Identification of the Unloaded Heart Configuration Including External Interactions

Abstract

Determining the unloaded, or reference, configuration of the heart is essential for developing patient-specific models that can accurately estimate in-vivo strains and stresses. Various strategies have been proposed to obtain this configuration, with inverse mechanics being a practical approach. The inverse mechanics method estimates the unloaded geometry from a deformed state and its known loads without the need for optimization procedures. However, for the resulting unloaded geometry to be physiologically meaningful, accurate boundary conditions are crucial. While spring boundary conditions are commonly applied to the epicardium to model interactions with surroundings, they do not account for the localized forces exerted by structures like the ribs and diaphragm. Since these external forces cannot be directly measured from medical images, we propose a novel approach that integrates them into the inverse mechanics formulation by penalizing large deformations. Using a series of test problems, we show this approach produces a reference configuration that closely matches the ground truth and improves circumferential strain estimations by an order of magnitude compared to standard inverse mechanics methods.

1. Introduction

Image-based computational modeling offers a powerful framework for studying disease mechanisms and evaluating therapeutic strategies on a patient-specific basis. However, generating these models presents a myriad of challenges, including the reconstruction of the cardiac geometry from medical images, selecting appropriate constitutive relationships and parameters, and defining boundary conditions (BCs) [3]. A key challenge is that image-derived cardiac geometries are subject to internal and external loads, making the unloaded configuration-against which deformations should be measured to compute true strain and stress-unknown. Identifying or defining this reference configuration is a critical step before simulating cardiac mechanics, as its choice directly influences strain calculations, affects stress distributions, and ultimately impacts the accuracy of cardiac simulation predictions [8].

To address this challenge, several approaches have been proposed in the literature. The most straightforward strategy defines the reference configuration by selecting a specific cardiac frame, such as diastasis [14], or end-systole [13]. However, since these image-derived geometries are not stress-free, they will lead to likely biases in strain and stress predictions. Other approaches approximate the effects of the loading in the deformed state by introducing a prestrain or a prestress. This type of strategy has shown positive results in the modeling of arteries [4, 10]. More advanced methods aim to solve an inverse problem and derive the reference configuration using the loaded geometry and known chamber pressures (typically at end-diastole). One such approach is the fixed-point method introduced by Sellier (2011) [15], which has been applied in the cardiac setting [12, 16] to estimate the reference configuration and optimize material parameters that best fit the end-diastolic pressure-volume relationship proposed by Klotz et al. [11]. Another option is the use of an inverse mechanics formulation [1, 8], which avoids iterative processes by directly estimating an unloaded geometry that, when inflated to the prescribed pressure, reproduces the exact image-derived deformed configuration.

All the strategies discussed above rely on defining a physiologically relevant boundary value problem, which typically includes specifying chamber pressures and epicardial BCs. In the methods that attempt to solve the inverse problem, inaccurate BCs can lead to errors in estimating the reference configuration. This was shown by Hadjicharalambous et al. (2021) [8], where the inclusion of the ribcage effects when solving the inverse mechanics significantly improved the reference configuration estimation in the porcine heart. However, as with the unloaded geometry, the forces exerted by external organs and structures on the heart cannot be directly measured from medical images. To overcome this limitation, we propose a novel strategy to estimate these forces based on how the heart would deform in their absence. Using simplified models that replicate passive heart deformation within the chest cavity, we demonstrate the robustness and accuracy of our method and show its potential for determining the reference configuration in the presence of unknown external forces.

2. Methods

2.1. Boundary Value Problem

Let us consider the reference cardiac domain ${\Omega }_0\in {\mathbb{R}}^2$ that deforms to its physical configuration $\Omega$. The deformation of a point in the reference configuration $\mathbf{X}$ is described by its displacement $\mathbf{u}$ such that the deformed position is $\mathbf{x}\left(\mathbf{X}\right)=\mathbf{X}+\mathbf{u}$. Here, we let $\mathbf{F}={\nabla }_{\mathbf{X}}\mathbf{u}+\mathbf{I}$ denote the deformation gradient tensor and $J$ its determinant.

In this work, we restrict the mechanics to the passive behavior of the myocardium as we assume that the starting loaded geometry is not under myocardial activation. The first Piola Kirchhoff stress tensor, $\mathbf{P}$, can be described as the sum of the hyperelastic material ${\mathbf{P}}_{\mathit{\text{hyp}}}$ and volumetric terms ${\mathbf{P}}_{\mathit{\text{vol}}}$,

$$\mathbf{P}(\mathbf{F},p)={\mathbf{P}}_{\mathit{\text{hyp}}}\left(\mathbf{F}\right)+{\mathbf{P}}_{\mathit{\text{vol}}}(\mathbf{F},p),$$ (1)

where $p$ is the hydrostatic pressure. Many options have been proposed in literature for ${\mathbf{P}}_{\mathit{\text{hyp}}}$. In this work, we will consider a simple Neohookean material, ${\mathbf{P}}_{\mathit{\text{hyp}}}^{\mathit{\text{NH}}}$ [2], and a reduced Holzapfel-Ogden law, ${\mathbf{P}}_{\mathit{\text{hyp}}}^{HO}$ [6, 9].

$$\begin{gathered} {\mathbf{P}}_{\mathit{\text{hyp}}}^{NH}=\mu J^{-1}\left(\mathbf{F}-\frac{\mathbf{F}:\mathbf{F}}{2}{\mathbf{F}}^{-T}\right); \\ {\mathbf{P}}_{\mathit{\text{hyp}}}^{HO}=a\mathbf{F}\text{exp}\left[b\left(I_1-3\right)\right]+2a_f\left(\mathbf{F}{f}_0\otimes {\mathbf{f}}_0\right)\text{exp}\left[b_f{\left(I_{4f}-1\right)}^2\right] \end{gathered}$$ (2)

where ${\mathbf{f}}_0$ is the vector describing the myocardium fiber direction. For ${\mathbf{P}}_{\mathit{\text{vol}}}$, we utilize a nearly-incompressible formulation [7],

$${\mathbf{P}}_{\mathit{\text{vol}}}=pJ{\mathbf{F}}^{-T}.$$ (3)

The boundary of ${\Omega }_0,{\Gamma }_0$, is divided into ${\Gamma }_0^D,{\Gamma }_0^{\mathit{\text{endo}}}$, and ${\Gamma }_0^{\mathit{\text{epi}}}$. ${\Gamma }_0^D$ denotes the boundaries subject to Dirichlet BC. The endocardial boundary is subject to pressure from the blood $\bar{p}$. The epicardium boundary ${\Gamma }_0^{\mathit{\text{epi}}}$ is either left traction-free or springs or Robin BCs are added to model the interaction with the pericardium and the surrounding organs [14]. In this work, we will consider the case of springs BCs with constant $k_{\mathit{\text{epi}}}$. Lastly, we consider that the interaction of the domain with external structures produces a traction ${\mathbf{t}}_{\mathit{\text{ext}}}$ on the epicardium. Section 2.2 details how ${\mathbf{t}}_{\mathit{\text{ext}}}$ is determined.

When we solve an inverse mechanics problem, we consider the reference domain ${\Omega }_0$ unknown, and we solve for the inverse displacement $\stackrel{\mathbf{ˆ}}{\mathbf{u}}=-\mathbf{u}$ such that $\mathbf{X}=\mathbf{x}+\stackrel{\mathbf{ˆ}}{\mathbf{u}}$. Assuming the inflation of the heart as a quasi-static problem, the strong formulation of the problem is: find $\stackrel{\mathbf{ˆ}}{\mathbf{u}}$ and $p$ such that,

$$\begin{array}{cc}\hfill {\nabla }_{\mathbf{X}}\cdot \mathbf{P}(\mathbf{F},p)=\mathbf{0}& \text{in}{\Omega }_0,\hfill \\ \hfill J-1-\frac{p}{K}=0& \text{in}{\Omega }_0,\hfill \\ \hfill \stackrel{\mathbf{ˆ}}{\mathbf{u}}=\mathbf{0}& \text{on}{\Gamma }_0^D.\hfill \end{array}$$ (4)

where $K$ is the bulk modulus. The problem is subject to the following conditions on the endocardium,

$$\mathbf{P}\mathbf{N}=-\bar{p}\mathbf{N}\text{on}{\Gamma }_0^{\mathit{\text{endo}}},$$ (5)

where $\mathbf{N}$ is the normal to the surface in the reference configuration. On the epicardium, we will consider one of the following BCs,

$$\mathbf{\text{PN}}=\mathbf{0}\text{on}{\Gamma }_0^{\mathit{\text{epi}}},\mathbf{\text{PN}}=k_{\mathit{\text{epi}}}\left(\mathbf{N}\cdot \mathbf{u}\right)\mathbf{N}\text{on}{\Gamma }_0^{\mathit{\text{epi}}},\mathbf{\text{PN}}={\mathbf{t}}_{ext}\text{on}{\Gamma }_0^{\mathit{\text{epi}}},$$ (6)

which will be denoted as free epicardium, springs, and external forces BCs, respectively.

2.2. Modeling the External Tractions

Many studies have modeled the influence of the surrounding structures of the heart either by adding a layer of elements that simulate the pericardium or using springs or Robin BCs applied to the entire epicardium [5, 14, 17]. However, short-axis images reveal that structures such as the ribs and diaphragm exert localized forces that shape the heart’s geometry, particularly affecting the RV (see Fig. 1). We believe that incorporating these localized forces is essential for accurately estimating the reference configuration.

Fig. 1.

Cardiac magnetic resonance short-axis slice showing the heart, the ribcage, and the diaphragm at end-diastole (ED) and end-systole (ES). Ribcage and diaphragm boundaries are delineated with a dashed line (ED) and a solid line (ES) for easier comparison.

Here we propose estimating external forces based on the principle that, if isolated, the global deformation of the heart (with uniform material properties) will be close to homogeneous, in a balloon-like manner. Any deviation from this expected deformation is attributed to interactions with surrounding tissues. Therefore, we consider finding an external traction, ${\mathbf{t}}_{\mathit{\text{ext}}}$, such that it can deform an isolated, nearly homogeneously inflated ventricle into the observed deformed configuration.

Given $n_{\mathit{\text{ext}}}$ external structures in contact with the heart, we propose that ${\mathbf{t}}_{\mathit{\text{ext}}}$ can be described as,

$${\mathbf{t}}_{ext}(\mathbf{x})=\sum _{i=1}^{n_{ext}}{T}_i{\varphi }_i(\mathbf{x})\mathbf{n}$$ (7)

where $\mathbf{n}$ is the surface normal on ${\Gamma }^{\mathit{\text{epi}}},T_i$ is the introduced unknown magnitude of the force produced by the structure $i$, and ${\varphi }_i\left(\mathbf{x}\right)$ is a basis function that has support only in areas where the external structure $i$ is observed to be in contact with the heart. We assume these forces are shear-free, and hence the reason ${\mathbf{t}}_{\mathit{\text{ext}}}$ is applied in the normal direction $\mathbf{n}$. This way, the problem reduces to finding $T_i$ for $i=1‥n_{\mathit{\text{ext}}}$. To find these values, when solving the inverse mechanics problem, we penalize displacements that divert from the average normal displacement of the epicardial surface, effectively homogenizing the heart deformation.

2.3. Synthetic Ground Truth Generation

For the examples presented in this work, we generated ground truth simulations by adding the following BC on the epicardium,

$$\mathbf{\text{PN}}={\mathbf{t}}_{\mathit{\text{wall}}}\text{on}{\Gamma }_0^{\mathit{\text{epi}}},$$ (8)

where,

$${\mathbf{t}}_{\mathit{\text{wall}}}=\left\{\begin{array}{cc}{k}_{\mathit{\text{wall}}}{\mathbf{N}}_{\mathit{\text{wall}}}{\left({\mathbf{N}}_{\mathit{\text{wall}}}\cdot \left(\mathbf{x}-{\mathbf{p}}_{\mathit{\text{wall}}}\right)\right)}^2& \text{if}{\mathbf{N}}_{\mathit{\text{wall}}}\cdot \left(\mathbf{x}-{\mathbf{p}}_{\mathit{\text{wall}}}\right)>0,\hfill \\ 0& \text{otherwise},\hfill \end{array}\right.$$ (9)

where ${\mathbf{N}}_{\mathit{\text{wall}}}$ is the unitary normal vector pointing outwards of the wall, ${\mathbf{p}}_{\mathit{\text{wall}}}$ is a point in the wall plane, and $k_{\mathit{\text{wall}}}$ is the stiffness of the wall. This BC simulates the no-shear force exerted by the surrounding organs. The value of $k_{\mathit{\text{wall}}}$ is chosen to be large to simulate the non-penetration of the heart past the wall plane.

3. Results

3.1. Simple Arc Test

To investigate the importance of accounting for the external forces when solving an inverse mechanics problem, we examine the inflation of a simple 2D arc geometry and a Neohookean material. To simulate the presence of the ribs and diaphragm, we add two stiff walls as shown in Fig. 2A. The parameters used in this simulation, including the position and normal for the left wall (the right wall is the reflection with respect to the Y axis), are shown in Table 1.

Fig. 2.

Inflation of a constrained arc. Each row shows the reference and deformed domain for (A) ground truth, (B) free epicardium, (C) springs with $k_{\mathit{\text{epi}}}=0.25\text{kPa}$, which delivered the best fit in a parameter sweep study (see Fig. 3C), and (D) external forces BCs. The physical configurations are colored by the circumferential stretch. Dashed lines are the ground truth reference, and solid lines are the starting reference configuration.

Table 1.

Parameters used in the simple arc test case.

$\bar{p}\left(\text{kPa}\right)$	$\mu \left(\text{kPa}\right)$	$K\left(\text{kPa}\right)$	${\text{N}}_{\mathit{\text{wall}}}$	${\text{p}}_{\mathit{\text{wall}}}\left(\text{mm}\right)$	$k_{\mathit{\text{wall}}}\left(\text{kPa}/{\text{mm}}^2\right)$
1	15	100	$(-0.82,0.57)$	$(-30,2.5)$	500

Using only the deformed configuration (shown in Fig. 2A), we first estimate the reference configuration using inverse mechanics considering the zero-traction and spring BCs (see Fig. 2B and C). Then, we use our proposed method to estimate the external forces (Fig 2D). Once the reference configuration was obtained, we inflated to the prescribed pressure and calculated the circumferential stretch. These results are shown in the last column of Fig. 2.

Figure 3A shows the location of the basis functions $\varphi$ used to calculate the external traction, ${\mathbf{t}}_{\mathit{\text{ext}}}$. A comparison of the ground truth force generated by the walls and the force estimated by penalizing large displacements is shown in Fig. 3B.

Fig. 3.

(A) Basis functions used to calculate the external force. Note that they are active only where walls are present. (B) Ground truth wall force and estimated force. (C) Reference configuration and stretch errors obtained using the different strategies. Open circle denotes the best $k_{\mathit{\text{ep}}i}$ values used in Fig. 2C. The marker × corresponds to the error obtained using a finer mesh (+ is used for the radial stretch error).

We also calculated the estimation error using two metrics: the error in the inverse displacement $\stackrel{\mathbf{ˆ}}{\mathbf{u}}$ and the error in the predicted circumferential, ${\lambda }^{\mathit{\text{circ}}}$, and radial stretch, ${\lambda }^{\mathit{\text{rad}}}$, when inflating the estimated unloaded configuration,

$$\text{ref.}\text{error}=\frac{‖\stackrel{\mathbf{ˆ}}{\mathbf{u}}-{\stackrel{\mathbf{ˆ}}{\mathbf{u}}}_{gt}‖}{‖{\stackrel{\mathbf{ˆ}}{\mathbf{u}}}_{gt}‖};(\mathit{\text{circ}},\mathit{\text{rad}})\text{stretch}\text{error}=\frac{‖{\lambda }^{\mathit{\text{circ}},\mathit{\text{rad}}}-{\lambda }_{gt}‖}{‖{\lambda }_{gt}‖},$$ (10)

where the subscript $gt$ denotes the ground truth field.

Since there are multiple possible values of $k_{\mathit{\text{epi}}}$, we performed a sweep to determine the value of $k_{\mathit{\text{epi}}}$ that best reproduces the ground truth. Figure 3C shows the errors in the inverse displacement, and in the circumferential and radial stretch for the different strategies considered. Furthermore, to make sure the results were independent of the mesh refinement levels, we rerun the simulations with a mesh about two times finer (1.5 mm vs 0.8 mm). The results of these simulations are shown with × and + markers for the circumferential and radial direction, respectively. The inverse displacement error went from 0.041 in the coarse mesh to 0.043 in the fine mesh, the circumferential stretch error from 0.0051 to 0.0036, and the radial stretch error from 0.0036 to 0.0027. Note that only the simulation with the best $k_{\mathit{\text{epi}}}$ parameter is shown.

Finally, we also used this test case to perform a sensitivity analysis of the basis position and width. The user will select these variables in the practical application of the method, so it is important to assess how the predictions change for different choices of these parameters. To do this analysis, we considered the central position and width of the basis as shown in Fig. 3A, perturbed them and quantified the error in circumferential stretch and inverse displacement prediction. The width perturbation is defined as a multiplier of the initial value. For the position, we defined the center of the basis functions using a normalized coordinate that is 0 in the left boundary and 1 in the right. The position perturbation is defined in this normalized coordinate. Figure 4A presents examples of different perturbations and the respective reference configuration prediction. Figure 4B and C show the error quantification for the different combinations of parameters, alongside the errors when using springs or a zero-Neumann BC.

Fig. 4.

Sensitivity analysis of the basis position and width. (A) Shows examples of the different perturbations. The left side of each example shows the basis function on the deformed configuration, while the right side shows the resulting reference configuration. The number of each example corresponds to the combination of parameters marked by the respective number in B and C. (B) Inverse displacement error for the difference combination of perturbations, the different spring constants, and the free epicardium BC. Note that the same colorbar range is used for all. (C) Same as (B) but for the circumferential stretch error.

3.2. Biventricular Slice

Next, we sought to study what happens in a biventricular (BV) geometry. We consider a 2D short-axis slice similar to the one presented in Fig. 1 and, similar to the previous simulations, added stiff walls to represent the diaphragm and ribs interactions. In this case, ${\Gamma }^{\mathit{\text{endo}}}={\Gamma }_{LV}^{\mathit{\text{endo}}}\cup {\Gamma }_{RV}^{\mathit{\text{endo}}}$ representing the left ventricle (LV) and the right ventricle (RV). The RV is pressurized to half the pressure value $\bar{p}$ of the LV. In this test case, we used the Holzapfel-Ogden law. The parameters used for these simulations are shown in Table 2. To avoid rigid body displacements, we fixed a small portion of the lateral LV wall, as shown in the first column of Fig. 5A. The ground truth results, alongside the obtained reference and deformed configuration and stretch fields for the external forces, springs, and free epicardium BCs are shown in Fig. 5A–D. The true external forces and the estimated ones are shown in Fig. 5E. The errors in inverse displacement and stretch prediction are shown in Fig. 5F.

Table 2.

Parameters used in the biventricular test case.

$\bar{p}\left(\text{kPa}\right)$	$a\left(\text{kPa}\right)$	$b(-)$	$a_f\left(\text{kPa}\right)$	$b_f(-)$	$K\left(\text{kPa}\right)$
1.2	4	5	10	5	100
	${\mathbf{N}}_{\mathit{\text{wall}}}^{\mathit{\text{ribs}}}$	${\mathbf{p}}_{\mathit{\text{wall}}}^{\mathit{\text{ribs}}}\left(\text{mm}\right)$	${\mathbf{N}}_{\mathit{\text{wall}}}^{\mathit{\text{diaph}}}$	${\mathbf{p}}_{\mathit{\text{wall}}}^{\mathit{\text{diaph}}}\left(\text{mm}\right)$	$k_{\mathit{\text{wall}}}\left(\frac{\text{kPa}}{{\text{mm}}^2}\right)$
	$(-0.93,0.37)$	$(-53.65,17.79)$	$(0.055,-0.998)$	$(-13.5,-32.17)$	500

Fig. 5.

Biventricular slice simulations. (A) Ground truth reference and deformed configuration. Reference estimation and stretch field results for the simulations using (B) external forces, (C) springs with $k_{\mathit{\text{epi}}}=0.1\text{kPa}$ (best reference configuration fit, as shown in (F)), and (D) free epicardium. (E) Shows a comparison between the ground truth and estimated external tractions. (F) Presents the errors in inverse displacement and stretch. Open circles denote the best fit of the springs BC.

4. Discussion

Accurately estimating the reference configuration of the heart is essential for predicting myocardial stress and strain. Through simple test cases, we demonstrated that while inverse mechanics can estimate the reference configuration, neglecting the true BCs results in an unloaded geometry that is inaccurate, resulting in unrealistic strain patterns. In this study, we present a methodology to estimate external forces by identifying the surfaces in contact with the surrounding tissues and minimizing the ventricular wall deformation by estimating external forces acting in those areas within the inverse mechanics problem. This approach leverages the principle that cardiac wall deformation should remain somewhat homogeneous unless altered by external forces.

Since we used the inverse mechanics framework, no matter the reference configuration estimated, the deform state always matched the known loaded geometry (see Fig. 2 and Fig. 5). However, we observed that neglecting the external forces in the inverse problem leads to exacerbated deformation in contact areas, resulting in regionally heightened stretches (see Fig. 2B and Fig. 5C). One approach to mitigate this effect is introducing springs at the epicardial boundary. Epicardial springs have been used in literature to model the interaction with the pericardium and the surrounding tissues [14, 17]. Since the ground truth reference configuration was known, we performed a parameter sweep to define the value of $k_{\mathit{\text{epi}}}$ that minimizes the estimation error. However, using this value still returns an unloaded geometry that no longer retains its original shape, and the reduced deformation caused by the springs leads to underestimated circumferential strains (Fig. 2C and Fig. 5D). Furthermore, determining the optimal value for $k_{\mathit{\text{epi}}}$ is not an option in a clinical setting since the ground truth $\mathbf{X}$ is not known. Therefore, although using springs BCs helps reduce exacerbated deformations, it will not return an accurate reference configuration and introduces another uncertainty to the modeling process. Moreover, the method we proposed to estimate external forces delivered an unloaded geometry that reduced the inverse displacement and stretch errors by about an order of magnitude compared to the best springs simulation in the simple arc test (Fig. 3C). In the biventricular test, the simulation with external forces showed errors that are lower than the best spring simulation, although the improvement was less than an order of magnitude. This is due to the LV having small deformations, which lowers the error norm (Fig. 3F). On that note, our simulation showed that the effect of the external forces is more relevant on the RV side, which can also be observed by the flattened walls in contact with the ribs and diaphragm in Fig. 1.

The accuracy of the reference configuration and stretch estimations achieved with the proposed strategy can be attributed to its ability to effectively capture the external traction, as demonstrated in Fig. 3C. In the simple arc test, the ground truth forces exhibit a profile similar to a quadratic form, which aligns well with our choice of a quadratic basis for defining ${\mathbf{t}}_{\mathit{\text{ext}}}$. Even in cases where the forces are more localized (see Fig. 5E), and the basis function support is chosen to be wider than the ground truth surface contact, our method still yields force profiles that reflect the effect of the external force. This translates into a more accurate estimation of the reference state and deformed stretch fields (Fig. 5B). It is important to note that our method estimates traction values that are different for each wall, with higher forces being predicted for the contacts that in reality are exerting more force (see Fig. 5E). This makes sense as, as shown in Fig. 5C, the heart walls that deform more in the free-inverse mechanics problem correlate to those where higher contact force is exerted, and that are therefore going to be more penalized by our method.

The main user-dependent component of the proposed method is the definition of the position and width of the basis functions. In Fig. 4B and C, we showed that variations in width will have a lower impact on the predicted circumferential stretch and inverse displacement. The position of the wall has a greater impact, but by looking at the examples presented in Fig. 4A, it is possible to see that in the cases that yield worst error values (examples 1, 2, 5, and 6), the wall is visible not centered on the flatten edge of the deformed configuration. Therefore, we believe that as long as the choice of basis position and width is within the visible contact area, using the proposed method will yield results that out-perform ignoring these local forces or considering springs in the whole epicardial boundary. Figure 1 shows that making a reasonable selection of the contact areas for the ribs and diaphragm is possible using clinical images, although strategies will need to be developed to translate this to 3D.

5. Conclusions

In this work, we presented a strategy to estimate the reference configuration considering the localized effect of the surrounding organs and structures. We demonstrated that our method delivers a reference estimation about an order of magnitude better than using spring or zero traction BCs. By using these simple examples, we showed that considering the external forces has huge repercussions in the estimated reference configuration. In the future, we would like to explore the robustness of the method using different test scenarios, experimental data, and by extending the formulation to three-dimensional biventricular geometries.