Wall

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

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

Wall object.

The Wall module describes a spherical wall (segment) with a wall volume \(V_{wall}\). It assumes that the myocardial tissue is deformable (soft) and incompressible. The input is its midwall area \(A_m\), which is calculated externally by e.g. the Chamber or TriSeg module. Using one or multiple Patch modules, midwall tension \(T_m\), stiffness \(dA_dT\), and transmural pressure \(p_{trans}\) are calculated.

Linearized One-Fiber Model and MultiPatch

The Wall module uses the linearized one-fiber model (Arts et al.[1]) to relate midwall area \(A_m\) to fiber strain \(\varepsilon_f\) and fiber stress \(\sigma_f\) to transmural pressure \(p_{trans}\). This module also includes the MultiPatch module (Walmsley et al.[2]), allowing for segmentation of the Wall by assuming equilibrium in tension \(T_m\) between all Patch objects. This equilibrium is iteratively obtained using the following three steps:

Step 1: The strain of the individual patches is unknown. Therefore, the zero-shortening length \(\varepsilon_{f,est} = \log(l_{si}+l_{se})\), which is equal to the sarcomere length \(l_s\) when \(A_{m,p}\) is constant in time, is used as first approximation.

Step 2: Based on the estimated strain \(\varepsilon_{f,est}\), the stiffness properties for each individual Patch object is calculated. Using the linearized one-fiber model, the tension in each Patch is estimated as

\[T_{m,est}=\frac{1}{2} \frac{V_{wall}}{A_m} \sigma_f\]

with \(A_{m,p,est}\) the midwall area of the patch, based on the strain of the patch \(\varepsilon_{f,p,est}\) and the Patch’s reference area \(A_{m,p,ref}\)

\[A_{m,p,est} = A_{m,p,ref} \cdot e^{2\varepsilon_{f,p,est}}\]

The total wall stiffness \(\frac{dA}{dT}\) and the linearized zero-tension wall area \(A_{m0}\) are calculated.

\[ \begin{align}\begin{aligned}\frac{dA_m}{dT_m}_{est} = \sum_{patches} \frac{dA_m}{dT_m}_{pc,est} = \sum_{patches} \frac{A_{m,p,est}}{ (0.25 \cdot \frac{V_{wall}}{A_m} \cdot \frac{d\sigma_f}{d\varepsilon_f})}\\A_{m0} = \sum_{patches} A_{m0,patch} = \sum_{patches} \left( A_{m,est} - T_{est} \cdot \frac{dA_m}{dT_m}_{est} \right)\end{aligned}\end{align} \]

Step 3 is to enforce tension balance by recalculating the Patch wall areas \(A_{m,p}\) based on the Wall’s linearized tension properties.

\[ \begin{align}\begin{aligned}T_m = (A_m - A_{m0}) / \frac{dA_m}{dT_m}_{est}\\A_{m,p} = A_{m0} + T_m \cdot \frac{dA_m}{dT_m}_{p,est}\end{aligned}\end{align} \]

Step 2 and Step 3 are repeated until no significant changes in estimated stiffness properties are found.

Derivation of the one-fiber model

More details on the one-fiber model can be found in (Arts et al.[1]).