# Population-based personalization of a 2D diffusion-based model of myocardial infarct

Anastasia Konik, Romain Deleat-Besson, Patrick Clarysse, Nicolas Duchateau

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Anastasia Konik, Romain Deleat-Besson, Patrick Clarysse, Nicolas Duchateau. Population-based personalization of a 2D diffusion-based model of myocardial infarct. International Conference on Functional Imaging and Modeling of the Heart (FIMH), 2025, Dallas, United States. hal-05046435

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# Population-based personalization of a 2D diffusion-based model of myocardial infarct

Anastasia Konik<fnref n="1" />[0009-0001-1916-7578], Romain Deleat-Besson<fnref n="1" />[0009-0009-4188-8169], Patrick Clarysse<fnref n="1" />[0000-0002-5495-7655], and Nicolas Duchateau<fnref n="1" />,<fnref n="2" />[0000-0001-8803-2004]

**Abstract.** Simple and effective infarct models are needed to enhance cardiac biophysical models or generate more accurate ground truth data for machine learning applications. This study presents a diffusion-based approach for modeling infarct shapes, applied to generate synthetic 2D infarct images. The diffusion is non-uniform, with different diffusion coefficients assigned to various sectors of the myocardium. We integrate a population-based approach to learn the model parameters and personalize them to a real population. This personalization method is relevant for models with randomness, such as our model, and is achieved through a gradient-free algorithm (CMA-ES), optimizing a distribution similarity metric between real and synthetic populations. We assess this approach on 2D infarct segmentations from LGE MR images and aligned to a reference geometry, from 117 patients with acute myocardial infarct. We generate 500 synthetic images and evaluate their characteristics using an attribute-based variational auto-encoder (AR-VAE), which captures a latent space with key infarct characteristics specifically disentangled on specific dimensions (transmurality, size, and orientation). Our model offers a simple, physically-based method to generate infarcts with diverse shapes. The evaluation shows that the synthetic population accurately reflects the distributions of the key infarct characteristics, with more sophisticated infarct shapes compared to previous simpler models also personalized with the population-based approach.

**Keywords:** Myocardial infarct · Personalization · Representation learning.

## 1 Introduction

Despite the significant progress in imaging myocardial damages [7], modeling of the shape of myocardial infarcts remains limited. Existing fast and simple models of infarct shapes (e.g., union of random spheres [5] or elliptical [18]) fail to capture the complex changes that occur in the heart during the infarct development. When personalized to a real population [10], their realism remains limited even for key geometrical characteristics such as transmurality (namely, to which extent the infarct affects the entire thickness of the heart wall). This gap represents a challenge, as better rendering of heterogeneities within the myocardium could benefit subsequent models of cardiac electrical propagation [1,15] or mechanics and help to generate more reliable ground truth data, e.g., for machine learning. Besides, models inspired by the progression of infarcts across the myocardium could be of value for longitudinal analyses.

The progression of myocardial infarct through the myocardium typically follows a front propagation pattern, starting from the endocardium and then expanding outward towards the epicardium [9,17]. This process is driven by several interrelated physical phenomena, including disruption of

<footnote n="1">¹ INSA-Lyon, Universite Claude Bernard Lyon 1, CNRS, Inserm, CREATIS, UMR 5220, U1294, Villeurbanne, France</footnote>

<footnote n="2">² Institut Universitaire de France (IUF), Paris, France</footnote>
<figure><img src="image_2.png" /><figcaption>**Fig. 1.** Examples of the variety of infarct shapes observed in the *real* population with an infarct size around 15% (median of the population) and a transmurality around 50%. Infarct patterns are displayed after alignment to a reference anatomy (which facilitates comparisons and the model personalization) and are not fully binary due to the interpolation of individual data to this reference anatomy (see Sec. 2.1 for details).</figcaption></figure>

blood flow (ischemia), of nutrient supply, and cellular death [19]. Complex models have been used to mimic this [11,19], incorporating detailed physiological processes, such as the reduction of the oxygen concentration modeling the cell death through diffusion-based mechanisms [19]. In [14], a concept similar to the front propagation pattern was described for the middle cerebral artery infarction, using electrical terms that are appropriate for modeling the behavior of neurons in the brain. For the heart, the propagation of the transmembrane voltage through the myocardium can be modeled using diffusion, as used in some electrophysiological models [15].

Cardiac diseases can actually be modeled at different scales [4]: cell-level, tissue level (mostly used for the myocardium) or a layered approach from the small to big scale. In this work, we focus on models of infarct shape to mimic physiological spreading processes at the tissue level in the infarcted myocardium. Figure 1 shows some examples of real infarcts from the dataset we studied, illustrating the range of morphological variations for infarcts with comparable size and transmurality.

Challenges consist of not only simulating more complex fronts of infarcts, but also evaluating their realism, especially when these models incorporate randomness to simulate the biological variability. For this purpose, a promising approach of model personalization at the population level was recently presented for simple models of infarct shape (union of spheres and elliptical) in 2D [13] and 3D [10]. This population-wise strategy contrasts with the widespread personalization to each individual's data, which works well for deterministic models. In these works, the evaluation of the models' realisms relied on representation learning to estimate a low-dimensional latent space encoding the main infarct characteristics and the reconstruction to the initial space for visualizing the complete infarct patterns encoded.

Here, we present a more sophisticated model of the myocardial infarct shape and evaluate it with the population-based personalization. We consider a diffusion-based approach to iteratively mimic the front propagation from endocardium to epicardium. We focused on a basic diffusion model, in 2D, as it strikes a balance between complexity and computational efficiency and allows personalization in feasible time. In the following, we provide a more detailed description of the model (Sec. 2.2), show how it is incorporated in the personalization pipeline, and evaluate the realism of the generated synthetic images (Sec. 3).
<figure><img src="image_3.png" /><figcaption>Fig. 2. (a) Schematic representation of the diffusion model. (b) 16 sectors corresponded to the maximum starting zone and used to define the diffusion process. (c) Propagation of the diffusion process with the starting zone in five sectors after thresholding, different colors were used for display purposes only and correspond to different 10, 30, 50, 70 iterations of the diffusion. (d) Final infarct after smoothing, dark blue corresponds to the healthy myocardium, yellow corresponds to the infarct.</figcaption></figure>

# 2 Methods

## 2.1 Data

We used the "8-day" real infarct segmentations from the MYOSAIQ public challenge<fnref n="1" />. They consist of endocardial, epicardial and lesion contours from 123 patients with acute myocardial infarct from the Minimalist Immediate Mechanical Intervention study (MIMI, Clinical Trials ID: NCT01360242) [2], segmented on late Gadolinium-enhanced images obtained up to 8 days after myocardial infarction. This dataset consists of stacks of 2D image slices, which were aligned to a 3D reference anatomy defined as a semi-ellipsoid, using the parameterization of the myocardium with circumferential, radial and z-axis coordinates as detailed in [6]. To facilitate comparisons and benefit from a larger population, infarct patterns were also rotated along the myocardial circumference to align with the left anterior descending (LAD) artery territory. In this work, we focus exclusively on the infarct segmentations from 123 patients with exploitable images and consider a single 2D mid-cavity slice per patient (provided it contains visible myocardial infarct), leading to 117 non-empty images of 80 × 80 pixels.

## 2.2 Diffusion-based model of infarct

We designed a model based on a finite difference schema of the heat equation. The infarct propagation is assumed to behave similarly to the heat diffusion, where the front of the infarct extends over time in a manner analogous to how heat disperses. The heat equation $\frac{\partial u}{\partial t} = \alpha \Delta u$, with $\Delta$ the Laplacian, is discretized with the following explicit finite difference schema (with the first order forward difference for the time and the second order central differences for the space):

$$u_{i,j}^{k+1} = \beta(u_{i+1,j}^{k} + u_{i-1,j}^{k} + u_{i,j+1}^{k} + u_{i,j-1}^{k}) + (1-4\beta)u_{i,j}^{k}, \quad (1)$$

where $\delta = \Delta x = \Delta y$, $\beta = \frac{\alpha \Delta t}{\delta^2}$ is the diffusion coefficient, $k$ stands for time discretization (number of iterations), $i$ and $j$ indexes are for 2D space discretization. To reduce computational time, we

<footnote n="1">¹ https://www.creatis.insa-lyon.fr/Challenge/myosaiq/</footnote>
apply this schema in the form of convolution with the 3 × 3 kernel K, in Cartesian coordinates:

$$ \mathbf{K} = \begin{pmatrix} 0 & \beta & 0 \\ \beta & 1 - 4\beta & \beta \\ 0 & \beta & 0 \end{pmatrix}. \qquad (2) $$

Note that we can approximate the real solution of the heat equation only when we have the stability of the schema, which is guaranteed for $\beta \le \frac{1}{4}$ [12]. If the diffusion process were uniform, a common diffusion coefficient $\beta$ would apply to each sector of the myocardium. Here, we mimic non-uniformity of the cardiac tissue by non-uniform diffusion, obtained by choosing different diffusion coefficients across different sectors of the myocardium. The starting zone of the diffusion process is randomly selected for each generated infarct, its value is restored to 1 at each iteration of the diffusion process, while the initial condition for the diffusion equation is 0 everywhere except the starting zone.

The starting zone is defined according to the distribution of infarct sizes in the real population. The myocardium is divided into 24 equally sized sectors, and the starting zone is set to spread between 2 and 16 sectors (the spread of the smallest and largest real infarcts), one of them including the location of the center of mass of the average real infarct. Of note, as the personalization in Sec. 2.3 requires a fixed number of parameters, it always optimizes the diffusion coefficients across 16 sectors, but no diffusion occurs in sectors that do not contain the starting zone.

The parameters of the model to be learned (personalized) in the diffusion model consist of:

1. The maximum diffusion coefficients $\beta_s^{max}$ (one for each sector $s$), which determine the maximum possible rate at which the infarct propagates from the starting zone on the endocardium; the diffusion coefficients for generations are sampled from the uniform distribution in the interval $[0, \beta_s^{max}]$.

2. The minimum $N_{min}$ and maximum $N_{max}$ possible numbers of iterations that control the duration of the diffusion process; the final numbers of iterations for generations are sampled from the interval $[N_{min}, N_{max}]$.

Once the diffusion process is terminated, we smooth out the obtained infarct profiles and threshold the image to only keep two distinct zones: healthy myocardium and infarcted tissue. To compute the convolutions, we use the built-in Python `convolve` from `scipy.ndimage`, with the extension that reflects the data at the edges of the last pixel. This convolution should be computed within a defined rectangular region: we chose the minimal rectangle over each sector. This simple and effective approach is an asset for iterations of the personalization process, compared to computing convolutions in the radial-circumferential domain, which would require mapping from Cartesian to radial-circumferential coordinates and backward. The main stages of the diffusion process are illustrated in Fig. 2.

## 2.3 Evaluation of the model

**Population-based model personalization.** As the proposed model incorporates randomness, the strategy of individual-based personalization is not straightforward. Performing a simple grid search for parameters would be time-consuming and would require to estimate the model parameters separately. However, with the population-based personalization approach, we can efficiently learn the optimal combination of distributions of diffusion coefficients and the number of iterations. Thus, to evaluate the realism of the generated synthetic infarcts compared to the real population,
we used the population-based personalization strategy proposed in [13]. It consists in optimizing
the model parameters to match the distributions of synthetic and real populations, as quantified by
the Kullback-Leibler (KL) divergence between the probability distributions derived from myocardial data vectors of both populations. The optimization is performed by the gradient-free algorithm Covariance Matrix Adaptation – Evolution Strategy (CMA-ES) [8]. It starts by initializing parameters and then iteratively updating their mean and covariance matrix based on the best-performing candidates. These best-performing candidates of parameters are selected on the basis of the loss function calculated after the generation of synthetic infarct images. After the update step, new parameters are generated by sampling from the updated distribution.

We adapted the Python implementation from [10]. To address constraints in the optimization,
we incorporated a strategy to resample until all parameters fall within the required interval. This
approach was suggested as a simple solution by the CMA-ES authors in their code tutorial. Due
to the potentially large number of parameters that need to be optimized, it was not feasible to
search for the best initialization parameters. Therefore, in this work, we determined reasonable
initialization based on experimentation.

**Population analysis with representation learning.** Once the population-based personalization is achieved, we estimate a low-dimensional latent space encoding the main trends in the data (for the real and synthetic populations separately) and reconstruct the infarct images encoded along its main dimensions to better interpret the main characteristics of both populations.

Dimensionality reduction is performed within the framework of variational auto-encoders. These models provide a latent space of lower dimensionality while simultaneously controlling the population distribution (generally with a Gaussian prior) and the reconstruction error of samples encoded and decoded. However, standard VAEs do not guarantee that the latent dimensions correspond to imaging or physiological attributes of interest (for example, infarct characteristics such as size or transmurality). We therefore used the AR-VAE [16], a variant of VAEs that performs disentangling for specific attributes chosen beforehand. It imposes that each identified attribute is explicitly ordered in a monotonic manner along a given (chosen) latent dimension. In our case, we chose geometrical characteristics of the infarcts:

- size, in % (the proportion of pixels of the myocardium where infarct is present);

- transmurality, in % (the weighted average proportion of infarcted pixels within the infarcted segments, after dividing the myocardium into sectors);

- orientation, in ° (the angle corresponding to the location of the infarct center of mass).

The encoding and decoding parts of the AR-VAE are performed by convolutional neural networks. However, convolutions over infarct images in Cartesian coordinates are not optimal given the circular geometry of the reference myocardium. We therefore specifically unfolded the myocardium image of each subject (splitting it at the opposite side of the infarct) so that the radial and circumferential coordinates correspond to the x-y axes, therefore allowing standard convolutions on the unfolded data. Padding with the border values was performed over 16 pixels to maintain the continuity of the data. Image dimensions before and after unfolding were 80 × 80 and 12 × 544 pixels, respectively.

The AR-VAE was implemented using Tensorflow 2.12.0 and Python and was trained on NVIDIA Quadro T1000 with 4 GB RAM. The encoder consisted of three convolutional layers (with LeakyReLu activation functions and max-pooling layers in-between), each containing 32, 64, and 128 feature channels, followed by a fully connected layer and a sampling layer for extracting the latent representation. The decoder closely mirrors the encoder's architecture.
The main variations in the real and synthetic populations were examined by reconstructing through the decoder infarct images corresponding to {-2, -1, 0, +1, +2} standard deviations along the dimensions of the latent space, in particular those corresponding to the chosen infarct attributes.

**Quantitative metric for image realism.** Finally, we compared the synthetic and real images with the Fréchet Inception Distance (FID), commonly used to quantify the realism of a population of synthetic images. A first baseline for comparison consisted of half of the real population (58 samples taken randomly, experiment repeated 100 times) compared to the other half. This was complemented by comparing these 58 samples to a synthetic population obtained by random sampling across a given latent space (here, obtained with PCA or the AR-VAE encoding the real population), and to the synthetic population generated by a given infarct model personalized to the real population (elliptical [18], union of spheres [5], and diffusion-based). In these real-synthetic configurations, we considered 500 synthetic samples and a random set of 58 samples (experiment repeated 100 times).

## 3 Experiments and results

**Personalization hyperparameters.** For the training procedure with 117 real images, the CMA-ES hyperparameters were empirically set to: 13 samples for the estimation of the covariance matrix, the maximum number of loss function evaluations to 260, the loss function threshold to $4 \times 10^{-3}$, the kernel bandwidth used for the KL divergence to 9, 40 samples generated at each new generation of images. The stabilization of the optimization was achieved according to the loss function. The time required for the personalization with this configuration was approximately 11 min.

**Representation learning hyperparameters.** We used the Adam optimizer with the learning rate of $4 \times 10^{-4}$ during 68 epochs until convergence with the batch size of 32 for the real population, and the learning rate of $2 \times 10^{-5}$ during 253 epochs with the same batch size for the synthetic population. Ponderations between the AR-VAE losses were set to $\alpha = 3$ for both populations (reconstruction term), $\beta = 10^{-1}$ for the real and $\beta = 7 \times 10^{-2}$ for the synthetic population (KL divergence regularization), $\gamma = 2$ and 1, respectively (attribute-based regularization, with $\delta = 20$ controlling the strength of the regularization, see details in [16]). We set the bottleneck of the network (the dimensionality of the latent space) to 8 dimensions, the first three corresponding to the retained infarct attributes. A full training required less than 15 min while inferring an image takes 20 ms.

As our primary focus is on analyzing the main variations in the population, without the need for further inference, the AR-VAE was trained on the whole set of real or synthetic cases, except 2 random cases kept for qualitatively checking the reconstruction quality.

**Evaluation of the personalized model.** For the final generation with the optimized parameters of the model, we generated 500 synthetic images whose distribution should match the real population.

Figure 3 shows examples of synthetic infarct images generated with the diffusion-based model after personalization. Those examples have global characteristics (transmurality and size) comparable to those from Fig. 1 for the real population. Nonetheless, they also illustrate the diversity of infarct shapes allowed by the diffusion-based model.
<figure><img src="image_4.png" /><figcaption>**Fig. 3.** Examples of the variety of infarct shapes observed in the synthetic population with characteristics in the same range than those of Fig. 1 (infarct size around 15% and transmurality around 50%).</figcaption></figure>

<table><caption>**Table 1.** Fréchet Inception Distance (FID) comparing real images (half of the population: N<sub>real</sub> = 58) to synthetic images (N<sub>synth</sub> = 58 or 500) generated by a given infarct model (last three columns). The other columns consist of baseline comparisons (marked with the * symbol): to the other half of real images, or to synthetic images generated by random sampling across a latent space obtained by PCA or AR-VAE. Average and standard deviation correspond to 100 repeated random samplings of the real samples.</caption><thead><tr><th rowspan="2">FID (↓)</th><th rowspan="2">real-real*</th><th colspan="4">real-synthetic</th></tr><tr><th>PCA*</th><th>AR-VAE*</th><th>Elliptical [18]</th><th>Iterative [5]</th><th>Diffusion (ours)</th></tr></thead><tbody><tr><td>N<sub>synth</sub> = 58</td><td>5.6 ± 0.3</td><td>8.1 ± 0.6</td><td>6.3 ± 0.4</td><td>22.6 ± 1.1</td><td>25.9 ± 1.1</td><td>18.7 ± 0.7</td></tr><tr><td>N<sub>synth</sub> = 500</td><td>-</td><td>6.8 ± 0.3</td><td>5.3 ± 0.3</td><td>21.7 ± 0.6</td><td>25.1 ± 0.6</td><td>17.6 ± 0.4</td></tr></tbody></table>

Figure 4 complements this with the analysis of the main modes of variation described in Sec. 2.3. The latent spaces for the real (Fig. 4a) and synthetic (Fig. 4b) populations show relevant ordering of the attribute-based dimensions (in this example, transmurality is imposed along dimension 0 while dimension 4 is free), even for the real population which contains substantially less samples. The modes of variation corresponding to the infarct attributes (first three rows in Fig. 4c and d) illustrate that these characteristics are rather well reproduced by the diffusion-based model. They also show additional variations of shapes resulting from the randomness of the diffusion coefficients in each sector, which is even more pronounced by looking at free dimensions (e.g., last row in Fig. 4d).

Finally, Tab. 1 reports the Fréchet Inception Distance (FID) to quantify the realism of the synthetic images compared to the real ones. The FID for synthetic images generated by PCA or AR-VAE are slightly higher than the real-real baseline, indicating that these images correspond to the real distribution, in the limits of the encoding and decoding capabilities of these algorithms (lower FID is obtained for the AR-VAE, which has better encoding and decoding compared to PCA). Our diffusion-based model has reasonably higher FID, confirming that our synthetic population does not fully match the real one, but is realistic. Higher FID are observed for the personalized elliptical and iterative (union of spheres) models, confirming they are less realistic. For comparison, the FID for these models without personalization but tuned based on our expertise<fnref n="2" /> were (with N<sub>synth</sub> = 500) 22.4±0.6 and 18.5±0.5, indicating that personalization (which is based on the KL divergence) may be suboptimal regarding other metrics. Of note, the FID involving 500 synthetic samples is slightly lower than the one involving 58 synthetic samples, which may be due to the better estimation of the population distribution (involved in the FID computations) with more samples.

<footnote n="2">² Values of the hyperparameters used: elliptical model (mean and std for the first and second axis): ((19,11);(11,11)) with personalization, ((22,16);(16,11)) with manual setting; union of spheres (number of iterations and maximal radius): (1,32) with personalization, (5,10) with manual setting.</footnote>
<figure><img src="image_5.png" /><figcaption>**Fig. 4.** Top: 2D views of the latent spaces for the real (a) and synthetic (b) populations, with transmurality constrained by the AR-VAE ordered along dimension 0. Bottom: main variations encoded along four dimensions for the real (c) and synthetic (d) populations, three dimensions being constrained to geometrical attributes of the infarct.</figcaption></figure>

# 4 Discussion and conclusion

In this work, we demonstrated the feasibility of a diffusion-based approach to better model the geometry of myocardial infarcts, in 2D, and personalize it to a real population with a population-based strategy tailored for models with randomness. Our model for infarct shapes is inspired by the propagation of infarcts through the myocardium to model infarct shapes, based on diffusion. Additionally, we applied a relevant representation learning strategy to evaluate the generated synthetic images while considering key geometrical characteristics of the infarcts.

The advantage of the presented model is its ability to better represent transmurality while offering more sophisticated infarct patterns compared to simpler models [10]; it also captures well the variations in infarct size and orientation in 2D, for the LAD territory. The diffusion-based model is more computationally intensive than simpler models based on ellipses or the union of spheres, requiring more time to simulate, but still much faster than complex physically-based models. Note
that generative deep learning models based on adversarial networks or diffusion allow generating
very realistic synthetic populations matching the distribution of real data. However, we focused on
modeling infarct shapes, in 2D, to demonstrate the relevance of the population-based personalization
in the context of physiologically-relevant simulations, whose use could be extended to more complex
biophysical models.

Population-based personalization can be a powerful tool for adjusting models with randomness to better represent real data. Although its initialization and evaluation may be challenging depending on the application, in particular due to the high number of learning parameters, this approach holds a great potential and can be extended to other models involving randomness, such as the generation of vascular trees [3].

The infarct modeling was done in Cartesian coordinates. Radial and circumferential coordinates could also be used, provided the myocardium is unfolded based on these coordinates. We preferred the simpler Cartesian coordinates, as it avoids additional folding-unfolding and interpolation operations. The modeling part can still be improved regarding the definition of the starting zone, the non-uniform propagation not necessarily through equally sized sectors, and can be extended to 3D and other infarct territories. It may also include a border zone, although this information is not explicitly available from the image segmentations. However, our modeling and personalization approach represent a step forward to synthesize realistic infarct shapes, which can be integrated with models of cardiac electrophysiology or mechanics that require myocardial heterogeneities, as well as methods that could generate synthetic cardiac images based on segmentations.

**Acknowledgments.** The authors acknowledge the support from the French National Research Agency (ANR) through the LABEX PRIMES of Université de Lyon (ANR-11-LABX-0063), from the Institut Universitaire de France, and from the Universite Lyon 1 (AAP SENS 2022). The authors are also grateful to the H2P platform<fnref n="3" /> (CREATIS, Villeurbanne, France) for the data storage and management support, and to Pierre Croisille, Magalie Viallon and Lorena Petrusca (CREATIS, CHU Saint Etienne, France) for providing the segmentations for the MIMI study from the MYOSAIQ dataset.

**Disclosure of Interests.** The authors have no competing interests to declare that are relevant to the content of this article.

**Data availability.** Code for the infarct models and their personalization is publicly available in the form of a ready-to-launch Jupyter notebook: https://github.com/creatis-myriad/2d-personalization-generated-data

**References**

1. Alfonso, B., David, K., Vicente, G., et al.: Fractional diffusion models of cardiac electrical propagation: role of structural heterogeneity in dispersion of repolarization. J R Soc Interface 11, 20140352 (2014)

2. Belle, L., Motreff, P., Mangin, L., et al.: Comparison of immediate with delayed stenting using the Minimalist Immediate Mechanical Intervention approach in acute ST-segment-elevation myocardial infarction: the MIMI Study. Circ Cardiovasc Interv 9, e003388 (2016)

3. Cury, L., Maso Talou, G., Younes-Ibrahim, M., et al.: Parallel generation of extensive vascular networks with application to an archetypal human kidney model. R Soc Open Sci 8, 210973 (2021)

4. van Doorn, E., Amesz, J., Sadeghi, A., et al.: Preclinical models of cardiac disease: a comprehensive overview for clinical scientists. Cardiovasc Eng Tech 15, 232–49 (2024)

<footnote n="3">³ http://humanheart-project.creatis.insa-lyon.fr/</footnote>
5. Duchateau, N., De Craene, M., Allain, P., et al.: Infarct localization from myocardial deformation: prediction and uncertainty quantification by regression from a low-dimensional space. IEEE Trans Med Imaging **35**, 2340–50 (2016)

6. Duchateau, N., Viallon, M., Petrusca, L., et al.: Pixel-wise statistical analysis of myocardial injury in STEMI patients with delayed enhancement MRI. Front Cardiovasc Med **10**, 1136760 (2023)

7. Gissler, M., Antiochos, P., Ge, Y., et al.: Cardiac magnetic resonance evaluation of LV remodeling post-myocardial infarction: prognosis, monitoring and trial endpoints. JACC Cardiovasc Imaging **17**, 1366–80 (2024)

8. Hansen, N., Ostermeier, A.: Adapting arbitrary normal mutation distributions in evolution strategies: the covariance matrix adaptation. Proc. ICEC pp. 312–7 (1996)

9. Hearse, D., Yellon, D.: The "Border zone" and myocardial protection: a time for reassessment? Acta Med Scand **651**, 37–46 (1981)

10. Konik, A., Clarysse, P., Duchateau, N.: Detailed evaluation of a population-wise personalization approach to generate synthetic myocardial infarct images. Pattern Recognit Lett **188**, 8–14 (2025)

11. Li, W.: Biomechanics of infarcted left ventricle: a review of modelling. Biomed Eng Lett **10**, 387–417 (2020)

12. Linge, S., Langtangen, H.: Diffusion equations. Finite Difference Computing with PDEs. Texts in Computational Science and Engineering, Springer. **16**, 207–322 (2017)

13. Mom, K., Clarysse, P., Duchateau, N.: Population-based personalization of geometric models of myocardial infarction. Proc. FIMH, LNCS **12738**, 3–11 (2021)

14. Nakamura, H., Strong, A., Dohmen, C., et al.: Spreading depolarizations cycle around and enlarge focal ischaemic brain lesions. Brain **133**, 1994–2006 (2010)

15. Nezlobinsky, T., Solovyova, O., Panfilov, A.: Anisotropic conduction in the myocardium due to fibrosis: the effect of texture on wave propagation. Sci Rep **10** (2020)

16. Pati, A., Lerch, A.: Attribute-based regularization of latent spaces for variational auto-encoders. Neural Comput Appl **33**, 4429–44 (2020)

17. Reimer, K., Lowe, J., Rasmussen, M., et al.: The wavefront phenomenon of ischemic cell death. 1. Myocardial infarct size vs duration of coronary occlusion in dogs. Circulation **56**, 786–94 (1977)

18. Rumindo, G., Duchateau, N., Croisille, P., et al.: Strain-based parameters for infarct localization: evaluation via a learning algorithm on a synthetic database of pathological hearts. Proc. FIMH, LNCS **10263**, 106–14 (2017)

19. Saez, P., Kuhl, E.: Computational modeling of acute myocardial infarction. Comput Methods Biomech Biomed Eng **19**, 1107–15 (2016)