Network Analysis of Coronary Circulation: II. Pulsatile Flow

Abstract

The blood flow in the coronary arteries is very pulsatile (i.e., time-dependent) with zero or even reversing (negative) flow in systole. The two major determinants of coronary flow pulsatility are the: (1) Pulsatile aortic pressure due to cyclic cardiac contraction, and (2) Myocardial/vascular interaction where the heart cyclically compresses the coronary vasculature within the myocardium. In order to understand coronary blood flow, each of these two effects must be understood in turn. We will start with the former in this section and the latter is described in a subsequent section.

Appendices

Appendix 1: Womersley Model (Huo & Kassab, 2006)

6.1.1 Governing Equations

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 ρ and viscosity μ are assumed constant.

Wave propagation in a tube is governed by wave equations for the pressure p(x,t) and volume rate of flow q(x,t). In order to derive the wave equations, the continuity (mass conversation) and momentum equations for tube flow may be written as follows:

$$ \frac{\partial A\left(x,t\right)}{\partial t}+\frac{\partial q\left(x,t\right)}{\partial x}=0 $$

(6.1)

$$ \frac{\partial {u}_{\mathrm{x}}\left(r,x,t\right)}{\partial t}+\frac{1}{\rho}\frac{\partial p\left(x,t\right)}{\partial x}=\frac{\mu }{\rho r}\frac{\partial }{\partial r}\left(r\frac{\partial {u}_{\mathrm{x}}\left(r,x,t\right)}{\partial r}\right) $$

(6.2)

It is noted that there are two equations and three independent variables (A(x, t), $q\left(x,t\right)=2\pi {\int}_0^{R(n)}{u}_x\left(r,x,t\right) rdr$, 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, the following holds:

$$ {\tau}_{\theta }=\frac{\left(p\left(x,t\right)-{p}_0\right)R(n)}{h(n)} $$

(6.3)

where τ_θ 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 the stress–strain relation is assumed linear, we may write:

$$ {\tau}_{\theta }=E(n)\frac{R^2-{R}^2(n)}{R^2(n)} $$

(6.4)

where E(n) is Young’s modulus for each order vessel and $\frac{R^2-{R}^2(n)}{R^2(n)}$ is the circumferential strain. If Eqs. (6.3) and (6.4) are combined, we obtain the following constitutive equation:

$$ p\left(x,t\right)-{p}_0=\frac{E(n)h(n)}{R(n)}\left(\frac{A\left(x,t\right)}{A(n)}-1\right) $$

(6.5)

where the radius is expressed in terms of cross-sectional area. At this point, a pressure–CSA relation is needed as provided in Chap. 3 (Kassab & Molloi, 2001). It is found that the differences in the P-CSA relation of vessels proximal to 1.5 mm in diameter (order 11, 10, 9) are relatively small (see Table 3.1 in Appendix 1, Chap. 3). Therefore, the mean of the P-CSA data for the 1.3 and 2.8 mm diameter vessels are adopted here. As determined by linear least squares fits of data, the experimental data may be expressed as follows:

$$ \frac{A\left(x,t\right)}{A(n)}=2.5\times {10}^{-6}\times \left(P\left(x,t\right)-{P}_0\right)+1.06 $$

(6.6)

which can be approximated as:

$$ P\left(x,t\right)-{P}_0=4.0\times {10}^5\times \left(\frac{A\left(x,t\right)}{A(n)}-1\right) $$

(6.7)

If Eqs. (6.5) and (6.7) are combined, we obtain $\frac{E(n)h(n)}{R(n)}=4.0\times {10}^5$. The thickness-to-radius ratio for the 1.3 and 2.8 mm vessels is very similar and is used to calculate the Young’s modulus E. When R(n) and h(n) are obtained from Guo and Kassab (Guo & Kassab, 2004; Appendix 3, Chap. 3), the static Young’s modulus E(n) is determined as ~8.0 × 10⁶ (dynes/cm²). In these simulations, it is 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 (Douglas & Greenfield, 1970; Gow, Schonfeld, & Patel, 1974) have investigated the dynamic elastic properties of canine coronary artery. Here, the ratio of E_dyn/E_stat (Gow et al., 1974) is used to adjust the dependence of Young’s modulus on frequency.

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

$$ {u}_x\left(r,x,t\right)={U}_x\left(r,x,\omega \right){e}^{i\omega t} $$

(6.8)

$$ p\left(x,t\right)=P\left(x,\omega \right){e}^{i\omega t} $$

(6.9)

$$ q\left(x,t\right)={Q}_x\left(x,\omega \right){e}^{i\omega t} $$

(6.10)

$$ A\left(x,t\right)=A\left(x,\omega \right){e}^{i\omega t} $$

(6.11)

where ω is the angular frequency, also P(x, ω), U_x(r, x, ω), Q_x(x, ω), and A(x, ω) may be simplified as P, U_x, Q, and A. If Eqs. (6.5)–(6.11) are substituted into Eqs. (6.1) and (6.2), we obtain:

$$ i\omega P\frac{A(n)R}{Eh}+\frac{\partial Q}{\partial x}=0 $$

(6.12)

and

$$ i\omega {U}_x+\frac{1}{\rho}\frac{\partial P}{\partial x}=\frac{\mu }{\rho r}\frac{\partial }{\partial r}\left(r\frac{\partial {U}_x}{\partial r}\right) $$

(6.13)

The velocity profile is first obtained by Womersley (1955) for pulsatile flow in a rigid tube as a solution to Eq. (6.13) in the form:

$$ {U}_x=\frac{1}{i\omega \rho}\left(-\frac{\partial P}{\partial x}\right)\left\{1-\frac{J_0\left[{i}^3{\alpha}^2r/R(n)\right]}{J_0\left({i}^3{\alpha}^2\right)}\right\} $$

(6.14)

where $\alpha =R(n)\sqrt{\omega \rho /\mu }$ is the Womersley number, and J₀ is zero-order Bessel function. The volume flow rate is the same as that obtained by Duan and Zamir (1992). Integrating the flow velocity over the cross-sectional area, the following wave equation is obtained:

$$ i\omega Q+\frac{A(n)}{\rho}\frac{\partial P}{\partial x}\left[1-{F}_{10}\left(\alpha \right)\right]=0 $$

(6.15)

where ${F}_{10}\left(\alpha \right)=\frac{2{J}_1\left({i}^{3/2}\alpha \right)}{i^{3/2}\alpha {J}_0\left({i}^{3/2}\alpha \right)}$, and J₁ is the first-order Bessel function. Finally, using Eqs. (6.12) and (6.15), the wave equations for the pressure P and volume rate of flow Q may be written as follows:

$$ i\omega P+\frac{c_0}{Y_0}\frac{\partial Q}{\partial x}=0 $$

(6.16)

$$ i\omega Q+\left[1-{F}_{10}\left(\alpha \right)\right]{c}_0{Y}_0\frac{\partial P}{\partial x}=0 $$

(6.17)

where ${c}_0=\sqrt{\frac{Eh}{\sqrt{\rho R}}}$ is the wave velocity without the viscous effect, and ${Y}_0=\frac{A(n)}{\rho {c}_0}$ is the characteristic admittance. Also we define Z₀ = 1/Y₀ as the characteristic impedance. If Eqs. (6.16) and (6.17) are combined, we obtain:

$$ {\omega}^2Q+\left[1-{F}_{10}\left(\alpha \right)\right]{c}_0^2\frac{\partial^2Q}{\partial {x}^2}=0\kern0.5em \mathrm{or}\kern0.5em {\omega}^2P+\left[1-{F}_{10}\left(\alpha \right)\right]{c}_0^2\frac{\partial^2P}{\partial {x}^2}=0 $$

(6.18)

When the viscous effect is incorporated into the wave velocity, the wave velocity can be defined as: $c=\sqrt{1-{F}_{10}\left(\alpha \right)}{c}_0$. Equation (6.18) can then be written as follows:

$$ {\omega}^2Q+{c}^2\frac{\partial^2Q}{\partial {x}^2}=0\kern1em \mathrm{or}\kern1em {\omega}^2P+{c}^2\frac{\partial^2P}{\partial {x}^2}=0 $$

(6.19)

Defining ${Y}_1={Y}_0\sqrt{1-{F}_{10}\left(\alpha \right)}$ and ${Z}_1={Z}_0/\sqrt{1-{F}_{10}\left(\alpha \right)}$, and solving Eqs. (6.16) and (6.19) yields:

$$ Q\left(x,\omega \right)=a\cos \left(\omega x/c\right)+b\sin \left(\omega x/c\right) $$

(6.20)

$$ P\left(x,\omega \right)={iZ}_1\left[-a\sin \left(\omega x/c\right)+b\cos \left(\omega x/c\right)\right] $$

(6.21)

where a and b are arbitrary constants of integration, $c=\sqrt{1-{F}_{10}\left(\alpha \right)}\cdot {c}_0$ $\left({c}_0,=,\sqrt{\frac{Eh}{\sqrt{\rho R}}}\right)$ is the wave velocity, ${Y}_0=\frac{A(n)}{\rho {c}_0}$ the characteristic admittance, Z₀ = 1/Y₀ the characteristic impedance, ${Y}_1={Y}_0\sqrt{1-{F}_{10}\left(\alpha \right)}$, and ${Z}_1={Z}_0/\sqrt{1-{F}_{10}\left(\alpha \right)}$. Following the transmission line method (TLM) (Christopoulos, 1995), the impedance and admittance can be defined as:

$$ Z\left(x,\omega \right)=\frac{P\left(x,\omega \right)}{Q\left(x,\omega \right)}=\frac{iZ_1\left[-a\sin \left(\omega x/c\right)+b\cos \left(\omega x/c\right)\right]}{a\cos \left(\omega x/c\right)+b\sin \left(\omega x/c\right)} $$

(6.22)

$$ Y\left(x,\omega \right)=\frac{1}{Z\left(x,\omega \right)} $$

(6.23)

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

$$ Z\left(0,\omega \right)=\frac{iZ_1b}{a} $$

(6.24)

and

$$ Z\left(L,\omega \right)=\frac{iZ_1\left[-a\sin \left(\omega L/c\right)+b\cos \left(\omega L/c\right)\right]}{a\cos \left(\omega L/c\right)+b\sin \left(\omega L/c\right)} $$

(6.25)

If Eqs. (6.24) and (6.25) are combined, we obtain:

$$ Z\left(0,\omega \right)=\frac{iZ_1\sin \left(\omega L/c\right)+Z\left(L,\omega \right)\cos \left(\omega L/c\right)}{\cos \left(\omega L/c\right)+{iY}_1Z\left(L,\omega \right)\sin \left(\omega L/c\right)} $$

(6.26)

Equation (6.26) is used to calculate the impedance/admittance in the entire coronary tree from inlet to the capillary vessels.

6.1.2 Method of Solution

The characteristic impedance, characteristic admittance, and velocity (including the viscous effect) are 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 anywhere in the current tree structure. Mass is conserved at each junction and pressure is continuous at the junction, which may be written as:

$$ Q\left(\mathrm{mother},\omega \right)=\sum Q\left(\mathrm{daughters},\omega \right) $$

(6.27)

$$ P\left(\mathrm{mother},\omega \right)=P\left(\mathrm{daughters},\omega \right) $$

(6.28)

From Eqs. (6.27) and (6.28), we may write:

$$ Y\left(L\left(\mathrm{mother}\right),\omega \right)=\sum Y\left[0\left(\mathrm{daughters}\right),\omega \right] $$

(6.29)

Once the terminal impedance/admittance of the first capillary is computed, we proceed backwards to iteratively calculate the impedance/admittance in the entire coronary tree by using Eq. (6.26) (obtained from Eqs (6.22) and (6.29)). The sawtooth pulsatile pressure produced experimentally by the piston pump is discretized by a Fourier transformation to determine the constants a and b in Eqs. (6.20) and (6.21). The flow and pressure are then calculated by using Eqs. (6.20) and (6.21).

Appendix 2: Hybrid 1D/Womersley Model (Huo & Kassab, 2007)

6.1.1 Governing Equations

The details of the mathematical derivations are outlined in Huo and Kassab (2007). Briefly, the governing equations for flow and pressure may be expressed through conservation of mass and momentum as:

$$ \frac{\partial A}{\partial t}+\frac{\partial q}{\partial x}=0 $$

(6.30)

$$ \frac{\partial q}{\partial t}+\frac{\partial }{\partial x}\left(\frac{4}{3}\ \frac{q^2}{A}\right)+\frac{A}{\rho}\frac{E_{\mathrm{stat}}{h}_0}{R_0{A}_0}\frac{\partial A}{\partial x}=-8\pi v\frac{q}{A}+v\frac{\partial^2q}{\partial {x}^2} $$

(6.31)

where A is the cross-sectional area of the vessel, q is the volumetric flow rate, ρ is the density, E_stat is the static Young’s modulus, h is the wall thickness, R₀, h₀, and A₀ are the original radius of the vessel, original wall thickness, and cross-sectional area, respectively, and v = μ/p is the kinematic viscosity. Equation (6.31) can be rewritten as follows (Huo & Kassab, 2007):

$$ \frac{\partial q}{\partial t}+\frac{8}{3}\frac{q}{A}\frac{\partial q}{\partial x}-\frac{4}{3}{\left(\frac{q}{A}\right)}^2\frac{\partial A}{\partial x}+\frac{A}{\rho}\frac{E_{\mathrm{stat}}{h}_0}{R_0{A}_0}\frac{\partial A}{\partial x}=-8\pi \nu \frac{q}{A}+v\frac{\partial^2q}{\partial {x}^2} $$

(6.32)

The constitutive equation for a coronary vessel can be expressed as Huo and Kassab (2006):

$$ p-{p}_0=\frac{E_{\mathrm{stat}}{h}_0}{R_0}\left(\frac{A}{A_0}-1\right) $$

(6.33)

where p and p₀ are the internal pressure and external pressures, respectively.

Equations (6.30), (6.32), and (6.33) are used to calculate the pulsatile blood flow in each segment of the larger arteries (e.g., the main trunk and primary branches as shown below in Figs. 6.14a, b). The viscosity is considered constant at a value of 1.1 cp to mimic the cardioplegic solution used in the experiments. Below, the relevant boundary conditions will be established in order to extend the model to the entire coronary arterial tree.

Fig. 6.14

Schematic representation of computational domains (main trunk and primary branches) in right coronary artery (RCA, a) and left anterior descending coronary (LAD) and left circumflex (LCx) arterial trees (b). (c) Schematic of branching angles. Reproduced from Huo and Kassab (2007) with permission

6.1.2 Boundary Conditions

The inlet pressure boundary condition was obtained from experimental measurements (Huo & Kassab, 2006). The impedance boundary conditions by (Olufsen, 1999, 2000) is adopted as the outlet boundary conditions at the terminals of 1D model, as shown in Figs. 6.14a, b. The Womersley’s theory as outlined in Appendix 1 is applied to the morphometric trees to represent the distal vascular beds for each of the outlets of the numerical domain. The impedance/admittance (Z(x, ω)/Y(x, ω)) is calculated at each outlet of the numerical domain. By inverse Fourier transformation, z(x, t)/y(x, t) may be obtained from Z(x, ω)/Y(x, ω). Using the convolution theorem, the new outflow boundary conditions may be obtained as follows:

$$ {\displaystyle \begin{array}{l}\kern1.2em p\left(x,t\right)={\int}_{t-T}^tq\left(x,\tau \right)z\left(x,t-\tau \right) d\tau \\ {}\mathrm{or}\\ {}\kern1.2em q\left(x,t\right)={\int}_{t-T}^tp\left(x,\tau \right)y\left(x,t-\tau \right) d\tau \end{array}} $$

(6.34)

Equation (6.34) was used to prescribe the pressure–flow boundary conditions at each outlet. For the junction boundary condition, mass is conserved at each junction:

$$ {Q}_{\mathrm{mother}}\left(x,t\right)=\sum {Q}_{\mathrm{daughter}}\left(x,t\right) $$

(6.35)

Vortices created at the bifurcations can result in loss of energy. When the effect of gravity is neglected, a loss coefficient K (Miller, 1990; Olufsen, 2000; Sherwin et al., 2003) can be incorporated into Bernoulli’s equation as follows:

$$ {\displaystyle \begin{array}{c}{P}_{\mathrm{daughter}}\left(x,t\right)={P}_{\mathrm{mother}}\left(x,\kern0.5em t\right)+\frac{\rho }{2}\left[{u}_x{\left(\mathrm{mother}\right)}^2-{u}_x{\left(\mathrm{daughter}\right)}^2\right]\\ {}-\frac{\mathrm{K}\rho }{2}{u}_x{\left(\mathrm{mother}\right)}^2\end{array}} $$

(6.36)

Equations (6.35) and (6.36) can be used to determine the pressure–flow relationship at each junction in the large coronary arteries.

6.1.3 Branching Angles

In order to calculate the loss coefficient, K, the optimum branching angles (Fig. 6.14c) are calculated using the formulations in Murray (1926) and Zamir (1978). The optimal branching angles are given as a function of area ratios as:

$$ \cos\;{\theta}_{\mathrm{BD}}=\frac{{\left(1+{\alpha}^3\right)}^{4/3}+1-{\alpha}^4}{2{\left(1+{\alpha}^3\right)}^{2/3}}\kern1.12em \mathrm{and}\kern1em \cos {\theta}_{\mathrm{BC}}=\frac{{\left(1+{\alpha}^3\right)}^{4/3}+{\alpha}^4-1}{2{\alpha}^2{\left(1+{\alpha}^3\right)}^{2/3}} $$

(6.37)

where

$$ \alpha =\frac{R_{0\mathrm{BC}}}{R_{0\mathrm{BD}}}\kern1em \mathrm{with}\kern1em {R}_{0\mathrm{BC}}<{R}_{0\mathrm{BD}} $$

(6.38)

R_0BC and R_0BD are the radii for vessel segments BC and BD (Fig. 6.14c), respectively. Once the optimum branching angles are calculated, the loss coefficients can be estimated from Miller (1990).

6.1.4 Material Parameters

The viscosity (μ) and density (ρ) of the solution are selected as 1.1 cp and 1 g/cm³, respectively, to mimic our experimental cardioplegic solution containing Albumin. The coronary wall thickness for every order is adopted from Guo and Kassab’s data ((2004); Chap. 3). The static Young’s modulus is calculated as ~7.0 × 10⁶ (dynes/cm²) as described in Huo and Kassab (2006).

6.1.5 Method of Solution

In order to solve the nonlinear hyperbolic 1D equations, the time-centered implicit (Trapezoidal) finite difference method (stable and second-order in both time and space) is adopted. The details of the mathematical derivations are outlined in Huo and Kassab (2007). The criterion for the convergence of the iteration between different meshes is set to 1 × 10⁻³ (norm-2 relative error). Finally, the mesh size (Δx) is selected as 0.05 cm and the time step (Δt) is set to 2.0 × 10⁻³. The run time for the FORTRAN program is approximately one hour using general LU decomposition method to solve the sparse matrix assembled in the largest computational domain on an AMD Opteron 240 computer. Even for a large sparse matrix with millions of equations, the matrix can be solved relatively fast by using the LU decomposition with partial pivoting and triangular system solvers through forward and back substitution (SuperLU_dist implemented in ANSI C, and MPI for communications).

Table 6.1 Various parameters for proximal LAD vessel segments in Fig. 6.14b

Appendix 3: Myocardial–Vessel Interaction (Algranati et al., 2010)

6.1.1 Network Reconstruction

Microvascular Network

Stochastic network reconstruction is based on statistical morphometric data (Kassab et al., 1999; Kassab & Fung, 1994; Kassab, Lin, & Fung, 1994; Kassab, Rider, et al., 1993). The first network (Fig. 6.15) features 174 segments: 20 arterioles, 123 capillaries, and 31 venules. Each network is supplied by an order 3 (17 μm diameter) arteriole (microvascular inlet) and two order −3 (30 μm diameter) venules (microvascular outlet). The volume of myocardium tissue supplied by this network (determined from the network dimensions) is 2.4 × 10⁻⁶ mL. The network capillary density is roughly 2800 capillaries/mm², consistent with measured data.

Fig. 6.15

A schematic of network reconstruction based on morphometric data (Chap. 2). Upper left panel: A general layout of transmural distribution of vasculature: Representative microvascular networks (rectangles) interconnected through arterial and venous trees, at four representative transmural layers. P_A, P_V are the inlet and outlet boundary pressures, respectively. Upper right panel: A symmetric arterial transmural tree. Length, diameter, and outlet flow conditions of both daughter vessels (denoted by asterisks) are mutually equal, except when daughter vessels feed different wall layers and are subject to different myocardial loading (bifurcations 1–3). Middle panel: A single microvascular network. A, V denote the microvascular arterial inlet and two venous outlets, respectively. Thin lines are capillaries while thicker ones are arterioles and venules. Broken bold lines are the perfused area boundaries. Thin arrows and full and blank circles represent capillaries that branch out in the network plane, and in the upward and downward directions, respectively. Bottom panel: Analog circuit for flow analysis in a single vessel segment. P_in, P_out are the segmental inlet and outlet pressures, respectively. P_IV and P_EV are intravascular and extravascular pressures, respectively. ℜ and C are the vessel (nonlinear) resistance and capacitance, respectively. Reproduced from (Algranati et al., 2010) by permission

Arterial and Venous Trees

To significantly reduce the computational load associated with a full-scale network (consisting of millions of vessel segments) but still retain realistic morphometric features, the reconstructed microvascular networks are placed at four representative transmural locations: at sub-epicardium (myocardial relative depth, MRD = 0.125), midwall (MRD = 0.325 and MRD = 0.625), and sub-endocardium (MRD = 0.875). These microvascular networks are interconnected and linked with the major epicardial vessels via intramyocardial arterial and venous tree-like networks, taken to be symmetric and dichotomous (Fig. 6.15) up to order 8 vessels. These are reconstructed based on the morphometric data (Kassab et al., 1994; Kassab, Imoto, et al., 1993), but assigned identical daughter vessels diameters, lengths, and outlet flow conditions at each bifurcation. The MRD of interconnecting vessels is assigned intermediate values, depending on their transmural location. This reconstructed network had 906 segments, representing the flow in four characteristic myocardial layers.

6.1.2 Mechanics of Vessel-in-Myocardium System

Following Vis, Sipkema, and Westerhof (1995), a simplified geometry is considered: each vessel is surrounded by a myocardial tissue of circular cross-section (Fig. 6.16e). The myocardial outer diameter is chosen to satisfy the measured 1:7 vessel-to-myocardium area ratio (Aliev et al., 2002; Spaan, 1991). Both tissues are considered incompressible and hyperelastic, having a common interface. Each vessel diameter is determined by stress analysis based on the force equilibrium equations of the two concentric cylinders (vessel wall and myocardial tissue) under the prescribed loading conditions of dynamic axial stretch λ_z (defined as the ratio between loaded and unloaded lengths, see below) and trans-luminal pressures ΔP. The model equations are presented below, followed by analysis of the vessel and myocardium reference configurations.

Fig. 6.16

Vessel loading configurations (a) Vessel configuration as determined from measurements. The cylinders are exposed to in situ axial stretch and trans-luminal pressure. (b) Externally unloaded: Vessel and myocardium cylinders are closed and share a common length due to tethering. (c) Untethered: both cylinders are closed but allowed to freely stretch/contract. (d) Stress free: The residual stresses in both cylinders are released by radial cut. (e) Loaded Configuration: as A, but stretch and pressure depend on the specific MVI mechanism studied. (f) Predicted and measured pressure–diameter relations of coronary artery: measured diameters in the passive heart, (Hamza et al., 2003), rectangles), normalized against diameters under zero pressure and stretch. Model predicted diameters similarly normalized, using passive (solid line) and active (dash-dot) myocardium material laws. Reproduced from Algranati et al. (2010) by permission

Model Equations

Kinematics: The axisymmetric mappings between each pair of configurations i and i+1 (Fig. 6.16 in cylindrical coordinates is (r, θ, z)_i → (r, θ, z)_i+1 prescribed by:

$$ {r}_{i+1}={r}_{i+1}\left({r}_j\right);\kern0.6em {\theta}_{i+1}=\left({OA}_{i+1}/{OA}_i\right)\cdot {\theta}_i;\kern0.72em {z}_{i+1}={\Lambda}_{i+1,i}\cdot {z}_i $$

(6.39)

where OA denotes the opening angle, L is the cylinder length and the stretch ratio Λ_i+1,i = L_i+1/L_i. The incompressibility constraint implies that:

$$ {r}_{i+1}=\sqrt{{\left({r}_{i+1}^{\mathrm{in}}\right)}^2+\left[{\left({r}_i\right)}^2-{\left({r}_i^{\mathrm{in}}\right)}^2\right]\cdot \left({OA}_i/{OA}_{i+1}\right)/{\Lambda}_{i+1,i}} $$

(6.40)

The Green-Lagrange strain is E = (F^TF − I)/2, where I is the unit matrix and F is the deformation gradient for each mapping between configurations. Assuming no twist, F is given by:

$$ \mathbf{F}=\operatorname{diag}\;\left(\partial {r}_{i+1}/\partial {r}_i\kern1em {\Lambda}_{i+1,i}\cdot {OA}_{i+1}/{OA}_i\kern1em {\Lambda}_{i+1,i}\right) $$

(6.41)

Explicit expressions for the deformation gradient of each mapping are given in Algranati et al. (2010).

Equilibrium Equations: The Cauchy stress tensor T is derived from the strain energy function W of each material (vessel wall and myocardium, see below) via the hyperelastic relationshipT =  − PI + F ⋅ (∂W/∂E) ⋅ F^T. The equilibrium equations in the circumferential and axial directions imply that the Cauchy stress components T_rθ and T_rz vanish. The radial force equilibrium equation is ∂T_rr/∂r + (T_rr − T_θθ)/r = 0. By applying the axial and radial equilibrium equations, the external axial force F_z and the trans-luminal pressure ΔP can be expressed in terms of the components T_ij of the tissue stress tensor T as follows (Humphrey, 2002):

$$ {F}_z=\pi {\int}_{r_i}^{r_o}\left(2{T}_{zz}-{T}_{rr}-{T}_{\theta \theta}\right) rdr;\kern1em \Delta P={\int}_{r_i}^{r_o}\frac{T_{\theta \theta}-{T}_{rr}}{r} dr $$

(6.42)

Vessel and Myocardium Constitutive Properties

Vessel wall mechanics is studied in the left anterior descending (LAD) artery under positive trans-luminal pressures (Wang et al., 2006) and represented by Fung-type exponential material law (Algranati et al., 2010). The corresponding parameters are estimated for 10 samples. The passive and active myocardial mechanics are previously studied on 7 samples and described by invariant based material laws (Lin & Yin, 1998).

To obtain a representative vessel/myocardium pair, each of the 10 vessel parameter sets is combined with each of the 7 parameter sets of the passive myocardial sample, and properties of each of the vessel/myocardium pairs is used as inputs to evaluate the passive in vivo pressure–diameter relationship. The pair having the best fit to the in situ data of swine large coronary arteries (Hamza et al., 2003) (Fig. 6.16f) is selected for further analysis of MVI mechanisms.

Reference Configurations

Stress analysis requires knowledge of the reference stress-free configuration of both vessel and myocardium. These references have to be estimated in order to determine the true states of stress and strain. The available data consists of statistics of the in situ diameter, length, and wall thickness (Guo & Kassab, 2004; Kassab et al., 1994; Kassab & Fung, 1994; Kassab, Rider, et al., 1993) taken under vasodilation, no myocardial activation, and fixed stretch and intravascular pressure (vessel configuration). The data however, do not account for transmural morphometric heterogeneity. To incorporate the latter, the reconstructed diameters are first modified by up to ±10% from the vessel cast values in a linear transmural manner to comply with the observed twice higher endocardial than epicardial flows in diastolic vasodilated hearts (Goto et al., 1991). To obtain the reference configurations, the vessel/myocardium equilibrium equations (Eq. 6.42) are solved subject to the relevant loading boundary conditions. The latter are as follows:

• 1. Vessel Configuration (Fig. 6.16a): Kassab and co-workers (Guo & Kassab, 2004; Kassab et al., 1994; Kassab & Fung, 1994; Kassab, Imoto, et al., 1993) measured vessel diameters under fixed cast pressure. The loading conditions in this configuration are ΔP^v + ΔP^m = casting pressure, where v and m superscripts denote vessel and myocardium, respectively. Each vessel’s specific cast pressure is taken from a steady-state coronary flow analysis.

• 2. Unloaded Configuration (Fig. 6.16b): The transition to this configuration is prescribed by the mapping from the vessel cast configuration (Fig. 6.16a). The loading conditions are ${F}_z^{\mathrm{v}}+{F}_z^{\mathrm{m}}=0;\kern1em \Delta {P}^{\mathrm{v}}+\Delta {P}^{\mathrm{m}}=0$. The axial stretch λ_z is taken to remain constant during this mapping.

• 3. Untethered Configuration (Fig. 6.16c): Unloaded coronary vessels are not stress-free. When myocardial tethering is removed, large epicardial arteries are found to shorten by 40% in swine (Wang et al., 2006). The untethered configuration is obtained upon mapping from the tethered unloaded configuration (Fig. 6.16b). The loading conditions in this untethered state are ${F}_z^{\mathrm{v}}=0;\kern1em {F}_z^{\mathrm{m}}=0;\kern1em \Delta {P}^{\mathrm{v}}+\Delta {P}^{\mathrm{m}}=0$, where it is assumed that the two cylinders maintain a common but stress-free interface.

• 4. Stress-Free (Reference) Configuration (Fig. 6.16d): The tethered-free vessel and myocardium are still not stress-free, but rather loaded by internal residual stress. Their magnitudes are quantified by the measured opening angles (OA) of the corresponding cylinders when cut open radially (Chap. 3). For coronary vessels, OA are specimen dependent. For the myocardium OA = 2.75 rad. (Lanir et al., 1996). The stress-free reference configuration is obtained by mapping between configurations A and D (Fig. 6.16). Stress and strain analysis is possible if the radii and stretch ratios at each configuration are known. Since they are not, they are solved by applying the vessel-in-myocardium model equations under the six loading boundary conditions listed above (one in section i, two in ii and three in iii), subject to the assumptions of incompressibility of and common interface between cylinders (Algranati et al., 2010a). The solution of this highly nonlinear system is obtained by MATLAB^® ga code for genetic algorithm search and MATLAB^® fsolve.

Loaded Vessel Diameter (Fig. 6.17e)

With the stress-free configuration determined, the vasodilated vessel diameters are evaluated (using the above MATLAB^® codes) to optimally satisfy force equilibrium (Eq. 6.42), subject to the vessel internal and external (MVI dependent) loading conditions. This is done for arteries, capillaries, and veins, under prescribed conditions of trans-luminal pressure, dynamic axial stretch, and myocardial activation. To obtain the corresponding in vivo diameters required for the network flow analysis, modifications are needed to account for the vessel dynamic axial stretch during the cardiac cycle, and for the autoregulatory myogenic response and myocardial state of activation, as discussed below.

Fig. 6.17

MVI mechanisms. (a) Varying elasticity (VE): flow is affected by the activation mediated changes in myocardial stiffness (represented by a spring); (b) Shortening-induced intracellular pressure (SIP): flow is regulated by the difference between intravascular and contraction-induced myocyte intracellular pressures (arrows); (c) Cavity-induced extracellular pressure (CEP): extravascular pressure (P_EV) is the interstitial pressure which varies linearly (Heineman & Grayson, 1985) from cavity pressure (LVP) at the endocardium to atmospheric pressure (P_atm) at the epicardium. Reproduced from Algranati et al. (2010) by permission

Dynamic Vessel Stretch (λ_z)

Vessels are dynamically stretched during the cardiac cycle, together with the surrounding myocardium. This stretch, added to the constant in situ tethering stretch, affects the vessel diameter. This effect can be evaluated by solving the vessel-in-myocardium model at each time point and for each vessel. To reduce computational load, the stretched diameter is evaluated by linear interpolation between the vessel highest and lowest levels of stretch. The maximum and mean errors associated with this interpolation are found to be 4% and <1% respectively, in all vessels at all pressures, in either passive or fully active myocardium. The dynamic vessel stretch depends on the vessel orientation. Vessels oriented along myocytes stretch in proportion to the myocytes stretch, whereas vessels perpendicular to myocytes stretch in proportion to myocytes thickening, and therefore shorten during myocyte elongation. Hence, penetrating vessels (orders 5–8) are assumed to stretch in proportion to the myocardial wall thickening, whereas capillaries (except for cross-connections) are assumed to stretch in proportion to the sarcomere stretch ratio. Other vessels are assumed to be randomly oriented, and therefore unaffected by myocardium contraction (i.e., no dynamic stretch).

Myocardial Activation

This is accounted for by calculating the loaded diameters under passive and fully active myocardium (D_p and D_a, respectively). The diameter under intermediate activation is determined as a linear interpolation between these two states.

Autoregulation

The entire complexity of tone regulation is not explicitly accounted for. Yet, in order to compare predictions with in vivo (autoregulated) data, arterioles diameters are reduced by 30% from their calculated dilated values, in line with data for coronary arteries of <200 μm diameter (Kuo et al., 1995).

The Computational Scheme

Equations (6.39)–(6.42) are used to evaluate the dependence of vessel diameter D on the cast value D_cast, and on the trans-luminal pressure ΔP. These values are presented as surfaces of D = D(D_cast, ΔP) for 12 cases: one for each combination of vessel type (artery, capillary, vein), lowest and highest axial stretch, and passive and active myocardium. Based on the experimental results of Hamza et al. (2003), the above response surfaces are fitted for each vessel by a sigmoid function expressed by:

$$ {D}_{\mathrm{loaded}}\left(\Delta P\right)=\frac{\left({D}_{+}-{D}_{-}\right)}{1+\left[\frac{D_{+}-{D}_0}{D_0-{D}_{-}}\right]\times \left[\exp \left(\ln \left(\frac{D_{+}-{D}_0}{D_0-{D}_{-}}\right)\times \left(\frac{-\Delta P}{\Delta {P}_{1/2}}\right)\right)\right]}+{D}_{-} $$

(6.43)

where D₊ and D₋ denote maximum inflation and deflation diameters, respectively, and ΔP_1/2 is the trans-luminal pressure corresponding to the average of D₊ and D₋. These parameters where estimated to best fit the diameter vs. pressure curve of each vessel. The maximum and mean errors associated with this fit relative to the exact solution is found to be 6% and <1%, respectively, in all vessels under all simulated loading conditions.

6.1.3 MVI Network Flow Analysis

Conservation of mass requires that the difference between in and out discharges of each vessel n (Fig. 6.15, bottom panel), ${Q}_{\mathrm{in}}^n$ and ${Q}_{\mathrm{out}}^n$, respectively, should equal the time derivative of the vessel’s volume (Vⁿ), i.e.:

$$ {Q}_{\mathrm{in}}^n(t)-{Q}_{\mathrm{out}}^n(t)=\frac{P_{\mathrm{in}}^n(t)-{P}_{\mathrm{IV}}^n(t)}{\Re^n(t)/2}+\frac{P_{\mathrm{out}}^n(t)-{P}_{\mathrm{IV}}^n(t)}{\Re^n(t)/2}=\frac{dV^n}{dt}=\frac{d}{dt}{\left(\frac{\pi {D}^2(t)L(t)}{4}\right)}^n $$

(6.44)

where ${P}_{\mathrm{in}}^n$ and ${P}_{\mathrm{out}}^n$ denote vessel inlet and outlet pressures, respectively. The hydraulic capacity, C(t), is defined as:

$$ C(t)\equiv \frac{V}{P_{\mathrm{IV}}(t)-{P}_{\mathrm{EV}}(t)}=\frac{\pi /4\ D{(t)}^2L(t)}{\Delta P} $$

(6.45)

Equation (6.45) extends the common definition of capacity in nonlinear intramyocardial pump models (e.g., C(t) ≡ d(Volume)/d(P_IV(t) − P_EV(t)) to the case in which changes in vessel lumen volume are caused not only by trans-luminal pressure, but also by changes in axial stretch and myocardial activation.

Flow in each vessel is analyzed using a three-element Windkessel model consisting of two identical nonlinear resistors and one nonlinear capacitor (Fig. 6.15 lower panel). This lumped segment flow model is previously validated (Jacobs et al., 2008) against a distributive vascular model (Fibich et al., 1993). At each network bifurcation, mass conservation implies that the sum of discharges Q^jk should vanish, i.e.:

$$ \sum \limits_{j=1}^3{Q}^{jk}=\sum \limits_{j=1}^3\frac{P_{\mathrm{IV}}^j-{P}_{\mathrm{bif}}^k}{\Re^j/2}=0\kern1em k=1,2,\dots, \mathrm{Bifurcation}\;\mathrm{No}. $$

(6.46)

Here ${P}_{\mathrm{IV}}^j$ denotes the intravascular pressure in each of the 3 vessels composing the kth bifurcation, and ${P}_{\mathrm{bif}}^k$ is the bifurcation pressure. The vessel hydraulic resistance ℜ is calculated from Poiseuille's law, i.e.:

$$ \Re (t)\equiv \frac{Q(t)}{P_{\mathrm{in}}(t)-{P}_{\mathrm{out}}(t)}=\frac{128\mu (t)L(t)}{\pi D{(t)}^4} $$

(6.47)

where L, D, and μ are the vessel length, diameter, and blood apparent viscosity, respectively. The latter is taken to vary with diameter, following Pries et al. (1994). The nonlinearity of ℜ(t) stems from the vessel elasticity, expressed by the dependence of the diameter D and length L on the MVI-dependent extravascular loading. Since each bifurcation pressure P_bif equals either P_in or P_out of the vessels forming that bifurcation, Eq. (6.46) can be combined with Eq. (6.44), resulting in a system of N nonlinear ordinary differential equations (N denotes the number of network vessels), which is iteratively solved using the MATLAB^® ode15s solver until satisfying periodicity condition. The solution process requires the boundary conditions of the network inlet and outlet pressure, as well as MVI-dependent extravascular loading based on the various conditions defined below.

Varying Elasticity (VE, Fig. 6.17a)

Contractility is assumed to affect coronary flow through activation-dependent changes of myocardial stiffness (Vis et al., 1995). To analyze this effect alone, the myocardium is modeled as a hyperelastic solid, and the effects of LV cavity pressure and associated extracellular (interstitial) pressure are ignored. Hence, the boundary conditions for this mechanism are: (1) vanishing extravascular pressure (P_EV = 0), and (2) linear dependence of vessels dynamic diameter on activation, namely:

$$ {P}_{\mathrm{EV}}=0;\kern1em D=\left[1-\mathrm{activation}\right]\times {D}_{\mathrm{p}}+\mathrm{activation}\times {D}_{\mathrm{a}} $$

(6.48)

where D_p and D_a are the vessel diameters under passive and fully active myocardium, respectively (Appendix “Mechanics of Vessel-in-Myocardium System”). This analysis extends the two time points (diastole and peak systole) study of Vis, Sipkema, and Westerhof (1997) to the entire cardiac cycle.

Shortening-Induced Intracellular Pressure (SIP, Fig. 6.17b)

Myocytes are modeled as membrane-contained fluid compartments that surround the vessels. During shortening, their thickening (lateral expansion of their membranes) is due to an internal pressure elevation. This intramyocyte pressure is transmitted to the vessel due to the impingement by the vessels which is based on data and a model (Rabbany et al., 1994) that showed a linear increase in pressure with contractile shortening. Hence, P_EV is proportional to myocytes shortening (expressed by their stretch ratio, SSR) through a scale factor α, with baseline value of 0.14 mmHg/% shortening, implying a systolic extravascular pressure of 15 mmHg in the intact tissue. This pressure is lower than the measured intracellular pressure in isolated myocytes (Rabbany et al., 1994), but in the range of measured extravascular pressure in intact papillary muscle (Heslinga, Allaart, Yin, & Westerhof, 1997). The baseline value of α is varied through a sensitivity analysis (see below). Here, the vessel response to pressure is unaffected by activation:

$$ {P}_{\mathrm{EV}}=\alpha \times \left[1- SSR(t)\right];\kern1em D={D}_{\mathrm{p}} $$

(6.49)

Cavity-Induced Extracellular (Interstitial) Pressure (CEP, Fig. 6.17c)

This mechanism relies on the underlying assumption of both the intramyocardial pump (Spaan et al., 1981) and the vascular waterfall (Downey & Kirk, 1975) models. P_EV is assumed to stem from the LVP alone and is taken to vary linearly (Heineman & Grayson, 1985) with transmural position (expressed by myocardial relative depth, MRD). As in the SIP mechanism, activation is not considered in this interaction mode, namely:

$$ {P}_{\mathrm{EV}}=\mathrm{MRD}\times \mathrm{LVP}(t);\kern1em D={D}_{\mathrm{p}} $$

(6.50)

CEP+VE

The extracellular pressure and varying elasticity are combined in this scenario. The material law of surrounding myocardium is activation dependent (as in VE) and the extracellular pressure is applied to the myocardium cylinder external surface, i.e.:

$$ {P}_{\mathrm{EV}}=\mathrm{MRD}\times \mathrm{LVP}(t);\kern1em D=\left[1-\mathrm{activation}\right]\times {D}_{\mathrm{p}}+\mathrm{activation}\times {D}_{\mathrm{a}} $$

(6.51)

CEP+SIP

Here, myocytes are assumed to contract within an LVP-derived pressurized interstitium. Hence, the assigned P_EV equals the algebraic sum of extracellular pressure and of the shortening-dependent intracellular pressure such that:

$$ {P}_{\mathrm{EV}}=\mathrm{MRD}\times \mathrm{LVP}(t)+\alpha \times \left[1-\mathrm{SSR}(t)\right];\kern1em D={D}_{\mathrm{p}} $$

(6.52)

A combination of both VE and SIP or of all three basic mechanisms are not considered since the extravascular myocytes can be considered either as a solid (as in the VE mechanism) or as fluid (as is with the SIP), but not as both together.

Effects of Dynamic Axial Stretch on the Flow

Under all tested mechanisms, the instantaneous diameter and length of each vessel are taken to be affected by the dynamic axial stretch as outlined in Appendix “Mechanics of Vessel-in-Myocardium System”.

Table 6.2 Comparison of various mechanisms (VEM-varying elasticity model; IPM-intramyocyte pressure model; CPM-cavity pressure model) with experimental observations

Appendix 4: Coronary Flow Regulation

6.1.1 The Network Structure

A morphological reconstruction of left circumflex (LCx) arterial tree (see Chap. 2) is used for a comprehensive flow analysis. A sub-endocardial subtree consisting of 400 vessels (orders 6 to 0) is pruned from the LCx coronary tree. The subtree contained 195 bifurcations, 3 trifurcations, and 79 terminal order 1 vessels. Diameters are randomly assigned to the tree vessel segments based on morphological statistical data of the measured diameters (Chap. 2) at 80 mmHg. Preliminary flow analysis revealed that this method of diameter assignment resulted in excessive heterogeneity in both perfusion and wall shear stress. An iterative diameter re-assignment is implemented to reduce flow heterogeneity to be in agreement with published data. Re-assignment of diameters is based on the assumption that the flow in each upstream network vessel should be proportional to the number of terminal vessels it perfuses. The iteration is terminated when perfusion heterogeneity (coefficient of variation, CV = SD/mean) is reduced to within the range of published data.

To study the effects of transmural location on the network flow, the same sub-endocardial subtree is “implanted” in the sub-epicardium layer. The network flow analysis is subjected to the sub-epicardium boundary conditions (i.e., inlet and outlet pressures and extravascular loading by the myocardium–vessel interaction, MVI).

6.1.2 Network Flow Analysis

The flow in each elastic vessel is modeled by a three-element Windkessel made of two nonlinear resistors in series and one parallel capacitor as shown in Figs. 6.18a, b. Flow resistance is governed by Poiseuille’s equation and vessel capacitance by the vessel pressure–diameter relationship (PDR). Network flow is solved by imposing flow continuity at each vessel midpoint and at each network junction, i.e., the net flow at each junction vanishes to conserve mass. This formulation resulted in a system of ordinary differential equations which are numerically solved as outlined below. The network dynamic flow is analyzed subject to boundary conditions of inlet and outlet pressures, and the extravascular loading by myocardial contractions.

Fig. 6.18

Flow models in a single vessel and vessel bifurcation. (a) Scheme of a single uniform cylindrical coronary vessel and its pressure boundary conditions: P_in, the inlet pressure; P_out, the outlet pressure; P_T, the surrounding tissue pressure. (b) Lumped three-element Windkessel model of a single vessel segment consisting of two nonlinear resistors connected in series and a parallel nonlinear capacitor. (c) Lumped model of a single bifurcation of a parent vessel into two daughters. Reproduced from Namani et al. (2018) by permission

Under the myogenic and shear regulation, network flow is iteratively solved by adjusting each vessel diameter according to the local pressure and flow. For cases of metabolic regulation set to achieve a certain level of perfusion to match the metabolic demand, coronary flow is solved by optimizing the distribution of metabolic activation which produced terminal flows similar to the desired perfusion level.

Flow in Single Vessel

Flow in a single vessel (Fig. 6.18a) is governed by Poiseuille relation as:

$$ Q(t)=\frac{\pi R{(t)}^4}{8\upmu \left(\mathrm{R}\right)}\frac{\left({P}_{\mathrm{in}}(t)-{P}_{\mathrm{out}}(t)\right)}{\mathrm{L}} $$

(6.53)

where Q(t) is the flow rate, R(t) is the vessel radius, (P_in(t)–P_out(t)) is the longitudinal input/output pressure drop, μ(R) is the dynamic viscosity which is a function of vessel radius, and L is the length of the vessel (Jacobs et al., 2008). The shear stress, τ, is given by:

$$ \tau =\frac{\left({P}_{\mathrm{in}}(t)-{P}_{\mathrm{out}}(t)\right)R(t)}{2L} $$

(6.54)

Flow in each vessel is simulated by a validated lumped three-element Windkessel model (Jacobs et al., 2008). Two nonlinear resistors (${\mathfrak{R}}_1$, ${\mathfrak{R}}_2$) in series are connected in parallel to a capacitor C (Fig. 6.18b). The capacitive element represents the pressure-induced volume change in each elastic vessel. The resistance $\mathfrak{R}$ (and conductance G), and the capacitance C of each vessel are:

$$ \mathrm{\Re}(t)=2{\mathrm{\Re}}_1(t)=2{\mathrm{\Re}}_2(t)=\frac{8\mu (t)L}{\pi R{(t)}^4},\kern1em G(t)=1/\mathrm{\Re}(t) $$

(6.55)

$$ C(t)=\frac{dV}{d\Delta P}=2\pi LR(t)\frac{dR}{d\Delta P} $$

(6.56)

The junction of the three elements in the single vessel (Fig. 6.18b) is the geometric center of the vessel with an unknown pressure, P_mid. The bifurcation node between vessels is the junction of three resistors, each belonging to a different vessel (Fig. 6.18c). The nodal pressure at the junction of the three vessels is P_node. The unknowns P_mid and P_node are solved based on conservation of mass.

The model is based on the following assumptions: (a) each vessel is of uniform cross-section and wall thickness, (b) flow in the vessels is laminar, (c) blood viscosity in each vessel is diameter dependent following measured data of the apparent viscosity in microvascular beds (Pries et al., 1994), (d) active and passive vessel properties are homogeneous in each vessel but vary between vessels, (e) dynamic extravascular pressure P_T is constant along each vessel but varies between vessels depending on their transmural location, and (f) active vessel response depends on the time-averaged pressure, flow, and metabolic signal.

Network Flow Analysis

Flow in the multiple vessel network is determined via the iterative solution of a system of ordinary differential equations (ODEs) based on the conditions of conservation of mass which requires that the net flow in each node be zero, and hence for a vessel midpoint as:

$$ {\displaystyle \begin{array}{l}{Q}_{\mathrm{in}}^i+{Q}_{\mathrm{out}}^i+\frac{dV^i}{dt}=\frac{P_{\mathrm{in}}^i(t)-{P}_{\mathrm{mid}}^i(t)}{\Re^i/2}+\frac{P_{\mathrm{out}}^i(t)-{P}_{\mathrm{mid}}^i(t)}{\Re^i/2}\\ {}-\frac{d}{dt}\left[\pi {\left({R}^i\right)}^2{L}^i\left({P}_{\mathrm{mid}}^i-{P}_{\mathrm{T}}^i\right)\right]=0;\kern1em i=1,2,3\dots \end{array}} $$

(6.57)

where ${P}_{\mathrm{T}}^i$ is the given input signal of the extravascular pressure which depends on the myocardial transmural wall location. The mass conservation at the midpoint in each vessel is given by:

$$ \frac{P_{\mathrm{in}}^i(t)-{P}_{\mathrm{mid}}^i(t)}{{\mathrm{\Re}}_1^i/2}+\frac{P_{\mathrm{out}}^i(t)-{P}_{\mathrm{mid}}^i(t)}{{\mathrm{\Re}}_2^i/2}={C}^i\frac{d\left({P}_{\mathrm{mid}}^i-{P}_{\mathrm{T}}^i\right)}{dt} $$

(6.58)

where P_in(t) is the vessel input pressure signal and P_out(t) is the vessel output pressure.

The net flow at a bifurcation or trifurcation between a mother vessel and daughter vessels at a designated “network node” is zero. Application of mass conservation at each node yields additional equations for the nodal pressures. For an ith vessel, which is neither source nor sink, mass balance at the vessel inlet (Fig. 6.18a) yields:

$$ \left({P}_{\mathrm{mid}}^{i,{n}_{-1}}-{P}_{\mathrm{in}}^i\right){G}^{i,{n}_{-1}}+\left({P}_{\mathrm{mid}}^i-{P}_{\mathrm{in}}^i\right){G}^i+\left({P}_{\mathrm{mid}}^{i,{n}_{-2}}-{P}_{\mathrm{in}}^i\right){G}^{i,{n}_{-2}}=0 $$

(6.59)

Hence, the inlet nodal pressure as a function of neighboring vessel pressures and conductance is given by:

$$ {P}_{\mathrm{in}}^i(t)=\frac{\kern0.5em {P}_{\mathrm{mid}}^i{G}^i+\kern0.5em {P}_{\mathrm{mid}}^{i,{n}_{-1}}{G}^{i,{n}_{-1}}+\kern0.5em {P}_{\mathrm{mid}}^{i,\kern0.5em {n}_{-2}}{G}^{i,{n}_{-2}}}{G^i+{G}^{i,{n}_{-1}}+{G}^{i,{n}_{-2}}} $$

(6.60)

Similarly, applying mass balance at the outlet node of the ith vessel gives:

$$ {P}_{\mathrm{out}}^i(t)=\frac{\kern0.5em {P}_{\mathrm{mid}}^i{G}_i+\kern0.5em {P}_{\mathrm{mid}}^{i,{n}_1}{G}^{i,{n}_1}+\kern0.5em {P}_{\mathrm{mid}}^{i,\kern0.5em {n}_2}{G}^{i,{n}_2}}{G^i+{G}^{i,{n}_1}+{G}^{i,{n}_2}} $$

(6.61)

For a source vessel, the inlet pressure Pⁱⁿ(t) and for a sink vessel, the outlet pressure, P^out(t) are prescribed boundary conditions. The governing equations are assembled for the network into a system of ordinary differential equations (ODEs) which is written in matrix form as:

$$ \frac{d{\boldsymbol{P}}_{\mathbf{mid}}}{dt}={\boldsymbol{AP}}_{\mathbf{mid}}+\boldsymbol{B} $$

(6.62)

Expressions for the A and B matrices are given below. The trifurcation node is the junction of four resistors, each belonging to a different vessel. Similar to a bifurcation, the nodal pressure (P_node) at the junction of the four elements is an unknown. The network flow is solved based on mass conservation at each vessel midpoint and at each vessel nodal junction.

The network structure matrix in Eq. (6.62) for a subtree is used to build the coefficient matrices A (n × n) and B (n × 1). The indices of the non-zero elements at an ith row in A correspond to the ith vessel properties and its neighbors. A vessel connected to bifurcating vessels at its origin and end has indices (i, n_-1), (i, n_-2), (i, n₁), and (i, n₂) for the mother, sister, and two daughters respectively. The corresponding conductances are: ${G_i}^{n_{-1}},{G_i}^{n_{-2}},{G_i}^{n_1},{G_i}^{n_2}$. A vessel with a trifurcating branch at its end has an additional daughter with index (i, n₃) with conductance${G_i}^{n_3}.$A vessel with a trifurcating vessel at its origin has an additional sister with index (i, n_-21).

For a vessel, i, connected to bifurcations at both ends, the elements of matrices A are as follows:

$$ \frac{dP_i^{\mathrm{mid}}}{dt}={A}_{i,{n}_{-1}}{P}_{i,{n}_{-1}}^{\mathrm{mid}}+{A}_{i,{n}_{-2}}{P}_{i,{n}_{-2}}^{\mathrm{mid}}+{A}_{i,i}{P}^{\mathrm{mid}}+{A}_{i,{n}_1}{P}_{i,{n}_1}^{\mathrm{mid}}+{A}_{i,{n}_2}{P}_{i,{n}_2}^{\mathrm{mid}}+{B}_i $$

(6.63)

$$ {A}_{i,i}=\frac{2{G}_i}{C_i}\left(\frac{G_i}{G_i+{G_i}^{n_{-1}}+{G_i}^{n_{-2}}}+\frac{G_i}{G_i+{G_i}^{n_1}+{G_i}^{n_2}}-2\right),\kern1em {A}_{i,{n}_{-1}}=\frac{2{G}_i}{C_i}\left(\frac{{G_i}^{n_{-1}}}{G_i+{G_i}^{n_{-1}}+{G_i}^{n_{-2}}}\right) $$

(6.64)

$$ {A}_{i,{n}_{-2}}=\frac{2{G}_i}{C_i}\left(\frac{{G_i}^{n_{-2}}}{G_i+{G_i}^{n_{-1}}+{G_i}^{n_{-2}}}\right),\kern1em {A}_{i,{n}_1}=\frac{2{G}_i}{C_i}\left(\frac{{G_i}^{n_1}}{G_i+{G_i}^{n_1}+{G_i}^{n_2}}\right),\kern1em {A}_{i,{n}_2}=\frac{2{G}_i}{C_i}\left(\frac{{G_i}^{n_2}}{G_i+{G_i}^{n_1}+{G_i}^{n_2}}\right), $$

(6.65)

where P^mid is the pressure in the vessel midpoint, and C is its capacity (Eq. 6.56).

For a vessel, i, connected to a bifurcation at its origin and a trifurcation at its end, the elements of matrices A are given by:

$$ {\displaystyle \begin{array}{ll}\frac{dP_i^{\mathrm{mid}}}{dt}=& {A}_{i,{n}_{-1}}{P}_{i,{n}_{-1}}^{\mathrm{mid}}+{A}_{i,{n}_{-2}}{P}_{i,{n}_{-2}}^{\mathrm{mid}}+{A}_{i,i}{P}^{\mathrm{mid}}\\ {}& +{A}_{i,{n}_1}{P}_{i,{n}_1}^{\mathrm{mid}}+{A}_{i,{n}_2}{P}_{i,{n}_2}^{\mathrm{mid}}+{A}_{i,{n}_3}{P}_{i,{n}_3}^{\mathrm{mid}}+{B}_i\end{array}} $$

(6.66)

$$ {A}_{i,i}=\frac{2{G}_i}{C_i}\left(\frac{G_i}{G_i+{G_i}^{n_{-1}}+{G_i}^{n_{-2}}}+\frac{G_i}{G_i+{G_i}^{n_1}+{G_i}^{n_2}+{G_i}^{n_3}}-2\right) $$

(6.67)

$$ {A}_{i,{n}_1}=\frac{2{G}_i}{C_i}\left(\frac{{G_i}^{n_1}}{G_i+{G_i}^{n_1}+{G_i}^{n_2}+{G_i}^{n_3}}\right),\kern1em {A}_{i,{n}_2}=\frac{2{G}_i}{C_i}\left(\frac{{G_i}^{n_2}}{G_i+{G_i}^{n_1}+{G_i}^{n_2}+{G_i}^{n_3}}\right) $$

(6.68)

For a vessel, i, connected to a trifurcation at its origin and a bifurcation at its end, the elements of matrices A are given by:

$$ {A}_{i,i}=\frac{2{G}_i}{C_i}\left(\frac{G_i}{G_i+{G_i}^{n_{-1}}+{G_i}^{n_{-2}}+{G_i}^{n_{-21}}}+\frac{G_i}{G_i+{G_i}^{n_1}+{G_i}^{n_2}}-2\right) $$

(6.69)

$$ {A}_{i,{n}_{-1}}=\frac{2{G}_i}{C_i}\left(\frac{{G_i}^{n_{-1}}}{G_i+{G_i}^{n_{-1}}+{G_i}^{n_{-2}}+{G_i}^{n_{-21}}}\right),\kern1em {A}_{i,{n}_{-2}}=\frac{2{G}_i}{C_i}\left(\frac{{G_i}^{n_{-2}}}{G_i+{G_i}^{n_{-1}}+{G_i}^{n_{-2}}+{G_i}^{n_{-21}}}\right) $$

(6.70)

If an ith vessel is terminating, the elements at each row in A are:

$$ {A}_{i,i}=\frac{2{G}_i}{C_i}\left(\frac{G_i}{G_i+{G_i}^{n_{-1}}+{G_i}^{n_{-2}}}-2\right),\kern1em {A}_{i,{n}_{-1}}=\frac{2{G}_i}{C_i}\left(\frac{{G_i}^{n_{-1}}}{G_i+{G_i}^{n_{-1}}+{G_i}^{n_{-2}}}\right) $$

(6.71)

$$ {A}_{i,{n}_{-2}}=\frac{2{G}_i}{C_i}\left(\frac{{G_i}^{n_{-2}}}{G_i+{G_i}^{n_{-1}}+{G_i}^{n_{-2}}}\right) $$

(6.72)

For the source vessel, the elements of A are as follows:

$$ {A}_{1,1}=\frac{2{G}_1}{C_1}\left(\frac{G_i}{G_1+{G_1}^{n_1}+{G_1}^{n_2}}-2\right),\kern1em {A}_{1,{n}_1}=\frac{2{G}_1}{C_1}\left(\frac{{G_i}^{n_1}}{G_1+{G_1}^{n_1}+{G_1}^{n_2}}\right),\kern1em {A}_{1,{n}_{-2}}=\frac{2{G}_1}{C_1}\left(\frac{{G_1}^{n_2}}{G_1+{G_1}^{n_1}+{G_1}^{n_2}}\right) $$

(6.73)

For all interior vessels, the elements of B are given by:

$$ {B}_i=\frac{dP_i^t}{dt} $$

(6.74)

For a terminal vessel, i, the elements at each row in B are:

$$ {B}_i=\frac{dP_i^t}{dt}+\frac{2{G}_i}{C_i}{P}^{\mathrm{out}} $$

(6.75)

For the source vessel, the elements of B are as follows:

$$ {B}_1=\frac{dP_1^t}{dt}+\frac{2{G}_1}{C_1}{P}^{\mathrm{in}} $$

(6.76)

Pⁱⁿ and P^out are the network input and output pressures, respectively.

Terminal Arteriole Flow

The flow in the pre-capillary arterioles, q_term, represents the cardiac perfusion. The flow must satisfy the myocyte demand for oxygen as represented by the target flow q_target. This is an input to the model which depends on the heart metabolic demand. Several levels of q_targetare used to represent a range of physiological activity levels.

Table 6.3 Parameters for the reference case in the model of comprehensive blood flow

6.1.3 Vascular Mechanical Properties

The Passive Vessel Properties

The passive vessel response consists of the intrinsic passive mechanics (given by the PDR) and the tethering effect of the myocardial tissue. The PDR of isolated in vitro passive vessels under positive trans-vascular pressure showed the radius to be a sigmoidal function of the trans-vascular pressure, ΔP (Young, Choy, Kassab, & Lanir, 2012) as:

$$ {R}_{\mathrm{p}}\left(\Delta P\right)={B}_{\mathrm{p}}+\frac{A_{\mathrm{p}}-{B}_{\mathrm{p}}}{\pi}\left[\frac{\pi }{2}+\mathit{\arctan}\;\left(\frac{\Delta P-{\phi}_{\mathrm{p}}}{C_{\mathrm{p}}}\right)\right] $$

(6.77)

where A_p, B_p are the asymptotical highest and lowest radii, respectively, ϕ_p is the trans-vascular pressure corresponding to the average of radii A_p and B_p, and C_p is the passive response bandwidth. The tethering model is described below and the data on the PDR constants for orders 1–6 vessels are listed in Table A.1A.

Coronary vessels are tethered to the surrounding myocardium by a network of short collagen struts (Borg & Caulfield, 1979; Caulfield & Borg, 1979). These struts prevent the vessels from collapse under negative trans-vascular pressure which occurs in the deeper myocardial layers (Kajiya et al., 2008). The importance of tethering is clearly seen when considering the wall tension balance under negative trans-vascular pressure in the absence of tethering. In the case of passive vessels, the tension balance is expressed by:

$$ \Delta P\cdot {R}_{\mathrm{reg}}={T}_{\mathrm{pas}} $$

(6.78)

The compliant untethered vessel wall can only sustain a very small negative transmural pressure before collapse. In the presence of active regulation, the wall smooth muscle cells contract thereby adding to the vessel tendency to collapse. Tethering prevents collapse by adding a positive pressure-like term to the combined tension balance equation, i.e.:

$$ \Delta P\cdot {R}_{\mathrm{reg}}+{T}_{\mathrm{teth}}={T}_{\mathrm{pas}}+{T}_{\mathrm{act}} $$

(6.79)

Two assumptions are adopted in the analysis. First, under negative trans-vascular pressure (ΔP < 0), the passive wall tension T_pas vanishes. Second, under positive trans-vascular pressure, the tension in the tethering struts increases quadratically with the gap between the zero-pressure radius (R₀) of the myocardial “tunnel” which embeds the vessel and that of the vessel (R_reg). This nonlinearity in the tension–gap relationship reflects the gradual recruitment of the non-uniformly undulated struts with stretch. Under positive transvascular pressure (ΔP < 0), the tension in the tethering struts is assumed to vanish while the vessel adheres to the myocardial tunnel. The struts tension is thus C_str (R₀ – R_reg)². This expression cannot be used for coronary vessels of all orders with the same value of C_str since it assumes that the struts density on the myocardial tunnel wall is independent of R₀. It is likely that the struts density is higher for smaller vessels with smaller perimeters and vice versa for larger vessels. Hence, the strut pressure-like stress per unit myocardial tunnel area can be expressed as:

$$ {P}_{\mathrm{str}}^{\mathrm{myo}}={C}_{\mathrm{str}}{\left({R}_0-{R}_{\mathrm{reg}}\right)}^2/{R}_0 $$

(6.80)

The strut density on the vessel wall under negative trans-vascular pressure (i.e., when R_reg < R₀) is higher than that on the myocardium by a factor of R₀/R_reg, so that the strut stress on the vessel wall is ${P}_{\mathrm{str}}^{\mathrm{vessel}}={C}_{\mathrm{str}}{\left({R}_0-{R}_{\mathrm{reg}}\right)}^2/{R}_{\mathrm{reg}}$. Finally, the contribution of the strut pressure-like stress to the vessel wall tension (“tethering tension”) is given by:

$$ {T}_{\mathrm{teth}}={P}_{\mathrm{str}}^{\mathrm{vessel}}\cdot {R}_{\mathrm{reg}}={C}_{\mathrm{str}}{\left({R}_0-{R}_{\mathrm{reg}}\right)}^2 $$

(6.81)

Table 6.4 (A) Order distribution of passive vessel parameters obtained by fitting Eq. (6.77) to the experimental data of pressure versus diameter

Under ex vivo conditions, the compliant isolated vessels collapse. In vivo, the vessel tethering to the surrounding myocardium supports it from collapse (see Chap. 3). To comply with these observations, the level of B_p is set to zero, and tethering is integrated into the model. The isolated vessel passive parameters (A_p, ϕ_p, and C_p) are re-estimated from the ex vivo experimental data on isolated vessels (Liao & Kuo, 1997) for vessel orders 5–7 under the constraint of B_p = 0. Parameters for vessel orders 0–1 are obtained by curve fitting Eq. (6.77) to in situ diameter–pressure data and parameters of order 10 are extracted from a sigmoidal fit of ex vivo data ( Chap. 3).

The passive parameters as a function of vessel radius across all orders are obtained by fitting a quadratic curve through all data points of the parameter, A_p, and linear fits to ϕ_p and C_p (Table 6.4A). The vessel radius is transformed to the radius under 80 mmHg pressure to be compatible with the in situ data (Kassab, Imoto, et al., 1993). Given a vessel radius, the fits are used to interpolate the passive material properties between orders 1 and 8 (Fig. 6.19).

Fig. 6.19

Distribution of vessel passive parameters A_p, φ_p, and C_p (Eq. 6.53) over the vessel radii (Kassab, Imoto, et al., 1993). The minimum vessel radius, B_p, is set to zero for all vessels. The data sources are listed in Table 6.4 for vessel order 1–10. Because the data are incomplete, values for in-between orders for which data are not available are interpolated based on a linear fit for A_p and quadratic fits for φ_p and C_p. The data sources a, b, and c are listed in the footnotes of Table 6.4. Reproduced from Namani et al. (2018) by permission

Active Vessel Properties

The active wall properties are determined by VSMCs which are affected predominantly by myogenic (pressure), flow (shear), and metabolic control mechanisms. Literature data on the vascular response to these mechanisms are primarily on the steady-state response (Liao & Kuo, 1997). There is paucity of data on the dynamics of coronary control and much of the transient response relates to changes in heart rate including the combined effects of multiple flow regulation mechanisms (Dankelman, Vergroesen, Han, & Spaan, 1992). The analysis of flow regulation, however, requires detailed knowledge of the dynamics of each flow control mechanism which is currently unavailable. In the absence of such data, the steady-state responses including the functional role of each control mechanism and of their interactions are considered whereas the mechanics of the vascular system and its interaction with the myocardium are dynamically analyzed under these quasi-steady state control conditions.

The radius of the active vessel varies dynamically under a cyclical trans-vascular pressure. The vessel response is determined by the active dynamic vessel wall stiffness. In vitro studies of isolated vessels (Halpern et al., 1978) have shown that the vessel dynamic stiffness is proportional to the active tension, T_act. This proportionality is in line with findings of in vitro isolated myocytes and myocyte culture (Campbell, Patel, & Moss, 2003; Lipowsky, Kovalcheck, & Zweifach, 1978; Yadid & Landesberg, 2010). It is thought that in the heart and skeletal muscle, both the stiffness and active tension are proportional to the number of attached actin–myosin cross-bridges. Since the contractile machinery in VSMC is actin–myosin as well, it is reasonable to adopt the same relationship for the arterial wall, and with the same proportionality constant. To this end, the total vessel wall tension is taken as the sum of the passive and active components. Under equilibrium, this wall tension balances the contributions of the trans-vascular pressure and of the tethering tension. Hence, the following can be obtained:

$$ \Delta \overline{P}\cdot {R}_{\mathrm{reg}}\left(\Delta \overline{P},A\right)+{\overline{T}}_{\mathrm{teth}}={\overline{T}}_{\mathrm{act}}+{\overline{T}}_{\mathrm{pas}} $$

(6.82)

The tension of the passive wall, T_pas, stems from the passive elements given by:

$$ {\overline{T}}_{\mathrm{p}\mathrm{as}}\left(\Delta \overline{P},A\right)=\Delta {P}_{\mathrm{p}}\left({R}_{\mathrm{reg}}\right){R}_{\mathrm{reg}}\left(\Delta \overline{P},A\right)={R}_{\mathrm{reg}}\left({\mathrm{c}}_{\mathrm{p}}\mathit{\tan}\left\{\left(\frac{\pi \Big({R}_{\mathrm{reg}}-{B}_{\mathrm{p}}}{A_{\mathrm{p}}-{B}_{\mathrm{p}}}\right)-\frac{\pi }{2}\right\}+{\phi}_{\mathrm{p}}\right) $$

(6.83)

where A_p, B_p, C_p, and ϕ_p are passive vessel parameters (B_p = 0).

The active tension is assumed to depend solely on the vessel circumference (or radius) and activation. This is due to the dependence of VSMC active tension in the vessel wall on the actin/myosin overlap which is directly related to the wall circumference or radius. Tethering does not affect the active tension–radius relationship. Hence, at a given passive tension, the force balance in Eq. (6.44) under T_teth = 0 determines the active tension as a function of both the R_reg and the activation level A.

The active constitutive relationship allows the determination of the passive vascular dynamic stiffness, k_pas, as follows:

$$ {\displaystyle \begin{array}{ll}{k}_{\mathrm{p}\mathrm{as}}\left({R}_{\mathrm{reg}}\right)& =\frac{d{\overline{T}}_{\mathrm{p}\mathrm{as}}}{2\pi \mathrm{dR}}=\frac{d\left(\Delta P.R\right)}{2\pi \mathrm{dR}}=\frac{1}{2\pi}\left(\frac{{\mathrm{R}}_{\mathrm{reg}}\left(\Delta \overline{P},\mathrm{A}\right)}{{\left.\frac{dR}{d\Delta P}\right|}_{\overline{\Delta P}}}+\Delta P\right)\\ {}& =\left\{\frac{{\mathrm{R}}_{\mathrm{reg}}\left(\Delta \overline{P},\mathrm{A}\right){\mathrm{c}}_{\mathrm{p}}}{2\left({A}_{\mathrm{p}}-{B}_{\mathrm{p}}\right)}\left(1+{\left(\mathit{\tan}\;\left(\frac{\pi \Big({\mathrm{R}}_{\mathrm{reg}}\left(\Delta \overline{P},\mathrm{A}\right)-{B}_{\mathrm{p}}}{A_{\mathrm{p}}-{B}_{\mathrm{p}}}\right)-\frac{\pi }{2}\right)}^2\right)\right)\\ {}& \left(+\frac{{\mathrm{c}}_{\mathrm{p}}\left\{\mathit{\tan}\left(\frac{\pi \Big({\mathrm{R}}_{\mathrm{reg}}\left(\Delta \overline{P},\mathrm{A}\right)-{B}_{\mathrm{p}}}{A_{\mathrm{p}}-{B}_{\mathrm{p}}}\right)-\frac{\pi }{2}\right\}+{\phi}_{\mathrm{p}}}{2\pi}\right\}\end{array}} $$

(6.84)

when the regulated radius (R_reg) is less than the passive zero-pressure radius (R₀), the tethering tension becomes effective. The tethering “stiffness” is defined similar to passive stiffness (from Eq. 6.84) as:

$$ {k}_{\mathrm{teth}}\left({R}_{\mathrm{reg}}\right)=-\frac{d\left({\overline{T}}_{\mathrm{teth}}\right)}{2\pi d\left({R}_{\mathrm{reg}}\right)}=\frac{C_{\mathrm{str}}}{\pi}\left({R}_0-{R}_{\mathrm{reg}}\right) $$

(6.85)

where the minus sign designates the decrease in vessel radius to an increase in tethering stiffness. Since the dynamic stiffness of the active vessel to stretch perturbations is found to be linearly proportional to the active tension (Halpern et al., 1978), the level is evaluated from their data as:

$$ {k}_{\mathrm{act}}\left(\Delta \overline{P},\mathrm{A}\right)={C}_1{\overline{T}}_{\mathrm{act}}\left(\Delta \overline{P},\mathrm{A}\right)+{C}_0 $$

(6.86)

where C₁ = 30.6 mm^-1 is the slope of the linear regression and C₀ = 4.85 kPa is the intercept. The total vascular dynamic stiffness is the sum of active, passive, and tethered stiffness components given by:

$$ k\left(\Delta \overline{P},\mathrm{A}\right)={k}_{\mathrm{act}}\left(\Delta \overline{P},\mathrm{A}\right)+{k}_{\mathrm{pas}}\left({R}_{\mathrm{reg}}\right)+{k}_{\mathrm{teth}}\left({R}_{\mathrm{reg}}\right) $$

(6.87)

Given the above expressions for each stiffness term, the total vascular dynamic stiffness can be evaluated from Eq. (6.88).

Similar to Eq. (6.84), the vessel compliance under dynamic loading of ΔP(t) around $\Delta \overline{P}$ is obtained from the vessel wall dynamic stiffness, k, as follows:

$$ {\left.\frac{dR}{d\Delta P}\right|}_{\Delta \overline{P},A}=\frac{R_{\mathrm{reg}}\left(\Delta \overline{P},A\right)}{2\pi k\left({R}_{\mathrm{reg}},A\right)-\Delta \overline{P}} $$

(6.88)

Wall Tension with No Tethering

The regulated radius as a function of average trans-vascular pressure is given by:

$$ {\widehat{R}}_{\mathrm{reg}}\left(\Delta \overline{P}\right)={\mathrm{R}}_{\mathrm{p}}\left(\Delta \overline{P}\right)-A\Delta {R}_{\mathrm{m}}\left(\Delta \overline{P}\right) $$

(6.89)

where the passive contribution of the average trans-vascular pressure derived as:

$$ \Delta {\overline{P}}_{\mathrm{p}\mathrm{as}}\left({\widehat{R}}_{\mathrm{reg}}\right)={\phi}_{\mathrm{p}}+{c}_{\mathrm{p}}\left\{\tan \left[\frac{\pi \left({\widehat{R}}_{\mathrm{reg}}-{B}_{\mathrm{p}}\right)}{A_{\mathrm{p}}-{B}_{\mathrm{p}}}-\frac{\pi }{2}\right]\right\} $$

(6.90)

The wall tension due to passive elements of the vessel wall as a function of the regulated radius is:

$$ {T}_{\mathrm{pas}}\left({\widehat{R}}_{\mathrm{reg}}\right)=\Delta {\overline{P}}_{\mathrm{pas}}\left({\widehat{R}}_{\mathrm{reg}}\right).{\widehat{R}}_{\mathrm{reg}} $$

(6.91)

The wall tension due to active elements is the difference between the total wall tension and the wall tension due to passive elements of the vessel wall and is given by:

$$ {T}_{\mathrm{act}}\left(A,{\widehat{R}}_{\mathrm{reg}}\right)=\Delta \overline{P}{\widehat{R}}_{\mathrm{reg}}-\Delta {\overline{P}}_{\mathrm{pas}}{\widehat{R}}_{\mathrm{reg}}={\widehat{R}}_{\mathrm{reg}}\left(\Delta \overline{P}-\Delta {\overline{P}}_{\mathrm{pas}}\right) $$

(6.92)

The wall tension contribution from the passive and active elements in the vessel wall are calculated with the above equations for experimental data of Liao and Kuo (1997). Since data are available at only some pressure values, sigmoidal models of pressure–diameter relationships of different order vessels are used to calculate the wall tensions.

For the case of active tension for a vessel with tethering, the constitutive equation is modified to include additional pressure on wall due to tethering as given by:

$$ \Delta {\overline{P}}_{\mathrm{total}}\left({R}_{\mathrm{reg}},A\right)=\Delta \overline{P}\left({R}_{\mathrm{reg}},A\right)+\frac{T_{\mathrm{teth}}\left({R}_{\mathrm{reg}}\right)}{R_{\mathrm{reg}}} $$

(6.93)

The regulated radius, ${R}_{\mathrm{reg}}\left(\overline{\Delta P},\mathrm{A}\right)$ is an unknown variable and is determined by the iterative solution of the force balance equation as:

$$ \Delta \overline{P}\cdot {R}_{\mathrm{reg}}\left(\Delta \overline{P},\mathrm{A}\right)+{T}_{\mathrm{teth}}\ \left({R}_{\mathrm{reg}}\right)={T}_{\mathrm{act}}\left({R}_{\mathrm{reg}}\right)+{T}_{\mathrm{pas}}\left({R}_{\mathrm{reg}},A\right) $$

(6.94)

Myogenic Regulation

The myogenic regulation results from VSMC contraction in response to local wall stress as determined by the trans-vascular pressure. The myogenic diameter reduction is expressed by a sigmoidal function of the time-averaged trans-vascular pressure $\varDelta \overline{P}$ (Young et al., 2012) as:

$$ \Delta {R}_{\mathrm{m}}\left(\Delta \overline{P}\right)=\frac{\rho_{\mathrm{m}}}{\pi}\left[\frac{\pi }{2}-\arctan \left({\left[\frac{\Delta \overline{P}-{\phi}_{\mathrm{m}}}{C_{\mathrm{m}}}\right]}^{2\mathrm{m}}\right)\right] $$

(6.95)

where ρ_m is the myogenic response amplitude, ϕ_m is the trans-vascular pressure under which the myogenic radius change is highest, C_m is the myogenic response bandwidth, and m is a shape factor. The parameters for Eq. (6.95) for the various vessel orders are given in Table 6.4.

The regulated vessel radius under quasi-static loading is taken to be a function of the total activation level, A, and of the mean trans-vascular pressure, $\varDelta \overline{P}$ (Liao & Kuo, 1997). The maximum myogenic reduction in radius ΔR_m under full activation (A = 1), R_reg, is attenuated with the total activation (A < 1) (Liao & Kuo, 1997) to yield the regulated radius as:

$$ {R}_{\mathrm{reg}}={R}_{\mathrm{reg}}\left(\Delta \overline{P},A\right)={R}_{\mathrm{p}}\left(\Delta \overline{P}\right)-A\Delta {R}_{\mathrm{m}}\left(\Delta \overline{P}\right) $$

(6.96)

where the total activation (A) due to myogenic, shear stress and metabolites is given by:

$$ A=\left(1-{F}_{\tau}\right)\left(1-{F}_{\mathrm{meta}}\right) $$

(6.97)

Expressions for F_τ and F_meta are listed below. The product form of Eq. (6.97) between the metabolic and flow regulations relates the respective residual activities (1 − F_τ)and (1 − F_meta). This form is mathematically identical to the physiologically based additive model proposed and experimentally validated by (Liao & Kuo, 1997).

The longitudinal distribution of myogenic parameters of vessel orders 5–7 is obtained by fitting Eq. (6.95) to ex vivo data under varying pressures (Liao & Kuo, 1997). Since capillaries (order 0) and large epicardial arteries (order 10) do not exhibit myogenic radius changes, the myogenic amplitude (ρ_m) of these vessels is taken as zero. The interpolated sigmoidal parameters of other order vessels are listed in Table 6.4B (Fig. 6.20). Due to the lack of experimental data for the myogenic amplitude, ρ_m, for vessel orders 1–5, the myogenic sensitivity curve is extrapolated from order 5 down to order 1 vessels assuming a constant shape factor m = 2, the level estimated from the in vitro data (Young et al., 2012). The myogenic sensitivity, ρ_m/R, is assumed to be the same for all order 1 vessels.

Fig. 6.20

Distribution of the vessel myogenic parameters ρm, φm, and C_m (Eq. 6.95) over their cast radii, R_80 mmHg (Kassab, Imoto, et al., 1993). The data sources are listed in Table 6.4. Reproduced from Namani et al. (2018) by permission

From Eq. (6.95), ϕ_m corresponds to $\Delta \overline{P}$of highest ΔR_m, the comparison of the ex vivo parameter estimates with the data of order 6 vessels showed that the myogenic diameter reduction ΔR_m reaches peak levels at a $\Delta \overline{P}$ which is different from the estimates of ϕ_m. Hence, ϕ_m for order 6 vessels is adjusted to the value of $\Delta \overline{P}$ at highest ΔR_m. Furthermore, since the ($\Delta {R}_{\mathrm{m}}\left(\Delta \overline{P}\right)$) relationship is symmetric around ϕ_m (Eq. 6.95), the estimates of C_m are adjusted to maintain this symmetry. These adjustments have an insignificant effect on the fit of Eq. (6.95) to the data. The passive and fully myogenic active (A = 1) vascular pressure–diameter relationship (PDR) are calculated from Eqs. (6.77) and (6.95). PDR distribution across the various vessel orders is presented in Fig. 6.21.

Fig. 6.21

Pressure–diameter relationship under passive and active vessel conditions. The model predicted vessel diameter, D, normalized by the vessel diameter under zero pressure, D₀, in the passive state (solid line) and under full myogenic state (dashed line), and the diameter reduction under full metabolic activation (dotted line), as functions of the trans-vascular pressure $\Delta \overline{P}$, for (a) small arteriole of order 5, (b) intermediate arteriole (I.A.) of order 6, (c) large arteriole (L.A.) of order 6, and (d) small artery of order 7. Corresponding data are the symbols (open circle, open square, and open triangle, respectively) from in vitro studies of isolated vessels (Liao & Kuo, 1997) as shown in Figure. Reproduced from Namani et al. (2018) by permission

Flow (Shear) Regulation

The shear regulation induces relaxation of the myogenic contracted vascular wall which is mediated by nitric oxide (NO) production by the endothelial cells in response to local wall shear stress. The shear fractional deactivation is taken to be dependent on the average shear stress, $\left|\overline{\tau}\right|\ \mathrm{and}\ \mathrm{is}\ \mathrm{expressed}\ \mathrm{by}$ (Liao & Kuo, 1997):

$$ {F}_{\tau }={F}_{\tau \max}\frac{\left|\overline{\tau}\right|}{K_{\tau }+\left|\overline{\tau}\right|} $$

(6.98)

where K_τ is the wall shear stress constant and F_τmax is the maximum deactivation due to wall shear stress. The parameters for Eq. (6.98) of various vessel orders are summarized in Table 6.4C (Fig. 6.22).

Fig. 6.22

Distribution of the vessel shear parameters F_τmax and Kτ (Eq. 6.57) over their cast radii, R₈₀ (Kassab, Imoto, et al., 1993). The data sources are listed in Table 6.4. Reproduced from Namani et al. (2018) by permission

Liao and Kuo (Liao & Kuo, 1997) pointed out that the values of their in vitro measured K_τ are too low due to the presence of hemoglobin which binds to NO and hence decreases the in vivo sensitivity to shear. In their flow analysis in an idealized symmetric network without MVI, they increased K_τ by a factor of 150 which allowed the vessels to respond to shear stress under physiological conditions. In the present network simulations, it is found that a factor of 15 is sufficient (Table 6.4C).

Metabolic Regulation

Early studies on the vasomotor response in the microcirculation observed that dilation spreads over a much larger area than can be explained by diffusion (Krogh, Harrop, & Rehberg, 1922). More recent studies established the predominant role of the endothelium layer in conducting vasodilatory stimulus (Emerson & Segal, 2000; Furchgott & Zawadzki, 1980; Looft-Wilson, Payne, & Segal, 2004) via cell-to-cell coupling (Larson, Kam, & Sheridan, 1983).

To establish the specific pathway by which coronary vessel diameter is regulated has been difficult due to redundancies in control pathways, difference between species, conflicting results in different studies, different (at times opposite) effects at rest versus during exercise (see review in (Duncker & Bache, 2008)), and at times opposite effects on vessels of different sizes (Gorman & Feigl, 2012). In humans, there are additional uncertainties due to inadequate control of the coronary endothelial state (Duncker & Bache, 2008). Notwithstanding these difficulties, it is established that coronary local metabolic control is not primarily due to adenosine, ATP-dependent K⁺ channels, NO, prostaglandins, and inhibition of endothelin (reviews in (Duncker & Bache, 2008; Tune, Gorman, & Feigl, 2004)). A number of mechanisms for the initiation and conduction of vasodilation have been proposed (Budel, Bartlett, & Segal, 2003; Doyle & Duling, 1997; Figueroa et al., 2007; Hoepfl, Rodenwaldt, Pohl, & De Wit, 2002; Looft-Wilson et al., 2004; Murrant & Sarelius, 2002; Rivers, 1997; Tallini et al., 2007; Xia & Duling, 1995). A specific pathway that has gained attention proposes that red blood cells (RBCs) may act as sensors of oxygen and thereby of the metabolic supply/demand imbalance (Ellsworth, 2000). Adenosine triphosphate (ATP) is found to be released from RBCs in response to hypoxia and hypercapnia (Bergfeld & Forrester, 1992; Ellsworth, Forrester, Ellis, & Dietrich, 1995). These conditions occur in the capillaries and venules under high metabolic demand when oxygen supply is lower than demand (Collins, McCullough, & Ellsworth, 1998; Farias III, Gorman, Savage, & Feigl, 2005; Gorman & Feigl, 2012). Venules are thus optimally positioned to monitor the metabolic state of the tissue (Jackson, 1987; Segal, 2005).

Based on a number of studies, the adenine nucleotides regulation mechanism is proposed (Farias et al., 2005; Gorman & Feigl, 2012; Gorman, Ogimoto, Savage, Jacobson, & Feigl, 2003; Gorman et al., 2010), where ATP released by RBCs in the venules under high metabolic demand is broken down to its metabolites, adenosine diphosphate (ADP) and adenosine monophosphate (AMP). All three adenine nucleotides are potent coronary vasodilators (Gorman et al., 2003). They bind to P1 (AMP) and P2 (ATP and ADP) purinergic receptors on the endothelial cells (Burnstock, 2007; Gorman et al., 2003) thereby stimulating endothelial synthesis of NO which interacts with the smooth muscle cells (SMCs) in the vessel walls to dilate the vessels thus reducing their resistance to flow (Sprague, Ellsworth, Stephenson, & Lonigro, 1996).

The vasodilatory signal is believed to be conducted (conducted response, CR) across the capillaries (Collins et al., 1998; Tigno, Ley, Pries, & Gaehtgens, 1989) to the endothelial cells of upstream arterial microvessels, likely via endothelial cells gap junctions (Collins et al., 1998; Domeier & Segal, 2007; Figueroa et al., 2007; Segal & Duling, 1987; Segal & Duling, 1989). The vasodilatory effect of CR is believed to decay exponentially with distance into the upstream arterioles (Delashaw & Duling, 1991; Hirst & Neild, 1978; Xia & Duling, 1995). Additional experimental support for the conducted response is found in studies in which ATP application inside small arterioles, outside capillaries, and inside venules, produced retrograde conducted vasodilatory response (Collins et al., 1998; Duza & Sarelius, 2003; McCullough, Collins, & Ellsworth, 1997).

In addition to the sustained and decaying CR, (Figueroa & Duling, 2008) found that short stimulation of Acetylcholine (ACh) evoked transient vasodilation that spread along the entire vessel length (up to 2 mm) without decay. This study (Namani et al., 2018) focuses on the steady-state effect of sustained metabolic demand. Since the characteristics of that non-decaying signal and its functional consequences for the entire network flow under sustained metabolic demand remain unclear, this mechanism is not included in the model.

Importantly, the CR is not restricted to the vasodilatory mechanism due to adenine nucleotides but is rather a generic framework which represents a range of possible vasodilatory signaling, spreading from the capillaries to upstream arterioles Extravascular synthesized metabolites such as muscle released ATP, adenosine, NO, and potassium which may diffuse radially into the arteriolar walls and relax the SMCs, however, are not considered. A theoretical analysis by Lo et al. (2003) showed that local responses alone provide insufficient flow regulation. Countercurrent exchange by diffusion of vasoactive arachidonic acid metabolites between paired venules and arterioles (Hammer, Ligon, & Hester, 2001) is not included since pairing and close alignment tend to be typical of larger and intermediate sized venules and arterioles. The major portion of resistance to flow, and therefore of flow regulation, resides in the smaller sized arterioles. These smaller vessels are highly affected by the CR signal due to their proximity to the venules.

The CR model (Arciero, Carlson, & Secomb, 2008) integrates the signaling effects along the various network pathways, from the pre-capillary arterioles to each specific upstream vessel. Hence, the metabolic activation ${F}_{\mathrm{meta}}^i$ in an upstream vessel i is expressed by:

$$ {F}_{\mathrm{meta}}^i=\sum \limits_{j=1}^{u^i}\frac{F_{\mathrm{mterm},j}^i{S}_j^i}{u^i};\kern1em {S}_j^i={e}^{-\frac{L_j^i}{L_0}};\kern1em i=1\dots n $$

(6.99)

where uⁱ is the number of terminal vessels fed by the upstream vessel i, ${S}_j^i$ is the strength of the response in an ith vessel conducted from its jth terminal vessel and L₀ is the decay characteristic length. The metabolic signal in the jth terminal order 1 vessel ${F}_{\mathrm{mterm},j}^i$ is a direct function of the local oxygen supply/demand imbalance. The conducted response is found to decay exponentially with the path length, ${L}_j^i$, towards the upstream vessels (Arciero et al., 2008; Delashaw & Duling, 1991; Goldman et al., 2012; Xia & Duling, 1995) with a characteristic length, L₀, which determines the rate of decay of the metabolic activation with path length (Eq. 6.99). The total metabolic activation, ${F}_{\mathrm{meta}}^i$, in an ith vessel is taken to be the average from all jth terminal vessels fed by that vessel. A reference value of L₀ = 1 mm is selected for the decay characteristic length. This value lies within the measured range of different vascular beds (0.15–2.5 mm, (Hald et al., 2012)).

The metabolic signal in each j terminal order 1 vessels, ${F}_{\mathrm{mterm},j}^i$, depends on the local demand/supply imbalance. In control theory, the control signal is the system desired output which in the coronary circulation is the requisite terminal arterioles perfusion which balances the metabolic O₂ demand. This is irrespective of the metabolites or mechanisms involved. Hence, although the terminal arterioles flow is not physiologically a sensed signal, its requisite level represents the metabolic demand regardless of the involved metabolite pathways. Our choice of a metabolic signal is supported by the findings that coronary flow correlates well with an increase in the coronary venous ATP concentration, and the latter correlates with the decline of venous PO₂ (Farias III et al., 2005). In the network flow analysis, the terminal vessel metabolic signal ${F}_{\mathrm{mterm},j}^i$ is either set to be constant in all terminal vessels or optimized in each terminal vessel to provide a set level of terminal flow.

Oxygen Demand and Target Terminal Flow

If the secondary contribution of dissolved oxygen on the total oxygen content are considered negligible, the oxygen mass balance is specified by:

$$ M={q}_{\mathrm{term}}{c}_0{H}_{\mathrm{D}}\left({S}_{\mathrm{a}}-{S}_{\mathrm{v}}\right) $$

(6.100)

where M is myocardial oxygen consumption for a single terminal arteriole, q_term is the flow in the terminal arterioles, H_D is the hematocrit, S_a is arterial oxygen saturation, S_v is the venous oxygen saturation, and c_o is the oxygen carrying capacity of RBCs. Hence, the following holds:

$$ {q}_{\mathrm{term}}=\frac{M}{c_0{H}_{\mathrm{D}}\left({S}_{\mathrm{a}}-{S}_{\mathrm{v}}\right)} $$

(6.101)

The values of c_0, H_D, and S_a are directly measurable and assumed to be constant (independent of M). The venous oxygen saturation S_v is a function of the oxygen consumption M, being dependent on oxygen mass balance, ATP release and transport, and the effect of sympathetic inputs on myocardial oxygen consumption. It is thus affected by the combined action of a feedback pathway signal that is determined by the level of plasma ATP in coronary venous blood, and by adrenergic open-loop (feedforward) signal that increases with exercise (Pradhan, Feigl, Gorman, Brengelmann, & Beard, 2016). Data has been measured by (Farias III et al., 2005) and (Gorman et al., 2010; Gorman, Tune, Richmond, & Feigl, 2000). Based on this relationship between M and S_v, the flow in terminal vessels q_term can be directly related to the oxygen consumption M.

Flow Regulation Time Constant

The time constant of the coronary vessel response to changes in pressure and flow (approximately 1.5 folds of t₅₀—the time required to establish half of the complete response) is found to be in the range of 15 s to minutes (Dankelman et al., 1992; Hoffman & Spaan, 1990; Mosher, Ross Jr., McFate, & Shaw, 1964; Tsoukias, Kavdia, & Popel, 2004). This response time constant is significantly higher than the cardiac period (~1 s). The stabilized system response can thus be considered as the time average over a cardiac cycle. Hence, the levels of regulated vessel radius (Eq. 6.89), of the active tension (Eq. 6.92) and of the active stiffness (Eq. 6.86) are formulated as functions of the time-averaged trans-vascular pressure.

Boundary Conditions

The coronary flow is determined by the myocardium–vessel interaction (MVI) which consists of the combined effect of the intramyocardial fluid pressure (IMP) and the shortening-induced intramyocyte pressure P_SIP. IMP varies with the myocardial relative depth (MRD) from the LV pressure at the endocardium to zero at the epicardium. Waveforms of the inlet pressure, P_in(t), outlet pressure, P_out(t), LV pressure, P_LV(t), and intramyocyte pressure, P_SIP(t), are input signals to the flow analysis (Fig. A.5). The P_out(t) signal is interpolated for different transmural locations based on predictions from simulation of the unregulated flow in an entire coronary network which included arterial and venous trees and four identical representative capillary networks, at relative myocardial depths (MRD) of 0.125, 0.375, 0.625, and 0.875 (Algranati et al., 2010). P_LV(t) waveform is taken from predictions based on a distributive LV mechanical model under resting heart rate (Kiyooka et al., 2005) of 75 BPM. Several considerations guided the choice of the P_in(t) signal for the sub-endocardial 400 vessel network. The first is the pressure drop from the aorta to the trunk vessel (order 6) of the subtree. On the other hand, there is a pressure increase due to the added intramyocyte pressure, P_SIP(t) which develops during contraction (Rabbany et al., 1989). Finally, P_in(t) must provide for sufficient flow perfusion in the terminal order 1 vessels in the range of measured flow of 0.4–2.0 × 10⁻³ mm³/s in systole and diastole (Tillmanns et al., 1974). Based on these considerations, P_in(t) is chosen to be 122/90 mmHg (with average ${\overline{P}}_{\mathrm{in}}$ = 100) in systole/diastole and the signal shape is adopted from (Algranati et al., 2010). P_out is assigned for each terminal vessel to be between the previously predicted sub-epicardium and sub-endocardium signals ${P}_{\mathrm{out}}^{\mathrm{subepi}}$ and ${P}_{\mathrm{out}}^{\mathrm{subendo}}$ (Fig. 6.23), depending on the transmural location of the vessel. In the absence of data on P_out under higher metabolic demands (i.e., higher q_target), the level for each vessel is kept the same under changes of q_target. The tissue pressure P_T(t) is derived based on the earlier analysis of unregulated coronary flow.

Fig. 6.23

The assigned pressure boundary conditions. P_in, the input pressure to the order 6 trunk vessel; ${P}_{\mathrm{out}}^{\mathrm{subendo}}$, the output pressure at the terminal order 1 vessels in a normalized myocardial depth of 0.875; ${P}_{\mathrm{out}}^{\mathrm{subepi}}$, the output pressure at the terminal order 1 vessels in a normalized myocardial depth of 0.125. Both ${P}_{\mathrm{out}}^{\mathrm{subendo}}$ and ${P}_{\mathrm{out}}^{\mathrm{subepi}}$ are adapted from previous analysis of unregulated coronary flow (Algranati et al., 2010). PLV pressure in LV chamber; PSIP intramyocyte pressure caused by their shortening. Reproduced from Namani et al. (2018) by permission

Time-Varying Vessel Radius

For flow analysis, Eq. (6.84) allows the calculation of the requisite vessel radius, R(t), along the cardiac cycle. In the embedded and tethered microvessel, the radius variations are likely small enough to retain just the first term in the Taylor series expansion of R(t). In this case, the following can be obtained:

$$ R(t)\simeq {R}_{\mathrm{reg}}+\frac{dR}{d\Delta P}\left(\Delta P(t)-\Delta \overline{P}\right) $$

(6.102)

where ΔP(t) is the time-varying trans-vascular pressure along the cardiac cycle.

Solution of Network Flow

Due to vessel elasticity and interaction with surrounding myocardium (MVI), the flow equations are highly nonlinear. Hence, network flow solution is an iterative solution of the system of ODEs (Eq. 6.62) subject to the respective boundary conditions. The matrices A and B are modified after each iteration which are continued to reach the desired convergence and periodicity conditions to within specified tolerances.

The numerical framework is used to first solve the passive network flow, followed by active regulation. Two different schemes are used to solve the autoregulated flow. In cases where metabolic regulation is absent (i.e., only myogenic and/or flow mechanisms are active—the solution for a passive vessel network is used as the initial guess to adjust each vessel diameter following the respective model equations, according to its predicted pressure and flow rate). When the metabolic regulation is active, the solution from passive network flow is iterated under a genetic algorithm search for the distribution across all the j terminal arterioles of the metabolic signals ${F}_{\mathrm{meta},j}^i$ (Eq. 6.99) which yield terminal flow rates close to q_target, up to within specified tolerance. The solution of a reference case is carried out with parameters listed in Table 6.3. It served as a baseline for the sensitivity analysis to compare predictions under a range of parameter levels.

Simulations are carried out to verify the flow periodicity condition, i.e., smoothness of the transition between nodal pressures from the end of one cardiac cycle to the start of the next. The smoothness tolerance is set to 0.075 mmHg. The convergence of the computational results is estimated based on the network flow solution, where the net inflow/outflow deviation at each time point (from the requisite zero level) is calculated at each vessel mid-node and at each coronary bifurcation. This is carried out for both the passive and regulated network flow under both steady as well as dynamic flow conditions. Convergence is satisfied when all flow deviations are <1% of the inflow to the respective node and vessel junctions.

6.1.4 Model Comparison with Flow Characteristics

Since detailed quantitative validation data are presently not available for individual vessels in vivo, model predictions are compared with global response patterns. To that end, four sets of simulations are performed under the following conditions: (Case 1) Full myogenic activation; (Case 2) Myogenic and shear activation with no metabolic activation; (Case 3) Myogenic, shear, and full metabolic activation of terminal order 1 vessels; and (Case 4) Myogenic, shear, and optimized metabolic activation in terminal order 1 vessels aimed to achieve target flow levels in the terminal arterioles. Case 1 is an arrested heart with the sole effect of pressure on flow control. The shear and metabolic signals are deactivated by setting all F_τ and all F_meta in Eq. (6.97) to zero. Case 2 is also an arrested heart with the effects of pressure and shear on flow control. The metabolic signal is deactivated by setting all F_meta in Eq. (6.97) to zero. Case 3 represents the full highest flow capability of the system in a beating heart. In Case 4, the optimal metabolic signals, ${F}_{\mathrm{mterm},j}^i$of all terminal order 1 vessels are targeted to obtain the least deviation of terminal perfusion level, ${q}_{\mathrm{term}}^j$, from the set target flow level (q_target).

The following model predictions are compared with observations: dispersion of the perfusion, transmural perfusion heterogeneity, the magnitude of the metabolic flow reserve (MFR), and autoregulatory response of the system flow.

Perfusion Dispersion

Coronary perfusion in the passive (unregulated) state has been shown to be highly heterogeneous (Austin Jr., Aldea, Coggins, Flynn, & Hoffman, 1990; Huo et al., 2009). Regulation is found to decrease flow dispersion (Austin Jr. et al., 1990). The predicted flow level and its dispersion (CV) are evaluated here under various regulation mechanisms with full (F_mterm = 1) and no (F_mterm = 0) metabolic activation, and under optimized metabolic activation for several levels of metabolic demand (i.e., target terminal vessel flow,q_target) and a number of input perfusion pressures.

Transmural Perfusion Heterogeneity

The effect of network transmural location on perfusion and its dispersion is analyzed as (1) terminal arterioles flow q_term under various regulation mechanisms and reference perfusion pressure, and (2) optimized flow under a few levels of metabolic demand and perfusion pressures. Transmural heterogeneity is also studied in terms of the metabolic flow reserve (MFR, see section below), the network total flow under different regulation mechanisms, and autoregulation of the network flow.

Metabolic Flow Reserve (MFR)

MFR is the flow reserve which results from flow regulation alone, without the effects of heart rate and activation waveform. It is estimated from the total flow in the active vessel network under myogenic, shear, and metabolic activations. The minimum flow is found under myogenic and shear activation and zero metabolic activation. The maximum flow is found under full metabolic activation of all order 1 terminal vessels. MFR is estimated as a function of the average input pressure (${\overline{P}}_{\mathrm{in}}$) in the range of 45 to 180 mmHg under the same waveform and outlet pressure signal (${\overline{P}}_{\mathrm{out}}$).

Flow Autoregulation

Autoregulation in coronary circulation under constant metabolic demand produces approximately constant network flow under an increase in perfusion pressure. It is studied with the metabolic regulation optimized to provide four q_target levels of 1.20, 1.50, 1.80, and 2.25 × 10⁻³ mm³/s. As one test of model validity, the subtree total flow Q is analyzed under changes in perfusion pressure for four levels of requisite terminal flow q_target. For comparison, the flow is analyzed also for the passive state, and under full (F_mterm = 1) and no (F_mterm = 0) metabolic activation.

Effect of Myocardial–Vessel Interaction (MVI)

The vessel loading by the myocardium (MVI) has three passive components (Algranati et al., 2010; Young et al., 2012): (1) chamber derived intramyocardial tissue pressure; (2) intracellular pressure in the contracting myocytes, and (3) effect of vessel tethering to the surrounding myocardium. The effects of MVI on the flow are analyzed in terms of the MFR (Namani et al., 2018), the significance of each regulation on the total network flow and on flow autoregulation (e.g., effect on the distribution of metabolic activation for various levels of metabolic demand and input perfusion pressure).

Order Dependence of the Metabolic Diameter Regulation

The network vessels increase their diameters in response to higher metabolic demand. The change in diameters is, however, non-uniform, reflecting the likewise non-uniform effect of the metabolic regulation on the different vessel orders.

Appendix 5

Effects of different regulations and of the transmural network location on the average terminal arterioles flow rates under a reference average input perfusion pressure of 100 mmHg

Flow control	Sub-endocardial		Sub-epicardial
	qterm (10−3 mm3/s)		qterm (10−3 mm3/s)
	Mean ± SD	CV	Mean ± SD	CV
Passive	2.78 ± 0.69	0.25	3.34 ± 0.81	0.24
Myogenic	0.29 ± 0.10	0.35	0.51 ± 0.23	0.46
Myogenic + Shear(all Fmterm = 0)	0.65 ± 0.19	0.29	0.91 ± 0.31	0.34
Myogenic + Shear + Full Metabolic Activation(all Fmterm = 1)	2.13 ± 0.61	0.29	2.50 ± 0.70	0.28

Appendix 6

Effects of the target terminal flow q_target, of the input pressure ${\overline{P}}_{\mathrm{in}}$,and the transmural network location, on the terminal arterioles flow rate and dispersion (coefficient of variation, CV), and on the terminal arterioles metabolic signal F_mterm, under optimized metabolic activation

q target	Sub-endocardium					Sub-epicardium
(10−3 mm3/s)	${\overline{P}}_{\mathrm{in}}$ (mmHg)	qterm (10-3 mm3/s)		F mterm		${\overline{P}}_{\mathrm{in}}$ (mmHg)	qterm (10-3 mm3/s)		F mterm
		Mean ± SD	CV	Mean ± SD	CV		Mean ± SD	CV	Mean ± SD	CV
1.20	75	1.14 ± 0.26	0.23	0.97 ± 0.08	0.08	75	1.24 ± 0.27	0.22	0.94 ± 0.12	0.12
1.20	100	1.34 ± 0.14	0.11	0.55 ± 0.24	0.44	100	1.44 ± 0.17	0.12	0.33 ± 0.27	0.82
1.20	135	1.48 ± 0.19	0.13	0.19 ± 0.23	1.23	135	2.25 ± 0.74	0.33	0.05 ± 0.16	3.23
1.20	165	2.15 ± 0.60	0.28	0.00 ± 0.00	0	165	4.29 ± 1.58	0.37	0.00 ± 0.00	0
1.50	75	1.18 ± 0.34	0.28	1.00 ± 0.00	0	75	1.32 ± 0.35	0.26	0.99 ± 0.05	0.05
1.50	100	1.67 ± 0.25	0.15	0.80 ± 0.20	0.25	100	1.72 ± 0.21	0.12	0.64 ± 0.25	0.40
1.50	135	1.78 ± 0.15	0.08	0.35 ± 0.25	0.72	135	2.34 ± 0.67	0.29	0.10 ± 0.22	2.12
1.50	165	2.15 ± 0.60	0.28	0.00 ± 0.00	0	165	4.29 ± 1.58	0.37	0.00 ± 0.00	0
1.80	75	1.18 ± 0.34	0.28	1.00 ± 0.00	0	75	1.34 ± 0.40	0.29	1.00 ± 0.00	0
1.80	100	1.91 ± 0.37	0.19	0.91 ± 0.14	0.16	100	2.03 ± 0.32	0.16	0.53 ± 0.25	0.47
1.80	135	2.05 ± 0.20	0.10	0.47 ± 0.25	0.53	135	2.19 ± 0.58	0.23	0.49 ± 0.27	1.43
1.80	165	2.25 ± 0.20	0.26	0.07 ± 0.18	2.69	165	4.29 ± 1.58	0.37	0.00 ± 0.00	0
2.25	75	1.18 ± 0.34	0.28	1.00 ± 0.00	0	75	1.34 ± 0.40	0.29	1.00 ± 0.00	0
2.25	100	2.08 ± 0.51	0.25	0.98 ± 0.06	0.06	100	2.33 ± 0.48	0.21	0.94 ± 0.13	0.14
2.25	135	2.53 ± 0.31	0.12	0.66 ± 0.24	0.36	135	2.82 ± 0.46	0.16	0.35 ± 0.31	0.86
2.25	165	2.74 ± 0.26	0.09	0.28 ± 0.25	0.91	165	4.29 ± 1.58	0.37	0.00 ± 0.00	0

1. Values of 0 and 1 for the metabolic signal CV imply that the metabolic regulation is exhausted

Appendix 7

Effect of myocardium–vessel interaction (MVI) on the metabolic flow reserve (MFR) in sub-endocardial and sub-epicardial networks under various levels of the mean input pressure ${\overline{P}}_{\mathrm{in}}$

${\overline{P}}_{\mathrm{in}}$ (mmHg)	Sub-endocardium		Sub-epicardium
	MFR (MVI)	MFR (no MVI)	MFR (MVI)
60	2.43	3.81	3.25
75	2.87	3.44	3.16
90	3.05	3.11	3.06
100	3.13	2.64	2.74
120	3.19	2.28	2.39
135	3.03	2.02	2.13
150	2.76	1.77	1.91
165	2.54	1.53	1.66
180	2.36	1.36	1.45

1. MFR is the ratio of network flow under full metabolic activation (all F_mterm = 1) to no metabolic activation (all F_mterm = 0). Reproduced from Namani et al. (2018) by permission