Investigating the reference domain influence in personalised models of cardiac mechanics

Abstract

A major concern in personalised models of heart mechanics is the unknown zero-pressure domain, a prerequisite for accurately predicting cardiac biomechanics. As the reference configuration cannot be captured by clinical data, studies often employ in-vivo frames which are unlikely to correspond to unloaded geometries. Alternatively, zero-pressure domain is approximated through inverse methodologies, which, however, entail assumptions pertaining to boundary conditions and material parameters. Both approaches are likely to introduce biases in estimated biomechanical properties; nevertheless, quantification of these effects is unattainable without ground-truth data. In this work, we assess the unloaded state influence on model-derived biomechanics, by employing an in-silico modelling framework relying on experimental data on porcine hearts. In-vivo images are used for model personalisation, while in-situ experiments provide a reliable approximation of the reference domain, creating a unique opportunity for a validation study. Personalised whole-cycle cardiac models are developed which employ different reference domains (image-derived, inversely estimated) and are compared against ground-truth model outcomes. Simulations are conducted with varying boundary conditions, to investigate the effect of data-derived constraints on model accuracy. Attention is given to modelling the influence of the ribcage on the epicardium, due to its close proximity to the heart in the porcine anatomy. Our results find merit in both approaches for dealing with the unknown reference domain, but also demonstrate differences in estimated biomechanical quantities such as material parameters, strains and stresses. Notably, they highlight the importance of a boundary condition accounting for the constraining influence of the ribcage, in forward and inverse biomechanical models.

Appendix

1.1 Registration of reference domain to in-vivo images

Fig. 10

Left panel shows the relative positions of the reference image and mesh, before and after registration, compared to the ED in-vivo frame, and the right panel shows the relative positions compared to the ES in-vivo frame. Top: Reference image (in colour) overlaid on top of the grey-scale in-vivo frame, before (left) and after (right) registration. Circles denote corresponding parts of the ribcage in the reference and in-vivo images, before and after registration. Bottom: Position of the reference mesh (in colour) relative to the ED or ES mesh (white), before (left) and after (right) registration

An essential step was to register the static reference image to the dynamic in-vivo images, in order to determine the relative position between the unloaded domain and in-vivo states. This is a key step for properly simulating the transition from reference domain to in-vivo images and vice-versa and for determining appropriate boundary conditions. However, as there is no known correspondence between the reference domain and a specific frame of the cardiac cycle, use of standard registration for cardiac images would not be possible.

Fig. 11

Behaviour of objective functions relying on cavity volumes (left) and distance (right) metrics, over the parameter space, in inverse simulations. The absolute value of the difference between the simulated cavity volume and $V_{Klotz}$ is presented in mL, while the RMSE between the estimated unloaded domain and the in-situ reference mesh is given in mm. White circles depict the combination of passive parameters providing the lowest error values, while the yellow circle corresponds to the parameter estimates obtained in the forward simulations’ framework

Instead, registration was performed based on nonmoving tissues, specifically the ribs. Segmentations of the ribcage in the end-diastolic in-vivo and reference images were used to create surface meshes, which were subsequently registered using the Iterative Closest Point Algorithm (ICP) (Chen and Medioni 1992).⁴ As the ribcage remains unchanged during the cardiac cycle, any in-vivo frame could had been used instead of the end-diastolic, without affecting the registration result. The resulting transformation information was then used to register the unloaded domain to the in-vivo frames. Figure 10 presents the original misalignment in the ribcage between the reference and in-vivo images, along with its correction after registration.

1.2 Investigation of passive parameters estimation for inverse model

The joint estimation scheme employed for acquiring passive parameters and reference domain for the inverse model (Sect. 2.3.4 and 3.2), relied on matching the cavity volume of the estimated unloaded domain to the Klotz estimate, $V_{Klotz}$. Initially, this was attempted through inverse simulations with sweeps over both passive parameters, with a and $a_f$ varying between [50, 5000] Pa. For each combination of passive parameters, the cavity volume of the resulting estimate of the unloaded domain was compared with the Klotz estimate. However, the volume error metric $V_d^{ref}$ exhibited multiple minima over the parameter space (Fig. 11), suggesting that is not possible to estimate both passive parameters and unloaded domain through this approach.

To alleviate this issue, the scheme proposed in Section 2.3.4 was performed, whereby the initial estimate of $a_f$ was acquired from the end-systolic model and parameter sweeps were performed only on a, leading to the estimates in Table 2. As the choice of $a_f=1326$ Pa was essentially arbitrary, the effect of a different selection for $a_f$ was investigated. Specifically, setting $a_f=2000$ Pa led to $a=600$ Pa and a maximum active tension of $\alpha _{max}=139$ kPa. These estimates are very similar to the values originally estimated for the inverse model ($a=800$ Pa, $\alpha _{max}=136$ kPa), suggesting that the choice of $a_f$ does not have a significant effect on remaining model parameters and model outcomes.

However, an important advantage of this study is that the reference domain is actually available. Accordingly, estimation does not need to rely only on the gross estimate $V_{Klotz}$ but could also take into account the actual geometry of the unloaded state. In this case, parameter estimates of the inverse model were acquired by matching the simulated reference domain to the in-situ unloaded domain. Estimation was performed by comparing the reference geometries resulting from parameter sweeps with the data-derived reference domain. However, these simulations were run with zero motion applied on the basal boundary, as it was assumed that the relative position between the reference and in-vivo images was not known. Accordingly, the distance errors were calculated after first registering each estimated unloaded geometry to the reference mesh, through point set registration with ICP.

Figure 11 demonstrates the behaviour of the RMSE over the parameter space, which presented a unique minimum. Interestingly, the parameters estimated through this inverse process ($a=500$ Pa, $a_f=1700$ Pa) were not very close to the parameters estimated with the forward process ($a=792$ Pa, $a_f=1200$ Pa) in the in-situ model. This discrepancy in parameters could be attributed to the difference in the boundary conditions employed. Specifically, the in-situ parameters were obtained with a prescribed motion on the basal plane, while inverse parameters were obtained with a fixed base plane.

To further investigate this issue, a separate set of forward simulations employing $\varOmega _{IN-SITU}$ as the unloaded domain was run. Specifically, following the boundary conditions of the inverse process, LV inflation simulations of the in-situ reference domain were run with zero displacement enforced weakly on the basal plane. The passive parameter estimates ($a=187$ Pa, $a_f=1876$ Pa) were then obtained using parameter sweeps and ICP registration to the end-diastolic domain, comparing more favourably to the parameters estimated with the inverse process ($a=500$ Pa, $a_f=1700$ Pa), as illustrated with a yellow circle in Fig. 11. Interestingly, when employing similar boundary conditions, the forward and inverse simulations led to similar model outcomes, strengthening the credibility of the modelling framework employed. It is also worth noting that the active tension was not substantially affected by the basal boundary condition in the in-situ model, giving a maximum value of $\alpha _{max}=133$ kPa for zero basal motion versus $\alpha _{max}=125$ kPa when displacement was prescribed.