Vessels biomechanics – A matter of fact Dept. of Applied Biotechnologies and Clinical Medicine D. Campanacci University of Bologna Professor Claudio Borghi – Doctor Marco Manca – Marco Manca, MD

Forecasting is difficult, especially when it the future. Niels Bohr ( ) father of quantum mechanics

The arterial wall is a layered structure with distinct sections identifiable as: Intima Media Adventitia

The wall is comprised of cells, elastin, and collagen, and the distribution of these elements varies from the inner wall to the outer wall, as well as along the vascular tree.

Second panel: Pressure changes in the transmission line model. Mean pressure is sustained in the transmission line, but falls steeply at its termination with small resistance vessels. Wave reflection at the junction causes higher pulsatile pressure changes at the end of the model than at its origin. Third panel: Resistance per unit length in the model. Fourth panel: Pressure waveform at the origin of the transmission line when aortic stiffness is low, as in adolescents. Fifth panel: Pressure waveform at the origin of the transmission line when aortic stiffness is high, as occurs with aging. Higher aortic pulse wave velocity causes the echo from wave reflection to move from diastole into systole, thus boosting aortic systolic pressure and reducing pressure during diastole. ORourke MF Journal of Biomechanics, 2002

X-ray diffraction patterns from dog carotid air-dried under pressure recorded with the incident beam along the radial (a) and circumferential (b) directions. The arrow in (b) indicates the 0.29 nm collagene reflection. Pallotti G et al. Journal of mechanics in medicine and biology, 2002

Arteries are: Nonlinear Anisotropic Viscoelastic A comprehensive constitutive equation remains an elusive goal

Incompressibility Incompressibility is a reasonable assumption in that biological tissues contain mostly water, which is incompressible at physiologic pressures: Vaishnav RN et al. Circulation Research, 1973

Residual stresses Blood vessels are formed in a dynamic environment, which gives rise to imbalances between the forces tending to extend the diameter, and length and the internal forces which tend to resist this extension. This imbalance is thought to drive the deposition of elastin and collagen which reduce the stress in the underlying tissue. Under these conditions it is not surprising that a residual stress state exists when the vessel is fully retracted and free from external traction. A longitudinal cut in an arterial section results in the unrolling of the artery. Bergel DH PhD Thesis - University of Londond, 1960.

Residual stresses Including residual stress in arterial models results in a more uniform distribution of stress, noticeably decreasing stress at the intimal layer, as compared to models without residual stress. Current models incorporate residual stress using a cut-open ring segment as the zero- stress state. In one approach, the opening angle is measured and included in the circumferential stretch ratio used to calculate strain. This method uses incompressibility to relate the radii and opening angle Humphrey JD et al. Annals of Biomedical Engineering, 2002.

Smooth muscles contractility Smooth muscle modifies with its contraction the dynamic properties of the arterial walls. One way of modeling smooth muscle contractility is the use of different functional forms for the active and passive circumferential stress: Rachev A et al. Annals of Biomedical Engineering, 1999 The total circumferential stress is the sum of the active and passive components.

Smooth muscles contractility Pressure waves recorded simultaneously in the proximal aorta and radial artery of a young man (A) under control conditions, and (B–D) with exercise or increasing intensity. ORourke MF Journal of Biomechanics, 2002

Pressure-Related Dynamic Wall Motion If (R, Θ, Z) and (r, θ, z) are the cylindrical polar coordinates of a body before and after deformation, wall motion caused by time-dependent internal pressure and axial force may be described as: Demiray H et al. Journal of Biomechanics, 1983

Strain Energy Density Functions Special attention is due to Strain Energy Density Functions. These are a convenient, commonly used way to derive a constitutive equation for a biological tissue. Strain energy is a function of the deformation gradient tensor, F, and various unknown parameters Green AE et al. Large Elastic Deformations. Oxford: Clarendon. 2nd ed., 1970

Pseudoelastic models The vessel is treated as one hyperelastic material in loading and another in unloading Fung YC et al. American Journal of Physiology - Heart and Circulatory Physiology, 1979 The approach is simple and captures vessel deformation. Pseudoelastic models form the backbone of other model types. The equation separately models loading or unloading data. Data multicollinearity leads to unstable parameters and protocol dependence if incremental loading methods are used.

Randomly elastic models The strain response is centered around a definite value that lies on a well-defined curve Brossollet LJ et al. Journal of Biomechanics, 1995 The equation models both loading and unloading data simultaneously Data are noisier than when separately modeling loading or unloading values

Poroelastic models The tissue is treated as a fluid-saturated porous medium. Due to their complexity, they are well suited for FEM implementation. Simon BR et al. Journal of Biomechanical Engineering, 1998 The equation models the fluid contribution to tissue properties The movement of fluid through the porous tissue may be irrelevant within the timescale of the experiment. Nonlinear models include many unknown parameters.

Viscoelastic models The strain response is a function of the stress history. Armentano RL et al. Circulation Research, 1995 The equation models observed viscoelastic behavior (creep, hysteresis, and stress relaxation) The response is usually modeled using a discrete number of elements. Data from one viscoelastic test may be insufficient to model characteristics from other viscoelastic tests

Applications Applications of blood vessel constitutive models include studies comparing healthy and diseased populations and studies that help with understanding or predicting the initiation, progression, and clinical treatment of diseases such as atherosclerosis.

One clinical treatment for stenosed vessels involves the use of stenting, whereby an expandable mesh tube is deployed at the stenosis site to reinforce the lumen. A goal of stenting studies is to understand the effect of the deployed stent on blood vessel properties and determine the resulting biological response. FEM has been shown to be a useful tool for the study of free stent expansion, balloon- stent interactions, and stent-wall interactions. In most approaches, simple linear-elastic models have been used for the artery, stent, and balloon. Stenting Holzapfel GA et al. Annals of Biomedical Engineering, 2002

Combined in vivo and in vitro experiments have been conducted in which a stent was harvested from a rabbit iliac artery in which it had been deployed an hour earlier. In a complementary experiment, a stent was deployed in a transparent glass tube to observe when and how the stent made contact with the tube and when and how the dilating balloon made contact with the tube. The observation and later modeling were aimed to describe how balloon-artery interactions contribute to vascular injury in an effort to recommend new stent designs and deployment protocols. Stenting Rogers C et al. Circulation Research, 1999

A drug-eluting stent is a device releasing into the bloodstream single or multiple bioactive agents that can deposit in or affect tissues adjacent to the stent. Drug can be simply linked to the stent surface, embedded and released from within polymer materials, or surrounded by and released through a carrier. Drug eluting stents Fattori R et al. Lancet, 2003

Conclusions The bioengineer has more to contribute to medicine than he/she ever has in the past. Clinician and engineer must be prepared to compromise, but to know where compromise is warranted, and where it is not. Mathematical modeling has to be driven not only by the need of achieving practical goals but also by the need to produce new consciousness in physicians approaching the clinics.