Mechanical Properties and Microstructure of the Coronary Vasculature

Abstract

An understanding of the mechanical properties of blood vessels is fundamental to understanding the hemodynamics of normal blood flow as well as the initiation and progression atherosclerosis (Vito & Dixon, 2003). The vascular mechanical properties largely stem from microstructural components, such as elastin and collagen fibers, smooth muscle cells, and ground substance (Azuma & Hasegawa, 1971; Azuma & Oka, 1971; Kassab & Molloi, 2001; Oka, 1972; Oka & Azuma, 1970; Vito & Dixon, 2003). Thus, the relation between the microstructure and macroscopic mechanical properties of the vessel is essential in both biomedical research and clinical practice. The accurate determination of microstructural deformation and stress, and in turn function of the blood vessel, has resulted in a new level of understanding of the blood vessel tissue.

Appendices

Appendix 1: Compliance and Distensibility

Table 3.1 Data on the diameter-compliance of the first several generations of the coronary arteries

Table 3.2 Comparison of diameter (D) distensibility data of various species

Appendix 2: Transmural Pressure–CSA Relation (Hamza et al., 2003)

The ΔP-CSA relationship for various vessels with diameter >0.5 mm is determined. The vessels are grouped in the following diameter ranges: 0.5–1.0 mm, 1.01–2.0 mm, and 2.01–3.5 mm, which roughly correspond to orders 9, 10, and 11, respectively. For each experiment, the ΔP-CSA measurements are taken for seven segments along the main LAD trunk and three segments along the side branches. The ΔP-CSA relationship in the range of −150 to +150 mmHg pressure difference are curve fitted using nonlinear regression, according to the following relationship:

$$ \mathrm{CSA}=\frac{\alpha }{1+{\mathrm{e}}^{\beta \left(\gamma -\Delta P\right)}}+\delta $$

(3.1)

where CSA is the cross-sectional area of the vessel at a given pressure difference (ΔP = intravascular pressure − box pressure) and α, β, γ, and δ are curve fit constants. Equation (3.1) can be expressed in terms of four physical constants as:

$$ \mathrm{CSA}=\frac{\left({\mathrm{CSA}}^{+}-{\mathrm{CSA}}^{-}\right)}{1+\left(\frac{{\mathrm{CSA}}^{+}-{\mathrm{CSA}}^0}{{\mathrm{CSA}}^0-{\mathrm{CSA}}^{-}}\right)\times \left({\mathrm{e}}^{\ln \left(\frac{{\mathrm{CSA}}^{+}-{\mathrm{CSA}}^0}{{\mathrm{CSA}}^0-{\mathrm{CSA}}^{-}}\right)\times \frac{-\Delta P}{\Delta {P}^{1/2}}}\right)}+{\mathrm{CSA}}^{-} $$

(3.2)

where CSA⁺ is the asymptotic value of CSA in the positive ΔP direction (below yield pressure where vessel may undergo plastic deformation and rupture); CSA⁻ is the asymptotic value of the CSA in the negative ΔP direction; CSA⁰ is the CSA value at ΔP = 0; and ΔP^1/2 is the pressure difference corresponding to the average of CSA⁺ and CSA⁻ (i.e. $\frac{{\mathrm{CSA}}^{+}+{\mathrm{CSA}}^{-}}{2}$). The empirical curve fit constants (α, β, γ, δ) are related to the physical constants (CSA⁺ and CSA⁻, CSA⁰, ΔP^1/2) as follows:

$$ {\displaystyle \begin{array}{c}{\mathrm{CSA}}^{+}=\alpha +\delta \\ {}{\mathrm{CSA}}^{-}=\delta \\ {}{\mathrm{CSA}}^0=\frac{\left[\alpha +\delta \left(1+{\mathrm{e}}^{\beta \gamma}\right)\right]}{1+{\mathrm{e}}^{\beta \gamma}}\\ {}\Delta {P}^{1/2}=\gamma \end{array}} $$

(3.3)

The volume data are defined similarly where CSA⁺, CSA⁻, and CSA⁰ are replaced with V⁺, V⁻, and V⁰, respectively, with similar definitions (see Table 3.3).

The compliance (C) of the coronary arteries is determined as the change in luminal dimension (ΔD, ΔCSA, or ΔV) per change in arterial pressure (ΔP_a), external pressure =0 mmHg, i.e., ΔP > 0 mmHg. In the negative pressure difference (ΔP < 0 mmHg) where the vessels are under compression, the compliance is not defined. The CSA-compliance for the first several generations of the LAD artery is calculated, as well as, the V-compliance of the total arterial tree (vessels >0.5 mm in diameter) as summarized in Table 3.4.

Table 3.3 Values for the empirical constants describing the ΔP-CSA relationship (Eq. 3.2) for the first several generations of the coronary left anterior descending (LAD) arteries

Table 3.4 Data for cross-sectional area (CSA)-compliance of the three largest orders of the left anterior descending (LAD) arteries

Appendix 3: Calculation of Transmural Strain (Guo et al., 2005)

If the circumference of a deformed vessel in the loaded state is designated by “C”, the circumferential deformation of a cylindrical can be described by Green strain as follows:

$$ {\varepsilon}_{\mathrm{i},\mathrm{o}}=\frac{1}{2}\left({\lambda}_{\mathrm{i},\mathrm{o}}^2-1\right) $$

(3.4)

where ${\lambda}_{\mathrm{i},\mathrm{o}}={C}_{\mathrm{i},\mathrm{o}}/{C}_{\mathrm{i},\mathrm{o}}^{\mathrm{zs}}$; C_i,o refers to the inner or outer circumference of the vessel in the loaded state and ${C}_{\mathrm{i},\mathrm{o}}^{\mathrm{zs}}$ refers to the corresponding inner or outer circumference in the zero-stress state. To assess the degree of non-uniformity of transmural strain, the ratio of outer to inner strain can be evaluated as:

$$ \frac{\varepsilon_{\mathrm{o}}}{\varepsilon_{\mathrm{i}}}=\left(\frac{C_{\mathrm{o}}^2-{C}_o^{{\mathrm{zs}}^2}}{C_{\mathrm{i}}^2-{C}_{\mathrm{i}}^{{\mathrm{zs}}^2}}\right){\left(\frac{C_{\mathrm{i}}^{\mathrm{zs}}}{C_{\mathrm{o}}^{\mathrm{zs}}}\right)}^2 $$

(3.5)

Hence, the product of the first and second terms of Eq. (3.5) gives the ratio of outer to inner Green strain. Equation (3.5) can be simplified if the quotient is considered in terms stretch ratio, λ, as:

$$ \frac{\lambda_{\mathrm{o}}}{\lambda_{\mathrm{i}}}=\left(\frac{C_{\mathrm{o}}}{C_{\mathrm{i}}}\right)\left(\frac{C_{\mathrm{i}}^{\mathrm{zs}}}{C_{\mathrm{o}}^{\mathrm{zs}}}\right) $$

(3.6)

The first and second terms become linearized and are easier to interrupt physically.

Theoretically, the intimal strain cannot equal to the adventitial strain when θ > 180°. This point can be simply illustrated if the deformation is considered in terms of stretch ratio as given by Eq. (3.6). The first term, ratio of outer to inner circumference in the loaded state, is physically always >1. The second term, ratio of inner to outer circumference in the zero-stress state, is <1 if θ < 180° and >1 if θ > 180°. Hence, when θ < 180° the product of the two terms (the first term is >1 and the second is <1) can be approximately equal to one and hence implies uniformity of strain. On the other hand, when θ > 180° both terms are >1 and hence their product must further deviate from unity. Hence, the strain cannot theoretically be transmurally uniform when θ > 180°. The experimental evidence for the non-uniformity is presented in Table 3.5.

Table 3.5 Comparison of inner (ε_i) and outer (ε_o) Green strains for the coronary arterial tree for different order numbers and ranges of opening angles

Appendix 4: Time Dependence of Opening Angle (Rehal et al., 2006)

The Kelvin model is comprised of a combination of linear springs (with spring constants μ₀ and μ₁) and a dashpot (with coefficient of viscosity η₁) as shown in Fig. 3.29. The stress–strain (σ–ε) equation for the Kelvin model can be stated as:

$$ \sigma +{\tau}_{\varepsilon}\frac{d\sigma}{d t}={E}_R\left(\varepsilon +{\tau}_{\sigma}\frac{d\varepsilon}{d t}\right) $$

(3.7a)

where μ₀ = E_R, μ₁ = (τ_σ − τ_ε) (μ₀/τ_ε), and η₁ = τ_ε μ₁. The corresponding creep solution of Eq. (3.7a) can be written as:

$$ \varepsilon (t)=\frac{1}{E_R}\left[1-\left(1-\frac{\tau_{\varepsilon }}{\tau_{\sigma }}\right){\mathrm{e}}^{-t/{\tau}_{\sigma }}\right]H(t) $$

(3.7b)

where H(t) is the heavy-side step function. Equation (3.7a) is solved for the creep recovery response (i.e., σ = 0):

$$ \varepsilon (t)={\varepsilon}_0{\mathrm{e}}^{\frac{-1}{\eta_1}\left(\frac{\mu_0{\mu}_1}{\mu_0+{\mu}_1}\right)t} $$

(3.7c)

where ε₀ is the initial strain. Experimental data of coronary arteries are obtained from the zero-stress state (measurements taken within 15–30 s after radial cut from loaded state, and followed for a period of 6 h). A total of 26 rings are examined from six hearts (256 ± 27.7 gm).

Fig. 3.29

A schematic of viscoelastic Kelvin model. F is the force acting in a spring, u is the displacement, and u′ is the velocity of displacement; μ₀ and μ₁ are spring constants and η₁ is the coefficient of viscosity of the dashpot

The material constants for the Kelvin model (E_R, τ_σ, and τ_ε) are determined from a curve fit of the strain or creep data with an equation of the form:

$$ \varepsilon (t)=A-B{\mathrm{e}}^{- Ct} $$

(3.7d)

This is analogous to Eq. (3.7b) with A = 1/E_R, B = (τ_σ − τ_ε)/(E_Rτ_σ) and C = 1/τ_σ. A nonlinear least squares yielded the values of the constants A, B, and C which are used to calculate E_R, τ_σ, and τ_ε as E_R = 1/A, τ_σ = 1/C, and τ_ε = (A − B)/CA. These values are then used to determine μ₀, μ₁, and η₁ as outlined above. These values are in turn used in Eq. (3.7c) to predict strain which is compared to the experimental data for the 6-h period.

The difference in opening angle for circumferential and axial loading are fitted with a nonlinear least squares fit of the form analogous to Eq. (3.7d):

$$ \varDelta \mathrm{OA}=\alpha \left(1-\beta {\mathrm{e}}^{-\chi t}\right) $$

(3.8a)

where α, β, and χ are empirical constants obtained from a nonlinear least squares fit. Equation (3.8a) can be expressed as:

$$ \varDelta \mathrm{OA}=\Delta {\mathrm{OA}}^{\infty }+\left(\Delta {\mathrm{OA}}^0-\Delta {\mathrm{OA}}^{\infty}\right){\mathrm{e}}^{-\ln 2\left(\frac{t}{t_{1/2}}\right)} $$

(3.8b)

where ΔOA^∞ = α, ΔOA⁰ = α(1 − β), and t_1/2 = ln 2/χ. ΔOA⁰ and ΔOA^∞ represent the difference in either opening angle or strain at t = 0 and t = ∞, respectively, and t_1/2 represents the time required for ΔOA to reach 50% of its final value. Table 3.6 summarizes the curve fit parameters for the circumferential and axial data.

Table 3.6 Circumferential and axial loading (difference in opening angle, ΔOA, between loaded and no-load states): empirical coefficients of nonlinear least squares fit of Eq. (3.7b)

Appendix 5: Morphology of Coronary Arteries and Veins

Table 3.7 Mean morphological measurements at zero-stress state and loaded state of orders 4–11 LAD arterial vessels (a) and orders −12 to −4 of coronary sinus vessels (b)

Appendix 6: Isovolumic Myography

The isovolumic system consists of a chamber with two connectors which bridge the blood vessel and rigid tubes. One tube connected to a 50 mL flask with physiological saline solution (PSS) and the flask is pressurized with a regulator to inflate the vessel to the desired pressure. Another tube is connected a solid state pressure transducer (SPR-524, Microtip catherter transducer, Millar Inc, Texas) to monitor the transmural pressure and a volume compensator is connected to compensate for water transport across the vessel wall. The outlet of the tube is blocked to achieve isovolumic conditions. The PSS aerated with mixed gas (22% O₂, 5% CO₂, balanced with 73% N₂) filled the chamber and tubes before vessel cannulation. A CCD (charge-coupled device) camera on a microscope transfered the image of the vessel to computer that digitized the external diameter of the vessel. Since the sample rate of digital conversion (200/s) is higher than the rate of change in the vessel during vasoreactivity, the diameter is easily tracked. The vessel is inflated to a physiologic pressure. Since the outlet is closed off, there is no flow in the vessel and the vessel is merely pressurized. To achieve isovolumic state, a clamp placed on the tube between the pressurized flask and the connector is closed and the PSS in the lumen of the vessel and tubes is sealed, i.e., constant volume. The vascular contraction or relaxation during chemical stimulation is characterized by significant changes of intraluminal pressure.

Although a fairly constant volume of the solution can be achieved in the lumen of vessel, it is not strictly constant since the PSS may be transported cross the vessel wall (water flux) driven by the transmural pressure. Although the rate of water flux is very small (<1 nL/min) and no visible reduction of diameter is seen during the duration of experiment (<1 h), a pressure drop (drop in baseline pressure) is still measurable (~0.6–3 mmHg/min). In order to stabilize the baseline pressure, a volume compensator is connected in parallel with the pressure transducer. The volume compensator is comprised of a gastight connector, a microsyrange (maximum volume: 25 μL), a microsyringe pump, and a microsyringe pump controller. The critera for the compensatory rate of the microsyringe pump controller is to maintain the transmural pressure at the desired baseline value (variation < ± 0.2 mmHg/min). There is no measurable change of vessel diameter during compansation. If the leak rate is >1 μL/min, the specimen is discarded as the vessel wall is damaged.

The circumferential tension (T) and stress (a) are computed based on the following:

$$ T=P\times {r}_{\mathrm{int}} $$

(3.9a)

$$ \sigma =\frac{P\times {r}_{\mathrm{int}}}{h} $$

(3.9b)

and

$$ {r}_{\mathrm{int}}=\sqrt{r_{\mathrm{ext}}^2-\frac{A_0}{\pi \lambda}} $$

(3.9c)

where P is intraluminal pressure measured by a pressure transducer and r_int is internal radius of blood vessel computed by the incompressibility assumption (Eq. 3.9c) from the external radius which is measured by diameter tracking system. h is wall thickness which is computed as the difference between r_ext and r_int. A₀ is the cross-sectional wall area of the vessel at the no-load state (zero intraluminal pressure) which is measured from the images of the arterial cross-sectional view. Finally, λ is the axial stretch ratio which is determined by the measurement of the comparison between the in vivo and ex vivo length between two markers on the outer vessel wall.

Dose–response vasoconstriction and vasodilatation in response to vasoconstrictors and vasodilators are carried out under isovolumic conditions. Phenylephrine (PE) is the vasoconstrictor and acetylcholine (ACh) is the vasodilator used for the arteries except for coronary artery. ACh is the vasoconstrictor and bradykinin (BK) is the vasodilator for the coronary artery. Briefly, the artery is stimulated to contract with a vasoconstrictor from 10⁻¹⁰ to 10⁻⁵ mole/L to determine the maximal dose at maximal contraction. Then, the artery is rinsed and equilibrated for 30 min. The artery is contracted with submaximal doses of the vasoconstrictor and relaxed with the vasodilator by a series of doses: 10⁻¹⁰ to 10⁻⁵ mole/L in the PSS. The relaxation resulted in the reduction of intraluminal pressure and circumferential tension which is computed using Eqs. (3.9a, 3.9b, and 3.9c). The calculation of percent relaxation (%R) is based on both intraluminal pressure (%R_P) and tension (%R_T) for comparsion:

$$ \%{R}_{\mathrm{P}}=\left({P}_{\mathrm{d}}-{P}_{\mathrm{i}}\right)/\left({P}_{\mathrm{max}}-{P}_{\mathrm{i}}\right)\times 100 $$

(3.10a)

$$ \%{R}_{\mathrm{T}}=\left({T}_{\mathrm{d}}-{T}_{\mathrm{i}}\right)/\left({T}_{\mathrm{max}}-{T}_{\mathrm{i}}\right)\times 100 $$

(3.10b)

and

$$ \%{R}_{\sigma }=\left({\sigma}_{\mathrm{d}}-{\sigma}_{\mathrm{i}}\right)/\left({\sigma}_{\mathrm{max}}-{\sigma}_{\mathrm{i}}\right)\times 100 $$

(3.10c)

where P_d, P_i, and P_max are the intraluminal pressures at each dose (P_d), inflation pressure (P_i), and maximum pressure (P_max) at 0 mole/L of ACh, respectively. T_d (σ_d), T_i (σ_i), and T_max (σ_max) are the circumferential tension (or stress) at every dose (T_d or σ_d), physiological level (T_i or σ_i), and maximum tension (T_max or σ_max) at 0 mole/L of ACh, respectively. To show the differences in various arteries, mid-tension and mid-stress are computed during vasorelaxation as the averages of maximal and minimal tension and stress, respectively.

Appendix 7: Morphology of Adventitia Fibers

Table 3.8 The layer-to-layer heterogeneity of mean width and area fraction of collagen and elastin fibers assumed as a linear function: P = αN + β, where N is the layer order and α and β are empirical curve fit parameters

Table 3.9 The mechanical loading–deformation relation of fiber (collagen and elastin) geometrical parameter is assumed as linear function: P = αλ_θ + β, where λ_θ is circumferential stretch ratio of vessels

Table 3.10 The relations between axial stretch ratio λ_z and fiber (collagen and elastin) geometric parameters (the orientation angle and waviness) are assumed as linear function: y = αx + β, determined by least squares method

Appendix 8: Morphology of Media Smooth Muscle Cells

Table 3.11 The distribution of geometrical parameters of the nucleus and vascular smooth muscle cells (VSMCs) of the media are fitted to a continuous normal distribution (or a bimodal normal distribution) $f(x)=\frac{1}{\sigma \sqrt{2\pi }}{\mathrm{e}}^{-\frac{{\left(x-\mu \right)}^2}{2{\sigma}^2}}$, where μ is the mean of the distribution, σ is standard deviation, and R² represents goodness of fit

Table 3.12 Nonlinear relations between geometrical parameters and distension pressures for smooth muscle cells (SMCs) and the nucleus obtained by curve fitting to a logarithmic function: y = a₁ + a₂Log(x)