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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. - PowerPoint PPT Presentation
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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.
– 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?
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 A-
vr [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
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]
Weak acids dissociate reversibly in aqueous solution, e.g.
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 Henderson-Hasselbalch equation
Mathematical formulation: mass action equations at steady state
The Henderson-Hasselbach equation
• Where does it come from? • What does it assume? • Parameters, variables. • Is this enough complexity, for what purpose?
So reaction
Translates to
The Henderson-Hasselbach equation
• 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
So reaction
Translates to
Plasma
Translates to1
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
Equations for situation (a) Equations for situation (b)
Known values – CO2(a), CO2(b), pK, pKA
Unknown values -
Four equations, seven unknowns – What are we missing?
Describe the experiment , with pictures and maths
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) Equations for situation (b)
Known values – 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
Is the model still adequate as a description of anaerobic metabolism?
Tissue (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 acidEquations for situation (a) Equations for situation (b)
Known values – CO2(a), pK, pKA, Atot
Unknown values -
Eleven equations, Eleven unknowns – we can solve
CO2(b),
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
NH - - - - - - - - N C C N C - - - - - - - - C - - - - - - - - COO3 -+
RH RH RHRH
HOHamino acid
Amino acid side chainsAmino end of chain
carboxyl endof chain
R
H
R
H+N
H+
H
H
N
H+
H
H
N
H+
COO
H
H-
Form 1 Form 2Form 2 Form 3
+
Form 1 Form 2
H
H H
CH2
C NH
CHN
HC
Fe++
O
O
1 i i+1 b
Oxygen binding site
Histidine side chainbinding to Feposition 87, chainsposition 92, chains
++
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
NH - - - - - - - - N C C N C - - - - - - - - C - - - - - - - - COO3 -+
RH RH RHRH
HOHamino acid
Amino acid side chainsAmino end of chain
carboxyl endof chain
R
H
R
H+N
H+
H
H
N
H+
H
H
N
H+
COO
H
H-
Form 1 Form 2Form 2 Form 3
+
Form 1 Form 2
H
H H
CH2
C NH
CHN
HC
Fe++
O
O
1 i i+1 b
Oxygen binding site
Histidine side chainbinding to Feposition 87, chainsposition 92, chains
++
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.
O2Haldane
Haldane
Why do we need this level of complexity – Bohr-Haldane effects.
O2Haldane
Haldane
CO2
So, if you want to simulate changes in O2 or CO2 in whole blood, you need Bohr-Haldane
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