36 research outputs found
On the Deformation of a Hyperelastic Tube Due to Steady Viscous Flow Within
In this chapter, we analyze the steady-state microscale fluid--structure
interaction (FSI) between a generalized Newtonian fluid and a hyperelastic
tube. Physiological flows, especially in hemodynamics, serve as primary
examples of such FSI phenomena. The small scale of the physical system renders
the flow field, under the power-law rheological model, amenable to a
closed-form solution using the lubrication approximation. On the other hand,
negligible shear stresses on the walls of a long vessel allow the structure to
be treated as a pressure vessel. The constitutive equation for the microtube is
prescribed via the strain energy functional for an incompressible, isotropic
Mooney--Rivlin material. We employ both the thin- and thick-walled formulations
of the pressure vessel theory, and derive the static relation between the
pressure load and the deformation of the structure. We harness the latter to
determine the flow rate--pressure drop relationship for non-Newtonian flow in
thin- and thick-walled soft hyperelastic microtubes. Through illustrative
examples, we discuss how a hyperelastic tube supports the same pressure load as
a linearly elastic tube with smaller deformation, thus requiring a higher
pressure drop across itself to maintain a fixed flow rate.Comment: 19 pages, 3 figures, Springer book class; v2: minor revisions, final
form of invited contribution to the Springer volume entitled "Dynamical
Processes in Generalized Continua and Structures" (in honour of Academician
D.I. Indeitsev), eds. H. Altenbach, A. Belyaev, V. A. Eremeyev, A. Krivtsov
and A. V. Porubo
On the Mechanics Underlying the Reservoir-Excess Separation in Systemic Arteries and their Implications for Pulse Wave Analysis
Several works have separated the pressure waveform p in systemic arteries into reservoir pr and excess pexc components, pĀ =Ā prĀ +Ā pexc, to improve pulse wave analysis, using windkessel models to calculate the reservoir pressure. However, the mechanics underlying this separation and the physical meaning of pr and pexc have not yet been established. They are studied here using the time-domain, inviscid and linear one-dimensional (1-D) equations of blood flow in elastic vessels. Solution of these equations in a distributed model of the 55 larger human arteries shows that pr calculated using a two-element windkessel model is space-independent and well approximated by the compliance-weighted space-average pressure of the arterial network. When arterial junctions are well-matched for the propagation of forward-travelling waves, pr calculated using a three-element windkessel model is space-dependent in systole and early diastole and is made of all the reflected waves originated at the terminal (peripheral) reflection sites, whereas pexc is the sum of the rest of the waves, which are obtained by propagating the left ventricular flow ejection without any peripheral reflection. In addition, new definitions of the reservoir and excess pressures from simultaneous pressure and flow measurements at an arbitrary location are proposed here. They provide valuable information for pulse wave analysis and overcome the limitations of the current two- and three-element windkessel models to calculate pr
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Optimal design of vascular stents using a network of 1D slender curved rods
In this manuscript we present a mathematical theory and a computational algorithm to study optimal design of mesh-like structures such as metallic stents by changing the stent strut thickness and width to optimize the overall stent compliance. The mathematical constrained optimization problem is to minimize the ācompliance functionalā over a closed and bounded set of constraints. The compliance functional is the stent's overall elastic energy. The constraints are the minimal and maximal strut thickness, and a given fixed volume of the stent material. We prove the existence of a minimizer, thereby proving that the constrained optimization problem has a solution. A numerical scheme based on an iteration procedure is introduced, and implemented within a Finite Element Method framework. Optimal design of three different stent prototypes is considered: (1) a single zig-zag ring, which can be found in many complex stent designs on the US market as a basic cell in the modular stent design, (2) a PalmazāSchatz type stent consisting of 6 zig-zag rings, and (3) a Cypher(TM) type stent consisting of zig-zag rings with sinusoidal connectors. Several interesting optimization solutions are found, some of which have already been implemented in the design of the currently available stents on the US market. The resulting computational algorithm is compared to a Genetic Algorithm, and it is shown that our computational approach outperforms the Genetic Algorithm in the following three key aspects: (1) computation time, (2) accuracy, and (3) maintaining the symmetry of the solution