Mathematical Modelling of Complex Biological Systems. A ... · “quasi fractal surface” which interacts with the outer environment, for example normal host cells and the immune

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Math for Living Systems Mathematical framework Modelling the immune competition Mathematical problems Simulations Perspectives References

Mathematical Modelling of Complex BiologicalSystems.

A Kinetic Theory Approach

MARCELLO DELITALA

[email protected] of Mathematics

Politecnico di Torino

Parma - May 17th, 2007

Mathematical Modelling of Complex Biological Systems. A Kinetic Theory Approach


Outline of the talk

1 Math for Living Systems

2 Mathematical framework

3 Modelling the immune competition

4 Mathematical problems

5 Simulations

6 Perspectives

7 References



Section One

A Mathematics for Living Systems





On the conceptual difficulties of dealing with the living matter :Models of the inert matters are occasionally used to describephenomena of the living matter without the necessary analysis oftheir substantial differences with respect to the behavior of the inertmatter. On the other hand, models of multicellular systems shouldinclude the expression of biological functions and their role toorganize the movement of cells, proliferating and destructive events,their ability to select evolutionary mutations and organize trendtowards equilibrium configurations which do not correspond to thoseobserve in the inert matter.




Hartwell , Nature 1999Biological systems are very different from the physical or chemical systemsanalyzed by statistical mechanics or hydrodynamics. Statistical mechanicstypically deals with systems containing many copies of a few interactingcomponents, whereas cells contain from millions to a few copies of each ofthousands of different components, each with very specific interactions.

Although living systems obey the laws of physics and chemistry, the notion offunction or purpose differentiate biology from other natural sciences.Organisms exist to reproduce, whereas, outside religious belief rocks andstars have no purpose. Selection for function has produced the living cell,with a unique set of properties which distinguish it from inanimate systems ofinteracting molecules. Cells exist far from thermal equilibrium by harvestingenergy from their environment.




E. Kant 1790 , da Critica della Ragion Pura, Living Systems: Specialstructures organized and with the ability to chase a purpose.

E. Schr odinger, P. Dirac - 1933 , What is Life?, I living systems have theability to extract entropy to keep their own at low levels.

R. May, Science 2003 In the physical sciences, mathematical theoryand experimental investigation have always marched together.Mathematics has been less intrusive in the life sciences, possiblybecause they have been until recently descriptive, lacking the invarianceprinciples and fundamental natural constants of physics.
















I Biological functions have the ability to modify conservation laws ofclassical mechanics due to the organized, somewhat intelligent, behavior ofbiological entities composing a living system.

II Systems in biology cannot be simply observed and interpreted at amacroscopic level . A system constituted by millions of cells shows at themacroscopic level only the output of the cooperative and organized behaviorswhich may not, or are not, singularly observed.

III Biological systems are different from the physical or chemicalsystems analyzed by statistical mechanics , which typically deals withsystems containing many copies of a few interacting components, whereascells contain from millions to a few copies of each of thousands of differentcomponents, each with specific interactions. If the number of components isreduced (technically to reduce complexity), then suitable collective behaviorshave to be properly identified.



The Multiscale representation

From Hartwell’s Theory of Modules to Multiscale

The molecular scale: The evolution of a cell is regulated by the genescontained in its nucleus. Receptors on the cell surface receive signals whichare then transduced to the cell nucleus, where various genes are activated orsuppressed. Particular signals can induce a cell to reproduce itself in theform of identical descendants - so-called clonal expansion, or to die anddisappear apparently without trace - so-called apoptosis or programmedcell death . Clonal expansion of tumor cells activates acompetitive-cooperative interaction between them and the cells of theimmune system. If the immune system is active and able to recognize theproliferating cells, then it may be able to destroy the incipient growth of cellswhich have lost their differentiation; otherwise, these cells may developprogressively.




The cellular scale: On the cellular level, models are proposed to simulatethe effects of the failure of programmed cell death and of the loss of celldifferentiation . If and when a tumor cell is recognized by immune cells acompetition starts which may end up either with the destruction of tumor cellsor with the inhibition and depression of the immune system. Cellularinteractions are regulated by signals emitted and perceived by cells throughcomplex reception and transduction processes. Therefore, the connection tothe afore-mentioned sub-cellular scale is evident. On the other hand, thedevelopment of tumor cells, if not suppressed by the immune system, tendstowards condensation into a solid form so that macroscopic features becomeimportant.




The macroscopic scale: After a suitable maturation time tumor cells startto condense and aggregate into an entity which eventually evolves as a“quasi fractal surface” which interacts with the outer environment, for examplenormal host cells and the immune system. These interactions usually occuron the surface and within a layer where angiogenesis (the process offormation of new blood vessels, induced by factors secreted by the tumor,and vital for tumor growth) takes place. Here, one has the overlap ofphenomena at the cellular level with typical macroscopic behavior such asdiffusion or, more generally, phenomena that can be related to the massbalance or evolution of macroscopic variables such as tumor size. In a laterstage, the tumor can be characterized by three zones: an external layerwhere environmental cells may penetrate and determine the detachment oftumor cells; an intermediate layer in which there are clusters of quiescenttumor cells; and the inner zone , which contains necrotic cells.




– Multiscale representation –




– From Molecular to Macroscopic –




• Generally, all scales are needed to represent real biological systems .Then, the idea of transferring straightforwardly the whole variety ofphenomena is definitely naive in absence of a constructive effort to reducecomplexity.

• Applied mathematicians often attempt to deal with the above complexityproblem by isolating a few phenomena to deal with them by suitabledifferential equations . On the other hand, interactions among varioussubsystems play a relevant role on the overall evolution, then the generalapproach should follow different lines.

• The theory of modules, proposed by Hartwell, can be profitably usedto deal with the above mentioned complexity problem . Indeed aconstructive interpretation of such a theory suggest to consider the collectivebehavior of systems of cells which have the ability of expressing certainbiological functions.




Definition 1. Biological systems can be regarded as an assembly ofsub-systems, each acting as a module with the ability of expressing a welldefined biological function. A sub-system (regarded as a module) is generallydefined at one scale only. It follows that a biological system is a network(system of systems) of interacting sub-systems each defined at differentscales.

Definition 2. A subsystem is an entity which has to be defined with referenceto the specific analysis under consideration.

Question 1. Which is the mathematical representation to be selected at eachscale towards modelling of a sub-system ?

Question 2. How different scales may interact in a network of severalinteracting sub-systems ?




X A mathematics for living systems.

Framework of the generalized kinetic theory

Modelling the immune competition.

Mathematical problems and qualitative analysis

Simulations, biological interpretation and parameter identification

Toward a multiscale representation



Section Two

Mathematical framework



• Equations of the mathematical kinetic theory for active particles , can beused to model large systems of interacting cells;

• Stochastic game theory , can be used to model gene expression andrelated biological functions which characterize models at the cellular scale;

• Equations at the tissue level are obtained by asymptotic limits of kineticmodels active particles, letting the intercellular distance tend to zero.

• Systems at each scale can be constituted by different interactingmodules . For instance, multicellular systems are constituted by differentpopulations each identified by the expression of a different biological function.

• A biological network is constituted by several interconnectedmodules which may be modelled at different scales.



Modelling at the Cellular Scale

On the Concept of Active Particle

• The system is constituted by a large number of interacting entities, calledactive particles organized into n interacting populations labelled by theindexes i = 1, . . . , n.

• The variable charged to describe the state of each particles is calledmicroscopic state, which is denoted by the variable w = {x , v , u},where x ∈ Dx is position, v ∈ Dv is mechanical state, e.g. linearvelocity, and u ∈ Du is the biological function or activity.

• The description of the overall state of the system is defined by thegeneralized one-particle distribution function

fi = fi(t ,w) = fi(t , x, v, u) , i = 1, . . . , p,

that is such that fi(t ,w) dw denotes the number of active particles whosestate, at time t , is in the interval [w,w + dw].




Macroscopic quantities

• The local density of the i th population is:

ni(t , x) =

∫Dv×Du

fi(t , x, v, u) dv du .

• The particle flow of the i th population is:

qi(t , x) =

∫Dv×Du

v fi(t , x, v, u) dv du ,

• The mean particle velocity of the i th population is:

Ei [v](t , x) =Aj [fi ](t , x)

ni(t , x),




Moments and Macroscopic Quantities

• The activation at the time t in the position x:

Aij = Aj [fi ](t , x) =

∫Dv×Du

uj fi(t , x, v, u) dv du ,

• The activation density:

Aij = Aj [fi ](t , x) =Aj [fi ](t , x)

ni(t , x)=

1ni(t , x)

∫Dv×Du

uj fi(t , x, v, u) dv du .




Classification of Microscopic Interactions

Short range binary interactions, related to the mutual actionsbetween the test or candidate and the field cell, when the test cellenters into the action domain of the field one. Their frequency is givenby the encounter rate ηij(v ,w)

Long range interactions, which refer to the action over the testcell applied by all field cells which are in the action domain of the test.




Conservative interactions modify the state of the interacting cells, butnot the size of the populations and are characterized by the transitiondensity function Bij(v ,w ; u)

Proliferating/destructive interactions imply the death or birth of cellsand are modelled by ψij(v ,w ; u) = µij(v ,w)δ(u − v)




The General Case

∂

∂tfi(t , x, v, u) + v · ∇x fi(t , x, v, u) +

n∑j=1

Fmij [f](t , x, u) [∇v fj(t , x, v, u)]

=n∑

j=1

∫Dv×Du×Dv×Du

cij |v∗ − v∗|Mij(v∗, v∗; v|u∗, u∗)

×Bij(u∗, u∗; u)fi(t , x, v∗, u∗)fj(t , x, v

∗, u∗) dv∗ dv∗ du∗ du∗

−fi(t , x, v, u)n∑

j=1

∫Dv×Du

cij |v − v∗|fj(t , x, v∗, u∗) dv∗ du∗

+fi(t , x, v, u)n∑

j=1

∫D

cij |v∗ − v∗|fj(t , x∗, v∗, u∗) dx∗ dv∗ du∗ .

The marginal densities :

f mi (t , x, v) =

∫fi(t , x, v, u) du , f b

i (t , u) =

∫fi(t , x, v, u) dx dv




A case, which is interesting for the applications refers to the spatialhomogeneity

fi(t , v, u) = f bi (t , u)Pi(v) ,

∫Dv

Pi(v)dv = 1 .

∂

∂tf bi (t , u) =

n∑j=1

η0ij

∫Du×Du

Bij(u∗, u∗; u)f b

i (t , u∗)f bj (t , u∗) du∗ du∗

−f bi (t , u)

n∑j=1

∫Du

η0ij

[1− µij(u, u

∗)]f bj (t , u∗) dv∗ du∗ ,

where

η0ij =

∫Dv×Dv

cij |z− v∗|Pi(z)Pj(v∗) dz dv∗ .




A First Remarkable Generalization (genetic mutations)

∂

∂tf bi (t , u) = Ci [f ] + Pi [f ] + Ti [f ]

∂

∂tf bi (t , u) =

n∑h=1

n∑k=1

η0hk

∫Du×Du

Bihk (u∗, u

∗; u)f bh (t , u∗)f b

k (t , u∗) du∗ du∗

−f bi (t , u)

n∑j=1

η0ij

∫Du

f bj (t , u∗) du∗

+f bi (t , u)

n∑j=1

η0ij

∫Du

µdij (u, u

∗) f bj (t , u∗) du∗

+n∑

h=1

n∑k=1

η0hk

∫Du×Du

µphk (i)(u∗, u

∗; u)f bh (t , u∗)f b

k (t , u∗) du∗ du∗




Interactions with Population Change µijk (v ,w ; u)

Genetic Mutation and Proliferation




A Second Remarkable Generalization (perturbation of space homogeneity)(∂

∂t+ v · ∇x

)fi(t , x, v, u) =

n∑h=1

n∑k=1

η0hk

∫D2

u

Bihk (u∗, u

∗; u)fh(t , x, v, u∗)fk (t , x, v, u∗) du∗ du∗

+n∑

h=1

n∑k=1

η0hk

∫D2

u

µphk (i)(u∗, u

∗; u)fh(t , x, v, u∗)fk (t , x, v, u∗) du∗ du∗

−fi(t , x, v, u)n∑

j=1

η0ij

∫Du

[1− µd

ij (u, u∗)

]fj(t , x, v, u

∗) du∗

+νn∑

j=1

∫Dv

[T (v, v∗)fi(t , x, v

∗, u)− T (v∗, v)fj(t , x, v, u)

]dv∗





X Framework of the generalized kinetic theory







Section Three

Modelling the immune competition



The Framework

The system is constituted by active particles .

The system is constituted by 2 interacting cell populations,environmental cells and immune cells. Each cell is characterized by acertain microscopic state .

Environmental cells (i = 1): natural state for negative values of u,abnormal state for positive values of uImmune cells (i = 2): inhibited state for negative values of u,activated state for positive values of u

The system is described by the generalized distribution functiondepending on the position, velocity and on the biological variable.

Here, in view of the the application proposed, we limit the presentationof the general kinetic framework in the spatially homogeneous case .



The generalized distribution function in the spatially homogeneouscase:

fi = fi(t , u) with i = 1, 2

the number of cells per unit volume whose state is, at time t , in theinterval [u, u + du].

The sizes of the populations, are recovered as momenta of thedistribution function.

nE1 (t) =

∫ 0

−∞f1(t , u) du , nT

1 (t) =

∫ ∞

0f1(t , u) du ,

nI2(t) =

∫ 0

−∞f2(t , u) du , nA

2 (t) =

∫ ∞

0f2(t , u) du .

The activations of the populations are recovered as first ordermomenta.



Evolution equations

The evolution equations are derived by a suitable balance between the rateof variation of the distribution function in the elementary volume of the statespace to the inlet and outlet flux of cells which enter or leave this volume dueto microscopic interactions.

∂fi∂t

(t , u) = Ji [f ](t , u) =2∑

j=1

(G∗ij − L∗ij + Gij − Lij

)[f ](t , u)

∂

∂tfi(t , u) =

2∑j=1

η0ij

∫Bij(v ,w ; u)fi(t , v)fj(t ,w) dv dw

+fi(t , u)2∑

j=1

η0ij

∫[µij(u,w)− 1]fj(t ,w) dw



Conservative interactions

Conservative interactions are described by a Gaussian distributionfunction

Bij(v ,w ; u) =1√2πsij

exp

{− (u −mij(v ,w))2

2sij

}

Interactions between cells of the first population.

v ,w ∈ R : m11 = v + α11

Tendency to degenerate

α11 characterizes the tendency of endothelial cells to degenerate.



Cells of the first population / cells of the second population.

v < 0 w ∈ R v ≥ 0 w < 0 : B12 = δ(u − v)

v ,w ≥ 0 : m12 = v − α12

Reduction of the degeneration

α12 characterizes the ability of the immune system to reduce the state ofabnormal cell.



Cells of the second population / cells of the first population.

v ∈ R w < 0 v < 0 w ≥ 0 : B21 = δ(u − v)

v ≥ 0 w ≥ 0 : m21 = v − α21

Immune inhibition

α21 characterizes the ability of abnormal cells to inhibit immune cells.

Interactions between cells of the second population

v ,w ∈ R : B22 = δ(u − v)



Proliferating/destructive interactions

Interactions between cells of the first population

µ11(u,w) = β11U[0,∞)(u)U(−∞,0)(w)

β11 characterizes the proliferating ability of abnormal cells

Cells of the first population / cells of the second population

µ12(u,w) = −β12U[0,∞)(u)U[0,∞)(w)

β12 characterizes the destructive ability of active immune cells

Cells of the second population / cells of the first population

µ21(u,w) = β21U[0,∞)(u)U[0,∞)(w)

β21 characterizes the proliferating ability of abnormal cells

Interactions between cells of the second population have a trivialoutput: µ22 = 0



The Model

The evolution equations , assuming the encounter rate equal to one, andwith variance going to zero

sij → 0 =⇒ Bij(v ,w ; u) = δ(u −mij(v ,w))

are:

∂f1∂t

(t , u) = n1(t)[f1(t , u − α11)− f1(t , u)]

+nA2 (t)f1(t , u + α12)U[0,∞)(u + α12)

+f1(t , u)[β11nE

1 (t)− (1 + β12)nA2 (t)

]U[0,∞)(u)

∂f2∂t

(t , u) = nT1 (t)

[f2(t , u + α21)U[0,∞)(u + α21)

+(β21 − 1)f2(t , u)U[0,∞)(u)]



The Model - Density equations

By “formal” integration of the evolution equation respectively on R+ andon R−:

∂nT1 (t)∂t

(t , u) = n1(t)∫ 0−α11

f1(t , u)du − nA2 (t)

∫ α120 f1(t , u)du

+nT1 (t)

[β11nE

1 (t)− β12nA2 (t)

]∂nE

1 (t)∂t

(t , u) = −n1(t)∫ 0−α11

f1(t , u)du + nA2 (t)

∫ α120 f1(t , u)du

∂nA2 (t)∂t

(t , u) = nT1 (t)

[β21nA

2 (t)−∫ α21

0 f2(t , u)du]

∂nI2(t)∂t

(t , u) = nT1 (t)

∫ α210 f2(t , u)du

We stress that these equations are not in a closed form , since f1(t , u)and f2(t , u) are unknown.



The Model - The parameters

Conservative parameters

α11 – tendency of endothelial cells to degenerateα12 – ability of the active immune cells to reduce the state ofabnormal cellsα21 – ability of abnormal cells to inhibit the active immune cells

Non-conservative parameters

β11 – proliferation rate of abnormal cells due to encounters withnormal endothelial cellsβ12 – ability of immune cells to destroy abnormal cellsβ21 – proliferation rate of immune cells due to encounters withabnormal cells



Modelling the immune competition. Examples

The general model can be properly particularized to describe differentphenomena related to the immune competition.

Model C : Conservative model. No proliferation or destruction occurwhile interactions modify only the biological function (all theβ-parameters are equal to zero).

Model P : Proliferating/destructive model. The distribution over thebiological variable is almost constant while proliferating/destructiveevents are predominant.

Three particular models: Model I, II, III . Two of the α-parameters areequal to zero and the third one is different from zero.



Model I: (α11 = 0, α12 = 0, α21 > 0)Cells do not degenerate (α11 = 0), i.e. negligible tendency todegenerate of normal cells. Abnormal cells are not contrasted byimmune cells (α12 = 0) and inhibit immune cells (α21 > 0).

Model II: (α11 > 0, α12 = 0, α21 = 0)Cells show a natural trend to degenerate (α11 > 0) not contrasted byimmune cells (α12 = 0). Abnormal cells cannot inhibit immune cells(α21 = 0), i.e. negligible ability of abnormal cells to inhibit active immunecells.

Model III: (α11 = 0, α12 > 0, α21 = 0)Cells do not degenerate (α11 = 0). Abnormal cells are contrasted byimmune cells (α12 > 0), while abnormal cells do not inhibit immune cells(α21 = 0).






X Modelling microscopic interactions.

X The evolution equations

X The model and some particularizations.






Section four

Mathematical problems



Well posedness of the initial value problem

∂f∂t

= N(f ),

f (t = 0, u) = f0(u),

• L1(R) is the Lebesgue space of measurable, real-valued functionsintegrable on R. The norm is denoted by ‖ · ‖1

• X = L1(R)× L1(R) = {f = (f1, f2) : f1 ∈ L1(R), f2 ∈ L1(R)} is theBanach space with the norm ‖ f ‖=‖ f1 ‖1 + ‖ f2 ‖1

• X+ = {f = (f1, f2) ∈ X : f1 ≥ 0, f2 ≥ 0} is the positive cone of X• Y = C([0,T ],X ) and Y+ = C([0,T ],X+) is the space of the functionscontinuous on [0,T ] with values, respectively, in a Banach space X andX+, with the norm: ‖ f ‖Y= supt∈[0,T ] ‖ f ‖



Local existence and uniqueness

For the general model, the following theorem holds true:

Theorem:

Let f0 ∈ X+.∃T > 0, a0 such that the initial value problem has a unique solutionf ∈ C([0,T ],X+).The solution f is positive , i.e. satisfies

f (t) ∈ X+, t ∈ [0,T ]

and‖f‖ ≤ a0‖f0‖, ∀t ∈ [0,T ]



Sketch of the Proof:f = M(f ) = f0(u) +

∫ t0 N(f )(s)ds = f0(u) + Ψ(g)(t)

M is a continuous map from Y into Y

‖ M(f ) ‖Y≤‖ f0 ‖ +C1T ‖ f ‖2Y

and it is contracting in Y

‖ M(f )−M(g) ‖Y≤ C1T (‖ f ‖Y + ‖ g ‖Y) ‖ f − g ‖Y⇒ Thus, by application of classical fixed point theorem , there exists aunique local solution f (t) of Theorem 1 on [0,T ].

The positivity of solutions , is shown applying again the fixed point Theoremin Y+.



Existence for large times

For the specific models, the following theorem holds true:

Theorem:

Let α11 = α12 = 0, or α12 = 0, α11 6= 0.∀T > 0 there exists a unique solution f ∈ C([0,T ],X ) of the model with theinitial datum f0 ∈ X+.The solution satisfies is positive

f (t) ∈ X+, ∀t ∈ [0,T ],

and boundedsup

t∈[0,T ]

f (t) ≤ CT

for some constant CT depending on T and on the initial data.=⇒ Initial value problems corresponding to models P, I, II have globalsolutions .



Sketch of the Proof:By equations of the densities,

nE1 = nE

1 (0)

∂n1

∂t(t , u) ≤ β11nE

1 (0)n1, ⇒ n1 ≤ n1(0)exp(β11nE1 (0)t)

⇒ the total number of abnormal cells is bounded on each finite interval [0,T ].

n2(t) ≤ n2(0)exp(β21n1(0)

β11nE1 (0)

(exp(β11nE1 (0)t)− 1)

)⇒ n2(t) is bounded on each finite time interval [0,T ].

CT = n1(0)exp(β11nE1 (0)T ) + n2(0)exp

(β21n1(0)

β11nE1 (0)

(exp(β11nE1 (0)T )− 1)

).



Asymptotic behavior

Theorems of the asymptotic behavior of each model are developed in thethesis.Setting to zero a subset of parameters, means selecting some biologicalphenomena with respect to others.In fact, different results are obtained, depending on the parameters set to beequal to zero, and the scenarios result complex and articulated depending onthe parameters.Theorems provide some information on the evolution of the densities (andthus of the population of cells in general) while simulations complete theseanalysis and provide the evolution of the distribution functions and thus of theevolution of the cells with their inner structure.



Model P: Asymptotic behavior

Model P is obtained setting equal to zero all the α-type parameters. Thedistribution over the biological variable is almost constant whileproliferating/destructive events are predominant.

Let define the critical immune density as

nA2C

= β∗, with β∗ =β11

β12nE

1 (0)

product between the initial number of environmental cells and the ratio of theproliferation rate of abnormal cells and the ability of immune cells to destroyabnormal ones.

The Theorem on the asymptotic behavior states that nA2 (t) ↑,

nE1 (t) = constant and, setting δ = β11nE

1 (0)− β12nA2 (0),

If nA2 (0) < nA

2C= β∗(δ ≥ 0) :

{∃t0 : nT

1 ↑ ,∀ t ∈ [0, t0] ; nT1 ↓ ,∀ t ∈ [t0,T ] ,

∀T > 0 .

If nA2 (0) ≥ nA

2C= β∗(δ ≤ 0) :

{nT

1 (t) ↓ ; ∃δ < 0 : nT1 (t) ≤ nT

1 (0)exp(δt) ,



Theorem:Consider the initial value problem for the Model P and define

δ = β11nE1 (0)− β12nA

2 (0).

If β21 = 0, nA2 = Cst , nE

1 = Cst and nT1 (t) = nT

1 (0)exp(δt),

if δ ≥ 0, then nT1 ↑

if δ < 0, then nT1 ↓

If β21 6= 0:

If β12 = 0, then nT1 ↑, nE

1 = Cst and n2 ↑If β12 6= 0, then nE

1 = Cst , nA2 ↑

If δ ≤ 0, then nT1 ↓ and nT

1 (t) ≤ nT1 (0)exp(δt)

If δ > 0 and nT1 (0) 6= 0, ∃t0: nT

1 ↑ in [0, t0] and nT1 ↓ in [t0, T ] ∀T > 0.



Sketch of the proof.From the densities equations, if β21 6= 0 and δ ≤ 0, thennT

1 (t) ≤ nT1 (0)exp(δt).

If δ > 0 (which implies β11 6= 0), T > 0 and nT1 (0) 6= 0, so nT

1 6= 0 and

∂nT1

∂t= 0 ⇔ nA

2 (t) =β11nE

1 (0)

β12.

Let t ∈ [0,T ], as δ > 0 and nA2 ↑

nA2 (0) <

β11nE1 (0)

β12= nA

2 (t) ≤ nA2 (T ) .

nA2 is continuous, increasing in [0,T ], then we get the existence of a unique

t0 ∈ [0,T ] such that

nA2 (t0) =

β11nE1 (0)

β12, t0 = (nA

2 )−1(β11nE

1 (0)

β12

).

And we get that nT1 ↑ in [0, t0] and nT

1 ↓ in [t0,T ].



Define

λ = (1 + β11)− β12nA

2 (0)

nE1 (0)

Theorem 2. Initial value problem for Model I.

t ↑⇒ nA2 (t) =

∫ ∞

0f2(t , u) du ↑ , nE

1 (t) =

∫ 0

−∞f1(t , u) du ↓

Moreover

• If β12 = 0 then nT1 (t) increases in time immune cells do not have destructive

ability.

• If β12 > 00, then nT1 (t) still grows in time, while estimate bounds can be

identified:

nT1 ≤ exp(λt)

(nT

1 (0) +(nE

1 (0))2

λ

)− (nE

1 (0))2

λ

The immune system is able to apply a control to the growth depending on thesign of the parameter δ



Theorem 3. Initial value problem for Model II

• If β12 = 0, (negligible destructive ability of immune cells ), then nT1 (t)

increases in time, nE1 = Cst and n2(t) satisfies the following

nA2 (t) ≥ nA

2 (0) exp[(β21 − 1)nT

1 (0)t]

Moreover, if β21 = 0, (negligible proliferation ability of immune cells)then nA2 (t)

decreases in time.

• If β12 > 0, then three cases are possible:

i) δ < 0:

? If β21 = 0, then nA2 decreases and if nT

1 (0) 6= 0 then, ∃t0 such that nT1

decreases in [0, t0] and increases in [t0,T ], ∀T > 0.



? If β21 > 0: If β11 = 0, then nT1 decreases and nA

2 (t) satisfies:nA

2 (t) ≤ nA2 (0)exp(β21nT

1 (0)t).∀T > 0, ∃β21 ∈ (0, 1) and r > 0 such that nA

2 (t) increases in [0,T ], if

supt∈[0,T ]

∫ α21

0f2(t , u)du ≤ r .

? If β11 > 0, then ∀T > 0, ∃n(0)1 , β(0)

11 , β(0)12 , such that if n1(0) ≤ n(0)

1 , β11 ≤ β(0)11

and β12 ≤ β(0)12 , then ∃β21 ∈ (0, 1) such that nT

1 decreases in [0,T ] andnA

2 (t) ≥ γ?

β21.

Moreover, if ∃γ(γ ≤ γ?) such that if supt∈[0,T ]

∫ α210 f2(t , u)du ≤ γ, then nA

2

increases in [0,T ]: nA2 (t) ≥ nA

2 (0).



ii) δ > 0

? If β21 = 0, then nA2 decreases and nT

1 increases.

? If β21 > 0, then ∀T > 0, ∃β(0)11 such that if β11 ≤ β

(0)11 , ∃β21 ∈ (0, 1) such that

nT1 increases in [0,T ] and nA

2 (t) ≤ γ?

β21.

Moreover, if ∃γ(γ ≥ γ?) such that supt∈[0,T ]

∫ α210 f2(t , u)du ≥ γ, then nA

2

decreases and nA2 (t) ≤ nA

2 (0).

iii) δ = 0

? β21 = 0, then nA2 decreases and nT

1 increases.

β21 > 0, then ∀T > 0, ∃n(0)1 , β

(0)11 and r > 0 such that if n1(0) ≤ n(0)

1 ,β11 ≤ β

(0)11 and sup

t∈[0,T ]

∫ α210 f2(t , u)du ≤ r , then nA

2 increases and nT1 decreases.





X Modelling the immune competition.

X Mathematical problems and qualitative analysis





Section five

Simulations



Simulations of Model P

nA2 (t) ↑ and if nA

2 (0) < nA2C

= β∗(δ ≥ 0) :

{∃t0 : nT

1 ↑ ,∀ t ∈ [0, t0] ; nT1 ↓ ,∀ t ∈ [t0,T ] ,

∀T > 0 .

nA2 (t) ↑ and if nA

2 (0) ≥ nA2C

= β∗(δ ≤ 0) :{

nT1 (t) ↓ ; ∃δ < 0 : nT

1 (t) ≤ nT1 (0)exp(δt) ,



Phase portrait for fixed values of β∗ and for fixed β21 > 0.The immune density increases, and two areas can be distinguished,depending on the initial values of immune density with respect to the criticalone, which differentiate the evolution.



Model P - Hints for parameters identification

Identify, on the basis of experimental data, the parameters directly related tothe growth ability of abnormal cells and to the defense ability of immune cells.Consistency with the assumptions of the proposed Model P,α11 = α12 = α21 = 0 and the experimental conditions.

Comparison between theoretical results and experimental data of growth ofa tumor in non treated mice (dots) and in irradiated ones (triangles). Prof. G.

Forni, University of Turin, Orbassano.



Simulations of Model I. α21 > 0

If δ > 0 :

∀T ≥ 0 : nT

1 (t) ↑ in [0,T ]

nA2 (t) ≤ γ∗

β21

10 20 30 40 50t

0.05

0.1

0.15

0.2

0.25

Density



Simulations of Model I. α21 > 0

If δ < 0 :

∀T > 0 : nT

1 (t) ↓ in [0,T ]

nA2 (t) ≥ γ∗

β21

10 20 30 40 50t

0.02

0.04

0.06

0.08

0.1

Density



Simulations of Model III. α12 > 0

θ = β11

(nE

1 (0)− 1β21

nA2 (0)

)≥ 0 .

10 20 30 40 50 60 70t

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Density



Simulations

0

0.25

0.5

0.75

1

u

0

0.2

0.4

0.6

f1

0

0.25

0.5

0.75u

0

0.25

0.5

0.75

1

u

0

0.25

0.5

0.75

1

f2

0

0.25

0.5

0.75u

Heterogeneity and progressive destruction of tumor cells Immune cellsremain active



Simulations

0

1

2

3

u

0

0.5

1

1.5

f1

0

1

2u

0

0.25

0.5

0.75

1

u

0

0.2

0.4

0.6

0.8

f2

0

0.25

0.5

0.75u

Heterogeneity and progression of tumor cells and progressive inhibition ofimmune cells





X Modelling the immune competition.

X Mathematical problems and qualitative analysis

X Simulations, biological interpretation and parameter identification




Section five

Perspectives Toward a Multiscale Representation



The Multiscale Representation

Macro from Micro

(∂

∂t+ v · ∇x

)f (t , x, v, u) =

(G[f ] + I[f ] + L[f ]

)f (t , x, v, u)

G[f ] = η0∫

D2u

B(u∗, u∗; u)f (t , x, v, u∗)f (t , x, v, u∗) du∗ du∗

−f (t , x, v, u)η0∫

Du

[1− µ(u, u∗)

]f (t , x, v, u∗) du∗

I[f ] = η0f (t , x, v, u)

∫Du

µ(u, u∗)f (t , x, v, u∗) du∗

L[f ] = ν

∫Dv

[T (v, v∗)f (t , x, v∗, u)− T (v∗, v)f (t , x, v, u)

]dv∗ ,




Stating some assumptions, there exists, in the continuous case, a boundedvelocity distribution M(v) > 0, independent of x and t , such that the detailedbalance:

T (v∗, v)M(v) = T (v, v∗)M(v∗)

holds. The flow produced by this equilibrium distribution vanishes, and M isnormalized: ∫

Dv

vM(v) dv = 0,∫

Dv

M(v) dv = 1 .

The kernel T (v, v∗) is bounded, and there exists a constant σ > 0 such that

T (v, v∗) ≥ σM, ∀(v, v∗) ∈ Dv × Dv , x ∈ IR 3, t > 0 .




ν =1ε, η = εq−1 , µ = εδ, q ≥ 1, δ ≥ 0 ,

yields (∂t + v · ∇x

)f (t , x , v , u)

=1ε

(L[f ](t , x , v , u) + εqG[f ](t , x , v , u) + εq+δI[f ](t , x , v , u)

),

The parameter ε is a time scale which here refers to the turning frequency.Let f be a solution to the kinetic equation and consider the following moments

ρ(t , x , u) =

∫V

f (t , x , v , u) dv , ρ(t , x , u)U(t , x , u) =

∫V

vf (t , x , v , u) dv .

Moreover, consider the following perturbation of the distribution

f (t , x , v , u) = Mρ(t,x,u),U(t,x,u)(v) + ε g(t , x , v , u) .




Case 1. δ ≥ 0, and q > 1: First order moments with respect to ε generate anthe hyperbolic system without source term:

∂ρ∂t +∇x(ρU) = 0 ,

∂(ρU)∂t +∇x(ρU ⊗ U + p) = 0 .

Case 2. δ 6= 0, and q = 1: In this case, in the first order with respect to ε, thefollowing hyperbolic system source term related to conservative interactionsis obtained:

∂ρ∂t +∇x(ρU) =

∫V G(Mρ,U ,Mρ,U)(v) dv ,

∂(ρU)∂t +∇x(ρU ⊗ U + p) =

∫V vG(Mρ,U ,Mρ,U)(v) dv .




Case 3. δ = 0, and q = 1. In this case, in the first order with respect to ε, thefollowing hyperbolic system source term related to both conservative andproliferating interactions is obtained:

∂ρ∂t +∇x(ρU) =

∫V G(Mρ,U ,Mρ,U)(t , x , v , u) dv

+∫

V I(Mρ,U ,Mρ,U)(t , x , v , u) dv ,

∂(ρU)∂t +∇x(ρU ⊗ U + p) =

∫V vG(Mρ,U ,Mρ,U)(v) dv

+∫

V vI(Mρ,U ,Mρ,U)(v) dv .

We observe that the influence of the turning operator L on the macroscopicequation only comes into play through the equilibrium state Mρ,U in thecomputation of the right-hand side and the pressure tensor P.




η = εq , γ µ = εr , ν =1εp,

where ε is a small parameter to be let tend to zero. In addition, the diffusionscale time τ = εt is used:

ε∂t fε(t , x, v) + v · ∇x fε(t , x, v) =1εpLfε + εq G(fε, fε) + εr I(fε, fε) ,

∂tρ(t , x) +∇x · 〈k(v)⊗ v · ∇xρ(t , x)〉 = 〈M2〉vn2, p = r = 1 ,

∂tρ(t , x) +∇x · 〈k(v)⊗ v · ∇xρ(t , x)〉 = 0, p = 1, r > 1 ,

∂tρ(t , x) = 〈M2〉vn2, p > 1, r = 1 ,

and∂tρ(t , x) = 0, p > 1, r > 1 .



Theoretical perspectives

Perspectives

Possible developments can be exploited both from mathematical andmedical point of view.

Developing suitable experiments and theoretical methods to obtain aneffective description of microscopic interactions . In fact, theassessment of microscopic interaction functions is based only on aphenomenological interpretation of physical reality. Therefore, theevolution equations have to be regarded as mathematical models ratherthan equations of a bio-mathematical theory.



Theoretical perspectives

• The mathematical structure we have seen above is a candidate to derivespecific models, still one has to stated that a biological-mathematical theoryhas been created.

• Possibly, it refers to a mathematical theory considering that a new class ofequations has been generated. Indeed, a rigorous framework is delivered forthe derivation of models, when a mathematical description of cell interactionscan be derived, by phenomenological interpretation, from empirical data.

• On the other hand, only when the above interactions are delivered by atheoretical interpretation delivered within the framework of biologicalsciences, then we may talk about a biological-mathematical theory.



To do

What has been done ...

The above described topics have been developed by several works of variousauthors - an updated bibliography will be distributed -, and through theparticipation to national and international projects. Among them,

PRIN: Modelli della teoria cinetica matematica per particelle attive nellostudio di sistemi complessi, 2005-2007

Research Training Network Project : ”Modelling mathematical methodsand computer simulations of tumor growth and therapy”, 2004-2007

CNR Strategic Project : Metodi e modelli matematici nello studio deifenomeni biologici, 2003

Research Training Network Project : ”Using Mathematical Modellingand Computer Simulation to Improve Cancer Therapy, 2000-20037.

Politecnico of Turin Project for Young Researchers: ”Developments ofgeneralized kinetic (Boltzmann) models in applied sciences”, 2002 -2003 [as Coordinator]



To do

...and what, hopefully, will be done

The author presented himself as a candidate, with the support of thePolitecnico of Turin, for a grant of the European Research Council (ERCStarting Grant). The research proposal, already submitted and waiting forevaluation, has the title

Mathematical Methods and Tools for the Modelling and Simulation of theOnset of Cancer, Immune Competition, and Therapies

or, briefly,

Mametomoc



To do

The research proposal - if approved - will spread over a five–years period. Itsmain goals are:

Development of computational toolboxes, founded on a strongmathematical framework, by which the complex phenomena oftumorigenesis and cancer evolution will be examined and modelled.

Training of two Pre-Docs and two Post-Docs in the Applied Mathematicsarea, which will interact with researchers in the filed of biology andcomputer sciences.

The Expected results of the project are:

Mathematical models to investigate the tumorigenesis, simulatingboth the immune response and the action of specific therapies.New mathematical methods and tools to develop qualitativeanalysis of the mathematical problems generated by thesefront-breaking models.Design and development of dedicated software tools, to be used bybiologists and clinicians during their experimental research.Training of experienced researchers to the objectives of the projectand of two early stage researchers up to the PhD title.Scientific Publications.



To do

Far beyond the expected results, it is to point out that the project has theambitious aim of putting into a mathematical framework several aspects ofbiological research in tumorigenesis - from molecular scale until aggregatematter - thus realizing that full ”multiscale approach” which, even if largelyinvoked, has been not still fully developed.

The team will collaborate with qualified young researchers, build up atEuropean level: as an example, it is foreseen as follows:

Name Country Afference Research FieldA. Bellouquid Maroc University of Safi Applied Mathematics

C. Dogbe France University of Paris VII Applied MathematicsK.A. Rejniak UK University of Dundee Computer sciences



Section six

References



References

N. Bellomo, A. Bellouquid, M. Delitala, Mathematical topics onthe modelling complex multicellular systems and tumor immunecells competition, Math. Mod. Meth. Appl. Sci., 14, 1683–1733,(2004).

A. Bellouquid and M. Delitala , (2006), Mathematical Modellingof Complex Biological Systems. A Kinetic Theory Approach ,(Birkhauser-Springer, Boston).

N. Bellomo, A. Bellouquid, J. Nieto, and J. Soler, MulticellularBiological Growing Systems - Hyperbolic Limits towardsMacroscopic Description, (2007), to appear.

M. Delitala and G. Forni, Modelling genetic mutation, onset ofprogressing cells and immune competition, (2007), to appear.


Documents

Mathematical Modelling of Complex Biological Systems. A ... · “quasi fractal surface” which interacts with the outer environment, for example normal host cells and the immune