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Posted: November 27th, 2021
Pharmacokinetics, an emerging field in BioPhysics and chemistry is the study of the time variation of drug and metabolite levels in various tissues and fluids of the body. Compartment models are used to interpret data. In our problem, we consider a simple blood-brain compartment model as shown in the figure below:
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k21
Input d(t) k12
K
where, Compartment 1 = Blood
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Compartment 2 = Brain
This model is made such that it can aid to help estimate dosage strengths of an orally administered antidepressant drug. The rate of movement of drug from compartment i to compartment ‘j’ is denoted by the rate constant kji and the rate at which the drug is removed from the blood is represented by the rate constant K. A pharmaceutical company must deal with many factors like dosage strengths that will aid a physician in determining a patient’s dosage in order to maintain the right concentration levels and also minimizing irritation and other side effects (Brannan 208).
If we assume that the drug is rapidly absorbed into the blood stream after it is introduced into the stomach, a mathematical representation of the dosage will be of a periodic square wave given as follows:
Based on our model and the equations we can solve the problems:
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1. If we let xj(t) be the amount of drug in milligrams in compartment ‘j’, j =1,2. The mass balance law states:
(i)
Using the mass balance law and the figure, we find:
System in Blood compartment:
System in Brain compartment:
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From (i) and the above equations, we can find the following:
(ii)
The systems above are the rates of drug over time in the compartments.
2. If we let ci(t) denote the concentration of the drug and Vi denote the apparent volume of distribution in compartment i, we can use the relation ci = xi/Vi in the equations of system (ii) to obtain:
(iii)
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Dividing the above systems by V1 and V2 respectively, we get :
3. Assuming x1(0) =0 and x2(0) =0, and the various parameters listed below:
k21
k12
K
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V1
V2
Tb
0.29/h
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0.31/h
0.16/h
6L
0.25L
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1h
and with the numerical simulation program Maple , we can perform simulations of the system with given parameters to recommend two different encapsulated dosage strengths A=RTb.
=> Guidelines to use for recommendation of drug dosage:
1) Target concentration level in the brain should be kept as close as possible between levels 10 mg/L and 30 mg/L and concentration fluctuation should not exceed 25% of the average of the steady-state response.
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2) Lower frequency of administration (once every 24 hours or once every 12 hours is best). Once every 9.5 hours is unacceptable and multiple doses are acceptable (i.e. taking two capsules every 4 hours)
Analysis: Drug usage of more than 4 times per day is unacceptable which makes maximum allowable dose to be 3, making 3 doses at 8 hours interval per day the best choice. We can then simulate from Tp = 8 to Tp = 12, 16 and 24.
From the numerical simulations obtained from Maple, we obtain the following data:
Tp(h)
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R (mg/h)
Steady-state variance
Comments
8
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4
9.04 mg/L to 12.5 mg/L
Below effective therapeutic concentration
8
5
We use peer-reviewed journals, academic databases, and reputable sources, verifying credibility for every citation. Your research is built on solid foundations. Source evaluation follows strict academic criteria for reliability and relevance. We avoid questionable websites and prioritize scholarly materials. Research study bay sourcing maintains the highest standards of academic integrity.
11.7 mg/L to 15.5 mg/L
8
6
14.4 mg/L to 19.2 mg/L
8
Our writers create concise, organized exam notes that highlight key concepts and improve retention. Study smarter with our targeted summaries. Exam notes focus on high-yield information to optimize your review time. Provide your syllabus for customized notes. Assessment help includes strategic study materials for exam success.
8
19.2 mg/L to 25.3 mg/L
8
9
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21.1 mg/L to 27.9 mg/L
8
10
23.2 mg/L to 31.2 mg/L
Above maximum therapeutic concentration
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12
5
10.9 mg/L to 6.5 mg/L
Below minimum therapeutic concentration
12
6
8.6 mg/L to 14.1 mg/L
We write concise, informative abstracts that capture your research essence and meet journal requirements. Our abstracts make strong first impressions. Abstract writing requires distillation of complex research into clear, compelling summaries. Include this in your order for publication-ready results. Research essay service abstracts are crafted for maximum academic impact.
Below minimum therapeutic concentration
12
7
8.32 mg/L to 15.1 mg/L
Below minimum therapeutic concentration
12
Our writers detail research methodology with clarity and academic rigor, justifying your approach and ensuring reproducibility. We strengthen your research design. Methodology sections require precise description of procedures and rationales. Provide your research plan for expert articulation. Ace my homework methodology with professional academic writing support.
8
10.6 mg/L to 18.3 mg/L
12
10
13.2 mg/L to 22.8 mg/L
12
13
17.9 mg/L to 30 mg/L
We handle both notes-bibliography and author-date Chicago systems, ensuring perfect formatting for humanities or sciences. Our expertise covers all variations. Chicago style offers flexibility that we navigate with precision for your discipline. Choose your preferred system in the order form. My assignment help includes comprehensive Chicago referencing proficiency.
16
10
9.11 mg/L to 19.5 mg/L
Sharp fluctuations; Below minimum therapeutic concentration
16
12
10.7 mg/L to 23.5 mg/L
Sharp fluctuations.
16
13
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11.5 mg/L to 25.4 mg/L
Sharp fluctuations.
16
14
12.5 mg/L to 27.3 mg/L
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Sharp fluctuations.
16
16
14.3mg/L — 31.4mg/L
Sharp fluctuations; Above maximum therapeutic concentration
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24
15
6.19mg/L — 24mg/L
Sharp fluctuations; Below minimum therapeutic concentration
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24
20
8.52mg/L — 32mg/L
Sharp fluctuations; Above maximum therapeutic concentration
Obtained corresponding Graphs from Maple and their respective Tp and R values are listed below:
Tp = 8, R = 4 Tp = 8, R = 5
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Tp = 8, R = 6 Tp = 8, R = 8
Tp = 8, R = 9 Tp = 8, R = 10
Tp = 12, R = 6 Tp = 12, R = 8
Tp = 12, R = 10 Tp = 12, R = 12
Tp = 12, R = 13
Tp=16, R=10 Tp=16, R=12
Tp=16, R=13 Tp=16, R=14
Tp=16, R=16
Tp=24, R=15 Tp=24, R=20
Some Comments:
When Tp= 8 and R = 4, the recommended dosage is below minimum therapeutic concentration range.
When Tp= 8 and R = 10 , the recommended dosage is above maximum therapeutic concentration range.
When Tp= 8 and R = 5 to 7, the recommended dosage is below effective therapeutic concentration range.
When Tp= 8 and R = 4, the recommended dosage is below therapeutic concentration range.
When Tp= 12 and R = 5 to 7, the recommended dosage is below minimum therapeutic concentration range.
When Tp= 16 and R = 12 to 14, sharp fluctuation is seen.
When Tp= 24 and R = 20, sharp fluctuation is seen and the recommended dosage is below therapeutic concentration range.
=>Calculation and Analysis of dosage strength ‘A’
Now we can calculate the dosage frequency for the remaining dosage frequency intervals of 8 hours and 12 hours:
(8 hour interval) (R being from 5 mg/h to 9 mg/h)
A = RTb = 5 mg/h x 1h= 5 mg
A = RTb = 9 mg/h x 1h= 9 mg
(12 hour interval) (R being from 8 mg/h to 13 mg/h)
A = RTb = 8 mg/h x 1h = 8 mg
A = RTb = 13 mg/h x 1h= 13 mg
4. From the simulation, we can know that it is best to skip the dose than to try to catch up and double the dose and ultimately overdose from the figures illustrated. If we assume the patient is at a 12 hour interval dose frequency, and R being 10mg/h, the following scenarios can be simulated:
Scenario: missed a dosage and skipped Scenario: missing a dosage & catching up
Analysis: From the scenarios simulations above, we can have a clear picture of what will go through the patient’s drug level.
In the 1st scenario, where the patient missed a dosage and skipped, the concentration level in the brain of the patient stays within the recommended level.
In the 2nd scenario, where the patient tries to catch up, the drug level will cross the recommended level and that also by a lot. Thus, skipping the dose is better than to catch up overdosing the drug level resulting in fatality.
5. Supposing the drug can be packaged in a timed-release form so that Tb = 8 hours and R also adjusted likewise, we get the following data from the Maple:
Tp(h)
R(mg/h)
Steady-state variance
Reasons
12
0.75
10.4mg/L — 13mg/L
12
1
13.9mg/L — 17mg/L
12
1.5
21mg/L — 25.5mg/L
12
1.75
24.5mg/L — 29.8mg/L
12
2
28.1mg/L — 34mg/L
Above maximum therapeutic concentration
16
1
9mg/L — 14.3mg/L
Below minimum therapeutic concentration
16
1.25
11.2mg/L — 17.7mg/L
16
1.5
13.6mg/L — 21.3mg/L
16
2
18.3mg/L — 28.4mg/L
16
2.25
20.5mg/L — 31.8mg/L
Above maximum therapeutic concentration
16
2.5
22.8mg/L — 35.4mg/L
Above maximum therapeutic concentration
24
2
8.7mg/L — 23.3mg/L
Sharp fluctuation
24
2.25
9.86mg/L — 25.9mg/L
Sharp fluctuation
24
2.5
10.9mg/L — 29mg/L
Sharp fluctuation
T=12, R=0.75
T=12, R=1
T=12, R=1.5
T=12, R=1.75
T=12, R=2
T=16, R=1 T=16, R=1.25
T=16, R=1.5 T=16, R=2
T=16, R=2.25 T=16, R=2.5
T=24, R=2 T=24, R=2.5
Analysis: If the drug can be packaged in a timed release form so that Tb = 8 and R is also adjusted likewise, we perform the simulations for the dosage of interval of a 12 hour frequency. We observe zero sharp fluctuations. Every graph seems to produce the concentration level within the recommended range of 10mg/L to 30mg/L when R is between 0.75 mg/h and 1.75 mg/h.
=>Calculation and Analysis of new dosage strength ‘A’
We can calculate the new strength level of the drugs as:
(12 hour frequency interval): A=RTb = 0.75 mg/h * 8h = 6mg
A=RTb = 1.75 mg/h * 8h = 14mg
Same analysis can be performed for 16 hour frequency interval. We observe zero sharp fluctuations and every graph produce the concentration level within the recommended range of 10mg/L to 30mg/L; R being in between 1.25mg/h and 2mg/h.
=>Calculation and Analysis of new dosage strength ‘A’
We can calculate the new strength level of the drugs as:
(16 hour frequency interval): A = RTb=1.25 mg/h * 8h = 10mg
A = RTb=2.00 mg/h * 8h = 16mg
Thus, this changes our recommendation.
Simulation Program Maple: We used the following code and simulated varying R and P values.
g :=t—> piecewise(0 <= t and t <= 1, R, 1 < t and t < P, 0, P < t and t < P+1, R, P+1 < t and t < 2*P, 0)
DEplot({diff(x(t), t) = (1/6)*g(t)+(1/6)*(.31*.25)*y(t)-x(t)*(.29+.16), diff(y(t), t) = (.29*6)*x(t)/(.25)-.31*y(t)}, [x(t), y(t)], t = 0 .. 40, x = 0 .. .50, y = 0 .. 80, scene = [t, y], [[x(0) = 0, y(0) = 0]], stepsize = .1, color = blue)
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