A Review on Atherosclerotic Biology, Wall Stiffness, Physics of Elasticity, and Its Ultrasound-Based Measurement

Abstract

Functional and structural changes in the common carotid artery are biomarkers for cardiovascular risk. Current methods for measuring functional changes include pulse wave velocity, compliance, distensibility, strain, stress, stiffness, and elasticity derived from arterial waveforms. The review is focused on the ultrasound-based carotid artery elasticity and stiffness measurements covering the physics of elasticity and linking it to biological evolution of arterial stiffness. The paper also presents evolution of plaque with a focus on the pathophysiologic cascade leading to arterial hardening. Using the concept of strain, and image-based elasticity, the paper then reviews the lumen diameter and carotid intima-media thickness measurements in combined temporal and spatial domains. Finally, the review presents the factors which influence the understanding of atherosclerotic disease formation and cardiovascular risk including arterial stiffness, tissue morphological characteristics, and image-based elasticity measurement.

Appendix A

$$ PWV = \frac{d}{t_2-{t}_1} = \frac{d}{\varDelta t}, $$

(1)

where d is the distance measured between carotid to femoral arterial sites.

$$ PWV=\sqrt{\frac{IMT \times {Y}_m}{2 \times \rho \times r}}. $$

(2)

Thus, Young’s elastic modulus Y _m is expressed as:

$$ {Y}_m = \frac{\left(2 \times \rho \times r \times PW{V}^2\right)}{IMT} $$

(3)

where IMT Y _m , ρ and r are intima-media thickness, YEM, density of fluid within the lumen and lumen radius, respectively.

$$ DD=\left(\frac{D_{\max }\ \hbox{--}\ {D}_{\min }}{D_{\min }}\right), $$

(4)

where D _max is the maximum artery diameter and D _min is the minimum artery diameter.

Peterson’s Elastic Module (E _p ) can be expressed as:

$$ {E}_p = \frac{PP}{DD}, $$

(5)

where PP is the pulse pressure and DD is the as in Eq. (4).

$$ {\upsigma}_{\mathrm{cs}} = \frac{PP \times {D}_{min}}{2 \times IM{T}_{D_{min}}}, $$

(6)

where σ_cs is the circumferential stress, $IM{T}_{D_{min}}$ is the IMT measured at the instance when arterial diameter was minimum in the far wall.

$$ {Y}_m = \frac{\mathrm{Stress}}{\mathrm{Strain}}=\frac{\upsigma_{\mathrm{cs}}}{DD}=\frac{\left[\frac{PP \times Dmin}{2 \times IM{T}_{D_{\min }}}\right]}{DD}=\left[\frac{PP}{DD}\right]\times \left[\frac{D_{\min }}{2\times IM{T}_{D_{\min }}}\right] $$

(7)

If ${E}_p=\frac{PP}{DD}$ (Peterson’s elastic modulus), then Y _m can be expressed in terms of Peterson’s elastic modulus (E _p ) as:

$$ {Y}_m=\frac{E_p \times {D}_{\min }}{2 \times IM{T}_{D_{\min }}}. $$

(8)

$$ C{C}_r = \frac{\left({D}_{\max}^2 - {D}_{\min}^2\right)}{PP}. $$

(9)

Distensibility DIS can be represented as:

$$ DIS = \frac{D{D}_{sr}}{PP} = \frac{C{C}_r}{D_{\min}^2}, $$

(10)

where DD _sr = $\left[\frac{\left({D}_{\max}^2 - {D}_{\min}^2\right)}{D_{\min}^2}\right]$, PP is the pulse pressure; D _max is the maximum diameter of artery and D _min is the minimum diameter of artery.

$$ C{C}_v=\frac{\pi }{4PP}\times {L}_d\times \left({D}_s^2-{D}_d^2\right) + \frac{\pi }{4PP}\times {L}_d\times {D}_s^2\times \frac{\varDelta L}{L_d}. $$

(11)

where, $\frac{\varDelta L}{L_d}$ is expressed in terms of IMT, LD and arterial compressibility. It is mathematically represented as in Eq. 12:

$$ \frac{\varDelta L}{L_d}=\left[\left(1-\delta \right)\times \frac{IM{T}_d \times \left({D}_d + IM{T}_d\right)}{IM{T}_s \times \left({D}_s + IM{T}_s\right)}\right]-1 $$

(12)

and L _d , D _s , D _d , PP, δ, IMT _d,  and IMT _s are diastolic length of cylindrical artery, systolic diameter, diastolic diameter, pulse pressure, compressibility factor, diastolic IMT and systolic IMT, respectively. δ can be expressed as:

$$ \delta =1-\frac{V_s}{V_d}, $$

(13)

where V _d  and V _s is volume of vessel wall in diastole, systole, respectively. V _s and V _d can be computed using Eq. 14 as:

$$ \left\{\begin{array}{l}{V}_s = \pi \times {L}_s \times IM{T}_s \times \left[{D}_s + IM{T}_s\right]\hfill \\ {}{V}_d = \pi \times {L}_d\times IM{T}_d\times \left[{D}_d+IM{T}_d\right]\hfill \end{array}\right. $$

(14)

where L _s is the systolic length of cylindrical artery, D _s and D _d are the diastolic and systolic diameter, respectively.