Chamber2022

Module summary

Tags: Chamber2022

Chamber2022 object.

Parameters

buckling: bool

Buckling function. If True, wall tension cannot be below zero.

Signals

V [m³]: array

Volume of the Cavity

p [Pa]: array

Pressure in the Node

Chamber

The one-fiber model [Arts 1991] simulates a pressurized cavity encapsulated by a wall composed of fibers, immersed in a soft incompressible material. Below, the model is presented briefly. The major assumption is that mechanical energy generated by myofibers dE_f is converted into pump function of the cavity dE_v. without any loss of energy. We define wall shell volume \(Vsh\), myofiber stress \(\sigma_f\), natural myofiber strain \(\epsilon_f\), cavity pressure p and midwall enclosed volume \(V_m\). It holds:

(1)\[V_sh \sigma_f d\epsilon_f = dE_f = dE_v = p dV_m (1)\]

Considering Vsh to be small relative to Vm,, a relative change in fiber strain equals one-third of the relative change in volume, resulting in:

(2)\[ d\epsilon_f = \frac{dV_m}{3V_m}\]

Combining the latter two equations, fiber stress is coupled to cavity pressure:

(3)\[ p=\sigma_f \frac{V_{sh}}{3V_m}\]

To accommodate inhomogeneity of mechanical properties in the wall, we modified our approach by inserting Patch-modules, forming together a wall. Linearized mechanical properties of the wall composed of patches are described with a given zero tension mid-wall area Aw0 and wall stiffness dt/dAw. For wall tension T it is found:

(4)\[ T = \frac{dt}{dA_m}(A_m-A_{m, 0})\]

In this equation, A_{m,0} and frac{dt}{dA_m} are calculated using the Wall2022 component. At low tension, myocardial fibers may buckle. This model has the option ‘buckling’, which replaces this equation with

(5)\[ T = \frac{dt}{dA_m}(\max(0, A_m-A_{m, 0}))\]

The Chamber and TriSeg-module are designed to render cavity pressure(s) as a function of cavity volume(s), given zero-tension area \(Aw0\) and wall stiffness \(dt/dAm\).

A chamber consists of a cavity with cavity volume Vc, wall volume Vw, zero tension midwall area Am0 and wall stiffness dt/dAm. For the primary derivation we use the assumption that the mid-wall surface is spherical with radius r. Using that the mid-wall surface encloses cavity volume and half wall volume, for mid-wall area Aw it holds:

(6)\[ A_m=(6 \sqrt\pi V_m) ^\frac{2}{3} \text{ with } V_m=V_c+\frac{1}{2} V_w\]

Using Eq.(5) wall tension T is determined. Wall tension T and pressure pc are related by the following equation:

(7)\[ T \cdot dA_m = p dV_m\]

Using \(A_m=4\pi r^2\) and \(V_m=4\pi r^3 / 3\), Eq. (7) renders cavity pressure pc :

(8)\[ p_c = \frac{2T}{r} \text{ with } r=(\frac{3V_m}{4\pi})^\frac{1}{3}\]

Since pressure pc is written as a function of Vc and Vw, just like with the one-fiber model, the thus calculated pressure is likely to be not very sensitive to differences in geometry, i.e., geometry may differ from spherical.

circadapt.components.cavity.Chamber2022(model)

Chamber2022 object.