3. Chapter 1
Introduction
1.1 Definitions
Pharmacokinetics is the study of how the body works on a drug. More specifically, it is the
study of the rates at which a drug is absorbed, distributed, metabolised and eliminated within
the body (Bourne, 2012).
Pharmacokinetic (PK) modelling, then, refers to the systems of equations used to model these
processes mathematically. The NONMEM software used in this project is the industry standard
software to run PK models and estimate parameter values.
Closely linked with PK modelling is pharmacodynamic modelling, which studies how a drug
works on the body (Lees, Cunningham and Elliott, 2004). PD modelling uses PK parameters
to estimate things such as the maximum and minimum concentration of drug in the plasma at
a time t. This links the dose to a required response, for example the plasma concentration of
drug. PD modelling is not discussed in this project, but is important to dose estimation.
Absorption is the movement of a drug into the bloodstream. Distribution is the movement
of a drug between the blood and various tissues such as the lung or liver. Elimination is the
movement of a drug out of the body, also referred to as excretion (Rayman et al., 2006).
Metabolism is the chemical alteration of a drug by the body. This occurs most often in the
liver and can make a drug inactive or alter its toxicity (Le, n.d.).
The rate at which elimination occurs is called the clearance rate and is defined as the amount
of drug eliminated from the body per unit of time (Bourne, 2012).
The volume of distribution is a theoretical value and refers to the total amount of fluid that
would be necessary to contain the initial dose of a drug in the same concentration as it is found
in the plasma (Lymn, 2007).
The dose of a drug is the amount of drug administered to a patient. This can be either as
a bolus dose, where the entire dose enters the body at once, or as a continuous IV drip. In this
project, only bolus doses are considered.
Covariates, or confounding factors are factors that could have an e↵ect on how the drug works
on an individual. These can include, but are not limited to, the sex, age and weight of a pa-
2
4. tient, how long they’ve been on a drug, their genetic make-up, or any other medications they
are taking at the same time.
They are important when the impact of a covariate is significant. For example, the anti-
coagulating drug Warfarin requires lower doses in older patients (Lane et al., 2011). Particularly
important is body weight - paediatric drug doses are often scaled down from adult doses by this
measure.
An adverse drug reaction is a side e↵ect that is assumed to be as a direct result of taking a drug.
The random e↵ect describes how much of the di↵erences in clearance rates and volume of
distribution among patients on the same drug are due to ’unexplained variation’ (Davidian,
2010). This is used to explain how, for example, two patients with identical confounding factors
can eliminate a drug at di↵erent speeds.
1.2 Background
When a clinician is prescribing a drug to a patient, what is important to consider is the rate at
which the patient will clear the medication from the body, and particularly from the plasma.
This will dictate the dosage that is required to reach a therapeutic e↵ect. As already discussed,
there are a number of confounding factors that might a↵ect this rate, for example body-weight
or age.
Consider a patient being prescribed an antibiotic. For the antibiotic to work (i.e. treat the
patient’s illness) its concentration in the plasma must reach above a certain level. The drug is
e↵ective for as long as it exceeds that level. Therefore, the aim of the clinician is to prescribe
the optimum dose to ensure the plasma concentration is above the therapeutic level for as long
as possible (Kajbaf, De Broe and Lalau, 2015).
At the same time, an excessively high concentration of the drug in the plasma can lead to
adverse drug reactions, which should be avoided. The range of values for which the drug will
work is known as the therapeutic range (Drugs.com, 2016).
Next consider two patients, A and B. Patient A has a relatively low clearance rate, while
Patient B has a relatively high clearance rate. In Patient A, the concentration of antibiotic in
the plasma is likely to stay within the therapeutic range for a longer period of time than in
Patient B.
As a consequence, the clinician should prescribe a higher dose to Patient B than to Patient
A in order for B to get a similar length of treatment. Patient A should also be monitored more
closely for adverse drug reactions as a result of their lower clearance rate.
This does not happen in practice, as it is time consuming to build individual models for each
patient requiring a drug, which is why research into dose estimation techniques is important.
Pharmacokinetic modelling is used to estimate the extent to which various confounding fac-
tors a↵ect the clearance rate and other parameters such as volume of distribution. Using this
and other predictive models, it can then be decided whether a patient requires a di↵erent dose
to the standard, and by how much (Clairambault, 2013).
3
5. In this way, the ultimate goal of PK modelling and dose estimation is to find the optimal
dosage for each individual patient which results in the e↵ective treatment of their disease while
avoiding most of the unpleasant symptoms associated with adverse drug reactions (Kajbaf, De
Broe and Lalau, 2015).
A number of di↵erent methods are available for estimating individual dosing, including regres-
sion. This project will look at the more novel technique based on stochastic control methods to
calculate the optimal dose for each individual patient, discussed in detail in Chapter 5.
1.3 Aims
The aims of this project are:
• To explore the di↵erential equations used in PK modelling.
• To estimate parameter values for the drugs Imatinib, Metformin and Lamotrigine using
NONMEM.
• To demonstrate the use of stochastic control methods in individualised dose estimation.
4
6. Chapter 2
Pharmacokinetic Modelling
2.1 Compartmental Modelling
The body can be modelled as a system of compartments (Bourne, 2012). For the simplest
possible model, let one compartment represent the plasma and a second compartment represent
peripheral tissue and muscle.
If a drug enters the body intravenously, it enters the plasma compartment directly and it
is assumed that the drug is instantly equally distributed there (The Hamner Institute, 2006).
For this, a one-compartment model can be used.
Figure 2.1: Dots represent drug concentration equally distributed throughout the body after an
intravenous injection (Niazi, 1979)
In compartmental modelling, the key assumption is linearity (Bourne, 2012). It is assumed that
the rate of elimination and the distribution of the drug follow first-order kinetics.
Therefore, the amount of drug leaving a compartment per unit of time is proportional to the
amount of drug in the compartment to begin with. This is in contrast to zero-order kinetics in
which the amount eliminated remains constant (Rao, 2011).
5
7. Example. Example adapted from (Rx Kinetics, n.d. a)
Consider a drug which clears the compartment at a rate of 10% of its volume per hour. Let the
initial concentration of the drug in the compartment be C0 = 200 mg/L. Letting t denote time,
the amount of drug in the compartment can be estimated using:
C0 ⇥ 0.1t
Hence over a period of 5 hours, the concentration of drug in the compartment can be calculated
as below.
Time Since Dose (Hrs) Concentration in Compartment (mg/L)
0 200
1 180
2 162
3 146
4 131
5 118
Table 2.1: Estimates of concentrations of drug in a compartment following first-order kinetics.
All values rounded to the nearest integer.
2.2 Bolus Injection Model
Recall the definition of a bolus dose as one which distributes the drug around the body instantly
and evenly. Note that a dose by injection will enter the bloodstream immediately.
Under these two conditions, it is therefore possible to model a bolus injection by a one-
compartment model (Rx Kinetics, n.d. b). In this case, that one compartment is the plasma.
Represent the compartment as a box. The amount of drug in the box at time t = 0 is equal to
the initial dose. At each time t after this, a percentage of the drug is leaving the box due to
the assumption of linearity.
Figure 2.2: Box diagram showing drug entrance and clearance of compartment in a one-
compartment model (Van Peer, 2007).
Let the percentage of drug eliminated from the box be represented by the elimination rate con-
stant ke.
Letting Cl denote the clearance rate and Vd the volume of distribution, then:
6
8. Cl = keVd (2.1)
From the definition of the volume of distribution as the volume of fluid required to hold the
initial dosage in the same concentration as it is found in the plasma, it is possible to express Vd
as
Vd =
D
C0
(2.2)
where D is the initial dose and C0 is the concentration of drug in the compartment at time
t = 0 (Huang, 2010).
Example. An adult is given a 200mg dose of Ibuprofen by bolus injection. It is known from
previous studies that the elimination rate constant is 0.6h 1 and that the concentration of drug
in the patient’s body at t = 0 is 40mg/L .
So D = 200, ke = 0.6 and C0 = 40.
By Eq. (2.2) ,
Vd =
200
40
= 5L
Hence by Eq. (2.1) ,
Cl = 0.6 ⇥ 5 = 3L/h
Next consider the rate of change of concentration of drug in the compartment over time, repre-
sented by (Bourne, 2012):
dC
dt
= keCt
The solution to this is:
dCt
dt
= keCt
)
dCt
Ct
= ket
)
Z
dCt
Ct
= ke
Z
dt
) ln(Ct) = ket + a ( a a constant)
) Ct = Ae ket
7
9. So at time t = 0 , then C0 = A
) Ct = C0e ket
(2.3)
Using Eq. (2.3) it is possible to estimate the concentration of a drug in the plasma at any time
after the initial dose by bolus injection.
Example. A patient is given a 200mg dose of a drug as a bolus injection. The elimination rate
constant is known to be 0.3h 1.
Then Ct = 200e 0.3t and the concentration of drug in the plasma over 24 hours can be cal-
culated as:
Time Since Dose (Hrs) Concentration in Plasma Compartment (mg/L)
0.5 172
1 148
2 109
6 33
12 5
24 0.15
Table 2.2: Estimations for concentration of drug in plasma at 6 points in time after bolus
injection.
From the table it can be seen that Ct is a monotonically decreasing function for this method of
dosing. After 1 day, the body has almost entirely cleared the drug.
The points can be plotted as a time-concentration curve:
Figure 2.3: Time concentration curve for example drug using points in Table 2.2 and the rela-
tionship between rate of change and concentration.
In Figure 2.3, plotting the concentration of drug against its rate of change results in a straight
line - reflecting the assumption of first-order kinetics in compartmental modelling (Bourne,
2001a).
8
10. 2.3 Oral Dose Model
When a drug is administered as a pill, it first passes through the gut before reaching the plasma
and, for some drugs, continuing into the peripheral tissue (Gateway Coalition, 2016). In the
PK sense, this remains a one-compartment model. When considering the problem through dif-
ferential equations, however, two compartments are needed.
Let the first compartment be an absorption compartment representing the gut, and let the
second compartment represent the plasma as before.
After a dose is taken, it enters the gut. It is eliminated from the gut and absorbed into the
plasma, where it is finally eliminated from the body.
Figure 2.4: Model for drug taken as oral pill, in which drug enters gut before being absorbed into
plasma compartment (Bourne, 2001b).
As before, there is an assumption that elimination follows first-order kinetics.
Let Ca(t) be the concentration of drug in the gut at time t and let D be the initial dose.
Then:
dCa
dt
= D kaCa
Hence after a dose D is administered, the amount of drug leaving the gut into the plasma is
equal to some proportion of the gut’s concentration, kaCa, where ka is the elimination rate
constant for the absorption compartment.
This amount - kaCa - enters the plasma compartment and is eliminated in the same way as in
a one-compartment model for bolus injection doses. Its rate of change of concentration can be
described then by:
dC
dt
= kaCa keC (2.4)
where C is the concentration of drug in the body at time t and ke the elimination rate constant
for the plasma compartment.
Solving Eq. (2.4) by Laplace transforms gives (Gateway Coalition, 2016):
Ct =
kaC0
Vd(ke ka)
(1 e kat
)
9
11. A time concentration curve for a drug administered as an oral dose might look as below:
Figure 2.5: Sketch of time concentration curve for oral dose of a drug (Twitchett and Grimsey,
2012)
Compare this to Fig. 2.3. For an oral dose, the concentration of drug in the plasma increases
initially as it is absorbed into the blood from the gut. This is in contrast to the curve for an
IV bolus dose, which begins at its peak concentration because the drug enters directly into the
bloodstream.
As the drug is eliminated from the plasma, the curve for both the oral dose and IV bolus
dose both decrease to a minimum concentration. The gradient of each curve tends to 0 as the
drug is eliminated - this reflects that the amount of drug eliminated at a time t is proportional
to the amount in the plasma immediately before.
The uses of these time concentration curves are discussed in Section (5.2). Note that these
formulae can be used to find algebraic solutions for the various parameters, but it is of more
use practically to find analytical solutions. An analytical solution gives much more information
about the model and its behaviour over time (Neumann, 2016). This is where the use of the
NONMEM software comes in.
2.4 Modelling with NONMEM
In order to run dose estimation using a PK model in NONMEM, it is first necessary to decide
which covariates should be included. The model should only include covariates that have a
significant impact on how the body works on a drug (seen in parameters like clearance rate).
Building a PK model in NONMEM is similar to the forward selection method used in building
multiple linear regression models. Here the model for clearance rate will be constructed.
The initial model is the control
Cl = ✓1 ⇥ exp( i)
where ✓1 represents the clearance rate of the standard patient and exp( i) is the random e↵ect
for a particular patient, both computed by NONMEM (Menon-Andersen et al., 2008).
10
12. The next model considers exactly one confounding factor, for example,
Cl = ✓1 ⇥ ✓age ⇥ exp( i)
Again, ✓age is given in the NONMEM output for each covariate model (Holford, 2009a).
Note that, unlike in multiple linear regression modelling, the PK model is log additive - ✓
values are multiplied by each other rather than added (Lane et al., 2011).
Here, the patient’s age will change the e↵ectiveness of the drug by
100(✓age 1)%
If this change is not statistically significant, then the covariate for age is removed from the model
and not used again. If it is significant, it is carried forward for testing in the multivariate model.
A multivariate model is tested if more than one single-covariate model is shown to signifi-
cantly a↵ect the clearance rate.
For example, if age and weight are both shown to have an e↵ect on the way the body works
on a drug when considered individually, the next step is to test whether there is still an e↵ect
when they are considered in combination. It could be that the change brought by one of the
covariates cancels out the e↵ect of the other - implying both covariates are explaining the same
proportion of variability - meaning the multivariate model can be discarded as insignificant and
only one covariate should be included in the model.
Thus to build the PK model, single-variate models for all possible confounding factors are
tested one-by-one, with significant covariates retained. Multivariate models are then tested for
the confounding factors that were not removed.
With NONMEM, the e↵ect of a covariate on the model can be assessed using the minimum
value of the objective function (Holford, 2009a). This is a measure of the fit of a model to the
observed data.
The statistical significance of the objective function value is assessed using a 2 test at the
1% level to 1 degree of freedom, the tabulated value for which is 6.635 (Lane et al., 2011).
If the objective function value for one model di↵ers from that of the control model by more than
6.635, it is said to be significant. The overall aim is to reduce the objective function (Widmer
et al., 2006).
11
13. Example. The minimum value of the objective function for the control model for a drug is
1300. Single-covariate models are run for 3 confounding factors: age, weight and sex. The
minimum values of the objective functions for these models are:
Model Objective Function Value (OBJ) 1300 OBJ
Control 1300 -
Age 1297 3
Weight 1299 1
Sex 1292 8
Table 2.3: Minimum values of objective function for control model and single-covariate models.
From the table, it is clear that the sex of a patient is significant, since
1300 1292 > 6.635
This is the only confounding factor for which the minimum value of the objective function is
more than 6.635 di↵erent from the control model, so the final model for this drug is:
Cl = ✓1 ⇥ ✓sex ⇥ exp( )
This model gives the population average only - the clearance rate of the average male or female
patient - until the value for a patient is included.
When this model is run through NONMEM, the value of ✓sex can be plugged in as below:
Example. Consider the drug in the previous example, for which the only significant confound-
ing factor was the sex of a patient. When the single variate model that includes the patient’s
sex is run through NONMEM, it is estimated that ✓1 = 35.6L/h and ✓sex = 1.14.
Hence for the average male patient (where ’average’ is relative to the patients in the study),
Cl = 35.6 ⇥ 1.14
= 40.584L/h
and for the average female patient
Cl = 35.6 ⇥ 1
= 35.6L/h
It can be seen, then, that the clearance rate for male patients is 14% higher than that of female
patients - reflecting the covariate coe cient of 1.14.
To account for variability, use the standard error values for PK parameters to form confidence
intervals. Standard error is a measure of how precise the values for ✓ values were, an indication of
12
14. how close the estimations to the true population clearance rates and volumes (Holford, 2009b).
The 95% confidence interval is found by
↵ ± 1.96 ⇥ s.e.(↵)
where:
• ↵ is the parameter being estimated
• 1.96 is the Normal value corresponding to the 95% significance level
• s.e.(✓) is the standard error of the parameter ↵
Example. The clearance rate of the drug in the previous example is given as ✓Cl = 35.6L/h
with standard error s.e.( ✓Cl ) = 2.4. The 95% confidence interval for the clearance rate is:
✓Cl ± 1.96 ⇥ s.e.(Cl)
= 35.6 ± 1.96 ⇥ 2.4
= (30.9, 40.3)
Hence it can be said with 95% certainty that the true clearance level of the average patient is
between 30.9 L/h and 40.3 L/h. This is before taking into account the 14% increase in clearance
rate for male patients.
13
15. Chapter 3
Results
3.1 Imatinib
The drug Imatinib is used to treat several types of cancers, in particular chronic myeloid
leukaemia (Peng et al., 2004). In this study, it was taken as an oral pill in doses of 400mg,
600mg or 800mg.
In this study, 4 confounding factors were of interest in modelling the drug:
• Age (years)
• Sex
• Weight (kg)
• Months a patient has been on treatment
Note that age, weight and months on drug were all standardised:
age ! age 48
weight !
weight
70
months ! months 48.7
so that a patient aged 52 and weighing 120kg would be recorded as having age 52 48 = 4
years and weight 120
70 = 1.71 kg.
The Imatinib study featured 83 patients whose characteristics are summarised on in Table
3.1.
14
16. Characteristic Patients
Plasma Samples
Total 714
Range per Individual (1, 16)
Dose (mg)
400 (%) 65 (76)
600 (%) 11 (13)
800 (%) 7 (7)
Sex
Male (%) 45 (54)
Female (%) 38 (46)
Age (years)
Mean 48.5
Standard Deviation 13.8
Body Weight (kg)
Mean 80.5
Standard Deviation 20.5
Months on Treatment
Median 24
Range (0.5, 74.4)
Table 3.1: Summary statistics for 83 patients on Imatinib study.
For each observation, the concentration of plasma can be plotted against time to produce the
following graph:
Figure 3.1: Plasma concentration of Imatinib over time.
There was insu cient data to plot a dose curve because the samples were taken opportunisti-
cally when patients attended the clinic, often several months apart. Therefore to estimate the
absorption rate constant, so a fixed value of ke = 0.8h 1 was used from a previously published
paper.
In total, 5 models were tested: the control model (with no covariates), and a single-variate
model for each confounding factor of interest.
15
17. Model Objective Function Value Di↵erence to Control Model
Control 9468.409 -
Age 9465.645 2.764
Sex 9459.89 8.519
Weight 9467.609 0.8
Months 9468.379 0.03
Table 3.2: To assess significance of covariates, compare objective function values of single-
variate model to control.
By the 2 test at the 1% level to 1 degree of freedom, only sex is a significant covariate:
8.519 > 6.635, where 6.635 is the tabulated value for the test.
Using values computed by NONMEM and the methods discussed in Section (2.4), the final
model for Imatinib is:
Objective Function Parameter Estimate Standard Error 95% Confidence Interval
CL(L/h) 11.8 0.596 (10.6, 13.0)
V (L) 648 199 (258, 1038)
ka 0.8 Fixed -
✓sex
Female 1
Male 1.21 0.0789 (1.06, 1.36)
Table 3.3: Final covariate model for Imatinib.
16
18. 3.2 Metformin
Metformin is a drug used to control blood sugar levels in people with Type 2 diabetes (Drugs.com,
2012). It is given as an oral pill in doses of 500mg, 850mg or 1000mg.
There were 75 patients in the Metformin study:
Characteristic Patients
Plasma Samples
Total 293
Range per Individual (2, 4)
Dose (mg)
500 (%) 26 (34)
850 (%) 17 (23)
1000 (%) 32 (43)
Sex
Male (%) 43 (57)
Female (%) 32 (43)
Age (years)
Mean 64.8
Standard Deviation 9.7
Body Weight (kg)
Mean 89.6
Standard Deviation 18.5
Table 3.4: Summary statistics for 75 patients on Metformin study.
The plasma concentration vs time plot for Metformin is:
Figure 3.2: Plasma concentration of Metformin over time.
The absorption rate constant was taken as 0.6h 1. There were 4 covariate models tests: one
each for sex, age, weight and BMI. None of these models were found to have a significant influ-
ence on the clearance rate or the volume of distribution of a patient, so the final model is the
17
19. control model:
Objective Function Parameter Estimate
CL(L/h) 52
V (L) 363
ka 0.6
Table 3.5: Final model for Metformin.
Denoting by i the random e↵ect for a patient i, estimates for the clearance rate and volume
of distribution of that patient are:
• Cl = 52 ⇥ exp( i)
• V = 363 ⇥ exp( i)
18
20. 3.3 Lamotrigine
Lamotrigine is a drug used to treat seizures, and as a mood-stabiliser in the treatment of bipolar
disorder (Malik, Arif and Hirsch, 2006).
Most patients on this study took two doses of Lamotrigine per observation. There was a lot of
variability in the combinations of doses patients took, although the most common combination
was to take a 100mg dose followed by a 50mg dose. In the summary table below, the proportion
of patients taking this combination has been recorded, with all other possibilities combined
under ’Other’.
There were 50 patients on the Lamotrigine study.
Characteristic Patients
Plasma Samples
Total 157
Range per Individual (1 , 4)
Dose (mg)
100, 50 (%) 94 (60%)
Other (%) 63 (40%)
Sex
Male (%) 26 (52)
Female (%) 24 (48)
Age (years)
Mean 39.7
Standard Deviation 16.3
Body Weight (kg)
Mean 74.8
Standard Deviation 16.5
Table 3.6: Summary statistics for 50 patients on Lamotrigine study.
The plasma concentration vs time plot for Lamotrigine is:
Figure 3.3: Plasma concentration of Lamotrigine over time.
The elimination rate constant was taken as 3.5h 1. There were 5 covariate models tested: one
19
21. each for sex, age, weight, and for the two genotypes SNP1 and SNP2. Of these, only weight
was found to be significant.
Note that weight was standardised and recorded as W = weight 71kg.
Objective Function Parameter Estimate Standard Error 95% Confidence Interval
CL(L/h) 2.38 0.165 (2.06, 2.70)
V (L) 1.29 0.996 (-0.66, 3.24)
ka 3.5 Fixed -
✓W 0.0163 per kg 0.00271 (0.01, 0.02)
Table 3.7: Final covariate model for Lamotrigine.
Letting i denote the random e↵ect for a patient i and letting Wi denote the standardised
weight for that same patient, their clearance and volume are found as:
• Cl = 2.38 ⇥ exp( i) ⇥ (1 + Wi)
• V = 1.29 ⇥ exp( i) ⇥ Wi
20
22. Chapter 4
Discussion
All three studies in this project involved opportunistic sampling. Measurements were taken as
and when it was possible, over a period of many months, rather than at regular intervals over
a 24 hour period under controlled conditions.
The impact of this can be seen in the three time-concentration plots (Fig. 3.1, Fig. 3.2 and
Fig. 3.3).
There is no discernible pattern in these graphs, especially compared to what is expected of
a time-concentration curve for an orally administered drug (see Fig. 2.5) and we cannot fit a
trend line over the plotted points.
PK analysis showed that a one-compartment model for oral dosing is appropriate for Imatinib
and Lamotrigine.
4.1 Imatinib
The final model for Imatinib says that the clearance rate of a male patient is 21% higher than
that of a female patient. The confidence interval indicates that the di↵erence between the clear-
ance of men and women is significant, as it does not cross 0. It is 95% certain that a man’s
clearance rate is between 6% and 36% higher than a woman’s.
It is interesting to note that, in the final model for volume - V = 648 ⇥ i, where i repre-
sents the random e↵ect for a patient i - the covariate coe cient 1.21 is not included for Imatinib.
Despite its exclusion, when the covariate model results are compared to those of the control
model, there is a di↵erence between the volume of distribution values in each.
Compare the model in Section (3.1) to the following control model for the same drug and
patient data set:
Objective Function Parameter Estimate Standard Error 95% Confidence Interval
CL(L/h) 13.1 0.503 (12.1, 14.1 )
V (L) 645 140 (371, 919)
ka 0.8 Fixed -
Table 4.1: Control model for Imatinib.
21
23. Under this model, the covariate coe cient for sex (1.21) is not taken into consideration.
In the control model, the volume of distribution is estimated as 645L, with a minimum possible
value of 371L and a maximum possible value of 919L. In the covariate model, these values are
648L, 258L and 1038L respectively.
The covariate model has a wider confidence interval. This implies that including the sex of
a patient in the calculations increases the variability of the result - it can no longer be said with
95% certainty that the volume lies within the same width range as before. This can be seen
in the standard error of the control model for volume being less than that of the covariate model.
Consider again the clearance rates in the covariate and the control models. When sex is ac-
counted for, the point estimate is 1.3 L/h lower than when it isn’t. Similarly, the minimum and
maximum clearance rates of the covariate model are both lower than respective values in the
control model.
This is because the control model considers the group of patients as a whole, whereas the
covariate model distinguishes between males and females. Since we know that male patients
have a clearance rate approximately 21% higher than females’, it can be concluded that these
higher values of males are what push up the estimates of the control model.
Conversely, it could be considered that the covariate model describes explicitly only the fe-
male patients, as for females the model for the population average is Cl = 11.8 ⇥ 1. This is
obviously a lower data set on average than if males were included too.
4.2 Metformin
Unlike Imatinib, there were no confounding factors found to have a significant e↵ect on the way
the body works on Metformin. It can be concluded that a patient’s clearance rate and volume
are not impacted by their sex, age, weight or BMI.
As a result, the final models for a patient’s clearance and volume involve only the popula-
tion estimate and an individual’s random e↵ect i. The standard errors for Metformin could
not be calculated so it is not possible to calculate a 95% confidence interval for each parameter.
The point estimate for clearance rate is 52L/h and for volume of distribution is 363L.
4.3 Lamotrigine
In the final model for Lamotrigine, the point estimate for the e↵ect of weight on how the body
works on a drug is 0.0163% per kg. If a patient puts on 1kg in the time between measurements,
their volume of distribution will have risen 0.0163% and their clearance rate by 1.0163% (from
(1 + Wi)).
There is 95% certainty that the amount by which parameters increase per kg is between 0.01%
22
24. at the lower limit, and 0.02% at the upper limit. The result is significant as the confidence
interval doesn’t cross 0, so it can be concluded that there is a definite increase in parameters
when a patient puts on weight.
This can have an e↵ect on the drug dose required by a patient, and is an example of where the
stochastic control methods discussed in Chapter 5 would be useful. If a patient goes through
sudden dramatic weight loss, the rate at which the drug is eliminated from their system will
decrease, which potentially raises the risk of adverse drug reactions.
Similarly, a patient who puts on a lot of weight over a short period of time may find their
usual dose of Lamotrigine does not have as a great a therapeutic e↵ect.
The point estimate for the volume of distribution is 1.29L, with a 95% certainty that the
true value lies between 0.66L and 3.24L. This is an interesting result as it implies, at the
lower limit, that a theoretical volume of 0.66L is required (multiplied by the random e↵ect
and weight of a patient) to hold the drug in the same concentration as it is found in the plasma.
Interpreting the results for the clearance rate is more straightforward. The point estimate
is 2.38L/h and it is 95% certain that the clearance rate is between 2.06L/h and 2.7L/h.
23
25. Chapter 5
Stochastic Control Methods
5.1 Introduction
When a clinician prescribes a drug to a patient, the dose is chosen based on the average PK
values of the patients who have been involved in studies of that drug (Munk, 2001).
For example, it might be known that for a certain antibiotic, patients who weigh less than
60kg clear the drug relatively slowly, while patients over 100kg do so quite quickly. The recom-
mended doses could therefore be:
• 200mg for patients weighing under 60kg
• 400mg for patients weighing between 60kg and 100kg
• 800mg for patients weighing above 100kg
So if a patient comes in who weights 75kg, the clinician will be recommended to prescribe a
400mg dose.
There are limitations to this approach. Consider a 75kg patient who has an uncommon geno-
type that increases their clearance rate by 30%. This could be a large enough rise that the
antibiotic clears the patient’s plasma too quickly to e↵ectively treat their disease. They would
therefore require a higher dose than the recommended amount.
By applying stochastic control methods, it is possible to calculate an individualised dose that
takes into account all significant confounding factors in a patient (Jeli↵e et at., 1998). This dose
can continually be updated as new data is recorded for a patient and avoids the prescription of
a dose too low (no treatment e↵ect) or too high (increased risk of side e↵ects) (Tod et al., 1997).
So far in this project, the way in which the concentration of a drug in the plasma changes
with time has been found using systems of di↵erential equations. This doesn’t account for noise,
that is, for the unexplained variation in a sample (up until now referred to as the random e↵ect).
Including noise in the model requires a system of discrete time stochastic di↵erential equations
instead, made possible due to dose administration happening at discrete time points rather than
continuously (Schumitzky, 1991). This reduces a lot of the uncertainty in the model caused by
unexplained variation between patients (Kristensen, Madsen and Ingwersen, 2005).
24
26. The aim now is to be able to solve these systems analytically to obtain better estimates for
the plasma concentration of a drug at a point k. One method for finding these solutions is to
use a Kalman Filter. (Faragher, 2012)
Kalman Filters find an estimate at stage k using a previous estimate for stage k 1 and
observations about current conditions and noise. (Esme, 2009)
Example. Adapted from (Levy, n.d.).
Consider a car decelerating such that it loses 10% of its speed per minute. Ignoring for the
moment all noise, if the car’s distance (m) from its starting point after k minutes is denoted by
xk, then the position of the car at any point can be defined as:
xk = (1 10%) ⇥ xk 1 = 0.9xk 1
Now observe that measurements taken as the car moves show that it has actually travelled a
distance zkm in k minutes. This is because noise - unexplained variation due to things like air
resistance and unexpected potholes - have changed the rate at which the car is decelerating by
some amount.
Letting the unexplained variation in the kth minute be denoted vk. Then:
zk = xk + vk
These two equations can be combined to give a recursive formula for the distance travelled by
the decelerating car:
xk = 0.9xk 1 + wk
where wk represents the noise a↵ecting the car’s movement.
Applying this to individualised dose estimation, there are several variables needed in order to
estimate the concentration over time:
• The inital estimate C0, which can be taken from a PD model.
• A noise vector wk representing unexplained variation (e.g. from errors in dosing times).
Other variables that can be included in a stochastic model include:
• A control variable uk to represent fixed values from the data set such as dose and sample
times.
• The vector to represent PK parameters like clearance and volume (Schumitzky, 1991).
Using C0, a better estimate for the concentration can be found for the subsequent state C1, and
again improved for C2 and so on, using information about the previous state and the random
e↵ect to get closer to the true solution (Kleeman, 1996).
Every patient has di↵erent random e↵ect sizes and PK parameters, so using a concentration
estimated using stochastic methods results in personalised dose amounts that can change over
time as more observations are made (as k increases).
25
27. Example
Consider a gene that is known to a↵ect how the body works on a particular drug. In the general
population, it is known that 97% of people have the wild type of the gene; 2% of people have 1
mutation of the gene; and 1% of people have 2 mutations.
A patient being prescribed the drug is chosen at random. The probabilities of the patient
having each type of drug are:
P(wild type) = 0.97
P(1 mutation) = 0.02
P(2 mutations) = 0.01
Until the patient’s genotype is known for certain - found from a blood test - a final model
cannot be built to calculate the best dose for them as an individual. However, it is possible to
build a model that takes into account all 3 possible genotypes and their probabilities, in order
to estimate an optimal dose.
For example, if this gene is the only significant covariate for the drug, and each mutation
from the wild type increases the clearance rate by 15%, the model for clearance would be
Cl = ✓1 ⇥
✓
(P(wild type) ⇥ 1) + (P(1 mutation) ⇥ 1.15) + (P(2 mutations) ⇥ 1.3)
◆
= ✓1 ⇥ (0.97 + 0.023 + 0.013)
= ✓1 ⇥ 1.006
multiplied by the random e↵ects value for the patient. Note that ✓1 still defines the point esti-
mate of the clearance rate as computed by NONMEM.
This model can be used to estimate the optimum first dose for the patient, as it accounts
for the uncertainty the clinician has over the patient’s true genotype.
Once the true genotype is known, the model can be refined as follows:
• If the patient has the wild type gene, Cl = ✓1 ⇥ 1 = ✓1
• If the patient has 1 mutation, Cl = ✓1 ⇥ 1.15
• If the patient has 2 mutations, Cl = ✓1 ⇥ 1.3
all multiplied by the random e↵ects value for the patient.
The models no longer include population probabilities as there is now su cient data about
the patient.
In this example, the only significant covariate was categorical and impossible to change over
time: the patient could either have wild type, or 1 or 2 mutations. For a continuous covariate
like age or weight, using stochastic control methods for individualised dose estimations becomes
26
28. an iterative process (Kleeman, 1996).
For instance, consider a patient who is prescribed the drug Warfarin. Currently, a patient
on Warfarin must monitor their levels of vitamin K daily to calculate the international nor-
malised ratio (INR). If this ratio jumps above or below a certain fixed value, the patient’s dose
of drug might be adjusted accordingly, to keep their response to treatment within the thera-
peutic range (Bpac, 2010).
What if the INR was included as a significant covariate in the clearance and volume model
as the genotype was in the previous example?
Using probabilities from the general population (say 80% of Warfarin patients have an INR
between 2.5 and 3, 12% above 3, and 8% below 2.5), a model can be constructed that allows a
clinician to prescribe an optimal dose to a patient. As the INR is monitored, it can be fed back
into the model to estimate a new optimal dose base on the patient’s data rather than population
estimates. The longer the INR is measured, the more data is collected and therefore the better
the dose estimate can be.
Cost Function
Of high importance to the application of stochastic control methods to individualised dose esti-
mation is an understanding of what the drug is supposed to do - its target (Jelli↵e et al., 1998).
Is the aim to keep the drug above a certain concentration in the plasma for a certain length
of time? Is it preferred that the drug reach a certain concentration once per dose? Or for the
total amount of drug in the plasma over the dosing period to exceed a certain value?
These aims can all be visualised in the time concentration curves of a drug administered orally
(see Section (2.2)). As the dose amount changes, so does the position of the curve, meaning its
minimum and maximum points shift along the y axis (Begg, 2008).
Figure 5.1: Time concentration curve for a drug which has therapeutic e↵ect if concentration in
plasma stays about 4mg/l.
Consider a drug which requires its concentration in the plasma to stay above a concentration of
27
29. 4mg/L at all times in order to have a therapeutic e↵ect. Then the part of the time-concentration
curve of the drug that is most important is its trough, labelled A in Fig. 5.1. This is the mini-
mum concentration (Mullan, 2006).
The ideal dose amount for a patient prescribed this drug, then, would be one for which the
trough is above 4mg/L as this is the lowest point of the curve. Di↵erent doses will naturally
have di↵erent troughs as the rate at which the concentration changes follows first-order kinetics
(see Section (2.1)).
For each of the three drugs studied in this project - Imatinib, Metformin and Lamotrigine
- it is the trough that is used to establish the optimal dose for individual patients.
Figure 5.2: Time concentration curve for a drug which has therapeutic e↵ect is concentration
reaches 12mg/l.
Alternatively, the peak concentration of drug in the plasma can be used (Mullan, 2006). In Fig.
5.2, the peak is a B and the concentration it must exceed is 12mg/L in order to be e↵ective. As
with the trough, it is possible for individual patients to have very di↵erent peak concentrations
despite both being prescribed the same dose.
Figure 5.3: Time concentration curve for a drug which requires the total amount in plasma over
the dosing interval exceed a certain value. Total amount indicated by shaded area under curve
(AUC).
28
30. Finally, if it is the total amount of drug in the plasma over the dosing period that is important,
then the area under the curve can be used as in Fig. 5.3 (Mullan, 2006). This takes into account
the peak, the trough, and the rate at which a drug is eliminated.
Example. Consider a patient prescribed an antibiotic administered as an IV bolus dose. For
the antibiotic to have a positive e↵ect, it must be in a concentration in the plasma that is con-
sistently above 3mg/L.
It is known that the minimum concentration of the drug (the trough of the time-concentration
curve) occurs at t = 12 hours and that the elimination rate constant for the patient is
ke = 0.44h 1
The equation for the concentration of drug in the plasma after t hours for an IV bolus drug is:
Ct = C0e ket
Use this to find the concentration of antibiotic in the patient’s plasma after 12 hour for 5 dose
amounts: 200mg, 400mg, 600mg, 800mg and 1000mg.
Dose Conc. at t = 12 ( mg/L )
200 1.01
400 2.03
600 3.06
800 4.07
1000 5.09
These can be represented graphically as in Fig. 5.4.
29
31. Figure 5.4: Sketch of time concentration curves for 5 doses, with dashed line at 3mg/L.
Here, 3 di↵erent doses will keep the concentration of antibiotic above the required 3mg/L:
600mg, 800mg, 1000mg. These are the three doses whose curves all have a trough falling above
the dashed target line. If the minimum concentration was the only restriction, any of these dose
amounts would su ce.
However, the best dose for the patient is the one which causes the fewest adverse drug re-
actions while providing e↵ective treatment. To account for this, all doses which fall outside the
safety range should be eliminated (Jelli↵e et al., 1998).
The safety range is the range of doses which provide therapeutic e↵ect but do not usually
coincide with harmful side e↵ects. If a dose of 1000mg lies outside the safety range for the
antibiotic in the example, it should not be prescribed, despite satisfying the condition of having
its minimum concentration above 3mg/L.
A patient’s optimum dose, it turns out, is that which has its trough closest to the target
value - in the case of the example, to 3mg/L. To identify the dose which gets the closest, the
cost function is used.
The cost function J(d) describes how far away a concentration is from the target when a patient
is given a dose d.
Let Cmin denote the minimum concentration of drug in the plasma. The ’distance’ of this
number from the target is:
J(d) = (Cmin Ctarget)2
The aim is to minimise J(d).
30
32. Note that when it is the peak of the curve that is of interest, Cmax is used in place of
Cmin. Similarly, when looking at the area under the curve, the cost function can be written as
J(d) = (AUCdose AUCtarget)2 where AUCd is the area when a particular dose is prescribed
and AUCtarget is the area for which the drug has the desired treatment e↵ect.
Example. For the data from the previous example, the cost functions are:
Dose d 200 400 600 800 1000
J(d) 3.96 0.94 0.004 1.14 4.37
Clearly the dose amount that has the smallest cost function value is 600mg, suggesting this
patient should be given 600mg of antibiotic by IV bolus to maintain a plasma concentration
of the drug above 3mg/L and therefore treat their illness with minimum adverse drug reactions.
This is reflected in Fig. 5.4, in which it is the green curve of the 600mg dose whose trough at
12 hours is the closest to the dashed line without dipping below it.
31
33. 5.2 Results
Individualised doses were computed for 3 patients per drug. Patients were chosen such that one
of the 3 had low parameter values relative to the population rates estimated by the PK model,
while another had high parameter values and the third fell roughly near the average.
For each drug, it is the minimum concentration that is of interest, so the trough is used.
The aim is to find the dose for which the di↵erence between a target trough and the plasma
concentration is minimal.
In the following results tables, any doses which fall outside the drug’s safety range are high-
lighted in red.
Imatinib
Recall that the elimination rate constant for Imatinib is taken as 0.8h 1.
There were two troughs targeted for this drug, measured in nanograms per millilitre: 1000ng/ml
and 1500ng/ml. The aim was to find the minimal doses necessary for the patient’s plasma con-
centration to remain above these.
Patient Number Volume Clearance Dose for Dose for
(L) (L/h) 1000ng/ml trough 1500 ng/ml trough
3 1160 0.01 300 600
6 873 0.01 200 400
76 274 0.06 900 1400
Table 5.1: Optimal doses for patients 3, 6 and 76.
Figure 5.5: Time concentration graphs for patients 3, 6 and 76.
32
34. Metformin
Recall that the elimination rate constant for Metformin is taken as 3.5h 1.
The two target troughs for this drug were 1500ng/ml and 2000ng/ml.
Patient Number Volume Clearance Dose for Dose for
(L) (L/h) 1500ng/ml trough 2000 ng/ml trough
1 423 0.01 1500 2000
3 555 0.01 2500 3500
76 179 0.06 500 500
Table 5.2: Optimal doses for patients 1, 3 and 76.
Figure 5.6: Time concentration graphs for patients 1, 3 and 76.
Lamotrigine
Recall that the elimination rate constant for Lamotrigine is taken as 3.5h 1.
This time there were three target troughs, measured in micrograms per millilitre: 3µ/ml,
4µ/ml and 5µ/ml.
Patient Number Volume Clearance Dose for Dose for Dose for
(L) (L/h) 3 µ/ml trough 4 µ/ml trough 5 µ /ml trough
1 129 0.02 200 250 325
11 1290 0.05 750 1000 1250
34 99 0.01 75 100 125
Table 5.3: Optimal doses for patients 1, 11 and 34.
33
35. Figure 5.7: Time concentration graphs for patients 1, 11 and 34.
5.3 Discussion
Imatinib
In order to maintain a concentration of Imatinib in their plasma over 1000ng/ml, Patient 3 re-
quires a dose of 300mg. For a minimum concentration of 1500ng/ml, that dose doubles to 600mg.
Compare these to the other two patients. Of the three, Patient 3 has by far the great-
est volume of distribution. They also have a low clearance rate at 0.01L/h. Patient 6, who has
the same clearance rate but a lower volume of distribution, requires only 200mg of Imatinib to
keep their plasma concentration above 1000ng/ml, and 400mg for 1500ng/ml.
Both these values are lower than the respective doses for Patient 3. This does not,
however, mean a lower volume correlates to a lower dose.
Patient 76 has the lowest volume of the three at 274L, but needs the biggest doses:
900mg for a 1000ng/L trough and 1400mg for 1500ng/ml. One possible explanation for 76’s
much higher dose requirements is their clearance rate: Imatinib leaves their system 6x quicker
than it does for 3 and 6.
In fact, 76 requires a higher dose of Imatinib - for both minimum concentrations - than
has been declared safe: 900 and 1400 fall outside the safety range for the drug.
In Fig. 5.5, it is notable that the bigger the dose, the greater the di↵erence between
minimum and maximum concentrations. For each patient, the lower dose (in green) has a
lower amplitude than the bigger dose (in red).
Metformin
As for Imatinib, two patients here have the same clearance rate at 0.01L/h. Of these, Patient
1 has the lower volume and also requires a lower dose of the drug to stay above the target
troughs. For a trough of 1500ng/ml, Patient 1’s optimal dose is 1500mg whereas Patient3’s is
34
36. 2000mg. Similarly, for a trough of 2000ng/ml, Patient 1’s dose is 2500mg while Patient’s 3’s
best dose is outside the safety range at 3500mg.
In this case, Patient 3 could maybe be prescribed the maximum possible dose within
the safety range in order to get as close as possible to the target minimum concentration, but
this would not give them the same therapeutic e↵ect as patients whose best doses fall within
the range.
The patient with the smallest optimal doses also has the lowest volume of distribution
and the lowest clearance rate. For a trough of 1500ng/ml, Patient 76 requires 500mg, which
also satisfies a minimum dose of 2000ng/ml.
In Fig. 5.6, Patient 76’s time-concentration curves are interesting. They both represent
the concentration of Metformin in 76’s plasma after a dose of 500mg, but the green curve shows
that the drug has not fully cleared the patient’s system before the next dose is administered
(Hartford, 2012), while the red curve shows the drug clearing too quickly. This comes as a
result of the di↵erent target troughs, with the green curve representing the 1500ng/ml.
Lamotrigine
The drug Lamotrigine was slightly di↵erent to Imatinib and Metformin in that patients usually
take 2 doses of di↵erent amounts in 1 observation (e.g. 100mg and then 50mg). The optimal
doses found using stochastic control methods were all in the form of a singular dose.
Patient 34, with the smallest volume and slowest clearance rate, had predictably the
smallest optimal doses. For a targeted minimum concentration of 3µ/ml, Patient 34’s best dose
was 75mg, increasing to 100mg for a trough of 4µ/ml and to 125mg for a trough of 5µ/ml.
The patient with the next smallest parameter values was Patient 1, who also had the
next smallest dose estimations: 200mg for 3µ/ml, 250mg for 4µ/ml and 325mg for 5µ/ml.
The di↵erences in doses for Patient 1 were not as uniform as for Patient 34, instead
getting bigger with each 1µ/ml increase in trough.
Finally, Patient 11 - with a clearance rate 5x that of Patient 34 and by far the largest
volume - has optimal doses outside the safety range for all three target troughs.
In Fig. 5.7 there is evidence of the same relationship between dose amount and ampli-
tude as seen for Imatinib: for each patient, the green curve (the smallest dose) has the smallest
di↵erence between maximum and minimum concentrations.
It is likely due to this relationship that Patient 11’s dose estimations are all amounts
deemed ’unsafe’, if the maximum concentrations for each dose is large enough to cause
toxicological side e↵ects.
35
37. Chapter 6
Conclusion
By applying stochastic control methods, then, it has been possible to find the ideal dose
amount for patients from the 3 studies in this project.
In order to use these methods, it was first necessary to compute the PK parameters -
clearance and volume. It was found that Imatinib and Lamotrigine supported the use of a one-
compartment model (for oral doses), while for Metformin it is assumed that a one-compartment
model is appropriate, but di culties in calculating standard errors meant a firm conclusion
could not be reached.
When optimal doses can be found for individuals, the advantage is the removal of risk
of over- or under-prescription of a drug. Currently, it is possible a patient will be prescribed a
dose that is too small to provide them with a treatment e↵ect, which is of little use to them but
still costs the NHS, or is prescribed a dose too large that then increases their risk of adverse
drug reactions. (Jeli↵e et al., 1998)
Using stochastic control methods, the clinician is able to pinpoint very quickly the best
dose for a patient - this reduces the need for trial and error dosing. Patients on Lamotrigine,
for example, are recommended one single optimal dose using these methods, as opposed to
having pairs of dose amounts to try for the same therapeutic e↵ect.
Clearly, then, the methods discussed in Chapter 5 of this project are of benefit to the
clinician and the patient.
36
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