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?

 k 1 HA  H + + A -  k -1 Forward velocity proportional to concentration HA v f  [HA] or v f = k 1 [HA] Reverse velocity proportional to concentration H + and A - v r  [H + ] [A - ] or v r = k -1 [H + ][A - ] NOTE: k 1 and k -1 are rate constants, defined as the fraction of mass transported in that direction per unit time e.g. k 1 = 0.5 /s ( or s -1 ) k 1 and k -1 describe the dynamic properties of the system. Mathematical formulation: mass action equations

 k 1 HA  H + + A -  k -1 At steady state the forward and reverse velocity is equivalent i.e. v f = v r or k 1 [HA] = k -1 [H + ][A - ] If k 1 /k -1 = Keq then K eq = [H + ][A - ] [HA] Weak acids dissociate reversibly in aqueous solution, e.g. Mathematical formulation: mass action equations at steady state

K eq = [H + ][A - ] [HA] Rearrange to give [H + ] = K eq [HA] [A - ] Taking logarithms gives log 10 [H + ] = log 10 K eq +log 10 [HA] [A - ] From the definition of pH pH = - log 10 [H + ], we get pH = pK + log 10 [A - ] [HA] Where pK is a new constant pK = -log 10 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 CO 2. - YES –For simulating what happens on changing CO 2 in plasma – NO So reaction Translates to

Plasma Translates to 1 2 These are called ”mass-action” equations

Can we simulate what happens, when we measure pH and CO 2 in a plasma sample and want to understand what happens if we change CO 2 ?

Can we solve when changing CO 2 Equations for situation (a)Equations for situation (b) Known values – CO 2(a ), CO 2(b), pK, pK A 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 CO 2 ?

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 CO 2 Equations for situation (a)Equations for situation (b) Known values – CO 2(a ), CO 2(b), pK, pK A, 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 CO 2

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 CO 2 remains constant These are called ”mass balance” equations.

Can we solve when adding strong acid Equations for situation (a)Equations for situation (b) Known values – CO 2(a ), pK, pK A, Atot Unknown values - Eleven equations, Eleven unknowns – we can solve CO 2 (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 CO 2 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)statedynamical system

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 CO 2 ? (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 CO 2 levels, addition of acid, changing O 2 levels, etc?

Components Plasma Erythrocyte bicarbonate Erythrocyte haemoglobin

Haemoglobin structure

Consider the protein without side chains

So one can write mass-action and mass balance for these.

Haemoglobin structure

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. O2O2 Haldane

Why do we need this level of complexity – Bohr-Haldane effects. O2O2 Haldane CO 2 So, if you want to simulate changes in O 2 or CO 2 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:

Procedure

Simulation of blood mixing From: Rees S.E et al, EJAP 2010, 108: