Modelling and Simulation of Biological Systems · Modelling and Simulation of Biological Systems....
Transcript of Modelling and Simulation of Biological Systems · Modelling and Simulation of Biological Systems....
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).