Background

The heart is an extraordinary organ. Its ultimate function is rather simple: to pump the blood throughout the body to supply oxygen and nutrients to the cells. How such functionality is achieved is however remarkably complex and poses significant challenges from a modeling and computational perspective. An electric stimulus, originating in the heart and indipendently from the central nervous system, travels across the heart to orchestrate and direct the mechanical contraction and relaxation, which in turn determine the pumping function.

From a clinical viewpoint, the job of a cardiologist is to study the patho-physiology of the heart, which in fact is the study of electric disorders (cardiac electrophysiology) and mechanical disfunction. In the heart, electric and mechanical function are tightly coupled: an electric disease (e.g., atrial fibrillation, tachycardia, bundle branch block) impedes a correct mechanical function which results into a pathological and potentially dangerous situation. Similarly, a mechanical or circulatory problem (hypertension, ischemia, infarct) can affect the electric function and lead to a vicious feedback affecting the overall functionality.

Diagnostically, the electrocardiogram (ECG) is a simple yet powerful tool to detect electric anomalies. Nonetheless, the ECG has acknowledged limitations, being hard to interpret in some situations also by a experienced cardiologist. ECG recording with high spatial coverage---the body surface potential mapping (BSPM), a vest composed by more than 200 electrodes---can significantly improve the diagnostic power, but they are difficult to summarize. In the so-called inverse problem of electrophysiology, these data is used to reconstruct the electric activity of the heart, enabling non-invasive diagnosis and therapy planning.

Mathematical modeling of cardiac electrophysiology is well established and already reaching the stage of clinical application. Commercial tools for solving the inverse problem of electrophysiology (ECG mapping) are nowadays available. Patient-specific modeling for optimal therapy delivery is, however, still in its infancy. One limitation is that current ECG mapping approaches is that they are imaging tools, without providing any valuable information to individualize patient-specific models. Another limitation is that being able to fix the electric pathology may not fix the mechanical function, which is ultimately the most relevant one. This is why coupled electro-mechanical models have started to emerge and being considered for in silico therapy planning.

The aim of this course is therefore to review these two important aspect of cardiac modeling: the inverse problem of electrophysiology, covered in the first part, and cardiac mechanics, in the second part. The inverse problem of electrophysiology comes in several flavors, each with its own benefits and idiosyncrasies. In the first lecture, after reviewing some basic facts of the ECG, we will study the modeling aspects of each formulation, their  mathematical characterization as an inverse problem and the numerical solution strategy. In the second lecture, we will cover inverse problems in a more general sense, showing why they are illposed and the consequence of this. Regularization is a standard approach to alleviate the illposedness, but it does not come for free. A concept of optimal regularization (in the sense of Pareto optimality) will be introduced.

The second part of the course is devoted to cardiac mechanics or, more broadly, tissue mechanics. Biological materials usually undergo large deformations, hence the geometrical description of kinematics is more complex and constitutive assumptions are genuinely nonlinear. Additionally, biological tissues, and specifically the cardiac tissue, are anisotropic. After reviewing the introductory concepts of continuum mechanics in the third lecture, we will focus on hyperelastic materials and the well-posedness of the equilibrium formulation. Weaker concepts of convexity, polyconvexity and rank-one convexity, are introduced, which are more suitable for continuum mechanics. The fourth lecture will also cover the numerical discretization, which particular emphasis on how to deal with the incompressibility constraint. The last lecture will shift the focus on inelastic behavior arising from the electro-mechanical coupling or from growth and remodeling of the heart. The concept of active materials and multiplicative decomposition is studied and applied to relevant examples.

If time permits, a third part on the fluid-structure interaction (FSI) problem is planned. The heart is an extremely efficient mechanical pump, capable of working effectively at high-pressure regime (systolic pressure) as well as low-pressure regime (diastolic pressure). FSI is particularly relevant for the valves: these thin structures embedded in the blood control the inflow and outflow from the ventricles very efficiently.

Schedule

• 23/10/2019
  This lecture is devoted to get familiar with the basic concepts of the *electrocardiogram* (ECG) and the inverse problem of electrocardiology.
  A brief historical perspective from Einthoven's seminal studies will be proposed. The ECG is the manifestation of the electric activity of the heart, and nowadays can be accurately detected on the chest with high temporal resolution. A quite natural question is whether is possible to reconstruct the cardiac activity from such measures. In mathematical terms, this problem represents an *inverse problem*. In the lecture we will formalize the forward and inverse problem, and sketch solution strategies. Multiple formulations have been proposed, based on the extracellular potential, the transmembrane potential or the activation map. Each formulation has pros and cons.
  
  
• 30/10/2019
  Illposedness of inverse problems. Interpretation. Regularization strategies. Tichonov, truncated SVD. KKT. Adjoint problem. Solution of the KKT system and preconditioning. Regularization in time.
  
  
• 06/11/2019
  Brief introduction to continuum mechanics. Kinematics. Clausius-Duhem inequality and hyperelastic materials. Coleman's formalism and conjugated variables. Objectivity. Symmetries and invariants. Transversely isotropic materials. Incompressibility.
  
  
• 13/11/2019
  Well-posedness, rank-one convexity, quasi-convexity, polyconvexity. Why full convexity of strain energy density function is not good. Lack of uniqueness: bifurcation. An example with Mooney-Rivlin material. Discretization and numerical solution. Three-field formulation, static condensation. Continuation  methods. Why hexahedral grid are better than tetrahedral grids in computational mechanics?

• 20/11/2019
  Active materials. Multiplicative decomposition and scale separation. Example application: tumour growth, morphogenesis, electro-mechanical coupling.
  
  
• 27/11/2019
  Boundary conditions and coupling to systemic circulation. Fluid-structure interaction problem. Solution strategies.
  
  
• 04/12/2019
  Valves. Immersed boundary method.
  
  
• 9-10/12/2019
  TRM forum.
  
  
• 11/12/2019
  Lab session.

References

• Keener J., Sneyd J., Mathematical Physiology. Cambridge university press, 2009
• Colli Franzone et al., Mathematical Cardiac Electrophysiology. Springer, 2014
• Katz, A.M., Physiology of the Heart, Lippincott W&W, 2010