drug modelling

7/28/2019 Drug modelling

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223MSE

ASSIGNMENT 3

March 14

2013Gerasimos Politis SID: 4083429


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Gerasimos Politis SID: 4083429

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Table of Contents

Formulating the Mathematical Problem ..........................................................................................................2

a) Formulation of the quantity of drug in the blood at time t

b) Formulation of the elimination rate of drug at time t

c) Calculation of the differential equation that expresses concentration at t=0

Solving the Mathematical Problem ..................................................................................................................3

(i) General solution to the differential equation satisfied by c(t)

(ii) Curve fitting and Matlab simulations for the calculation of the particular solution of concentration

(iii) Estimate the Half-time course of the drug

(iv) Determining the apparent volume of distribution and the average concentration at t=0

(v) Calculation of the time where Theophylline concentration in the blood equals 12 (vi) The rate of intravenous required to maintain the theophylline concentration in the blood at 12 mg/l


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Formulating the Mathematical Problem

According to the assumptions which our model is based I was able to create the following

formulas.

a) The first formula representing the relation among the quantity of drug, q(t) mg ; the

concentration in the blood, c(t) mg/l, as well as the the apparent volume of distribution, V

litres, is:

b) The second formula representing the quantity removed from the body during a short time

interval which is proportional to the total quantity present at the start of the interval and also

the length of the time interval, the positive constant of proportionality being , is:

As the equation becomes :

Where the minus sign (-) represents the fact that the quantity of drug in the blood is

decreasing, therefore the rate of change must be negative.

c) At time t=0 a dose of D mg is injected into a patient that was previously free of the drug.

Using equation (1):


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Solving the Mathematical Problem

i. To find the general solution to the differential equation satisfied by c(t) we first use (2).

Integrating with respect to t gives:

()

Where is a constant.

since

Therefore, the general solution to the differential equation is:

By using equations (1) and (3) we can deduce that the general solution to the differential

equation satisfied by c(t) is :

ii. In order to establish the actual relationship between c(t) and t, we will insert the data into

Matlab .

X = [1 2 3 4 5 6 7 8];

Y = [16 14 11.5 10 9 8 6.3 5.8];

The X matrix represents the time in hours and the Y matrix the concentration in mg/L.


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Then using the command cftool a blank window comes up.

In order to insert our data we click on the Data box.


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Once it opens we select the X matrix as the X Data and the Y matrix as theY Data; and we type a

Data Set name. Then we click on the Create data set box.

On the previously blank window our data appear.


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In order to fit a curve to our data we click on the Fittting box and on the top left we click on the

New fit box. Afterwards we name our fit in the Fit name box, we leave the Data set box the same

and then we choose the Type of fit. In our case we need an exponential fit and specifically we

choose a formula in the form a*exp(b*x)and click apply. Once we have done that the results fora

and bas well as the R-square, which indicates the goodness of fit of our graph, appear in the field

below.


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Finally, the following graph will appear.

Furthermore, we can plot the residuals by clicking on the Analysis box and then click-select the

Evaluate fit at Xi and For function boxes and select the confidence bounds desired and then

click-select the Plot results and Plot data set boxes.


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The graph produced from the table above is:

The fact that our data are within the 99.5% prediction bounds means that the fit is fairly accurate.

Having proved that the Matlabs results are accurate by 99.5% we can claim that the particular

solution to the differential equation satisfied by c(t) is:


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iii. he half-life of the drug is:

iv. From question (ii) we can notice that the average concentration of theophylline at t= 0 is:

Therefore, using (1) the patients (70 kg) apparent volume of distribution is:

According to (Abou-Auda 2008), who has published an article on theophylline, the apparent

volume of distribution for a 70kg patient should be and for a 70kg

patient with cystic fibrosis should be .

The difference in our estimation of the volume of distribution might be the result of not

taking into consideration several factors of major importance such as habits (Eg. smoking),

age, weight, the interaction with other drugs, underlying diseases and many more.


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v. In order to find at what time was the theophylline concentration in this patients blood equal

to 12mg/l :

vi. The theophylline concentration, starting from the time indicated in part (v) and afterwards

must be maintained at 12 mg/l.

We can easily understand that equation (2) represents the rate of elimination of the quantity

initially injected into the body. Therefore, calculating the removal rate () at the timeindicated in part (v) would show us at what rate is the quantity of into blood removed.

The rate of intravenous infusion that the patient should be injected in order for the

concentration to remain constant must be equal to the absolute value of the elimination

rate.

Using equations (1) and (2), the rate of intravenous infusion (in mg/hour) required to

maintain the theophylline concentration in this patients blood at 12 mg/lis :

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References

Abou-Auda H. (2008) Theophylline [online] avalaible at [LastAccessed: 13 March 2013]

Lamb R.G (n.d.) Time Course of Drug Action [online] avalaible at [Last Accessed: 12 March 2013]
http://faculty.ksu.edu.sa/hisham/Documents/PHCL462/THEOPHYLLINE_2008.pdfhttp://www2.courses.vcu.edu/ptxed/m2/powerpoint/download/Lamb%20Time%20Course%20Drug%20Action.PDFhttp://www2.courses.vcu.edu/ptxed/m2/powerpoint/download/Lamb%20Time%20Course%20Drug%20Action.PDFhttp://www2.courses.vcu.edu/ptxed/m2/powerpoint/download/Lamb%20Time%20Course%20Drug%20Action.PDFhttp://www2.courses.vcu.edu/ptxed/m2/powerpoint/download/Lamb%20Time%20Course%20Drug%20Action.PDFhttp://faculty.ksu.edu.sa/hisham/Documents/PHCL462/THEOPHYLLINE_2008.pdf