Introduction

The human heart is composed of four chambers alternating expansion and contraction phases to pump blood to systemic and pulmonary circulation. Cardiac motion is determined by the interplay of four biophysical domains: i) electrophysiology, ii) tissue mechanics, iii) atrio-ventricular fluid dynamics and iv) metabolism. During every heartbeat, mechanical contraction is triggered by myocyte electrical activation and modulated by cell stretching and by the availability of oxygen and metabolic substrates. The increase in tension during contraction determines an increase of pressure causing blood flow from atria to ventricles and from ventricles to pulmonary and systemic circulation. Blood inflow in the chambers is then driven by elastic recoil of myocytes that causes the expansion of the cavity. The alteration of homeostasis for any of these aspects reflects on cardiac function and motion.
Therefore, the analysis of cardiac motion is of paramount importance for the diagnosis of heart failure and the prediction of pathological evolution and anatomical remodelling. Clinically, cardiac function is mainly classified based on ejection fraction computed as the ratio between the chamber internal volume variations over the heartbeat (stroke volume) and the maximum volume (end diastolic volume). This global metric is used for discriminating failing from healthy hearts, although there is strong evidence on the limitations of this approach (Cikes and Solomon, 2016). If we focus on the left ventricle for example, ejection fraction is in fact the global result of the interplay of deformation mechanisms that happen in the longitudinal direction, related to the valve motion in the direction of the long axis, and in the circumferential and radial ones, related to the thickening of the cardiac muscle during myocyte contraction. It has been suggested that ejection fraction might be one of the last metrics to evolve in relation of pathological alterations of the heart. In fact, in the early stages of pathology progression, impaired longitudinal motion of the ventricles might be compensated by an increase in contraction in the circumferential direction leading to preserved ejection fraction. Therefore, it has been proposed to use myocardial motion and deformation as descriptors of local cardiac function (Bijnens et al., 2009, 2012).
The correct inference of time-varying local motion is a much harder task than the segmentation of the inner chamber volume since it has to take accurately infer motion and displacements at pixel levels and, possibly, combine multi-contrast images in the process. For deriving detailed evaluations of cardiac motion, we need efficient and robust ways of analysing, integrating and processing patient data. Corral-Acero et al. (2020) suggest that the tools we need should come from the combination of mechanistic and statistical models. The following sections will provide a general overview of mechanistic and statistical models that will conclude with a discussion on the possible synergy between these two categories with examples on applications that already exploited this principle.

Mechanistic models

Mechanistic models represent our knowledge of physiology and physics in terms of mathematical formulations developed over the years and adapted from clinical observations. They are highly interpretable and their parameters are associated to corresponding physical, biological and chemical aspects of the problem. Their expressivity and fidelity to physiology are intrinsically limited by our understanding of the problem and by the assumption upon which the formulation is based. Also, they are usually function of parameters that need to be tailored to each specific patient. Examples of classical mechanistic approaches include the electromechanical and mechanical models for cardiac function (Buoso et al., 2021) and blood flow simulations using the Navier-Stokes equations (Buoso et al. 2019). Although several lumped parameter models have been developed and successfully applied to cardiac applications, the most popular electromechanical models are usually based on the solution of partial differential equations on detailed anatomical models, where microstructure and tissue properties are defined in each location. A complete mechanistic cardiac model is usually composed of anatomical, microstructural, mechanical, electrophysiological and fluid dynamic models. Due to the large computational cost, one of more of these fields are often modelled using simplified or lumped parameter models.

Anatomical models: The anatomy represents the computational domain of the mechanistic model. It can be a single chamber, usually the left-ventricle, or the full heart model. For patient-specific application, the anatomy is usually generated from the segmentation masks of the myocardium from patient exams and it usually requires manual user intervention to ensure the shape can be used in a numerical model. Recent approaches aimed at automatizing the process by fitting statistical shape models to the masks from the segmentation networks (Ouzir et al. 2017; Romaszko et al. 2021; Joyce et al., 2022). These shape models are obtained from the low-rank anatomical representation of a dataset of geometries that have been manually processed and prepared for simulations and they provide a prior that restricts the network output to the variability of the shape model (Buoso et al., 2021). Anatomical statistical models can therefore provide an input to mechanistic models and they also allow to generate new shapes that can be used to generate synthetic datasets.

Microstructural models: Microstructural tissue orientation in the heart is often prescribed using rule-based models fitted from histological representations. The most common approaches impose linear helix angles variations with transmural distance of myocyte orientations moving from epicardium to endocardium where helix angles are prescribed (Beyar and Sideman, 1984; Potse et al., 2006). In some more recent applications, Laplace equations with Dirichlet boundary conditions are used to model the variation of the microstructure taking into account the heart shape (Doste et al., 2019). However, these approaches determine a partial mismatch between the real patient condition and the microstructural model with an impact on electrophysiological simulation results (Rodríguez-Cantano et al., 2019). Recently, few works have focused on approaches to merge high-resolution myocyte orientations from in-vivo and ex-vivo magnetic resonance imaging. Sack et al. (2018) have shown that including subject-specific, high-resolution myocyte orientation in biomechanical models enables to reduce cross-fiber contraction and to obtain physiologically realistic cardiac simulations. While their approach relied on ex-vivo imaging, in-vivo conditions are much more challenging due to limited resolution and coverage. For this conditions, statistical models generated from datasets of ex-vivo cardiac in-vivo diffusion tensor imaging have been proposed to augment in-vivo data with prior knowledge of myocyte distribution (Stimm et al, 2020).

Electrophysiological models: Cellular electrical activity is being described using highly detailed cell models formulated coupling ordinary differential equations describing the kinetics of the ion-channels on the cell membranes (Hodgkin and Huxley, 1952.) These are made available at http://www.cellml.org. Currently, representing the full myocardium considering models describing features at molecular level is out of reach due to its computational cost. Alternative approaches, computationally more efficient, treat the myocardium as a continuum. Individual electrical evolution of single cells is assumed to happen at a much smaller scale than those of the variables of interest, such as voltage gradients. The latter are described with phenomenological models defining only the interactions at the larger scales. It is the case of the bidomain model that, despite the very large computational cost, was found to be able to accurately replicate physiological and pathological electrical activation patterns (Henriquez, 1993). The bidomain model is very often reduced to the monodomain equation, further simplifying the interaction between intra-cellular and extracellular fields (Baillargeon et al., 2014). Further simplifications to the monodomain models are usually adopted in practice to reduce even more the computational cost: in the Eikonal models the spatial and local transient due to the action potential upstroke are neglected and tissue activation is simulated with a step function (Tomlinson et al., 2002). Recent alternative approaches to reduce the computational cost electrophysiological simulations are based on deep-neural networks (Fresca et al., 2020).
An important aspect for cardiac activation is the Purkinje fiber network that is a network of fast-conducting cells providing rapid depolarization of endocardium tissue. While this is important in the physiological description, it is usually neglected or approximated with very simplified models due to the impossibility of imaging such structures for patient specific cases.

Mechanical model: The passive mechanical response of myocardial tissue is usually described using constitutive material models, among which those from Holzapfel and Ogden (2009) and Guccione et al. (1991) are generally considered the standards in the field. In these formulations, material behaviour is determined by coefficients defining tissue shear properties of collagen matrix and fibres. The total mechanical response is obtained by including the effect of active contraction from the myocyte which is computed from the electrophysiological fields associated to that cardiac phase. These constitutive models are phenomenological models that describe the macroscopical changes in mechanical configurations using relationships fitted to selected measurements. As in the electrophysiological domain, this allows to reduce the computational cost with respect to microscopical models defining the behaviour of isolated myocytes. Of paramount importance for the mechanical problem is the correct definition of the boundary conditions to the problems, i.e the loads and constraints to the motion. The heart is surrounded by a pericardial sac, which provides fixture in space of the myocardium (Pfaller et al., 2019). This constrain to cardiac motion is often neglected, but is important to model correctly epicardial motion. It can be represented by modelling a fibrous surface with a high tensile stiffness surrounding the heart, or by prescribing on the epicardium displacement-dependent loads that oppose to normal and/or tangent motion. In many applications, only one of the chambers of the heart is modelled, usually the left ventricle. In these cases, the effect of the surrounding missing structures is usually neglected or represented by additional loads acting at the interface. Usually, at the endocardium, a uniform pressure is imposed, which can be derived by invasive measurements or by coupling with lumped parameters the systemic and pulmonary circulation models (Buoso et al., 2021). Alternatively, a full fluid-structure interaction model can be used to account for the implicit feedback of the blood flow structures in the cardiac chambers and the myocardial wall (Santiago et al.,2018).

Statistical models

Statistical models embed the knowledge that can be derived from available data. They autonomously identify correlations and intrinsic features from the observations without specific user inputs, but they are limited by the amount and the quality of data available. We broadly collect here all approaches where no explicit mathematical description is provided, but it this is rather learned by the method in a supervised/unsupervised way. Currently, the majority of machine learning methods belong to this class.
Unsupervised statistical models are usually deployed to identify data representations that can provide useful and new insights into the dataset. Examples are variational autoencoders that are usually used to identify unknown non-linear dimensionality reduction manifolds. These are extensions of classical linear dimensionality reduction approaches such as principal component analysis (PCA) and proper orthogonal decomposition (POD).
Supervised approaches aim at identifying a function mapping the input data, e.g., images or signals, to output labels provided by the user. For input data with large dimensionality, supervised approaches often exploit dimensionality reduction approaches which facilitates the regression/classification tasks and make them more robust to noise. In many applications two or more statistical models are combined together to increase the robustness of the predictions (Ouzir et al. 2017; Romaszko et al. 2021; Joyce et al., 2022). There is a large variability in the applications of statistical models including segmentation (Peng et al. 2016, Leiner et al. 2019), scar identification and myocardial tissue characterization (Fahmy et al., 2018)

Sinergy between mechanistic and statistical models

The use of mechanistic models is very limited in clinical practice and they are mostly confined to research and academic settings. The most severe drawbacks of these methods are the long computational time and the need of calibrating their parameters to each patient of interest. Statistical models are, instead, more widespread in clinical applications and they have been quite successful in extracting global parameters from images (Petitjean and Dacher 2011). Since they require sufficient availability of training data, they are still limited in the prediction of pathological evolution due to the lack of large datasets of longitudinal investigations.

There are at least three ways in which we could exploit the synergy between mechanistic and statistical models: i) synthetic data generation, ii) data-driven simulations, iii) digital twin personalization.

Synthetic data generation: training of supervised statistical models, especially deep learning methods, relies on large annotated datasets. For many applications, such as left ventricular segmentations, we have now available several datasets with clinical annotations that can be used to train segmentation algorithms. In many other cases, such as for the denoising and analysis of blood flow in PC-MRI, or the tracking of the myocardium in tagged-MRI, it is almost impossible to provide ground truth annotations pixelwise. Additionally, labels are usually not based on the underlying ground truth, which is unknown. They are, therefore, affected by noise, limited resolution, partial-volume effect and inter/intra operator variability. Finally, these datasets are often not balanced with the risk of determining strong biases in the predictions of the metrics of interest. It is possible to couple mechanistic and statistical models to generate tuples of synthetic ground truth data and the corresponding images to be used in supervised learning approaches. For examples on the topic check out the abstract 3962 (Buoso et al., Biophysical-model based synthesis of cine CMR of healthy and pathological left-ventricular function) and abstract 5072 (Dirix et al., Synthesis of 4D flow tensor MRI of patient-specific turbulent flow).

Data-drive simulation: statistical models can be used in combination with mechanistic models to provide priors to the output of the simulation. It is possible, for example, to constrain motion prediction from a biomechanical model to comply with the characteristics features expected from the displacement field of the myocardium this can be done using statistical models based on POD, for example, as in Buoso et al., 2021. Additionally, combining deep-learning with physical descriptions of the problem, it is possible to personalize model parameters to the particular case of interest (Raissi et al., 2017)

Digital twin personalization: mechanistic and statistical model classes could be combined to build a digital twin of a patient overcoming the limitations of both: mechanistic models provide a robust framework to integrate and augment clinical data but they need to be calibrated to the patient using the available measurements. Statistical model can robustly infer the required parameters from clinical data, feed them into mechanistic models allow to predict and extrapolate relevant metrics and outcomes. In this view, it is not either-or model selection, but it is rather important to understand benefits and limitations of each class to make sure data is processed and integrated correctly.

Conclusion

Currently, inference on MRI data is mostly compressed down to a few global labels and the majority of data in the exam is discarded. Local motion information, however, could provide predictive biomarkers for the detection of early stages of cardiac pathologies. Mechanistic and statistical models offer a fundamental tool to process and analyse MRI data and develop robust methods for the the inference of time-varying three dimensional fields from medical images. Data extracted from the images using statistical models, could then be combined with mechanistic models to derive patients’ digital twins and shift the current clinical reasoning, focused on population statistics, towards precision medicine, accounting for patient’s individual variability when delivering personalized treatments.

Acknowledgements

No acknowledgement found.