Gabriel WittumG-CSC

University of Frankfurt

Modelling and Simulation of

Biological Systems

Gabriel WittumG-CSC

University of Frankfurt

1

Modelling and Simulation

Gabriel WittumG-CSC

University of Frankfurt

Modelling and Simulation

Gabriel WittumG-CSC

University of Frankfurt

Aim

• Quantitative understanding of biosystems

• Relation of form and function

• Scale interaction

• Application in pharmacy and medicine

Gabriel WittumG-CSC

University of Frankfurt

Model Properties

1. Answer, requires question

Gabriel WittumG-CSC

University of Frankfurt

Model Properties

1. Answer, requires question

2. Complexity:

As simple as possible, as complex as necessary

in order to answer the question asked

Gabriel WittumG-CSC

University of Frankfurt

Model Properties

1. Answer, requires question

2. Complexity:

As simple as possible, as complex as necessary

3. Approximation of reality - contains errors

Gabriel WittumG-CSC

University of Frankfurt

Model Properties

1. Answer, requires question

2. Complexity:

As simple as possible, as complex as necessary

3. Approximation of reality - contains errors

4. Reliability - prognostic quality

Gabriel WittumG-CSC

University of Frankfurt

Reliability

• Model based on first principles

Conservation laws: Balance of

mass, energy, momentum, angular momentum

Maxwell‘s equations

Gabriel WittumG-CSC

University of Frankfurt

Prognostic Quality

• Given an initial state and boundary conditions

Task: Describe the state of the system at a later time

Gabriel WittumG-CSC

University of Frankfurt

History

• “Modellum”

wooden models used by artists and architects in Renaissance to demonstrate their ideas to sponsors

idea: show the essence of an object by abstraction

Gabriel WittumG-CSC

University of Frankfurt

History

Mathematical abstraction goes back to Pythagoras (500 b.c.)

Gabriel WittumG-CSC

University of Frankfurt

History

Description of harmony in musics

Gabriel WittumG-CSC

University of Frankfurt

History

• sounds nice, if the length of the strings generating two sounds are harmonic, i.e. integral multiples of each other

q1L1 = q2L2

• the smaller q1 and q2 are, the nicer the interval sounds

Gabriel WittumG-CSC

University of Frankfurt

Pythagoras Idea

• Describe the world by numbers

• Everything is number!

=> mathematical abstraction

Gabriel WittumG-CSC

University of Frankfurt

Diffusion

Gabriel WittumG-CSC

University of Frankfurt

Diffusion

• Let c(x,t) describe the concentration of a substance in space and time inside a given volume V.

• Let there be no sources and sinks in V and the substance be incompressible, i.e. no change of density.

Gabriel WittumG-CSC

University of Frankfurt

Mass Balance

• Temporal change of concentration in V must be effected by flux across the boundary of V

• Balance�

V

dc

dt= −

�

∂V

−→j ·−→n

Gabriel WittumG-CSC

University of Frankfurt

Constitutive Law

• Closing the model, i.e. formulating it in one unknown quantity

• Relation between flux j and concentration c needed => constitutive law

Gabriel WittumG-CSC

University of Frankfurt

Constitutive Law

• Closing the model, i.e. formulating it in one unknown quantity

• Relation between flux j and concentration c needed => constitutive law

• Fick‘s first law�j = −D∇u

Gabriel WittumG-CSC

University of Frankfurt

Adolf Fick

• German Physiologist

1829 - 1901

• Professor in medicine Zürich and Würzburg

• Mathematical description of biological processes

Gabriel WittumG-CSC

University of Frankfurt

Fick‘s First Law

• D “diffusivity”

• 2d tensor, i.e. 3x3 matrix

• symmetric and positive definite

�j = −D∇u

Gabriel WittumG-CSC

University of Frankfurt

Fick‘s First Law

• Described by A. Fick in 1855 merely phenomenological derivation

• Confirmed by A. Einstein, 1906 by homogenization of Brownian motion

�j = −D∇u

Gabriel WittumG-CSC

University of Frankfurt

Diffusion• Using the constitutive law

• Using Gauß theorem of integration

�

V

dc

dt= −

�

∂V

�j · �n =�

∂VD ·∇xc(�x, t) · �n

�

V

dc

dt=

�

VdivD∇xc(�x, t)

Gabriel WittumG-CSC

University of Frankfurt

Diffusion

• In differential form

dc

dt= divD∇xc(�x, t)

Gabriel WittumG-CSC

University of Frankfurt

Definition: Model etc.• A mathematical object is called a model, if it is

used in the context of Fig. 1.1.1 to describe a real process.

• A process is something going on in time described by it‘s state.

• The state of a process is a function u(t,…)

• Biological processes happen in physical space: described by u(t,x,…)

Gabriel WittumG-CSC

University of Frankfurt

Biosystems Difficulties

• complex geometries and processes

• cells are intelligent

• process details often unknown

• experts often have wrong imagination

• life scientists lack deeper understanding of math.

Gabriel WittumG-CSC

University of Frankfurt

Modelling Steps

• Model morphology

• Geometry

• Materials

• Model processes

Gabriel WittumG-CSC

University of Frankfurt

Ink on Blotting Paper

t=0 t=T>0

c(t=0,x) c(T,x)

Gabriel WittumG-CSC

University of Frankfurt

1.1 Modeling GeometryDomain Ω = {(x,y), 0 < x, y < 1}

Gabriel WittumG-CSC

University of Frankfurt

1.2 Modelling MaterialsDiffusivity D of the blotting paper (direction dependent)

Gabriel WittumG-CSC

University of Frankfurt

2. Process: DiffusionTransport of ink by diffusion in the paper c(t,x).

Gabriel WittumG-CSC

University of Frankfurt

2. Process: DiffusionTransport of ink by diffusion in the paper c(t,x).