disproportionate distribution of red blood cells and plasma at arteriolar bifurcations (17, 18). For a given fractional blood flow, the smaller branch was found to receive more red blood cells than the larger branch. This model leads to the larger heterogeneity of blood flow in the smaller orders, as reflected by the standard deviations shown in Fig. 5.

#### Hct and WSS.

WSS increases with a decrease of diameter from larger arteries to smaller arterioles in the diameter range of 30–1,000 μm (Fig. 3). The consistency between computational results and experimental measurements from fluorescence microangiography (25) depicts an inverse relation between WSS and vessel diameter in epicardial and transmural subnetworks. The area expansion ratio has a value of unity in the corresponding subnetworks (9). This implies a uniform flow velocity in the subnetworks, which has been validated by experimental measurements (25) and theoretical analysis (14). WSS in 10- to 30-μm-diameter arterioles is relatively uniform, however (Fig. 3). The same trend has been reported in <30-μm-diameter mesenteric arterioles (see Fig. 2 in Ref. 21). This is due to the increase of the area expansion ratio with the decrease of diameter in the perfusion subnetwork (9).

Furthermore, the Hct-induced redistribution of WSS is much larger in the epicardial subnetwork than in the transmural and perfusion subnetworks (Fig. 4). Because of the Fahraeus-Lindqvist effect, the dependence of viscosity on Hct is weaker in the transmural and perfusion subnetworks than in the epicardial subnetwork, and thus an increase of feed Hct causes a larger increase of larger- than smaller-vessel viscosity (Fig. 5). Figure 5 also shows a uniform flow rate change in each order. Since WSS = (32μQ̇)/π*D*^{3} (where Q̇ is the volumetric blood flow rate and *D* and μ are the diameter and coefficient of viscosity, respectively), it is determined by the product of Q̇ and μ when the diameter is constant. Figure 5 shows relative constancy of |Q̇ × μ| in orders 0–6 and a larger change of |Q̇ × μ| in orders ≥7. Therefore, WSS is redistributed, increasing in the larger vessels and remaining unchanged in the smaller vessels (Fig. 4*A*).

#### Critique of the model.

Many physiological and chemical factors, such as myocardial metabolism, shear-induced nitric oxide release, and pulsatility, regulate blood flow and, hence, WSS. Here, we implement a quasistatic hemodynamic analysis to study the effect of Hct in the entire coronary arterial tree of an arrested, vasodilated heart, which is the first step. The in vivo interaction of pulsatility, vasodilation, and heartbeat can be studied as an extension of the present model platform.

#### Significance of the model.

This is the first full anatomic model of the entire coronary arterial tree that evaluates the roles of vessel distensibility and blood rheology on the pressure-flow relation and WSS. Although distensibility of the blood vessels and blood rheology may, in principle, contribute to the nonlinearity of the pressure-flow relation, the model establishes the former as the major determinant. The latter, on the other hand, is a major determinant of flow resistance (smaller vessels) and WSS (larger vessels). Although the present model does not account for the vasoactivity of vessels, it does serve as a platform to incorporate these and other biological and physiological phenomena. As such, the model will eventually allow us to understand such phenomena as flow overload and iron overload (with multiple red blood cell transfusions), which are general risks of intravenous infusion and transfusion (24, 26). Other applications to chronic high altitude (increased Hct) or anemia (decreased Hct) will also be possible when additional realism is added to a fully integrated model of the coronary circulation.

## APPENDIX

### Mathematical Model

#### Governing equations.

We assume that the volumetric blood flow through a blood vessel is laminar, steady, and free from end effects (8). The classic Poiseuille's law can be used to describe the local pressure-flow relation in a cylindrical tube as (1) where Q̇ is the volumetric blood flow rate, *x* is the axial coordinate measured from the entrance section of each vessel, and *D* and μ are the local diameter and coefficient of viscosity, respectively.

#### Fahraeus-Lindqvist effect.

The relative apparent viscosity in a vessel segment in vivo, previously reported (20), can be written as (2) where μ_{vivo} and H_{D} are viscosity and discharge Hct, respectively. The apparent viscosity equals the product of relative apparent viscosity and 1.3 cP (the viscosity of plasma). μ*_{0.45} and the exponent *C* are defined as follows (3) (4) The units for μ_{vivo} and *D* are cP and μm, respectively. *Equation 2* reflects the Fahraeus-Lindqvist effect.

#### Phase-separation effect.

To consider the phase-separation effect, Pries et al. (17) studied the distribution of erythrocytes at microvascular bifurcations. The fraction of the erythrocyte flow and volumetric blood flow from the mother vessel to a daughter vessel is defined as FQ̇_{E} = q̇_{daughter}/q̇_{mother} and FQ̇_{B} = Q̇_{daughter}/Q̇_{mother}, respectively, where Q̇ and q̇ represent the volumetric blood flow and erythrocyte flow, respectively. An empirical relation (18) has been developed to describe the distribution of volumetric blood flow and erythrocyte flow at an individual bifurcation, which can be written as (5) where logit(FQ̇_{E}) = ln[FQ̇_{E}/(1 − FQ̇_{E})]. *A*, *B*, and *X*_{0} can be written as (6) (7) (8) *X*_{0} is the minimal fractional blood flow required to draw erythrocytes into the daughter branch, *B* is the nonlinearity of the relation between FQ̇_{E} and FQ̇_{B}, and *A* is the difference between the relations derived for the two daughter vessels. For FQ̇_{B} < *X*_{0}, FQ̇_{E} = 0; for FQ̇_{B} > 1 − *X*_{0}, FQ̇_{E} = 1. For *A*, *B*, and *X*_{0}, units are μm^{−1}; for *D*, units are μm.

#### Compliance effect.

The elastic deformation of a vessel segment can be described by (9) where *D* is the diameter at a given intravascular pressure P, *D** is the diameter corresponding to the reference pressure P*, and α is the diameter-compliance of the vessel.

### Method of Solution

*Equations 1*–*9* are solved using an iterative method as the change of inlet pressure, represented as (*n* = 1,...,*N*), where reference inlet pressure (*P*_{inlet}*) and pressure change (Δ*P**) are set to 100 and 2 mmHg, respectively. From *Eq. 1*, the volumetric flow rate Q̇_{ij}^{n+1} (at pressure step *n* + 1) in a vessel between any two nodes, represented by *i* and *j*, is given in terms of the pressure differential Δ*P*_{ij}^{n+1} and vessel conductance *G*_{ij}^{n+1} as (10) where Δ*P*_{ij}^{n+1} = *P*_{i}^{n+1} − *P*_{j}^{n+1}. From *Eqs. 2*–*9*, the vessel conductance *G*_{ij}^{n+1} can be calculated on the basis of the parameter at pressure step *n*, which can be written as (11) and (12) where *D*_{ij}^{n} and *L*_{ij} are the diameter and length at pressure step *n*, respectively, in the vessel between nodes *i* and *j*. The dynamic viscosity (μ_{ij}^{n}) is determined by substitution of the current values of diameter (*D*_{ij}^{n}) and Hct (H_{Dij}^{n}) in Eqs. *2*–*4*. *P*_{ij}^{n} = (*P*_{i}^{n} + P_{j}^{n})/2 represents the mean pressure in the vessel. By conservation of mass, we have the following equation (13) where *m*_{j} is the number of vessels converging at the *j*th node. Volumetric flow into a node is considered positive, and flow out of a node is negative for any branch. From *Eqs. 10* and *13*, a set of linear algebraic equations in pressure for *M* nodes in the coronary arterial tree may be written as follows (14) Once conductance and suitable boundary conditions are specified, the final global matrix formulation may be written as (15) where **G** is the matrix of conductance, **P** is the column vector of the unknown nodal pressures, and **G _{B}P_{B}** is the column vector of the conductance times the boundary pressure of their attached vessels. The large sparse matrix

**G**with millions of equations is solved by using the LU decomposition with partial pivoting and triangular system solvers through forward and back substitution (SuperLU_dist, which is implemented in ANSI C, and MPI for communications). Once the pressure at every node is calculated, the flow in each segment can be determined from

*Eq. 10*.

It is known in *Eqs. 11* and *12* that conductance, diameter, viscosity, and discharge Hct in each vessel at pressure step *n* + 1 are determined on the basis of the previous pressure step *n*. Therefore, a new iterative procedure must be carried out in each pressure step to satisfy convergence criteria. The conductance G_{ij}^{(n+1)(m+1)} of vessels can be calculated using *Eq. 11* with the current diameter *D*_{ij}^{(n+1)m} and viscosity μ_{ij}^{(n+1)m} (*m* = 0,...,*M*). Once the blood flow Q̇_{ij}^{(n+1)(m+1)} is calculated, the erythrocyte flow q̇_{ij}^{(n+1)(m+1)} can be obtained from *Eqs. 5*–*8*. When q̇_{ij}^{(n+1)(m+1)} and Q̇_{ij}^{(n+1)(m+1)} are calculated, the discharge Hct H_{Dij}^{(n+1)(m+1)} = [q̇_{ij}^{(n+1)(m+1)}/Q̇_{ij}^{(n+1)(m+1)}] can be updated. The criterion of convergence is relative error of volumetric flow rates <0.1%. The initial pressure (*n* = 0) at the inlet was set to 100 mmHg, because the morphometric data for the coronary arterial tree were obtained at 100 mmHg (10). The initial Hct in the arterial tree was set to the inlet feed Hct, which varies from 0.25 to 0.65 for different simulations. The change of pressure (ΔP*) between two pressure steps was set to 2 mmHg. The pressure at each outlet (first capillary segment) was fixed at 26 mmHg.

The product of normalized diameter-compliance (Table 1) and vessel diameter was used to determine the diameter-compliance in each vessel of the entire arterial tree. The vessel diameter was updated in each pressure step and between pressure steps. The pressure-flow relation was calculated in the pressure range of 60–140 mmHg, because there is a linear relation between pressure and diameter in the physiological range of interest (13).

## GRANTS

This research is supported in part by National Heart, Lung, and Blood Institute Grants HL-055554-11 and HL-084529 (G. S. Kassab) and American Heart Association Scientist Development Grant 0830181N (Y. Huo).

- Copyright © 2009 the American Physiological Society