A few mathematical tools for biology and medicine Thierry GOUDON - PowerPoint PPT Presentation

A few mathematical tools for biology and medicine Thierry GOUDON COFFEE–Sophia Antipolis Thierry Goudon

A few words on the scientific landscape ◮ D. Barbolosi, A. Iliadis, F. Hubert... (Marseille): tumor growth, chemiotherapy, toxicity ◮ Th. Colin (Bordeaux) team MONC, start-up Nenuphar: image analysis and prediction of tumor growth ◮ J. Clairambault, M. Doumic, D. Drasdo & B. Perthame, team MAMBA (Paris): chronotherapy, tumor growth ◮ GDR MaMoVi, T. Lepoutre ◮ CEMRACS 2018, Biological and medical applications ◮ French American Innovation Day : Math & Medicine, Houston, March 2018 with Marc Garbey

Objectives of the mathematical modeling: why could it be useful ? ◮ To reproduce the evolution of tumors (with equations and simulations...) ◮ To anticipate the evolution of tumors ◮ To anticipate the action of drugs or therapies ◮ To optimize the action of drugs or therapies For instance: after resection of a primary tumor, why is a preventative chemiotherapy useful ? Difficulties for imaging techniques to detect micro-metastases : can we predict in silico the growth of residual tumors ?

Basic ODE models N ( t ) : number of individuals (cells...) in a population, at time t . Variations due to gain (birth) and loss (death) d d t N ( t ) = ( λ − µ ) N ( t ) . The solution is N ( t ) = e ( λ − µ ) t N 0 : exponential growth if λ > µ , exponential extinction if λ < µ (cf. Malthus, end of XVIIIth century).

The discrete viewpoint = + ( λ − µ ) h X n X n + 1 X n ���� � �� � � �� � pop. at time t n pop. at time t n + 1 Variations due to gain/loss h =time interval of observation= t n + 1 − t n . Here the rate of variation τ n = X n + 1 − X n is constant. X n We can rewrite X n = ( 1 + ( λ − µ ) h ) n X 0 ; with t = nh , h → 0... it yields the exponential law. This is also the approach of Numerical Schemes with the “Stability” issue : the condition h ≤ ( µ − λ ) − 1 enforces positivity when µ > λ .

(Slightly) more complex models ◮ Verhulst : the more important the population, the stronger the loss rate/the lesser the gain rate � 1 − X ( t ) � � � d 1 − X n d t X ( t ) = a X ( t ) , X n + 1 = X n + ha X n . K K The population grows when X < K , decays when X > K . ◮ Gompertz : the growth rate itself obeys the exponential law: τ ( t ) = X ′ ( t ) X ( t ) = d � X ( t ) � d t ln b satisfies d d t τ = − a τ. d � b � It yields d t X ( t ) = aX ( t ) ln or X ( t ) X n + 1 = X n + haX n ln ( b / X n ) .

ODE models 1.4 exp Verhulst 1.2 Gompertz 1 0.8 X(t) 0.6 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 t

To incorporate the action of drugs d d t X ( t ) = X ( t ) R ( X ( t )) − X ( t ) c ( t ) � �� � � �� � your favorite ODE model acts against the growth Take into account toxicity (pharmaco-kinetic/dynamic) ◮ c ( t ) = G ( t , u ) with u =given dose. ◮ Admissible set K = { u s. t. F ( t , u ) ≤ C } . � �� � action on healthy cells � � ◮ Optimize min 0 ≤ t ≤ T X u ( t ) min . u ∈ K ���� keeps the patient alive

Bolus d � � d t X ( t ) = X ( t ) R ( X ( t )) − c ( t ) Question: How can we find t �→ c ( t ) that minimizes X ( T ) with T =a certain degradation time... under a constraint on the dose � T c ( s ) d s ≤ C M . 0 ◮ Bang-bang strategy : to give C M at time T Go1 ◮ In fact c ǫ ( t ) = C M ǫ 1 T − ǫ ≤ t ≤ T , with 0 < ǫ ≪ 1. d X ( t ) = ¯ ¯ X ( t ) R (¯ ◮ As ǫ → 0: X ( t )) and d t X ( T + ) = X ( T − ) e − C M .

A multi-dimensional model: Leslie’s system, population structured by age x j = # of individuals with age j ∈ { 1 , ..., N } f j > 0 fertility rate, t j + 1 , j transition rate j → j + 1.   f 1 f 2 f n     t 2 , 1 0 0       0 t 3 , 2 L =             0 0 t n , n − 1 0 The model says: Le X ( k + 1 ) = LX ( k ) with X ( k ) = ( x ( k ) , ..., x ( k ) n ) 1

Leslie’s system ctn’d λ ∈ C is an eigenvalue iff there exists x � = 0 such that Lx = λ x . Here L has a very specific property: L n has strictly positive entries ( primitive matrix ). Perron-Frobenius theorem : µ = max {| λ | , λ eigenvalue of L } is an eigenvalue, it can be associated to a vector ¯ X with non negative components. It governs the asymptotic behavior k →∞ µ k C ( X ( 0 ) ) ¯ X ( k ) ∼ X . d Similar conclusion for the differential system d t X = ( L − I ) X : blow up of the population if µ > 1, extinction of µ < 1.

What did we learn ? Large time asymptotics often corresponds to observable behaviors. We can try to exhibit structure properties (here L is a primitive matrix) that govern this behavior. We have efficient numerical procedures to compute directly the leading eigenpair, and thus to have direct access to the asymptotic state.

Towards PDE models: diffusion Diffusion ∂ t u = ∂ 2 xx u Heat eq. 2 1.8 1.6 1.4 1.2 u(t,x) 1 0.8 0.6 0.4 0.2 0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 x ◮ the solution is spread ◮ the solution becomes instantaneously smooth

Towards PDE models: transport Diffusion ∂ t u + ∂ x ( cu ) = 0 (here c > 0 constant) UpW Scheme vs. Exact solution 1 1 0.8 0.8 0.6 0.6 uexact(t,x) u(t,x) 0.4 0.4 0.2 0.2 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 x x ◮ the solution is... transported (finite speed) ◮ the solution keeps its regularity (singularities) Note: numerical schemes solve instead ∂ t u + ∂ x ( cu ) = ǫ∂ 2 xx u , 0 < ǫ ≪ 1, hence regularization effect of numerical nature.

K. Iwata, K. Kawasaki, N. Shigesada’s model for tumor growth A Dynamical Model for the Growth and Size Distribution of Multiple Metastatic Tumor, Journal of Theoretical Biology, 2000. ◮ Population of tumors structured by their size x � b a ρ ( t , x ) d x = # of cells with size in [ a , b ] at time t . ◮ The growth rate depends on the size according to governed by Gompertz’ law g ( x ) ◮ Each tumor produces new (small) cells with a rate β ( x ) . ◮ McKendrick–Von Foerster type equation:  ∂ t ρ + ∂ x ( g ( x ) ρ ) = 0 , for t ≥ 0 and 1 < x < b ,    ρ ( 0 , x ) = δ x = 1 , � b  g ( 1 ) ρ ( t , 1 ) = β ( x ) ρ ( t , x ) d x + β ( x p ( t )) .   1 � b � , β ( x ) = mx α , (typiquement α = 2 / 3) g ( x ) = ax ln x

we have a formula for the solution... By using Laplace’s transform ∞ � � λ k / a − 1 1 1 − ln x 1 a � e λ k t ρ ( t , x ) = c ( λ k ) , mb α ln b ln b x k = 1 where the λ k ’s are the roots of ∞ m λ k = F ( 1 , λ k 1 a � ( α ln b ) n , a + 1 ; α ln b ) = ( λ k a + 1 ) . . . ( λ k a + n ) n = 0 and ∞ ( − α ln b ) n � c ( λ k ) = a + n ) 2 . n !( λ k n = 0

Eigenvalue problem  ∂  ∂ x ( g ( x ) N ( x )) + λ N ( x ) = 0 ,     � b    g ( 1 ) N ( 1 ) = β ( y ) N ( y ) dy ,     1  − g ( x ) ∂ ∂ x Φ( x ) + λ Φ( x ) = Φ( 1 ) β ( x ) ,       � b � b    λ > 0 , N ( x ) ≥ 0 , Φ( x ) ≥ 0 , N Φ = 1 , N = 1 .     1 1 There exists a unique triple ( N , λ 0 , Φ) solution of this problem, with N ( x ) > 0, Φ( x ) > 0 An infinite-dimensional version of the Perron-Frabenius theorem...

Consequences 1. A conservation law: � b � b Φ( x ) ρ ( x , t ) e − λ 0 t dx = Φ( x ) ρ 0 ( x ) dx 1 1 2. Estimates in L ∞ : if cN ( x ) ≤ ρ 0 ( x ) ≤ CN ( x ) then cN ( x ) ≤ ρ ( t , x ) e − λ t ≤ CN ( x ) 3. and in L p (Φ( x ) N ( x ) dx ) . 4. The asymptotic behavior is driven by the leading eigenpair t →∞ C e λ t N ( x ) . ρ ( t , x ) ∼

Simulation (A. Devys’ Phd thesis) Difficulties: ◮ the growth rate g vanishes at x = b , ◮ b ≫ 1, ◮ the source at x = 1 is “large”. For t ≥ 2000 jours ≃ 5 . 5 years the asymptotic profile is a fair approximation of the solution. Comparison with Tubbiana’s experimental data and incoroporation of the action of treatments by the research group in Marseille

AABG model for tumor growth Interacting populations, structuration in size and space ◮ size-structured tumor concentration ( t , z ) �→ T ( t , z ) . The mass of the tumor changes due to natural growth + cell division . ◮ a bath of passive cells that become either immune ( ( t , x ) �→ E ( t , x ) ) or “collaborative” ( ( t , x ) �→ B ( t , x ) ). The model involves two distinct length scales; it assumes “scale separation: z ≪ x ”. It also means that we neglect some fine scale phenomena...

AABG model: evolution of tumor cells ∂ t T + ∂ z ( VT ) = Q ( T ) − m ( E , T ) � �� � � �� � “transport”=growth cell division - destruction by immune cells Q ( T )( t , z ) = 4 K ( 2 z ) T ( 2 z ) − K ( z ) T ( z ) (binary division) The operator Q increases the number of tumoral cells � ∞ µ 0 ( t ) = f ( t , z ) d z , but does not change the total mass of 0 � ∞ the tumor µ 1 ( t ) = zf ( t , z ) d z . 0 If m ( E , T ) = 0, we get � ∞ d d t µ 0 ( t ) = K ( z ) T ( t , z ) d z ≥ 0 , 0 d d t µ 1 ( t ) = V µ 0 ( t ) ≥ 0 .