Network Analysis of Coronary Circulation: I. Steady-State Flow

Abstract

Coronary heart disease is one of the most prevalent health problems affecting humans around the world. Mechanically, the main problem in coronary heart disease is that there is insufficient blood supply to the heart muscle to fulfill the cardiac metabolic needs. Therefore, the heart fails as an adequate blood pump. Despite the magnitude of this health problem, the coronary circulation system remains poorly understood, as clinical work has largely focused on the diseased vessels (atherosclerosis, hypertension, hypercholesteremia, diabetes, etc.) rather than on the dynamics of the healthy coronary blood flow.

Appendices

Appendix 1: Asymmetric Coronary Tree Model (Kassab et al., 1997)

The asymmetric model (Fig. 5.3) simulates the morphometric data of connectivity matrix (Tables 2.9 and 2.10, Appendix 2 in Chap. 2). Each element shown in Fig. 5.3 may represent one or more elements in parallel. The number of possible pathways for each element of Fig. 5.2 increases exponentially towards the capillary vessels (Kassab et al., 1997). A realistic analysis consistent with the morphometric data must also incorporate the dispersion of diameters and lengths of various orders. In a real flow, the inflow from the aortic sinus is non-uniformly distributed to the parallel vessels of order 10 because all order 10 elements do not have the same diameter and length and offsprings. An idealization is made such that vessels at a given mean element connectivity (Fig. 5.2) are considered parallel and hence have the same diameter and lengths consistent with the mean morphometric data.

Despite the sophistication of the asymmetric model, it still does not satisfy all of the statistical data measured previously (Kassab, Rider, et al., 1993). For example, only one tree topology, corresponding to the mean connectivity matrix, is considered which ignores the standard deviations of the connectivity matrix. Furthermore, the connectivity matrix shows a small number of vessels of order n branching from vessels of order n that the symmetric model does not consider. Finally, the asymmetric circuit is not a bifurcating tree model and cannot satisfy the statistics of the segment-to-element ratios (S/E) reported in Kassab, Rider, et al. (1993). The asymmetric model includes the assumption that the S/E = 1 for all orders of vessels that is not corroborated by experimental measurements (Fig. 2.9, Chap. 2). The analysis also includes the assumption that certain elements are grouped in parallel definition of the equivalent conductance G_eq, which simplifies the problem considerably.

Once the branching pattern and vascular geometry of the full arterial network are generated, a steady-state network analysis can be performed (Kassab et al., 1997; Mittal, Zhou, Linares, et al., 2005). Briefly, if the cylindrical vessel is considered rigid, long and slender, under laminar and steady flow, the Poiseuille’s law for a Newtonian fluid can be stated as:

$$ {Q}_{ij}=\frac{\pi }{128}\Delta {P}_{ij}{G}_{ij} $$

(5.1)

where Q_ij is the volumetric flow, in a vessel between any two nodes, represented by i and j. ΔP_ij is the pressure differential given by ΔP_ij = P_i − P_j, and vessel conductance, G_ij, is given by ${G}_{ij}=\frac{D_{ij}^4}{\mu_{ij}{L}_{ij}}$ where D_ij, L_ij, and μ_ij are the diameter, length, and viscosity, respectively, between nodes i and j. The variation of viscosity with vessel diameter is given by Pries et al. (1994) as:

$$ \mu =\left[1+\left(6\cdot {\mathrm{e}}^{-0.085D}+3.2-2.44{\mathrm{e}}^{-0.06{D}^{0.645}}-1\right)\cdot {\left(\frac{D}{D-1.1}\right)}^2\right]\cdot {\left(\frac{D}{D-1.1}\right)}^2 $$

(5.2)

where D is the vessel diameter.

Two or more vessels emanate from the jth node anywhere in the tree with the number of vessels converging at the jth node being m_j. By conservation of mass, the following must hold:

$$ \sum \limits_{i=1}^{m_j}{Q}_{ij}=0 $$

(5.3)

where the volumetric flow into a node is considered positive and flow out of a node is negative for any branch. From Eqs. (5.1), (5.2), and (5.3), a set of linear algebraic equations in pressure for M nodes in the network is obtained as:

$$ \sum \limits_{i=1}^{m_j}\left[{P}_i-{P}_j\right]{G}_{ij}=0 $$

(5.4)

The set of equations represented by Eq. (5.4) reduce to a set of simultaneous linear algebraic terms for the nodal pressures once the conductances are evaluated from the geometry, and suitable boundary conditions are specified. In matrix form, this set of equations is GP = G_BP_B where G is the n × n matrix of conductances (idealized to ~850 for LCCA), P is a 1 × n column vector of the unknown nodal pressures, and G_BP_B is the column vector of the conductances times the boundary pressures of their attached vessels, respectively. Boundary conditions are prescribed by assigning an inlet pressure of 100 mmHg and a uniform pressure of 25 mmHg at the outlet of the first capillary segment. Since matrix G is a very sparse matrix, it can be represented in a reduced form for optimal memory utilization. This system of equations can be solved to determine the pressure values at all internal nodes of the arterial tree. The pressure drops as well as the corresponding flows can be subsequently calculated.

5.1.1 Symmetric Model

This is an analytical model that simulates the mean statistical data (assumes all standard deviations are 0) but replaces the connectivity matrix by a diagonal matrix whose non-vanishing components in row m and column m + 1 are the branching ratios of the number of the elements of order m divided by the number of elements of order m + 1. Physically, it is equivalent to assuming that all the vessel elements in any order are in parallel, and the blood pressures at all the junctions between specific orders of vessels are equal. In this simplified circuit, the flow in each element of order n obeys Poiseuille’s formula as given by Eq. (5.1). There are N, elements of order n in parallel. If the total flow is Q_T, then q_n, in each vessel of order n is Q_T/N_n. Equation (5.1) then yields the pressure drop. The pressure at the Valsalva sinus, P₁₁ at n = 11 being given (e.g., an inlet diastolic coronary artery pressure of 100 mmHg similar to asymmetric model), P₁₀, P₉, … P_0a (0a refers to an arteriolar capillary) can be computed in turn. Using the mean morphometric data, the pressure profile can be obtained under the assumptions that the pressure at the first bifurcation of the capillary bed, P_0a, is a constant with a value of 25 mmHg as noted for the asymmetric model.

Appendix 2: Steady Laminar Flow in an Elastic Tube (Kassab, 2001)

If the distensibility of the blood vessels is known, the mechanics of the blood vessel can be coupled to the mechanics of blood flow to yield a pressure–flow relation for each vessel segment. This can be demonstrated for the cylindrical coronary arteries as follows: assume that the tube is long and slender, that the flow is laminar and steady, that the disturbances due to entry and exit are negligible, and that the deformed tube remains smooth and slender. These assumptions permit the use of Poiseuille’s law for a Newtonian fluid that can be stated as:

$$ dP/ dx=\left(128\mu /\pi {D}^4\right)Q $$

(5.5)

where P is the pressure, x is the axial coordinate, Q is the volume-flow rate and D, L, and μ are the diameter, length, and viscosity, respectively. In a stationary, non-permeable tube, Q is a constant throughout the length of the tube. The tube diameter is a function of x because of the elastic deformation. Pressure–diameter data (Chap. 3) show that, in the physiological pressure range, the elastic deformation can be described approximately by a linear relationship as:

$$ D-{D}^{\ast }=\alpha \left(P-{P}^{\ast}\right) $$

(5.6)

where D is the diameter at a given intravascular pressure P, D^* is the diameter corresponding to a pressure P^* and α is the compliance constant of the vessel (Kassab et al., 1999). Using Eq. (5.6), differentiation yields:

$$ dP/ dx= dP/ dD\kern1em dD/ dx=1/\alpha dD/ dx $$

(5.7)

On substituting Eq. (5.6) into Eq. (5.5) and rearranging terms, we obtain the following:

$$ {D}^4 dD=\left(128\mu \alpha Q/\pi \right)\; dx $$

(5.8)

Since the right-hand side term is a constant independent of x, we obtain the integrated result:

$$ {D}^5(x)=\left(640\mu \alpha Q/\pi \right)x+{D}^5(0) $$

(5.9)

The integration constant can be determined by the boundary condition at the entry section of the capillary, that when x = 0, D(x) = D(0). Putting x = L, at the exit section of a capillary, in Eq. (5.9) yields

$$ {D}^5(L)-{D}^5(0)=640{\mu}_{\mathrm{app}}\alpha QL/\pi $$

(5.10)

We now seek an approximate expression of Eq. (5.10) when D(L) − D(0) is small, i.e., the vessel compliance is small. Letting D(L) = D(0) + ε, expanding the left-hand side of Eq. (5.10) in power series of ε, and retaining only terms up to ε², we obtain the approximation:

$$ \left[D(L)-D(0)\right]\left\{1+2\left[D(L)-D(0)\right]/D(0)\right\}=\left(128{\mu}_{\mathrm{app}}\alpha LQ\right)/\left(\pi {D}^4(0)\right) $$

(5.11)

Using Eq. (5.6) first at x = L and then at x = 0 and subtracting, we have:

$$ D(L)-D(0)=\alpha \left[P(L)-P(0)\right] $$

(5.12)

Combining Eqs. (5.11) and (5.12), and writing D₀ for D(0), we obtain:

$$ \Delta P+\left(2\alpha /{D}_0\right)\Delta {P}^2=\left(128{\mu}_{\mathrm{app}} LQ/\pi {D}_0^4\right) $$

(5.13)

where ΔP = P(L) − P(0). The solution to Eq. (5.13) takes the form:

$$ \Delta {P}_n=\left[-{D}_n+{\left({D}_n^2+8{\alpha}_n\Delta {P}_{\mathrm{p}n}/{D}_n\right)}^{1/2}\right]/4{\alpha}_n $$

(5.14)

where ΔP^p is the Poiseuille’s pressure drop as given by the right-hand side of Eq. (5.13) and applies to each arterial vessel of order n. It is noted that when the compliance is zero (rigid vessel), the pressure drop corresponds to that given by Poiseuille’s equation. When the compliance is non-zero, however, the pressure drop is smaller than that given by Poiseuille’s equation and varies for various orders of vessels.

Appendix 3: Models of Blood Rheology (Huo & Kassab, 2009)

Fahraeus–Lindqvist effect

The relative apparent viscosity in a vessel segment in vivo, previously reported (Pries et al., 1994), can be written as:

$$ {\mu}_{\mathrm{vivo}}=\left[1+\left({\mu}_{0.45}^{\ast }-1\right)\cdot \frac{{\left(1-{H}_{\mathrm{D}}\right)}^C-1}{{\left(1-0.45\right)}^C-1}\cdot {\left(\frac{D}{D-1.1}\right)}^2\right]\cdot {\left(\frac{D}{D-1.1}\right)}^2 $$

(5.15)

where μ_vivo and H_D are the viscosity and discharge hematocrit (Hct), respectively. The apparent viscosity equals to the product of relative apparent viscosity and 1.3 cp (the viscosity of plasma). ${\mu}_{0.45}^{\ast }$ and exponent C are defined as follows:

$$ {\mu}_{0.45}^{\ast }=6\cdot {\mathrm{e}}^{-0.085D}+3.2-2.44\cdot {\mathrm{e}}^{-0.06{D}^{0.645}} $$

(5.16)

$$ C=\left(0.8+{\mathrm{e}}^{-0075D}\right)\cdot \left(-1+\frac{1}{1+{10}^{-11}\cdot {D}^{12}}\right)+\frac{1}{1+{10}^{-11}\cdot {D}^{12}} $$

(5.17)

The units for μ_vivo and D are cP and μm, respectively. Equation (5.15) reflects the Fahraeus–Lindqvist effect.

Phase-separation effect

In order to consider the phase-separation effect, Pries et al. (1989) have studied the distribution of erythrocyte at microvascular bifurcations. The fraction of the erythrocyte flow and volumetric blood flow from the mother vessel to a daughter vessel is defined as ${FQ}_{\mathrm{E}}=\frac{q_{\mathrm{daughter}}}{q_{\mathrm{mother}}}$ and ${FQ}_{\mathrm{B}}=\frac{Q_{\mathrm{daughter}}}{Q_{\mathrm{mother}}}$, respectively. Here, capital Q and small q represent the volumetric blood flow and erythrocyte flow, respectively. An empirical relation (Pries et al., 1990) has been developed to describe the distribution of volumetric blood flow and erythrocyte flow at an individual bifurcation, which can be written as:

$$ \mathrm{Logit}\left({FQ}_{\mathrm{E}}\right)=A+B\cdot \mathrm{Logit}\left(\frac{FQ_{\mathrm{B}}-{X}_0}{1-2{X}_0}\right) $$

(5.18)

where $\mathrm{Logit}\left({FQ}_{\mathrm{E}}\right)=\ln \left(\frac{FQ_{\mathrm{E}}}{1-{FQ}_{\mathrm{E}}}\right)$. A, B, and X₀ can be written as:

$$ A=\frac{-6.96\ln \left(\frac{D_{\mathrm{left}\kern0.17em \mathrm{daughter}}}{D_{\mathrm{right}\kern0.17em \mathrm{daughter}}}\right)}{D_{\mathrm{mother}}} $$

(5.19)

$$ B=1+6.98\left(\frac{1-{H}_{\mathrm{D}}}{D_{\mathrm{mother}}}\right) $$

(5.20)

$$ {X}_0=\frac{0.4}{D_{\mathrm{mother}}} $$

(5.21)

X₀ is the minimal fractional blood flow required to draw erythrocyte into the daughter branch, B is the nonlinearity of the relation between FQ_E and FQ_B, and A is the difference between the relations derived for the two daughter vessels. For FQ_B < X₀, FQ_E is equal to zero and for FQ_B > 1 − X₀, FQ_E equals to 1. A, B, and X₀ have units of μm⁻¹ and diameter D has a unit of μm.

Appendix 4: Compliance of Entire Coronary Arterial Tree (Huo & Kassab, 2009)

Table 5.1 Diameter-compliance and normalized diameter-compliance in different diameter-defined Strahler orders of pig heart

Appendix 5: Elliptical Tube Representation of Coronary Veins (Kassab et al., 1994)

At low normal venous pressures, the cross section of a vein can be approximated by an ellipse. Relative to a set of rectangular Cartesian coordinates x and y with origin located at the center, the parametric equations of the ellipse with a semi-major axis a and a semi-minor axis b are as follows:

$$ x=a\ \cos \theta; \kern1em y=b\ \sin \theta $$

(5.22)

The cross-sectional area is given by:

$$ {\displaystyle \begin{array}{l}\mathrm{Area}=1/2\oint \left(x\ dy+y\ dx\right)\\ {}\mathrm{Area}=1/2{\int}_0^{2\pi}\left( ab\ {\cos}^2\theta + ab\ {\sin}^2\theta \right) d\theta \\ {}\mathrm{Area}=\pi ab\end{array}} $$

(5.23)

An equivalent circle of radius R_e will have the same area if the following holds:

$$ \pi {R}_{\mathrm{e}}^2=\pi\ ab $$

(5.24a)

or

$$ {R}_{\mathrm{e}}={(ab)}^{1/2}=a{\left(b/a\right)}^{1/2} $$

(5.24b)

Kassab et al. (1994) showed that (a/b) varies from 1.25 to 1.96 for vessels between orders −1 and −12. Hence R_e, varies between 0.714a and 0.895a.

The circumferential length of the ellipse is given by:

$$ {\displaystyle \begin{array}{l}\mathrm{Circumference}=\oint {\left({dx}^2+{dy}^2\right)}^{1/2}\\ {}\mathrm{Circumference}={\int}_0^{2\pi }{\left({a}^2{\sin}^2\theta +{b}^2{\cos}^2\theta \right)}^{1/2} d\theta \end{array}} $$

(5.25)

which is an elliptical integral involving a and b. It may be argued that the cross-sectional shape of a vein is sensitive to internal pressure, especially if the transmural pressure is negative, when the compression of the wall causes elastic instability and buckling, whereas the circumferential length remains constant in the buckling process. Hence the circumferential length is a more stable parameter than the major and minor axes a and b. It is, however, difficult to measure this length, and its value depends on the smoothness of the wall and any irregularity.

The parameters relevant to the flow can be derived from the Navier–Stokes equation. For a steady longitudinal flow of a Newtonian viscous fluid in a long cylindrical tube of elliptical cross section subjected to a constant pressure gradient. In analogy to the exact solution of flow in a circular cylinder, the velocity profile (u):

$$ u=2U\left[1-{\left(x/a\right)}^2-{\left(y/b\right)}^2\right] $$

(5.26)

Equation (5.26) satisfies the Navier–Stokes equation and the boundary condition that u is zero on the elliptical wall described by Eq. (5.22). U is the mean velocity over the cross section. With Eq. (5.26), the Navier–Stokes equation yields the following:

$$ dP/ dx=-4\mu U\left[\left({a}^2+{b}^2\right)/\left({a}^2{b}^2\right)\right] $$

(5.27)

where μ is the coefficient of viscosity of the fluid, x is the length along the longitudinal axis of the tube, and dP/dx is the pressure gradient. Then the volume rate of flow (Q) is given by:

$$ Q=\mathrm{Area}\ U=\pi\ ab\ U=-\pi /4\mu \left[\left({a}^3{b}^3\right)/\left({a}^2+{b}^2\right)\right] dP/ dx $$

(5.28)

The blood flow conductance is given by the following coefficient:

$$ \mathrm{Conductance}=\pi /4\mu L\left[\left({a}^3{b}^3\right)/\left({a}^2+{b}^2\right)\right] $$

(5.29)

where L is the length of the tube. The resistance to flow is given by the inverse of conductance as:

$$ \mathrm{Resistance}=4\mu L/\pi \left[\left({a}^2+{b}^2\right)/\left({a}^3{b}^3\right)\right] $$

(5.30)

Equations (5.26)–(5.30) show that a and the ratio b/a are the most important parameters of venous blood flow in which the Womersley number is <1.

Finally, if the cross section is very narrow, the normal cross section may be better approximated as a rectangular slit rather than an ellipse. If h represents the thickness of the slit and the tube is rigid, then dP/dx is related to the mean flow velocity by the equation

$$ dP/ dx=-\mu {Uh}^{-2}F $$

(5.31)

in which F is a number that depends on the structure of the slit, red cell dimensions, and hematocrit. In this case, the elasticity of the tube becomes very important to hemodynamics.