RESEARCH SUMMARY SEMESTER 3
All modules
Radboud University, Nijmegen
Made by: Georgia Graat
,Research summary Q5 + Q6
Modelling systemic therapy
When administering drug treatment for cancer, you
have to find a dose that is more harmful for the
tumor than for the body cells. You have to determine
the maximum dose the average patient can survive
therefore (MTD = maximum tolerable dose) and use
this as a standard dose. The dose limiting toxicity
(DLT) is the point where the dose is too high for a
patient and thus leads to symptoms.
To determine the MTD you start by using a dose level
that is randomly chosen. You start with a very small amount of patients, test how many DLT cases
occur and then change the dose to a higher or lower number to eventually determine the correct
MTD with the least amount of DLT cases. You keep testing until you reach the optimal dose level
which you measured in a large group of patients.
Chemotherapy and targeted therapy might differ quite a bit in this optimal dose area. In
chemotherapy, the most efficacious dose is also the MTD and this is again closely related to the
toxicity. Targeted therapies on the other hand may have the efficacious dose way lower than the
MTD and will therefore also not come close to the toxic dose. Another problem is that the optimal
dose differs substantially between patients, so you only work with averages.
To model cancer therapy, it is needed to determine the drug concentration at
site of action. There are several steps in this model-guided dose finding.
1. Find the concentration-effect relationship in tumor-bearing animals
and confirm this in humans. This can be graphed through concentration
against inhibition activity and you want this graph as high as possible.
2. Then the plasma concentration can be
translated to effectiveness. To do this, the
pharmacokinetics in humans should be established
and appropriate dosing schemes for further clinical
development should be stimulated. A graph of inhibition activity over time
is plotted for the average human for different doses.
3. The last step is to build a pharmacokinetic model in which the dose
for a specific individual can be calculated. This can be done by making an
individual model-guided dose finding.
a. Administer a starting dose to the model of the average human.
b. Calculate the plasma concentration of the drug and estimate effectiveness from the
measured concentration.
c. If concentration is too high or too low, use the model to fit this patient’s parameters
to measured concentration and estimate optimal dose.
d. Measure plasma concentration again with the different dose.
e. Do the loop several times until you find the optimal dose for the specific patient.
In order to individualize the dose of an anti-tumor drug, it is important to understand the relation
between dose, systemic exposure and relation between systemic exposure and effect. With all
parameters about exposure and effect in a model, this can be used to predict the optimal dose.
Made by: Georgia Graat
,For modelling cancer therapy drugs, not only the most effective dose is important. Everolimus for
example has serious side effects and all with a grade 2 or higher (pneumonitis, metabolic event or
neutropenia) mean stopping the therapy until symptoms improve and then reducing the dose.
Systemic exposure means the whole body is exposed to the drug. This exposure (or amount of drug
in the body) is correlated with efficacy (tumor growth inhibition) but also linearly correlated with
toxicity (side effects).
Models have different aspects that you need to bear in mind. First of all, it is a static or dynamic
system equation. Static means the equation is very simple and does not have a time factor, meaning
the outcome is only dependent on the input at that time. The dynamic system equation does have a
time factor and is thus dependent on the input at that time and from the past. Essential in a dynamic
model is that the variables are not constant, but vary with time. Most things in life are dynamic
models.
A system can also be linear or non-linear. A linear
system has the output twice as large when the input is
twice as large, while a non-linear system is everything
that doesn’t do this. Almost all more complex equations
are non-linear. The right graph shows a linear line. You
can see that the concentration is twice as low when the
time is twice as high which is due to the velocity efflux
being dependent off the concentration. The left graph might seem linear, but actually only has a
constant excretion velocity over time. This means the concentration decreases with a certain amount
every t, which you can see when drawing lines in the middle of the concentration to the time line. It
is also possible to draw a new graph with the concentration in half as starting dose, in which you can
see that only the right graph will be half the concentration that it was at first at every t. In most
cases, it is realistic to assume linear kinetics, because the elimination is proportional to the
concentration.
In this dynamic model we will estimate the time course of the concentration in a compartment
model. Translating a research question into an R script also goes in steps.
1. Find the system/differential equation
An equation is always set up as change = in – out. The
change is determined as the change in amount of drug in
the compartment over time (mg/hour), not the change in
concentration!
The amount of drug in the compartment is equal to the
volume times the concentration. So the change in amount is the change in volume
times concentration. Because the volume is a known number, you can put this in
front, meaning you only have to determine the change (or derivative) of the
concentration. The efflux is proportional to the clearance times the concentration,
which is a linear kinetics equation. The system equation is eventually a differential equation (with a
derivative) that described the relation between the dose velocity and concentration.
Another thing is finding the equilibrium or stationary solution. In that case, the
derivative of the differential equation will always equal 0. With this in mind, it is
possible to calculate the concentration at which this equilibrium occurs.
Made by: Georgia Graat
, 2. Identify the parameters and variables
The parameters are the properties of the body that are measurable and always stay
the same (clearance and volume). The variables on the other hand are states, which
depend on the changes in the body over time (dose and concentration).
3. Create a loop in R and compute the c(t + Δt) at each step
a. This means you also have to find an expression for c(t + Δt)
On the right is the most simple version of an R model with the c(t + Δt) not
computed yet. In red are the parameters, values you determine and initial values.
The brackets cause the loop to repeat itself over all time instances and all steps will
be repeated for 1 – nsteps. V and k can be filled in for a specific patient.
The initial value of c[1] is D/V, because this stands for the concentration at time 1. The
c(t + Δt) is also a variable that needs to be calculated over and over again. The Δc is
always the derivative of the concentration times the Δt. The derivative of the
concentration can be computed from the system equation. You have to fill in this
derivative * deltaT in the red part of the model as ‘deltaC’ and the blue loop will be: c[i+1] <- c[i]
+ deltaC.
The model will eventually look something like on the left. This is a model in which the
D(t) is high at t=0 but is D(t)=0 after t>0 (a single pill for example). The concentration of
the drug will only decrease over time. This also means an equilibrium can’t be calculated
in this case. For that, you need multiple or a constant D(t) at different times to
eventually (at 4/5 times the half-life) reach a steady state. Equilibrium entails a constant
average concentration, which happens at c = D/k. The steady state will be reached
sooner with a larger derivative (larger k or smaller V).
A larger k will mean the clearance is higher and thus the model of the time course of the
concentration will decrease faster.
More complex models, like the drug concentration in the body, have multiple compartment models,
because the drug passes different parts of the body. The concentration of a drug only has a narrow
therapeutic window and the time course of this concentration is a dynamic process. This also means
you have multiple systems equations that will all have their own parameters and variables. In a multi-
compartment model we have to apply “change = in – out” for each compartment separately, which
results in one differential equation per compartment.
On the right, a two compartment model can be
seen. In both cases, the change in amount of drug
is still input – output, but now you have more
variables to take into account. As can be seen,
only the D is the input in the V1 compartment
(which makes sense as it enters the body there)
and the rest are computed as efflux with k * c(t).
The clearance k is always from one compartment to the other, which can be described with numbers.
The concentration in the efflux is always the one from where the drug is coming from, not where it is
going!
The ratio between c1 and c2 in the two compartment model can be determined with the differential
equation, but only in steady state. In this state, the derivative of c2 for example is 0, so k12c1 = k21c2
so the c1/c2 = k21/k12. When you measure these clearances, you will know the ratio.
Made by: Georgia Graat
