Wall

circadapt.components.wall.Wall(...[, objects])

Wall object.

Module summary

Wall object.

Parameters

Am_dead: float: Non-conctractile wall area.
n_patch: int: Number of patches in the wall. On increase, the last patch will be copied. Wall volume and area are distributed equally over the patches. On decrease, the last patches in the list will be removed.

Signals

Signals are arrays. Each point in the array represents a point in time with step-size controlled by the solver.

Am [m ²]: array: Wall area
Am0 [m ²]: array: Linearized stress-free wall area
Cm [1/m]: array: Wall curvature
dA_dT [m/N]: array: Stress-tension derivative
p_trans [Pa]: array: Transmural pressure
T [Nm]: array: Tension
Vm [m ³]: array: Mid-wall volume
V_wall [m ³]: double: Wall volume

One-Fiber Model

The myocardial tissue module Patch takes as its inputs mid-wall segment tension \(T_m\), mid-wall curvature \(C_m\), and time \(t\), and outputs fibre stiffness \(dA_m/dT\) and zero-tension reference area \(A_m0\). Each initialized module object consists of two state variables, namely the contractile element length \(l_{si}\) and the contractility curve \(C\). Myocardial tissue is deformable (soft) and incompressible. Sarcomere mechanics are assumed homogeneous in a single segment, meaning one single sarcomere represents the entire segment. According to the one-fiber model (Lumens et al.[1]), tension is related to wall stress by

(1)\[T_{\text{m}} \approx {\frac{{V_{\text{w}} \sigma_{\text{f}} }}{{2A_{\text{m}} }}}\left( {1 + {\frac{{z^{2} }}{3}} + {\frac{{z^{4} }}{5}}} \right)\quad {\text{with}}\quad \, \sigma_{\text{f}} = f(\varepsilon_{\text{f}} )\]

Therefore, \(dA_m/dT\) is given by

(2)\[ \begin{align}\begin{aligned}\frac{dA}{dT} = 1/\frac{dT}{dA}\\\frac{dT}{dA} = \frac{1}{4} \frac{V_{wall}}{A_m^2} (1 + \frac{z^2}{3} + \frac{z^4}{5}) \cdot \frac{d\sigma_f}{d\varepsilon_f}\end{aligned}\end{align} \]

and \(A_{m,0}\) is given by

(3)\[A_{m, 0} = A_m - T \cdot \frac{dA}{dT}\]

One Fiber Derivation

“The relation between fiber stress \(\sigma_f\) and cavity pressure \(p\) in a thick-walled rotationally-symmetric geometry is found by integration of pressure increments over a sufficient number of thin fitting shells. It is assumed that fiber stress is homogeneous in the thick wall and that the fibers are directed parallel to the isobaric surfaces.” Arts et al.[2].

\[\frac{dp}{dV} = \frac{-\sigma_f }{3V}\]