Multiscale modelling of the aortic media Marek Netu sil September - - PowerPoint PPT Presentation

Multiscale modelling of the aortic media

Marek Netuˇ sil September 1st, 2016

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Motivation

”Cardiovascular diseases are the leading causes of death and disability in the world. Although a large proportion of CVDs is preventable they continue to rise mainly because preventive measures are inadequate.”

• Global atlas on cardiovascular disease prevention and control. Geneva: World

Health Organization in collaboration with the World Heart Federation and the World Stroke Organization.

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Atherosclerosis

Figure: Evolution of atherosclerosis

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Aneurysm

Figure: Aortic aneurysm

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Goals of the tissue modeling These diseases are closely connected to the material properties of the vessel wall of interest and the mechanical processes it undergoes. Methods of physics and mathematics can help us to understand these material properties and mechanical processes. improve therapies. create non-invasive diagnostic procedures. reduce cost of the treatment.

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Modelling

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Modelling

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Modelling

ρf ˙ φf + DivM = 0 (1 − φ0

f )ρ0 s ¨

us + φf ρf ˙ vf + Div[Ps(∇us)] = 0 in Ω × (0, T) FM ρf = JKF −T(−∇p − ρf F T ˙ vf ) in Ω × (0, T) M = JF −1ρf φf (vf − ˙ us) in Ω × (0, T) J ≥ 1 − φ0

f

in Ω × (0, T)

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Structure of an aortic wall

Figure: Left - main layers of the wall. Right - tunica media.

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Simplifying assumptions

Simplified geometry (note to myself: draw a picture!)

◮ Structure of tunica media is significantly simplified ◮ Aorta is assumed to have a perfect cylindrical shape

Hyperelasticity/Porohyperelasticity

◮ Presence of a viscous ground substance composed of water, mucopolysacharides,

proteoglycans and many other constituents

Residual stress aren’t/are present

◮ Constituents have different natural configurations due to different growth and

turn-over time scales.

Stationary vs Nonstationary case No growth or remodelling

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Cylindrical coordinates

Natural geometrical framework to work in. BUT What is a deformation gradient or a divergence of a two-point tensor in curvilinear coordinates? Literature on the formulation of continuum mechanics in curvilinear coordinates is not easy to find.

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Homogenization

Mathematicaly formal way of deriving macroscopic properties from a microscopic description. Example:

• Ω

A x ε

• ∇uε(x) · ∇v(x)dx =
• Ω

f (x)v(x)dx ∀v ∈ H1

0(Ω)

We know that uε ⇀ u0 in H1

0(Ω). What equation does u0 solve?

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Three steps: Discretization Linearization Homogenization

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FEniCS Project and CBC.Twist

The FEniCS Project is a collection of free software with an extensive list of features for automated and efficient solution of partial differential equations. Among the main components of this tool there are the C++/Python problem solving environment DOLFIN and the Unified Form Language (UFL) that allows a user-friendly and almost mathematical formulation of the problem. CBC.Twist is an elasticity solver build on top of FEniCS: Easy implementation of nonlinear elastic problems. Robust Newton solver. Under construction:

◮ Curvilinear coordinates. ◮ Homogenization. ◮ Arc-length continuation Marek Netuˇ sil Multiscale modelling of the aortic media September 1st, 2016 13 / 16

Drawbacks

Lack of information about the structure and histology of the tissue. No way of using patient specific data. Computationaly very demanding. Difficult to ”sell” to engineers.

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Why bother?

To map the current state of knowledge in the field of cardiovascular biomechanics and histology. To motivate the measurements and experiments to fill the gaps. To encourage to leave the comfort zone of well known models and frameworks. To analyse qualitative properties of the aortic wall.

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Thank you for your attention.

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