“Although living systems obey the laws of physics and chemistry,the notion of function or purpose

differentiate biology from other natural sciences”Hartwell, Hopfield, Leibner, Murray, Nature 402, c47-c52 (1999)

Luigi Preziosi

calvino.polito.it/~preziosi

Mathematical Models inMedicine and Biology

Identificationof therapeutic

protocols

Clinicalpractice

Clinical trials

In vitro models

In vivo models

Simulations(in silicomodels)

Mathematical models

Modelling cycle in medicine

Deduction of the mathematical model

Simulation ofunknown cases

and newphenomena

Independent variables

Dependentvariables

Parameters

Qualitative analysis

Simulation of known cases

Experimental validation

No

Yes

Modelling cycle for medicine

Phenomenological observation

N(t) o r(t)Number of cells on a Petri dish

r(x, t) Spatial distribution of cells

N(t, a)Distribution of cells over the cell cycle

N(t), M(t)Different cells on a Petri dish

Choice of variables

Population in a country

r(x, t) Spatial localization of the population

N(t, a)Age-structured population

M(t), F(t), B(t), A(t)Sexual population

M(t, a), F(t, a)Age-structured sexual population

N(t) o r(t)

Choice of variables

Population in a country

Natural birth and death rates

Diffusion of diseases

Immigration

Other equation for thediffusion of the disease ??

N(t) o r(t)

Dependence ??

Moving

r(x, t) Spatial localization of the population

Introducing sex or age ??

Dependence ??

Mating

Choice of effects and parameters

“All models are a simplification, an idealization, and consequently a falsification, of reality….’’

Alan Turing

“…It is to be hoped that the features retained for discussion are those of the greatest importance in the present state of knowledge.”

“All models are wrong…some are useful”Box & Draper

“All decisions are based on models…and all models are wrong”

Sterman

“Although knowledge is incomplete, nonetheless decisions have to be made. Modeling…takes place in the effort to plan clinical trials or understand their results. Formal modeling should improve that effort, but cautious consideration of the assumptions is demanded”

Day, Shackness & Peters

volume ratio

Volu

me

of

the

sam

ple

Volu

me

of

the

con

stit

uen

t

Macroscopic and individual based models

j

u

Diffusion equation

density of cells changes

because of birth/death

because of influx/outflux

j

Diffusion equation

Fick’s law

If D is constant

# cells in the node at time t = # cells in the node at preceding time

# cells coming from neighbours+

# cells going to neighbours_

Diffusion equation

DyDx(i,j+1)

(i,j-1)

(i-1,j) (i+1,j)(i,j)

# cells in the node at time t = # cells in the node at preceding time

# cells coming from neighbours+

# cells going to neighbours_

Original at ocw.mit.edu/ans7870/1/1.061/f04/animation/walk2.avi

Diffusion equation

Random walk

Dx j - 1 j + 1j

j + 2j - 2

Constant coefficients

If

Dx → 0

Looking here

If

Dx → 0

Looking in the middle

If

Dx → 0

Looking in the middle

If

Dx → 0

Looking ahead

If

Looking ahead

Dx → 0

Anisotropic diffusion

=

# cells coming from neighbours

# cells in the node at time t # cells in the node at preceding time

+

# cells going to neighbours_

DyDx(i,j+1)

(i,j-1)

(i-1,j) (i+1,j)(i,j)

Original at ocw.mit.edu/ans7870/1/1.061/f04/animation/ANISO.AVI

Anisotropic diffusion

=

# cells coming from neighbours

# cells in the node at time t # cells in the node at preceding time

+

# cells going to neighbours_

DyDx(i,j+1)

(i,j-1)

(i-1,j) (i+1,j)(i,j)

Original at ocw.mit.edu/ans7870/1/1.061/f04/animation/ANISO.AVI

Anisotropic diffusion

=

# cells coming from neighbours

# cells in the node at time t # cells in the node at preceding time

+

# cells going to neighbours_

DyDx(i,j+1)

(i,j-1)

(i-1,j) (i+1,j)(i,j)

Original at ocw.mit.edu/ans7870/1/1.061/f04/animation/ANISO.AVI

Reaction-diffusion equation

=

# cells coming from neighbours

# cells in the node at time t # cells in the node at preceding time

+

# cells going to neighbours_

# generated (born) in the node+

# degradated (dead) in the node _

u changesin time

because diffuses

because of birth/death

# cells in the node at time t

# advected with velocity v _because it is

advected

=

# cells coming from neighbours

# cells in the node at preceding time

+

# cells going to neighbours_

# generated (born) in the node+

# degradated (dead) in the node _Original at

ocw.mit.edu/ans7870/1/1.061/f04/animation/ch6_cont.avi

Advection-reaction-diffusion equation

Breast tumour

Original at: www.maths.dundee.ac.uk/mbg/

Diffusion models

K. Swanson

K. Swanson

Lobectomy

Detectable border

Not visible on imaging

Sagittal, coronal and axial virtual MRIs from a continuous motion picture of the recurrence of a “favorable” glioblastoma (unusually small, 1 cm in diameter, and unusually far forward in frontal polar cortex) following extensive resection of most of the frontal lobe. Thick black contours represent the edge of the gadolinium-enhanced T1-weighted MRI (T1Gd) signal. Red: high glioma cell density; Blue: low density.

Loboctomy

Not visible on imaging

Detectable border

Sagittal, coronal and axial virtual MRIs from a continuous motion picture of the recurrence of a “favorable” glioblastoma (unusually small, 1 cm in diameter, and unusually far forward in frontal polar cortex) following extensive resection of most of the frontal lobe. Thick black contours represent the edge of the gadolinium-enhanced T1-weighted MRI (T1Gd) signal. Red: high glioma cell density; Blue: low density.

Growth of a glioblastoma

6-drug cocktail : 6-thioguanine, procarbazine, dibromodulcitol,

CCNN, 5-fluorouracil, hydroxyurea

Cycle of chemioterapy: Sagittal section (with legenda)

Horizontal section (without legenda)

Chemotherapy against glioblastoma

Convection-enhanced delivery

• ~ 10 6 cells• Maximum diameter ~ 2 mm• Necrotic core• Quiescent region• Periferic proliferation• “Nutrient” diffusion limit

Spheroid from V-79 Chinese hamster lung cells Folkman & Hochberg, Exp Med. 138:745-753 (73)

Multicellular spheroids

nutrientconcentration

r

proliferation threshold

proliferating rim

border valuenutrient diffuses in the tumour through its border and is consumed inside

Only tumour cells in 1D

proliferating rim

For R >> -1/2

Only tumour cells in 1D

nutrientconcentration

r

proliferation threshold

border valuenutrient diffuses in the tumour through its border and is consumed inside

proliferating rim

For R >> -1/2

Only tumour cells in 1D

nutrientconcentration

r

proliferation threshold

border valuenutrient diffuses in the tumour through its border and is consumed inside

proliferation threshold

necrosis threshold

necrotic corequiescent region proliferating rim

then added • growth promoting factors • growth inhibitory factors• ...

= (u,x)

nutrient diffuses in the tumour through its border and is consumed inside

r

border value

Only tumour cells in 1D

• 1-D Tumour (Cartesian)

• Cylindrical Tumour

R

• Tumour cord and Krogh cylinder

Exercises

= 0

Degenerate diffusion

Linear Diffusionwith ill-posed region

Degenerate diffusion equation

with aggregation

If