Peter Limkilde
Odsherreds Gymnasium
May 2008
Mathematical modelling
Mathematical modelling project in upper secondary school (3rd grade, students
age 16-19 years) Odsherreds Gymnasium, Denmark, spring 2008
Introduction
Mathematical modelling is part of the Danish upper secondary curriculum in mathematics. So it was
quite obvious to plan for a modelling project in my 3rd grade class in upper secondary at
Odsherreds Gymnasium, Denmark. But modelling is interesting for another reason: the cognitive
skills of the students might be related to how they understand the concept of modelling.
According to Piaget1 many people's thinking develops in a qualitative way between the ages 11 and
16. In the terminology of Piaget the change is from “concrete operational thinking” to “formal
operational thinking”. Concrete operations are here thought processes, which the child performs on
his or hers perceptions. Important characteristics are that the child can cope only with a limited
number of variables2, and a model is seen as the organization of reality by 1:1 correspondence3. By
contrast Formal operational thinking can handle multi-variable problems. In the early formal stage a
model is taken as true, not a hypothesis, so this level does not allow critical comparison of
alternative formal models, whereas in the late formal stage the child can actively search for an
explanatory model, extend one that is given, and compare alternative models for how they account
for the same data3.
The students had already in 1st grade worked with simple models based on linear, power and
exponential functions. In this project the students have to work with differential equations and are to
be confronted with the situation, that several different models can be used to model the same data.
This hopefully might contribute to some cognitive push in the development from early formal
operational thinking towards late formal thinking. The use of authentic data was considered to be a
factor of motivation.
1
Piaget. J. (2002), The Psychology of Intelligence, Routlegde London and New York
Adey, P., Shayer, M. & Yates, C. (1995): Thinking Science, 2.ed. Nelson
3
Shayer, M. & Adey, P. (1981): Towards a Science of Science Teaching. Heineman
2
page 1/8
Peter Limkilde
Odsherreds Gymnasium
May 2008
Mathematical modelling
The project
The project included 10 lessons each 45 min. The students worked with mathematical modelling
including authentic data-samples from H. Lundbeck A/S4
The program included:
1) A short introduction to one and two- compartment models.
2) The observed data and a related written correspondence between the teacher and H. Lundbeck,
see Enclosure 1.
3) The student works in small groups (3-4 students). They have to set up a spreadsheet and fit the
observed data using a one-compartment or a multi-compartment model. In the process they should
estimate the magnitude of the relevant parameters.
4) Each group gives an oral presentation of their findings. + discussion between groups
5) Each group gives in a written report; which is corrected and commented by the teacher.
Evaluation
The translation from differential equations to spreadsheet layout caused the students some trouble,
which we found was due to conceptual difficulties concerning the concept “rate of change” rather
than technical problems with the spreadsheet “language”. The oral presentations were repeated at a
seminar open to the public (“The Day of Research”), and were followed by a good discussion
between the students and the audience (upper secondary teachers, university teachers, upper
secondary students). The discussion showed that the students really had considered details of the
situation where two different models can fit the same data. The authentic data did in fact enhance
the motivation, which was notably higher than normal. This project did not train the early part of
the modelling process (translation from “the real situation” to “mathematical model”), nor did it
focus upon the use of modelling as a means of learning mathematical concepts, which therefore
might be worthwhile to dwell more upon in a future project.
Peter Limkilde
Odsherreds Gymnasium, 2008
/. Enclosures next pages
References:
1. Piaget. J. (2002), The Psychology of Intelligence, Routlegde London and New York
2.Adey, P., Shayer, M. & Yates, C. (1995): Thinking Science, 2.ed. Nelson
3.Shayer, M. & Adey, P. (1981): Towards a Science of Science Teaching. Heineman
Additional litterature:
Adey, P. & Shayer, M. (1994): Really raising standards. Routledge
Wylam, H. & Shayer, M. (1980): CSMS Science Reasoning Tasks. NFER
4
H. Lundbeck A/S is an international pharmaceutical company engaged in the research and development, production,
marketing and sale of drugs for the treatment of psychiatric and neurological disorders.
page 2/8
Peter Limkilde
Odsherreds Gymnasium
May 2008
Mathematical modelling
ENCLOSURE 1
Correspondence concerning the data from H. Lundbeck:
Dear ...
Please find enclosed two sets of data from H. Lundbeck. The data consist of observations of the
concentration in a person after oral intake of medication ”X” and ”Y”. Both medications are antidepressive agents in pills.
Several mathematical models can be used to fit the data. In connection to models based on
differential equations the student should consider:
1) How to describe the absorption through the wall of intestine into the blood?
2) Will the medicine distribute itself into one or several compartments?
3) How to describe the elimination of the medication?
These considerations are all treated in detail in the book by Gabrielsson and Weiner, as mentioned
earlier, especially in the chapter "Pharmakokinetic Concepts" ("1 compartment models", "multicompartment models"). This chapter includes many models in form of differential equations and in
analytical form.
A common model of the rate of absorption in oral medication can be written in the form:
dX/dt=k1*Exp(-k2*t). X refers to amount of material. You will get a good fit to data-sample 1 by
combining this absorption model with a one-compartment model. In the same way you will get a
good fit to data-sample 2 by using a two-compartment model. Try this out at first and experiment
further. The students should maybe start out with the (naïve) assumption that the whole of the pill is
instantaneously transferred to the blood (without delay). Hereafter the medication is eliminated at a
rate that is proportional to the amount of material in the blood. Plotting this model will reveal a
rather poor fit to the data. The students could then consider why?
Best wishes
…
Typical 1-compartment
TIME
(h)
CONC (ng/mL)
0
0
1
6,54
2
12,5
3
17,1
4
27,2
6
27,8
8
26,9
12
25
24
19,6
36
21,4
48
16,9
72
15,5
96
10,8
120
8,66
168
6,15
216
4,9
Typical 2-compartment
TIME
(h)
CONC (ng/mL)
0
0
1
3,99
2
13,8
3
19,4
4
19,6
6
20,9
8
16,2
12
14
24
8,87
36
5,9
48
4,58
72
2,42
96
1,39
120
0,779
page 3/8
Peter Limkilde
Odsherreds Gymnasium
May 2008
Mathematical modelling
Dear …
The data looks fine; I have two questions in order to understand the data:
1) Are the data reflecting a single pill taken at time t = 0 or are several pills taken with regular
intervals? (e.g. 24 h)
2) What is the size (mass) of the pill? (e.g. 5 mg?). Do you have to fit the size of the pill as an
unknown parameter? I assume that the volume of the blood-compartment in a human is about 5
liters.
By the way, I think that the pill itself could be considered as a compartment slowly released into the
blood circulation
Best wishes
…
Dear ….
1) It is only one pill taken at time t = 0.
2) Normally you will always know the dose (D) of the active agent, so you don’t have to fit this
parameter. But in the actual case I have no knowledge of D, but you could estimate it to about 20
mg. Please be aware of the fact that the corresponding volume V is only an “apparent volume of
distribution”, this means that it cannot be set equal to the blood volume of the actual person but has
to be fitted from the data. Depending on the affinity of the actual medication for the different
compartments you may obtain very large “apparent volumes of distribution” (e.g. 500 L)
3) Yes you could consider the pill as a compartment slowly released into the blood circulation
Best wishes
…
page 4/8
Peter Limkilde
Odsherreds Gymnasium
May 2008
Mathematical modelling
ENCLOSURE 2
4 different compartment-models: Medication in blood and brain
Model 1
Blood
y(t)
k6
Secretion through sweat,
urine or faeces
We want to follow the concentration, y(t), of the medication in the blood [ng/ml] as a function of
time, t [h], elapsed after the intake of the pill.
The blood is considered to be a compartment (the box) containing well mixed material. Material
(medication) can be removed through a drain or a sink (sweat, urine or faeces). We wish to account
for the medication in the compartment by the concentration (y)
The rate of in/outflow depends on the concentration of the medication already present in the
compartment and the flow-constant k6 as can be seen in equation (1):
y : the concentration of medication in the blood [ng/ml]
(1) y '  k6 y
Task: Set up a spreadsheet with cells for y (0) and k6 and columns for the relevant variables (t, y)
and a column for the observed data. Change the value of the constant k6 and the initial size of the
pill, p (0) , until the calculated values fits the observed data. Observe that y (0)  0 (there is no
medication in blood when the treatment start). Do you consider the resultant model to be a good
model? Explain your case in an oral presentation.
page 5/8
Peter Limkilde
Odsherreds Gymnasium
May 2008
Mathematical modelling
Model 2
Pill
Blood
k0
p(t)
y(t)
k6
Secretion through sweat,
urine or faeces
We want to follow the concentration, y(t), of the medication in the blood [ng/ml] as a function of
time, t [h], elapsed after the intake of the pill.
The pill and the blood are considered to be compartments, each containing well mixed material.
Compartments are represented by boxes and the connection between the compartments are
represented by arrows. Every compartment (that is every box) has a number of connections leading
to the box (inflows) and a number of arrows leading from the box (outflows). Material (medication)
can either flow from one compartment to another, it can be added from the outside through a
source, or it can be removed through a drain or a sink. We wish to account for the medication in
each compartment by the concentration (p and y).
The rate of in/outflow depends on the concentration of the medication already present in the
compartments and the flow-constants k0 and k6 as can be seen in equation (1)
p : the concentration of medication in the stomach/lower gastrointestinal tract (the pill) [ng/ml]
y : the concentration of medication in the blood [ng/ml]
(1) y '  k0  p  k6  y
Task: Set up a spreadsheet with cells for p(0), y (0) , k0 and k6 and columns for the relevant variables
(t, p and y ) and a column for the observed data. Change the values of the constants k0, k6 and the
initial size of the pill, p (0) , until the calculated values fits the observed data. Observe that y (0)  0
(there is no medication in the blood when the treatment start). Do you consider the resultant model
to be a good model? Explain your case in an oral presentation.
page 6/8
Peter Limkilde
Odsherreds Gymnasium
May 2008
Mathematical modelling
Model 3
Blood
Brain
k1
y(t)
k4
k3
k2
k6
z(t)
k5
Secretion through sweat,
urine or faeces
We want to follow the concentration, y(t), of the medication in the blood [ng/ml] as a function of
time, t [h], elapsed after the intake of the pill.
The pill, the blood and the brain are considered to be compartments, each containing well mixed
material. Compartments are represented by boxes and the connections between the compartments
are represented by arrows. Every compartment (that is every box) has a number of connections
leading to the box (inflows) and a number of arrows leading from the box (outflows). Material
(medication) can either flow from one compartment to another, it can be added from the outside
through a source, or it can be removed through a drain or a sink. We wish to account for the
medication in each compartment by the concentration (y and z).
The rate of in/outflow depends on the concentration of the medication already present in the
compartments and the flow-constants k0, k1, k2, k3, k4, k5 and k6 as can be seen in the equations (1)
and (2). (Please notice that k5  k1  k3 and k6  k 2  k 4 )
y : the concentration of medication in the blood [ng/ml]
z : the concentration of medication in the brain [ng/ml]
(1) y '  k1 y  k4 z
(2) z '  k3 y  k 2 z
Task: Set up a spreadsheet with cells for y (0) , z (0) , k0, k1, k2, k3, and k4 and columns for the
relevant variables (t, y and z) and a column for the observed data. Change the values of the
constants k0, k1, k2, k3, k4 and the initial concentration in the blood, y (0) , until the calculated values
fits the observed data. Observe that y (0)  0 and z (0)  0 (there is no medication in blood and
brain when the treatment start). Do you consider the resultant model to be a good model? Explain
your case in an oral presentation.
page 7/8
Peter Limkilde
Odsherreds Gymnasium
May 2008
Mathematical modelling
Model 4
Pill
Blood
Brain
k0
p(t)
k1
y(t)
k4
k3
k2
z(t)
k5
k6
Secretion through sweat,
urine or faeces
We want to follow the concentration, y(t), of the medication in the blood [ng/ml] as a function of
time, t [h], elapsed after the intake of the pill.
The pill, the blood and the brain are considered to be compartments, each containing well mixed
material. Compartments are represented by boxes and the connections between the compartments
are represented by arrows. Every compartment (that is every box) has a number of connections
leading to the box (inflows) and a number of arrows leading from the box (outflows). Material
(medication) can either flow from one compartment to another, it can be added from the outside
through a source, or it can be removed through a drain or a sink. We wish to account for the
medication in each compartment by the concentration (p, y and z).
The rate of in/outflow depends on the concentration of the medication already present in the
compartments and the flow-constants k0, k1, k2, k3, k4, k5 and k6 as can be seen in the equations (1)
and (2). (Please notice that k5  k1  k3 and k6  k 2  k 4 )
:
p : the concentration of medication in the stomach/lower gastrointestinal tract (the pill) [ng/ml]
y : the concentration of medication in the blood [ng/ml]
z : the concentration of medication in the brain [ng/ml]
(1) y '  k0  p  k1 y  k4 z
(2) z '  k3 y  k 2 z
Task: Set up a spreadsheet with cells for p(0), y (0) , z (0) , k0, k1, k2, k3, and k4 and columns for the
relevant variables (t, p, y and z) and a column for the observed data. Change the values of the
constants k0, k1, k2, k3, k4 and the initial size of the pill, p (0) , until the calculated values fits the
observed data. Observe that y (0)  0 and z (0)  0 (there is no medication in blood and brain when
the treatment start). Do you consider the resultant model to be a good model? Explain your case in
an oral presentation.
page 8/8