Mathématiques pour la biologie et la santé · Biofluides, biomécanique Les enjeux Dans le domaine de la biomécanique médicale, le monde industriel et médical se trouve

Mathematiques pour labiologie et la sante

Benoıt Perthame, Ecole Normale Superieure

Des enjeux pour l’avenir

Une source de problemes mathematiques nouveaux

3 exemples : • Biomecanique

.. • Modeles de croissance de tissus (et cancer)

.. • Evolution adaptative

Biofluides, biomecanique

Les enjeux

Dans le domaine de la biomecanique medicale, le monde industriel

et medical se trouve confronte a des besoins de modelisation et

simulation.

Par exemple :

• design optimal de valves cardiaques ou de stents

• traitements individualises


Les enjeux

Dans le domaine de la biomecanique medicale, le monde industriel

et medical se trouve confronte a des besoins de modelisation et

simulation.

Par exemple :

• design optimal de valves cardiaques ou de stents

• traitements individualises




Des questions mathematiques difficiles etactuelles

• Enjeux numeriques et algorithmiques majeurs

• Interactions fluides/structures (surfaces deformables)

• Theorie et specificite des materiaux vivants

• Aerosols∂

∂tu + u · ∇u +∇p = ∆u +

∫(v − u)f(t, x, v)dv

∂

∂tf(t, x, v) + v · ∇f + divv(u− v)f = 0.


• Comment concevoir la cascade de divisions bronchiques

! "l

"0

"i

! i~

Modelisation de croissances tumorales

Les enjeux

Dans le traitement des cancers : optimisation therapeutique

individualisee.

Comprendre un phenomene fondamentalement multiechelle.


de l’echelle macroscopique

Cancer metastatique chez le dinosaure (Rotschild M., Witzke B. J. and

Hershkovitz I., The Lancet 1999)


a l’echelle de la molecule et de la cellule

Cycline Cycline DD

Cycline Cycline EE

Cycline Cycline AA

Cycline Cycline BB

S

G1

G2M

Origine de la prolifOrigine de la proliféération : le cycle cellulaireration : le cycle cellulaire

Contrôle physiologique ou thérapeutique externe :

- transitions entre phases (G1/S, G2/M, M/G1)

- taux d’apoptose dans les phases

Le cycle de division cellulaire : les transitions sont liees a l’expression de proteines

qui sont les cibles de chimiotherapies


pieces. We model the migration of cells by the Metropolisalgorithm and a proper definition of time scales. A cell inisolation performs a random-walk-like movement while inthe neighborhood of other cells it tends to move into thedirection which minimizes the free energy. We quantify themigration activity of a cell by its diffusion constant D inisolation. We perform a number of successive migration andorientation trials between two successive growth trials. Thetrials are accepted with probability min(1,exp{!!V/FT}.FT is a parameter that controls the cell activity: it may becompared to the thermal energy kBT in fluids (kB: Boltzmanconstant, T : temperature). Together with the choice of stepsizes for growth, orientation change, and migration, ouralgorithm mimics a multi-cellular configuration changingwith time. The step sizes are chosen in such a way, that thesimulation reflects a realistic growth scenario. (The detailsof our model are explained in [8])We recently used this single-cell based model to studytumor spheroid growth in liquid suspension [8], which hasbeen extensively studied experimentally [21], [14] (for anoverview of tumor growth models, please see Ref. [8] andreferences therein). Here, we study growing tumors in atissue-like medium composed of cells to analyze the influ-ence of an embedding medium on the tumor morphology(for a simulation example, see Fig. 1)

Fig. 1. Typical simulated tumor growth scenario. Red: embeddingcells, white: cells of the expanding clone. The embedding cells areinitially placed on the nodes of a square lattice and subsequentlyrelaxed before the growth of the embedded clone is started.

The embedding medium was modeled as non-dividingcells with the same parameters as the dividing cell clonewith the following exceptions: (1) ”motX” within the nameof the dataset denotes that D " D/X with D being theDiffusion constant mentioned above, (2) except of the dataset”id100_mot1_adh” the embedding cells do not adhere.The id-value refers to the initial distance of embedding cellswhich is l for id100 and l = 1.2 (= 120/100) for id120.For selected parameter sets, we have validated that the resultsdo not change if we replace the embedding cells by granularparticles with the same physical properties, but with only

passive movement (i.e., no capability to migrate actively).Experiments to validate our findings can thus be easilyconducted in in-vitro studies with an experimental settingsimilar to that in Ref. [10] by growing tumor spheroids in agranular embedding medium.

IV. GEOMETRIC SHAPE PROPERTIES

A. Morphological Operators

All datasets were given as binary 3D images, which aregenerally defined as the quadruple P = (Z3,m,n,B), whereevery element of Z3 is a point (voxel) in P. The set B # Z3

is the image foreground, or the object, whereas Z3 \B is thebackground. The neighborhood relation between the voxelsis given by m and n with m being the connectivity of objectvoxels and n the connectivity of the background. To avoidtopological paradoxa, only the following combinations arepossible: (6,26), (26,6), (6,18) and (18,6) [15].

Morphological operators are well-known in image proces-sing. Erosion and Dilation are in fact binary convolutionswith a mask describing the background-connectivity of avoxel [11]. The Hit-Miss-Operator extracts specific featuresof a binary image. For morphological Thinning, this operatoris used with a set of masks, where each mask is applied tothe original image, and all resulting images undergo a logicalOR-operation and will be subtracted from the original image[15].

B. Distance Transform

The distance field of a binary digital image is a discretescalar field of the same size with the property, that eachvalue of the scalar field specifies the shortest distance ofthe voxel to the boundary of the object. The signed distancetransform contains negative values for distances outside theobject. Distance transforms using the L1 or L" metrics can becomputed using Erosion for successive border generation andlabeling of the removed voxels until the object is completelyremoved [11]. The computation of the Euclidean distancetransform is described in [19].

C. Medial Axis Transform and Skeletonization

In a continuous space, the medial axis of an object is theset of points, which are the centers of maximally inscribedspheres. A sphere is maximally inscribed, if it touches theobject boundary in at least two points, if it lies completelywithin the object, and if there is no larger sphere withthe same properties. The skeleton of a binary object is acompact representation of its geometry and shape. It is asubset of the object with three properties [17]: (1) topologicalequivalence, (2) thinness, and (3) central location within theobject. Topological equivalence implies that the medial axishas the same number of connected components, enclosedbackground regions and holes as the original object.

In discrete space, the medial axis can be approximated byiterative Thinning as described in [15].

Photo de spheroide tumoral. Simulation numerique (par D. Drasdo)


Objets mathematiques

-) Modeles stochastiques (au echelles fines, ’individus centres’)

dXi(t) = v(X)dt + dWi(t), E[τ ] =?

-) Modeles differentiels

∂

∂tp(t, a, x) +

∂

∂ap(t, a, x) +

∂

∂x[v(a, x)p(t, a, x)] + r

(a, x, [p(t)]

)p(t, a, x) = 0

p(t, a = 0, x) =∫

B(a, x, y)p(t, a, y, x) dy da

-) Comprendre les relations, simuler a la bonne echelle


Ces objets mathematiques permettent

• de valider des explications heuristiques,

• d’acceder a des valeurs quantitatives

Fondements de la biologie : Evolution darwinienne

Enjeux socetiaux : comprehension conceptuelle

• Maupertuis (1698-1759) Observe l’evolution des especes

• Lamarck Tentative d’explication

• Darwin (1859) Mutations-Selections expliquent l’evolution

• Nirenberg (1961–...) Code genetique

• Modeles mathematiques (1990)

(Probabilite, Eq. Diff., ’Evolution game theory’...)


• Maynard-Smith : strategie evolutionnaire (equilibres comme en

economie, en theorie des jeux)

maxR(x, E) E = environnement, depend des traits x

• Descriptions individu centree (stochastiques) νt =I(t)∑i=1

δXi(t)

• Echelle de la population (Eq. de Lotka-Volterra)

∂

∂tn(t, x)−∆n = n(t, x) R

(x, [n(t)]

).


−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

2

4

6

8

10

12

14

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

2

4

6

8

10

12

14

Monte-Carlo simulations by M. Gauduchon (individual based model)

Les defits pour les mathematiques

• Une demande forte des sciences du vivant et de la sante

• Accompagner des changements radicaux pour les societes

(genetique, neurosciences...)

• Une science du vivant qui evolue tres rapidement

• Des equipes organisees de longue date dans de nombreux pays

• Tous les objets mathematiques sont concernes

• Des formalismes et questions nouvelles (risques eleves)

Les defits pour les mathematiques

• Une demande forte des sciences du vivant et de la sante

• Accompagner des changements radicaux pour les societes

(genetique, neurosciences...)

• Une science du vivant qui evolue tres rapidement

• Des equipes organisees de longue date dans de nombreux pays

• Tous les objets mathematiques sont concernes

• Des formalismes et questions nouvelles (risques eleves)

Les atouts de la fondation

• Expertise (modelisation, simulation numerique, traitement des

donnees)

• Tradition de relations industrielles et interdisciplinaires

• Domaine attractif pour les jeunes generations

• Soutien theorique fort

Documents

Mathématiques pour la biologie et la santé · Biofluides, biomécanique Les enjeux Dans le domaine de la biomécanique médicale, le monde industriel et médical se trouve