`Constitutive Fitting`

*Module:* `tutorials.04_EM_tissue.07_constitutive_fitting.run`

*Section author: Laura Marx <laura.marx@medunigraz.at>*

Fitting of a passive mechanical material model to the Klotz end-diastolic pressure-volume relation (EDPVR).

As clinical data of the EDPVR is limited the computational method proposed by [Klotz2007] is used, which enables prediction of the EDPVR by a single measured volume-pressure pair.

According to Klotz et al. the stress-free volume of the left ventricle (LV) can be determined empirically by

(56)

where and is a measured volume-pressure pair at end diastole. Further, they showed that the EDPVR can be described by the following power law

(57)

where is cavity pressure in mmHg, is cavity volume in ml, and the constants and can be defined by the relations

(58)

is the cavity volume at a pressure of 30 mmHg, given by

(59)

where and were determined empirically as 27.78 mmHg and 2.76 respectively.

To avoid a singularity in equation (58) and were reposed to

(60)

for greater than 22 mmHg. is determined analytically as

(61)

As the myocardium is treated as nearly incompressible the strain energy function , see equation (62), is split into its decoupled form consisting of a volumetric (volume changing) and isochoric part (volume preserving). is composed of the bulk modulus and a penalty term related to the Jacobian J, see equation (63). Thus, volume change can be penalized and compressibility can be controlled.

(62)

(63)

The isochoric strain-energy function is composed of the transversely isotropic constitutive model of Guccione et al. [Guccione1991] consisting of four material parameters, see equation (64).

(64)

, , and are material parameters accounting for the different mechanical behavior of the tissue along the fibres (f), across the transverse planes (t) and the fibre-transverse shear planes. , , , , and are projections of the Green-Lagrange strain tensor in fibre (f), sheet(s) and sheet normal (n) direction.

Fitting is done by matching cavity volume of the LV in the stress free configuration, approximated during unloading, and EDPVR with those of the Klotz relation (see Fig. 142). is set to 650 and for the material parameters , and the default values 1.627, 3.580 and 18.480 found by [Guccione1991] are used respectively. The parameter C is then varied to achieve the best fit to the Klotz Relation.

A simple ring model, assumed to be in stress free configuration, is passively inflated by running the following command in the directory tutorial/inflate:

```
./run.py --meshtype ring --pressure 2.0 --fast
```

where **meshtype** indicates the type of mesh chosen, **pressure** the maximum pressure to inflate to in kPa and **fast** sets
the resolution and material to fast-solving values.

Switching to the generated file, containing the data files **endo.nbc_p.dat** and **endo.vol.dat** with obtained pressure
and volume traces, and calling

```
cavplot endo.nbc_p.dat endo.vol.dat --klotz --pvloop
```

generates a plot (see Fig. 143) containing the Klotz EDPVR (**klotz**) and the pressure-volume trace
from the data files (**pvloop**).

References

[Klotz2007] | Klotz, S. and Dickstein, M.L. and Burkhoff, D.,
A computational method of prediction of the end-diastolic pressure-volume relationship by single beat (2007),
Nature Protocols, Volume 2 |

[Guccione1991] | (1, 2) Guccione, J.L. and McCulloch, A.D. and Waldman, L.K.,
Passive material properties of intact ventricular myocardium determined from a cylindrical model (1991),
J Biomech Eng, Volume 113 |