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Modelling tutorial – ESCTAIC 2012
Stephen E. Rees
Center for Model-based Medical Decision Support, Aalborg
University, Denmark
Tutorial Purpose and content
• To provide an understanding of the principles of
mathematical modelling, some of the terminology,
and the issues related to clinical application.
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Dynamic verses steady state conditions.
Parameters or variables.
State variables, what are they? why are they useful?
Complexity, is bigger always better?
Application, modelling is fun but the purpose must be the
focus.
– To illustrate these issue we will consider the acid-base
chemistry of blood.
The Henderson-Hasselbach equation
Questions that can be asked to this (or any) model
• Where does it come from?
• What does it assume?
• Parameters, variables.
• Is this enough complexity, for what purpose?
Mathematical formulation: mass action equations
 k1
HA  H+ + A k-1
Forward velocity proportional to concentration HA
vf  [HA]
or
vf = k1 [HA]
Reverse velocity proportional to concentration H+ and Avr  [H+] [A-]
or
vr = k-1 [H+][A-]
NOTE: k1 and k-1 are rate constants, defined as the fraction of mass
transported in that direction per unit time
e.g. k1 = 0.5 /s ( or s-1)
k1 and k-1 describe the dynamic properties of the system.
Mathematical formulation: mass action equations
at steady state
Weak acids dissociate reversibly in aqueous solution, e.g.
 k1
HA  H+ + A k-1
At steady state the forward and reverse velocity is equivalent i.e.
vf = vr or
k1 [HA] = k-1 [H+][A-]
If k1/k-1 = Keq then
Keq =
[H+][A-]
[HA]
Mathematical formulation: mass action equations
at steady state
Keq =
[H+][A-]
[HA]
Rearrange to give
[H+] = Keq [HA]
[A-]
Taking logarithms gives
log10[H+] = log10 Keq +log10 [HA]
[A-]
From the definition of pH
pH = - log10 [H+], we get
pH = pK + log10 [A-]
[HA]
Where pK is a new constant pK = -log10 Keq
The HendersonHasselbalch equation
The Henderson-Hasselbach equation
So reaction
Translates to
•
•
•
•
Where does it come from?
What does it assume?
Parameters, variables.
Is this enough complexity, for what purpose?
The Henderson-Hasselbach equation
So reaction
Translates to
•
•
•
•
Where does it come from? – mass conservation.
What does it assume? – steady state
Parameters, variables. pK (parameter)
Is this enough complexity, for what purpose?
– For calculating
from pH and CO2. - YES
– For simulating what happens on changing CO2 in
plasma – NO
Plasma
Translates to
1
2
These are called ”mass-action” equations
Can we simulate what happens, when we measure pH
and CO2 in a plasma sample and want to understand
what happens if we change CO2?
Can we solve when changing CO2
Describe the experiment , with pictures and maths
Equations for situation (a)
Known values –
Equations for situation (b)
CO2(a), CO2(b), pK, pKA
Unknown values -
Four equations, seven unknowns – What are we missing?
Are there any physical constraints when we change
only CO2 ?
Mass balance equations.
• The total concentration of protein, phosphate etc
(Atot) remains constant.
• The total buffer base (BB) remains constant
These are called ”mass balance” equations.
Can we solve when changing CO2
Equations for situation (a)
Known values –
Equations for situation (b)
CO2(a), CO2(b), pK, pKA, Atot
Unknown values -
Eight equations, eight unknowns – Now we can solve
So plasma can be modelled as
For the situation when we are interested in changing CO2
Plasma
Tissue (anaerobic
metabolism)
Is the model still adequate as a description of anaerobic
metabolism?
Lets re-visit our assumptions
• The total concentration of protein, phosphate etc
(Atot) remains constant.
• The total buffer base (BB) remains constant
These are called ”mass balance” equations.
Lets re-visit our assumptions
• The total concentration of protein, phosphate etc
(Atot) remains constant.
• The total buffer base (BB) remains constant
• For a closed system, the total CO2 remains
constant
These are called ”mass balance” equations.
Can we solve when adding strong acid
Equations for situation (a)
Known values –
Equations for situation (b)
CO2(a), pK, pKA, Atot
Unknown values CO2(b),
Eleven equations, Eleven unknowns – we can solve
Plasma
So plasma can be modelled as
One mass-action per chemical reaction, one mass-balance per component.
Plasma - components, reactions, math.
One mass-action per chemical reaction, one mass-balance per component.
So plasma can be modelled as
For the situations when we are interested in changing CO2 or changing strong
acid or base concentration
So – The ”correctness” of a model depends on what we want to do with it!
How much do we need to know to know
everything about plasma?
5 equations, 8 unknowns – This means that values of 3 variables is enough to
completely understand plasma (not all combinations work), i.e. We need 3 state
variables. Not any variables, one for each component of plasma.
State variables
• A state variable is one of the set of variables
that describe the "state" of a dynamical system.
Intuitively, the state of a system describes
enough about the system to determine its future
behaviour. (from Wikipedia)
How much do we need to know to know
everything about plasma?
5 equations, 8 unknowns – This means that values of 3
variables is enough to completely understand plasma, i.e. We
need 3 state variables. Which to choose depends upon the
experiment we wish to simulate.
Exercise: Which variables are appropriate in
the following experiments?
1. We measure a sample of plasma and want to
simulate what will happen if we change CO2?
(Assume we know Atot)
2. We measure a sample of plasma and want to
simulate non-selective (i.e. non-charge
dependent, Atot) removal of plasma protein?
3. We measure two different samples of plasma
and want to simulate what happens when we
mix them?
So plasma can be modelled as
Is this enough to simulate what happens in blood – changing
CO2 levels, addition of acid, changing O2 levels, etc?
Components
Plasma
Erythrocyte bicarbonate
Erythrocyte haemoglobin
Haemoglobin structure
amino acid
H O H H
carboxyl end
of chain
H
+
NH3 - - - - - - - - N C C N C - - - - - - - - C - - - - - - - - COO
1
H
RH
i
i+1
RH
RH
b
RH
CH2
C NH
HC
Amino acid side chains
Amino end of chain
H
H N
+
H
Form 1
H
H N
+
H
Form 2
H
+
-
H N COO
H+
Form 3
CH
++
R
Fe
H
H+
O
Form 2
Histidine side chain
binding to Fe+ +
position 87,  chains
position 92,  chains
N
R
Form 1
-
O
Oxygen binding site
Consider the protein without side chains
Consider the protein without side chains
So one can write mass-action and mass
balance for these.
Haemoglobin structure
amino acid
H O H H
carboxyl end
of chain
H
+
NH3 - - - - - - - - N C C N C - - - - - - - - C - - - - - - - - COO
1
H
RH
i
i+1
RH
RH
b
RH
CH2
C NH
HC
Amino acid side chains
Amino end of chain
H
H N
+
H
Form 1
H
H N
+
H
Form 2
H
+
-
H N COO
H+
Form 3
CH
++
R
Fe
H
H+
O
Form 2
Histidine side chain
binding to Fe+ +
position 87,  chains
position 92,  chains
N
R
Form 1
-
O
Oxygen binding site
Consider the protein side chains
So one can write mass-action and mass
balance for these.
Why do we need this level of complexity –
Bohr-Haldane effects.
Haldane
Haldane
O2
Why do we need this level of complexity –
Bohr-Haldane effects.
CO2
Haldane
Haldane
So, if you want to simulate changes in O2 or
CO2 in whole blood, you need Bohr-Haldane
O2
The full model of blood
Tutorial Purpose and content
• To provide an understanding of the principles of
mathematical modelling, some of the terminology,
and the issues related to clinical application.
–
–
–
–
–
Dynamic verses steady state conditions.
Parameters or variables.
State variables, what are they? why are they useful?
Complexity, is bigger always better?
Application, modelling is fun but the purpose must be the
focus.
– To illustrate these issue we will consider the acid-base
chemistry of blood.
Summary, conclusions
• To provide an understanding of the principles of
mathematical modelling, some of the terminology,
and the issues related to clinical application.
– Dynamic verses steady state conditions.
• Are the dynamic of the system interesting to our problem?
– Parameters or variables.
• What can we estimate? What is constant?
– State variables, what are they? why are they useful?
• What variables usefully and completely describe the current state?
– Complexity, is bigger always better?
• How many parameters do we need?
– Application, modelling is fun but the purpose must be the
focus.
• This must drive complexity, otherwise it is purely academic.
Simulation of blood mixing
From: Rees S.E et al, EJAP 2010, 108:483-494
Procedure
Simulation of blood mixing
From: Rees S.E et al, EJAP 2010, 108:483-494