# Chamber2022

## Module summary

Chamber2022 object.

### Parameters

- buckling: bool
Buckling function. If True, wall tension cannot be below zero.

### Signals

- V [m
^{3}]: 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:

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:

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

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:

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

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:

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

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

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.

Chamber2022 object. |