From the Law of Laplace to FEM-based stress estimation

Cardiac-Work seminar
Rikshopitalet, November 30th 2023

Henrik Finsberg
Senior Research Engineer
Simula Research Laboratory

Goal

How well does the law of Laplace approximate the stresses in the heart (left ventricle)

• and what can we gain from a using a finite element-based model

The law of Laplace does not take into account

• regional differences
• anisotropy (myocardium behaves differently in fiber and cross fiber direction)

What is the law of Laplace?

The law of laplace states that

Woods, Robert H. "A few applications of a physical theorem to membranes in the human body in a state of tension." Journal of anatomy and physiology 26.Pt 3 (1892): 362.

How can we use the Law of Laplace?

• Hypertension → ↑ pressure → ↑ wall stress

• Concentric hypertrophy → ↑ wall thickness → ↓ wall stress

• So the heart can compensate increased wall stress by thickening the wall
• Increased wall stress → remodeling → heart failure

The finite element method (FEM) is a way to solve equations on complex geometries

We can test this in a finite element model

We can also test this law using different radii

With different widths

Or with different pressures

• But let us try to make our spherical ventricle a bit more realistic

Lets try to add some fibers in our geometry

And let us first make the fibers circumferential

• We will now compute the stress in the direction of the fibers

Stress has nine components

We orient a cube to so that fibers are aligned with one of the axes.

Taken from 10.1186/s12968-016-0258-x under [CC BY 4.0 DD](https://creativecommons.org/licenses/by/4.0/) and from https://en.wikipedia.org/wiki/Cauchy_stress_tensor#/media/File:Components_stress_tensor_cartesian.svg under [CC BY-SA 3.0 DEED](https://creativecommons.org/licenses/by-sa/3.0/deed.en)

Comparing circumferential stress with Laplace

Comparing fiber stress with Laplace

The myocardium is nonlinear and anisotropic

In the calculations so far we have assumed that the heart is linear elastic, isotropic and undergoes small deformation (e.g steel). However, the myocardium is nonlinear, anisotropic and undergoes large deformations.

Using an anisotropic material model

The left ventricle is not a sphere

We can redo the analysis using ellipsoids with different widths and ratios between the short and long axis

We fix the radius along the short axis and vary the width and the long axis radius

Using an anisotropic material model and an ellipsoidal geometry

With FEM-based models we can alo compute the stresses in each AHA segment

In FEM we can fit clinical data to patient specific geometries

In summary

• Laplace law is good for intuition about how pressure, wall thickness and radius is related to the wall stress

• But, it does not capture

  • anisotropy
  • regional differences

• Models based on finite element method can be fitted to clinical data which allows for patient-specific stress estimations

• Slides and material

  • Code and material: https://github.com/finsberg/stress-fem-vs-simple
  • Slides: https://finsberg.github.io/stress-fem-vs-simple/