# Workshop - Modèles mathématiques pour la santé

### List of Speakers

Tomás Alarcón (ICREA, Centre de Recerca Matemàtica, Barcelona)

Cédric Galusinski (Université de Toulon)

Jacques Le Pendu (INSERM, Université de Nantes)

Kent-Andre Mardal (University of Oslo)

Vuk Milisic (Université Paris 13)

Luigi Preziosi (Politecnico di Torino)

Ricardo Ruiz Baier (University of Oxford)

Christian Schmeiser (University of Vienna)

Thomas Stiehl (Universität Heidelberg)

Nicolas Vauchelet (Université Paris 13)

### Contributed talks

Monika Twarogowska (ENS Lyon)

Fatima Mroué (Centrale Nantes/U. Libanaise)

Jonathan Stéphano (U. de Nice)

### Booklet

# Schedule

## Wednesday June 20

- 09h-10h: Welcome
- 10h-10h10 : Opening
- 10h10-11h00 : Cédric Galusinski
- 11h00-11h20 : coffee break
- 11h20-12h10 : Kent-Andre Mardal
- 12h20-14h00 : Lunch
- 14h00-14h50 : Ricardo Ruiz Baier
- 14h50-15h20 : Monika Twarogowska
- 15h20-15h50 : Emmanuel Bakare
- 15h50-16h20 : Coffee break
- 16h20-17h10 : Christian Schmeiser
- 20h00-23h30 : Gala dinner

## Tuesday June 21

- 9h30-10h20 : Tomás Alarcón
- 10h20-10h50 : Fatima Mroué
- 10h50-11h10 : Coffee break
- 11h10-12h00 : Jacques Le Pendu
- 12h00-14h00 : Lunch
- 14h00-14h50 : Luigi Preziosi
- 14h50-15h40 : Vuk Milisic
- 15h40-16h10 : Coffee break
- 16h10-16h40 : Jonathan Stephano
- 17h00-18h00 : Colloquium: Julie Delon (salle des séminaires)

## Friday June 22

- 9h30-10h20 : Thomas Stiehl
- 10h20-11h : Coffee break
- 11h00-12h00 : Nicolas Vauchelet
- 12h00-14h00 : Lunch

# Titles and abstracts

## Tomás Alarcón (ICREA, Centre de Recerca Matemàtica, Barcelona)

### Unlocking the pluripotent phenotype: A multiscale model of the epigenetic regulation of cell fate and plasticity

Understanding the control of epigenetic regulation is key to explain and modify the aging process. Because histone-modifying enzymes are sensitive to shifts in availability of cofactors (e.g. metabolites), cellular epigenetic states may be tied to changing conditions associated with cofactor variability. The aim of this study is to analyse the relationships between cofactor fluctuations, epigenetic landscapes, and cell state transitions. Using Approximate Bayesian Computation, we generate an ensemble of epigenetic regulation (ER) systems whose heterogeneity reflects variability in cofactor pools used by histone modifiers. The heterogeneity of epigenetic metabolites, which operates as regulator of the kinetic parameters promoting/preventing histone modifications, stochastically drives phenotypic variability. The ensemble of ER configurations reveals the occurrence of distinct epi-states within the ensemble. Whereas resilient states maintain large epigenetic barriers refractory to reprogramming cellular identity, plastic states lower these barriers, and increase the sensitivity to reprogramming. Moreover, fine-tuning of cofactor levels redirects plastic epigenetic states to re-enter epigenetic resilience, and vice versa. Our ensemble model agrees with a model of metabolism-responsive loss of epigenetic resilience as a cellular aging mechanism. Our findings support the notion that cellular aging, and its reversal, might result from stochastic translation of metabolic inputs into resilient/plastic cell states via ER systems.

## Cédric Galusinski (Université de Toulon)

### 3D Vessel reconstruction from partial or full C-T scan data

In collaboration with A. Al Moussawi and C. Nguyen.

Abstract: The goal of this talk is first to to introduce the 3D reconstruction of blood vessels from a limited number of 2D transversal cuts obtained from scanners (C-T scans). This is motivated by the fact that data can be missing. The difficulty of this work is to connect the blood vessels between some widely spaced cuts. We identify the vessels on each transversal cut as a mass to be transported along a graph which allows to determine the bifurcation points of vessels. Specifically, we are interested in branching transportation to model an optimized graph associated to the network of vessels. Adapted cost functions are then selected. We are then able to reconstruct the 3D vessels identified as the zero level of a 3D level set function whose 2D transversal cuts fit to data (where they are known).

In a second part, when the whole scanners data are available,a global reconstruction is proposed with reconstruction improvements in order to overcome low resolution of C-T scan at the vessel scale and to overcome patient motion during the scan. Vessel deformations are also presented.

## Jacques Le Pendu (CRCINA, Inserm, Université d’Angers, Université de Nantes)

### Norovirus infection and vaccine development

Noroviruses represent the major single cause of gastroenteritis worldwide. They are responsible for a disease that is generally self-limited but can be severe and life-threatening in the elderly, in immunocompromized patients and in young children of developing countries. They are small non-enveloped viruses that can evolve rapidly and are equipped with a single capsid protein that may be subject to epochal evolution of epidemiologically dominant strains, similar to Influenza virus. To infect cells, the capsid protein attaches to carbohydrates exposed at the surface of cells lining the small intestine. The relative affinity of virus strains that undergo epochal evolution has been associated with their epidemiological impact, suggesting that immune-escape variants may become more pathogenic. Vaccine development is in progress. However, the results from the first trials in volunteers indicate that the extant vaccines are imperfect, only decreasing the severity of symptoms, but not the infection rate. The evolution of the virus under immune pressure in natural conditions and in vaccine conditions therefore should be evaluated in order to determine under which conditions an imperfect vaccine may lead to increased pathogenicity of the virus in the non-immunized. To this aim a modelling effort is ongoing in collaboration with researchers from the Jean Leray laboratory in Nantes.

## Kent-Andre Mardal (University of Oslo)

### Mathematical modeling of the glymphatic system

The newly proposed glymphatic system offers a potential explanation
for how the brain (which mostly lack a lymphatic system) clears waste.
As malfunctioning waste clearance seems to be a main problem in
diseases such as Alzheimer, where accumulation of amyloid-beta plaques
is one of the hallmark features, an understanding of this process may
have huge potential. The glymphatic system remains controversial. It
is a biomechanical theory that links the transport between
the cerebrospinal fluid, the peri- and paravascular spaces
that surrounds the blood vessels and the extracellular matrix
and has as such been the subject of subject of many recent modeling efforts.

In this talk we present an overview of mathematical models
for the glymphatic system and include our own results in this
type of modeling.

## Vuk Milisic (Université Paris 13)

### Mathematical analysis of adhesion forces in the filament based lamellipodium model

In collaboration with Dietmar Oelz(University of Queensland)

In this talk we present the mechanical model of the lamellipodial actin‐cytoskeleton meshwork. The model is derived starting from the microscopic description of mechanical properties of filaments and cross‐links and also of the life‐cycle of cross‐linker molecules [8, 7, 1, 2]. We introduce a simplified system of equations that accounts for adhesions created by a single point on which we apply a force. We present the adimensionalisation that led to a singular limit motivating our mathematical study. Then we explain the mathematical setting and results already published [3, 4, 5]. In the last part we present the latest developments : we introduce the space dependence and show how to include it in the model, we sum up asymptotic results available in this context and we give new results for the fully coupled system with unbounded non‐linear off‐rates [6].

References

[1] A. Manhart, D. Oelz, C. Schmeiser, and N. Sfakianakis. An extended Filament Based Lamellipodium Model produces various moving cell shapes in the presence of chemotactic signals. J. Theoret. Biol., 382:244–258, 2015.

[2] A. Manhart, D. Oelz, C. Schmeiser, and N. Sfakianakis. Numerical treatment of the filament‐based lamellipodium model (FBLM). In Modeling cellular systems, volume 11 of Contrib. Math. Comput. Sci., pages 141– 159. Springer, Cham, 2017.

[3]V. Milisic and D. Oelz. On the asymptotic regime of a model for friction mediated by transient elastic linkages. J. Math. Pures Appl. (9), 96(5):484–501, 2011.

[4]V. Milisic and D. Oelz. On a structured model for the load dependent reaction kinetics of transient elastic linkages. SIAM J. Math. Anal., 47(3):2104–2121, 2015.

[5]V. Milisic and D. Oelz. Tear‐off versus global existence for a structured model of adhesion mediated by transient elastic linkages. Commun. Math. Sci., 14(5):1353–1372, 2016.

[6]V. Milisic and D. Oelz. Space dependent adhesion forces mediated by transient elastic linkages : new convergence and global existence results, 2017.

[7]D. Oelz and C. Schmeiser. Derivation of a model for symmetric lamellipodia with instantaneous crosslink turnover. Archive for Rational Mechanics and Analysis, 198(3):963–980, 2010. [8]D. Oelz, C. Schmeiser, and V. Small. Modelling of the actin‐cytoskeleton in symmetric lamellipodial fragments. Cell Adhesion and Migration, 2:117–126, 2008.

## Luigi Preziosi (Politecnico di Torino)

### Multi-level mathematical models for cell migration in dense fibrous environments

Cell-extracellular matrix interaction and the mechanical properties of cell nucleus have been demonstrated to play a fundamental role in cell movement across fibre networks and micro-channels and then in the spread of cancer metastases.

The lectures will be aimed at presenting several mathematical models dealing with such a problem, starting from modelling cell adhesion mechanics to the inclusion of influence of nucleus stiffness in the motion of cells, through continuum mechanics, kinetic models and individual cell-based models.

## Ricardo Ruiz Baier (University of Oxford)

### Mixed formulations for coupled diffusion-stress systems and some applications in biomechanics

In this talk we will present an overview of a family of mixed-primal and mixed-mixed partial differential equations governing the interaction between the deformation of elastic and hyperelastic bodies and nonlinear reaction-diffusion mechanisms. We will address unique solvability of the coupled problems in the context of fixed-point operators, and will discuss the construction of suitable mixed finite element schemes for their numerical approximation. This general formalism will be then used to solve specific applicative problems, related for instance, to the modelling of cardiac electromechanics. Our approach in turn suggests a natural way of incorporating so-called mechano-electric feedback effects.

## Christian Schmeiser (University of Vienna)

### Simulation of cell-cell interaction by the Filament Based Lamellipodium Model (FBLM)

The FBLM, a two-dimensional anisotropic two-phase continuum model for the dynamics of the actin network in the lamellipodium, can be used to describe cell-cell interaction in monolayers spread on flat substrates. First attempts in this direction will be presented, taking into account steric repulsion as well as cell-cell adhesion. (joint work with A. Brunk, D. Peurichard, and N. Sfakianakis).

## Thomas Stiehl (Universität Heidelberg)

### Understanding clonal dynamics in blood cancer - insights from mathematical modeling

Acute leukemias are cancerous diseases of the blood forming (hematopoietic) system. The leukemic cell bulk is derived from a small and heterogeneous population of leukemic stem cells. Upon expansion, the leukemic cells out-compete healthy blood production which results in severe clinical symptoms.

To study the interaction of leukemic and healthy cells, we propose mathematical models of hierarchical cell populations. Cell competition and selection are mediated by various biologically inspired feedback mechanisms. The models relate disease dynamics to basic cell parameters, such as proliferation rate (number of cell divisions per unit of time) and self-renewal fraction (probability that a progeny of a stem cell is again a stem cell). Motivated by the recent findings, we extend the models to take into account competition of multiple leukemic clones. Depending on the posed questions, we use different mathematical approaches to study clonal selection in presence and absence of therapy. These include nonlinear ordinary differential equations, integro-differential equations and stochastic simulations.

A combination of mathematical analysis, computer simulations and patient data analysis provides insights into the following clinically relevant questions: (1) Which mechanisms allow leukemic cells to out-compete their benign counterparts? (2) How do leukemic stem cell parameters (proliferation rate and self-renewal fraction) affect the clinical course and patient prognosis? (3) How do leukemic stem cell parameters affect clonal competition? Which parameter combinations confer selective advantages? (4) How do leukemic stem cell parameters impact on the topology of the clonal hierarchy? (5) How do nonlinear feedback signals affect disease dynamics and treatment response?

The talk is based on joint works with Anna Marciniak-Czochra, Jan-Erik Busse (Institute of Applied Mathematics, Heidelberg University), Anthony D. Ho, Natalia Baran and Christoph Lutz (Heidelberg University Hospital).

## Nicolas Vauchelet (Université Paris 13)

### Mathematical modeling of the spread of Wolbachia for dengue control

Summary : Bacteria Wolbachia has gain a lot of attention since scientists discover that infected mosquitoes with this bacteria cease to transmit some disease like dengue, chikungunya and Zika. Moreover, this bacteria is maternally transmitted from mother to offsprings. Then a strategy of control of dengue transmission consists in releasing Wolbachia infected mosquitoes in the aim to replace to natural population of mosquitoes by infected mosquitoes. In this work, we are concerned with the spatial spread of Wolbachia infected mosquitoes into a host population. We focus on the two following questions: How the spatial repartition of the releases will influence the spread of the bacteria into the population ? Once the spread is initiated, is it possible that environmental characteristics stop the spread ?