Biomechanics

Abstract

There is no doubt that one of the most significant health problems facing people around the world is vascular disease that compromises perfusion of vital organs (e.g., heart, brain, etc.). Abnormal mechanical stresses and deformation of blood vessels have been identified as key culprits in the initiation and progression of vascular disease. To understand the blood circulation through blood vessels, one must consider the blood, the blood vessel wall, the tissue surrounding the vessel wall, the geometry of the vascular system, and the driving forces from pumping of the heart. Blood vessels are remarkable organs that nurture organisms, transport many enzymes and hormones, contain blood cells that flow or clot when needed, and transport oxygen and carbon dioxide between the lungs and the cells of the tissues. Physiologists study these important functions of the vasculature as they relate to the functioning of the body. Bioengineers apply engineering principles to understand biological systems. For the bioengineer, the understanding of the biomechanics of circulation is a central focus to explain vascular health and disease.

Appendices

Appendix 1: Derivation of Circumferential Stress (Laplace’s Law) and Longitudinal Stress in a Vessel

If a body is in equilibrium, every part of it is in equilibrium. To determine the internal reaction stresses, one can cut free certain parts of the body and examine their conditions of an equilibrium. Consider a cylindrical vessel subjected to an internal pressure P_i, as shown in Fig. 1.5a below, where cuts are made in different planes. The blood pressure induces stress in the vessel wall. Under equilibrium conditions, the force in the vessel wall in the circumferential direction 2τ_θ(r_o − r_i)L, is balanced by the force in the vessel lumen contributed by the pressure 2Lr_iP_i, as shown in Fig. 1.5b. Hence, the equilibrium equation in the circumferential direction is given by:

$$ {\tau}_{\theta }={P}_{\mathrm{i}}{r}_{\mathrm{i}}/\left({r}_{\mathrm{o}}-{r}_{\mathrm{i}}\right) $$

(1.1)

where τ_θ is circumferential stress, r_i is internal radius, and r_o is outer radius of vessel.

Fig. 1.5

A pressurized cylindrical tube. (a) An infinitesimal element of the longitudinal and circumferential cylindrical tube showing the radial, longitudinal and circumferential directions. (b) A free-body diagram of half of the tube cut parallel to the central axis. (c) A free-body diagram of the tube cut perpendicular to the central axis. Reproduced from Gregersen and Kassab (1996) with permission

The longitudinal stress in a vessel wall can be determined based on the equilibrium of forces in the longitudinal direction. The product of the longitudinal stress and the cross-sectional area of the vessel wall is the force that balances the total longitudinal force acting on the vessel as shown in Fig. 1.5c. The longitudinal force in the vessel wall ${\tau}_z\pi \left({r}_{\mathrm{o}}^2-{r}_{\mathrm{i}}^2\right)$ is balanced only by the pressure component ${P}_{\mathrm{i}}\pi {r}_{\mathrm{i}}^2$ as the external pressure is assumed to be zero. Thus, the desired equation follows:

$$ {\tau}_z={P}_{\mathrm{i}}{r}_{\mathrm{i}}^2/\left({r}_{\mathrm{o}}^2-{r}_{\mathrm{i}}^2\right) $$

(1.2)

where τ_z is the longitudinal stress. If the wall thickness-radius ratio is small, so that r_o = r_i = r and r_o − r_i = h, then these equations are simplified to

$$ {\tau}_{\theta }={P}_{\mathrm{i}}r/h,\kern2em {\tau}_z={P}_{\mathrm{i}}r/2h. $$

(1.3)

Appendix 2: Constitutive Equation of a Homogeneous, Isotropic, and Linear Elastic Solid (Hooke’s Law)

The constitutive equations of a solid that consists of a homogeneous, isotropic, linearly elastic material contain only two material constants given by Hooke’s law:

$$ {\varepsilon}_{ij}=\frac{1}{2\mu}\left[\frac{-\lambda }{3\lambda +2\mu }{\tau}_{ij}{\delta}_{ij}+{\tau}_{ij}\right] $$

(1.4)

where i and j are indices ranging from integers 1 to 3. The ith index denotes the component in the ith direction whereas the jth index denotes the surface perpendicular to the jth direction. The repetition of an index in a term denotes a summation with respect to that index over its range.

Several special cases will be considered:

Special Case 1

A uniaxial state of stress with the non-zero stress-component τ₁₁ corresponding to the x₁-direction. From Eq. (1.4), the following holds:

$$ {\varepsilon}_{11}=\frac{\lambda +\mu }{\mu \left(3\lambda +2\mu \right)}{\tau}_{11},\kern1em {\varepsilon}_{22}={\varepsilon}_{33}=\frac{-\lambda }{2\mu \left(3\lambda +2\mu \right)}{\tau}_{11} $$

(1.5)

The coefficient between stress and strain can be expressed as:

$$ E=\mu \frac{\left(3\lambda +2\mu \right)}{\lambda +\mu } $$

(1.6)

which is called the elastic modulus or Young’s modulus. This module can be readily measured in a uniaxial tension test. The ratio −ε₂₂/ε₁₁ =  − ε₃₃/ε₁₁ is given by:

$$ v=\frac{\lambda }{2\left(\lambda +\mu \right)} $$

(1.7)

which is known as Poisson’s ratio. Poisson’s ratio is a measure of the lateral contraction (extension) produced by an axial tension (compression).

Special Case 2

Consider a state of plane stress in pure shear in which the only non-zero component of the stress tensor is τ₁₂ = τ₂₁ ≠ 0. The corresponding non-zero strain component is given by

$$ {\varepsilon}_{12}=\frac{1}{2\mu }{\tau}_{12} $$

(1.8)

The ratio of shearing stress τ₁₂ and the corresponding change 2ε₁₂ of an initially right material angle is known as the shear modulus. In the engineering literature, the symbol G is widely used to describe the shear modulus.

Special Case 3

Finally, consider a hydrostatic state of stress, given by:

$$ {\tau}_{ij}=-p{\delta}_{ij} $$

(1.9)

If this result is combined with Eq. (1.4), the following holds:

$$ {\varepsilon}_{ij}=\frac{-1}{3\lambda +2\mu }p{\delta}_{ij} $$

(1.10)

Setting i = j, and summing on j, one finds:

$$ -p=K{\varepsilon}_{ii} $$

(1.11)

where ε_ii = ε₁₁ + ε₂₂ + ε₃₃ and proportionality constant given by:

$$ K=\lambda +\frac{2}{3}\mu $$

(1.12)

known as the bulk modulus. The bulk modulus is a measure of the compressibility of the solid. For an incompressible material, K and hence λ are unbounded (→∞). If a hydrostatic pressure is to be accompanied by a volume decrease, then the bulk modulus K must be positive. Since shearing should occur in the direction of the shearing stress, the following constrains holds for the parameters:

$$ \infty \ge \left(\lambda +\frac{2}{3}\mu \right)>0,\kern2.00em \infty >\mu >0 $$

(1.13a)

The corresponding bounds on E and v are given by:

$$ 0<E\le 3\mu, \kern2em -1<v\le \frac{1}{2} $$

(1.13b)

Hence, a simple tension is always accompanied by an extension. On the other hand, in a direction normal to the direction of this tension, a contraction may take place.

Appendix 3: Equations for Fluids and Solids

The governing equations for the fluid domain are the Navier–Stokes (NS) and continuity equations (Fefferman, 2000):

$$ {\displaystyle \begin{array}{c}\frac{\partial \overrightarrow{V}}{\partial t}+\overrightarrow{V}\cdot \overrightarrow{\nabla}\overrightarrow{V}+\frac{\overrightarrow{\nabla}p}{\rho }-2\frac{\eta }{\rho}\overrightarrow{\nabla}\cdot D=\overrightarrow{0}\\ {}\overrightarrow{\nabla}\cdot \overrightarrow{V}=0\end{array}} $$

where V is fluid velocity, P is fluid pressure, ρ is fluid mass density, η is fluid dynamic viscosity, $\overrightarrow{\nabla}$ is the gradient operator, and D is the fluid rate of deformation tensor.

The governing equations for the solid are the momentum and equilibrium equations (Fung, 1997):

$$ {\displaystyle \begin{array}{lll}\rho {a}_i-{\sigma}_{ij,j}-\rho {f}_i=0& \mathrm{in}& {}^s\Omega (t)\\ {}{\sigma}_{ij}{n}_j-{t}_i=0& \mathrm{on}& {}^s\Gamma (t)\end{array}} $$

where ^sΩ(t) is the structural domain at time t, t₁ is surface traction vector, σ_ij is stress of the solid, and a_i is the acceleration of the material point along the ith direction.