Contents

Boundary Conditions

Contents

Boundary Conditions

Choosing the correct boundary conditions for the model is essential, and the choice should mimic what is observed in reality. To physiologically constrain the ventricle in a correct way is difficult, and different approaches has been proposed. The boundary condition at the endocardium is typically modeled as a Neumann boundary condition, representing the endocardial blood pressure. For the left ventricle we have

\[\begin{align} \sigma \mathbf{n} = -p_{\mathrm{lv}} \mathbf{n}, \; \mathbf{x} \in \partial \omega_{ \text{endo LV}}, \end{align}\]

and for the right ventricle, lv is substituted with rv. This condition has a negative sign because the unit normal \(\mathbf{N}\) is pointing out of the domain, while the pressure is acting into the domain. Note that this condition is imposed on the current configuration, and to utilize the Lagrangian formulation we can pull back this condition to the reference configuration to obtain

\[\begin{align} \mathbf{P}\mathbf{N} &= -p_{\mathrm{lv}} J\mathbf{F}^{-T} \cdot \mathbf{N}, \; \mathbf{X} \in \partial \Omega_{\text{endo LV}} \end{align}\]

Likewise, it is common to enforce a Neumann boundary condition on the epicardium,

\[\begin{align} \mathbf{P}\mathbf{N} &= -p_{\mathrm{epi}} J\mathbf{F}^{-T} \cdot \mathbf{N}, \; \mathbf{X} \in \partial \Omega_{\text{epi}}. \end{align}\]

However, the pressure \(p_{\mathrm{epi}}\) is often set to zero as a simplification.

There exist a variety of boundary conditions at the base. It is common to constrain the longitudinal motion of base, even though it is observed in cardiac images that the apex tend to be more fixed than the base. A recent study shows that taking into account the base movement is important to capture the correct geometrical shape [Palit et al., 2016]. However, this has not been done in the studies in this thesis. Fixing the longitudinal motion at the base is enforced through a Dirichlet boundary condition,

\[\begin{align} u_1 = 0, \; \mathbf{X} \in \partial \Omega_{\text{base}}, \end{align}\]

where \(u_1\) is the longitudinal component of the displacement \(\mathbf{u} = (u_1, u_2, u_3)\). To apply this type of condition, it is easiest if the base is flat and located at a prescribed location, for example in the \(x= 0\) plane. Constraining the longitudinal motion of the base alone is not enough since the ventricle is free to move in the basal plane. In order to anchor the geometry it is possible to fix the movement of the base in all directions

\[\begin{align} \mathbf{u} = \mathbf{0}, \; \mathbf{X} \in \partial \Omega_{\text{base}}, \end{align}\]

or fixing the endocardial or epicardial ring

\[\begin{split}\begin{align} \mathbf{u} &= \mathbf{0}, \; \mathbf{X} \in \Gamma_{\mathrm{endo}} \\ \mathbf{u} &= \mathbf{0}, \; \mathbf{X} \in \Gamma_{\mathrm{epi}}. \end{align}\end{split}\]

Another approach which is used in this thesis is to impose a Robin type boundary condition at the base

\[\begin{align} \mathbf{P} \mathbf{N} + k \mathbf{u} = \mathbf{0}, \; \mathbf{X} \in \partial \Omega_{\text{base}}, \end{align}\]

or at the epicardium to mimic the pericardium

\[\begin{align} \mathbf{P} \mathbf{N} + k \mathbf{u} = \mathbf{0}, \; \mathbf{X} \in \partial \Omega_{\text{epi}}. \end{align}\]

Here \(k\) can be seen as the stiffness of a spring that limits the movement. The limiting cases, \(k = 0\) and \(k \rightarrow \infty\) represent free and fixed boundary respectively. More complex boundary conditions to mimic the pericardium are also possible [Fritz et al., 2014], but not considered in this thesis. An overview of the location of the different boundaries for the bi-ventricular geometry is illustrated in Fig. 1.

_images/boundaries.png

Fig. 1 Illustration of the different boundaries in a bi-ventricular domain.

References

FWS+14

Thomas Fritz, Christian Wieners, Gunnar Seemann, Henning Steen, and Olaf Dössel. Simulation of the contraction of the ventricles in a human heart model including atria and pericardium. Biomechanics and modeling in mechanobiology, 13(3):627–641, 2014.

PFB+16

Arnab Palit, Pasquale Franciosa, Sunil K Bhudia, Theodoros N Arvanitis, Glen A Turley, and Mark A Williams. Passive diastolic modelling of human ventricles: effects of base movement and geometrical heterogeneity. Journal of Biomechanics, 2016.

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