Influence of Valve Size, Orientation and Downstream Geometry of an Aortic BMHV on Leaflet Motion and Clinically Used Valve Performance Parameters

Abstract

The aim of this study was to reconcile some of our own previous work and the work of others to generate a physiologically realistic numerical simulation environment that allows to virtually assess the performance of BMHVs. The model incorporates: (i) a left ventricular deformable model to generate a physiological inflow to the aortic valve; (ii) a patient-specific aortic geometry (root, arch and descending aorta); (iii) physiological pressure and flow boundary conditions. We particularly studied the influence of downstream geometry, valve size and orientation on leaflet kinematics and functional indices used in clinical routine. Compared to the straight tube geometry, the patient-specific aorta leads to a significant asynchronous movement of the valve, especially during the closing of the valve. The anterior leaflet starts to close first, impacts the casing at the closed position and remains in this position. At the same time, the posterior leaflet impacts the pivoting mechanisms at the fully open position. At the end of systole, this leaflet subsequently accelerates to the closed position, impacting the casing with an angular velocity of approximately −477 rad/s. The valve size greatly influences the transvalvular pressure gradient (TPG), but does not change the overall leaflet kinematics. This is in contrast to changes in valve orientation, where changing valve orientation induces large differences in leaflet kinematics, but the TPG remains approximately the same.

Appendices

Appendix A: Three-Element Windkessel Model

At the outlet of the descending aorta, a physiological blood pressure profile is imposed using a three-element windkessel model and the patient-specific MR flows in the descending aorta. The windkessel consists of three elements: the resistance R, the characteristic impedance Z ₀ and the compliance C. Appropriate values for these parameters are calculated as follows.

The resistance R is calculated by

$$R = \frac{{p_{\text{mean}} }}{{Q_{\text{mean}} }} = \frac{{96\;{\text{mmHg}}}}{{84\;{\text{mL}}/{\text{s}}}} = 1.143\;{\text{mmHg}}/({\text{mL}}/{\text{s}})$$

(A1)

wherein Q _mean and p _mean represent, respectively, the mean patient-specific MR flow and the mean pressure level in the descending aorta (assessed proximal to renal artery branching).

The characteristic impedance Z ₀ is obtained from the following equation:

$$Z_{0} = \frac{\rho \cdot c}{A} = \frac{{1000\;{\text{kg}}/{\text{m}}^{3} \cdot 5\;{\text{m}}/{\text{s}}}}{{3.2558 \cdot 10^{4} {\text{m}}^{2} }} = 0.115\;{\text{mmHg}}/({\text{mL}}/{\text{s}})$$

(A2)

The parameter ρ is the density of blood, c is the pulse wave velocity, and A is the outlet area of the descending aorta.

The compliance C is set equal to 1 mL/mmHg, as is commonly done in literature.

All the parameter values agree with (Kim et al.18; Xiong28).

The resulting blood pressure profile is visualized in Fig. 4b.

Appendix B: Left Ventricular Volume Change

In this section, the equations that describe the volume of the LV during the diastolic expansion are derived.

Since no volume changes of the LV can be obtained from the MR dataset, the diastolic expansion of the LV needs to be modelled. The following data are available from the patient-specific MR dataset, as already discussed in “Fluid–Structure Interaction of the BMHV”: the systolic flow rate (of which SV _fwd is calculated), the ESV (obtained from the patient-specific LV geometry) and the heart rate. From these parameters, the diastolic phase can be modelled by adapting a statistical diastolic LV expansion model with these data. Hereafter, this is discussed in more detail.

The volume of the LV during diastolic filling is solved piecewise by three functions (V ₁(t), V ₂(t) and V ₃(t)), representing, respectively, the RF-phase, the SF-phase and the AC-phase of the LV:

$$\left\{ \begin{gathered} V_{1} \left( t \right) = a_{4} t^{4} + a_{3} t^{3} + a_{2} t^{2} + a_{1} t + a_{0} \hfill \\ V_{2} \left( t \right) = b_{3} t^{3} + b_{2} t^{2} + b_{1} t + b_{0} \hfill \\ V_{3} \left( t \right) = c_{4} t^{4} + c_{3} t^{3} + c_{2} t^{2} + c_{1} t + c_{0} \hfill \\ \end{gathered} \right.$$

(B1)

In order to obtain the appropriate values for the 14 unknowns (a _i , b _i and c _i ), the following equations are solved, based on experimentally parameters obtained in14:

Isovolumetric relaxation:

$$\left\{ \begin{gathered} V_{1} \left( {t_{\text{ESV}} } \right) = {\text{ESV}} \hfill \\ \frac{{dV_{1} }}{dt}\left( {t_{\text{ESV}} } \right) = 0 \hfill \\ \end{gathered} \right.$$

(B2)

Peak rapid filling (pRF):

$$\left\{ \begin{gathered} \frac{{dV_{1} }}{dt}\left( {t_{{p{\text{RF}}}} } \right) = 5.25 \cdot SV_{\text{fwd}} \hfill \\ \frac{{d^{2} V_{1} }}{{dt^{2} }}\left( {t_{{p{\text{RF}}}} } \right) = 0 \hfill \\ \end{gathered} \right.$$

(B3)

End of rapid filling (eRF):

$$\left\{ \begin{gathered} V_{1} \left( {t_{{e{\text{RF}}}} } \right) = V_{2} \left( {t_{{e{\text{RF}}}} } \right) \hfill \\ \frac{{dV_{1} }}{dt}\left( {t_{{e{\text{RF}}}} } \right) = \frac{{dV_{2} }}{dt}\left( {t_{{e{\text{RF}}}} } \right) \hfill \\ \end{gathered} \right.$$

(B4)

Peak slow filling (pSF):

$$\left\{ \begin{gathered} \frac{{dV_{2} }}{dt}\left( {t_{{p{\text{SF}}}} } \right) = 2 \hfill \\ \frac{{d^{2} V_{2} }}{{dt^{2} }}\left( {t_{{p{\text{SF}}}} } \right) = 0 \hfill \\ \end{gathered} \right.$$

(B5)

End of slow filling (eSF):

$$\left\{ \begin{gathered} V_{2} \left( {t_{{e{\text{SF}}}} } \right) = V_{3} \left( {t_{{e{\text{SF}}}} } \right) \hfill \\ \frac{{dV_{2} }}{dt}\left( {t_{{e{\text{SF}}}} } \right) = \frac{{dV_{3} }}{dt}\left( {t_{{e{\text{SF}}}} } \right) \hfill \\ \end{gathered} \right.$$

(B6)

Peak atrial contraction (pAC):

$$\left\{ \begin{gathered} \frac{{dV_{3} }}{dt}\left( {t_{{p{\text{AC}}}} } \right) = 2.1 \cdot SV_{\text{fwd}} \hfill \\ \frac{{d^{2} V_{3} }}{{dt^{2} }}\left( {t_{{p{\text{AC}}}} } \right) = 0 \hfill \\ \end{gathered} \right.$$

(B7)

End of diastole:

$$\left\{ \begin{gathered} V_{3} \left( {t_{\text{EDV}} } \right) = {\text{ESV}} + SV_{\text{fwd}} \hfill \\ \frac{{dV_{3} }}{dt}\left( {t_{\text{EDV}} } \right) = 0 \hfill \\ \end{gathered} \right.$$

(B8)

Moreover, since a time cycle of 0.96 s is used in,14 the time levels of14 need to be scaled to the period of 0.84 s. The used values are:

$$\left\{ \begin{gathered} t = t_{\text{ESV}} = 0.32s \hfill \\ t = t_{{p{\text{RF}}}} = 0.4375s \hfill \\ t = t_{{e{\text{RF}}}} = 0.5075s \hfill \\ t = t_{{p{\text{SF}}}} = 0.63s \hfill \\ t = t_{{e{\text{SF}}}} = 0.735s \hfill \\ t = t_{{p{\text{AC}}}} = 0.7525s \hfill \\ t = t_{\text{EDV}} = 0.84s \hfill \\ \end{gathered} \right.$$

(B8)

Solving previous equations results in the diastolic expansion as visualized in Figs. 4c–4d.