This is my work (layout edited to better suit a web environment) for Project Four in Chapter Three of the Brannan and Boyce Differential Equations text, full title and authors as:
- Differential Equations with Boundary Value Problems: An Introduction to Modern Methods and Applications
- James R. Brannan (Clemson University), William E. Boyce (Rensselaer Polytechnic Institute)
- Copyright 2010
The project concerns a two compartment (blood and brain) model of the time-variant concentration of an anti-depressant drug in each compartment. I won’t post the problem text in its entirety, though I believe the work-flow speaks for the problem statement and if you have the text available then you of course have the problem statement. Many thanks to the guys at codecogs and their outstanding online LaTeX equation editor.
DISCLAIMER: I do not have a solutions manual to the text. The veracity of this report should, like everything you encounter, be subjected to your careful scrutiny. I’m just one student, after all. For all you know, this entire report could be systemically flawed.
0.) Project Exposition
Pharmacokinetics studies time-variant drug and metabolite levels in the body. A simple compartment model concerning an antidepressant drug within the brain and blood is detailed below:
The various symbols are explained:
|Symbol||Classification and Units||Explanation|
|x1,x2||Function; milligrams||Represents mass of drug in respective compartment at any time t.|
|V1,V2||Constant; liters||The volume of distribution.|
|c1,c2||Function; milligrams per liter||Represents the concentration of drug in respective compartment at any time t.|
|k12,k21||Constant; 1/hour||A rate constant; describes rate at which drug travels from one compartment to another.|
|K||Constant; 1/hour||Rate at which drug is released from bloodstream.|
|R||Constant*; milligrams/hour||Rate of uptake of drug into bloodstream.|
|Tb||Constant; hours||Time for complete uptake of drug into bloodstream.|
|Tp||Constant*; hours||Time between doses.|
*Experimentally determined via numerical simulation
The ultimate goal is to establish an appropriate prescription regimen with respect to some given parameters in the project statement. Numerical simulation is employed to develop said regimen by systematically experimenting with R, Tp, and Tb values.
1.) Compartment Model Using Mass Balance Law
The mass balance law is as such:
dxi/dt = compartment i input rate – compartment i output rate
Using the above illustration that details the inputs and outputs in each compartment, we arrive at the following:
This system of equations describes the change in mass of the drug residing in each compartment, with respect to time (The bloodstream for x1 and the brain for x2).
2.) Conversion From a Mass System to a Concentration System
Knowing the amount of a drug in the system at any given time is not as important as its concentration per unit volume within the volume of distribution (blood, brain, etc). Concentration, i.e. density, is defined as:
The related rates of change with respect to time, knowing that V is a constant, are simply:
Defining xi in terms of ci and Vi is simple:
Using both of these relations, we can transform the mass system into a concentration system.
For the bloodstream:
For the brain:
3.0) Recommended Dosages
The following table of values for the various constants is given:
|.29/h||.31/h||.16/h||6L||.25 L||1 h|
An encapsulated dosage strength A with units mg is defined as:
The following parameters are given:
- The concentration in the brain must be as close as possible to constant levels between 10mg/L and 30mg/L. Fluctuations shall not exceed 25% of the steady-state response.
- Lower frequencies of administration are preferrable (every 12 to 24 hours). 9.5 hours or less between every dose is an unacceptable frequency. Taking multiple pills at once is permitted.
3.1) Experimental Methodology
The following methodology is used in collecting numerically simulated data:
1.) Utilizing ODE Architect software, populate the equation box as such:
c1'=-(.16+.29)*c1 + (.29*.25*c2)/6 +(1/6)*SqWave(t,Tp,Tb)*R c2'=(6*.29*c1)/.25 - (.31*c2)
where Tp will be the experimental time between doses, Tb the given “1” hour in the above table, and R the experimental rate of uptake. c1 is the concentration in the bloodstream and c2 the concentration in the brain.
2.) Populate the initial conditions as such:
x1(0) and x2(0) are both given to be zero in the problem statement. Hence, ci(t)=xi(t)/Vi gives c1(0)=0 and c2(0)=0
3.) Populate the integration conditions as such:
Tp is the experimental time between doses. i.e., if you are testing various R values for a dosage frequency of 12 hours, then the interval should be 12 hours as well. This will ease simulation of missed dosages and doubled dosages. The selected number of points is arbitrary, but should be large enough to permit an accurate description of function behaviour (at least 100).
4.) Systematically vary the Tp and R values, testing 12, 16, and 24 hour frequencies with various rates of uptake, up to an appropriate stopping level (once levels are within 2mg of toxicity).
5.) For each instance of Tp and R values, record the minimum and maximum concentration within the brain during steady-state response, any fluctuation above 30mg or below 10mg within the brain, and any sharp fluctuations (greater than 25% of the steady-state response) in concentration in either the bloodstream or the brain.
3.2) Data Collection
|Tp (h)||R (mg/h)||Steady-State Variance||Notes|
|12||8||10mg/L to 19mg/L|
|12||10||12mg/L to 21mg/L|
|12||12||14mg/L to 26mg/L|
|12||13||15.5mg/L to 28mg/L|
|12||14||17mg/L to 30mg/L|
|16||10||10mg/L to 25mg/L||Sharp fluctuations|
|16||12||10mg/L to 26mg/L||Sharp fluctuations|
|16||14||11mg/L to 28mg/L||Sharp fluctuations|
|16||16||11mg/L to 29mg/L||Sharp fluctuations|
|16||18||12mg/L to 31mg/L||Sharp fluctuations|
|24||10||2mg/L to 15mg/L||Sharp fluctuations, dips below minimum therapeutic concentration|
|24||12||4mg/L to 18mg/L||Sharp fluctuations, dips below minimum therapeutic concentration|
|24||14||4mg/L to 21mg/L||Sharp fluctuations, dips below minimum therapeutic concentration|
|24||16||5mg/L to 24mg/L||Sharp fluctuations, dips below minimum therapeutic concentration|
|24||18||6mg/L to 27mg/L||Sharp fluctuations, dips below minimum therapeutic concentration|
|24||20||7mg/L to 30mg/L||Sharp fluctuations, dips below minimum therapeutic concentration|
3.3) Data Analysis
For 12 hour dosage frequencies, no sharp fluctuations are observed and the steady-state response is always within the 10mg/L to 30mg/L range. An appropriate rate of uptake is between 8mg/hour and 13 mg/hour. An appropriate dosage is consequently between 8mg and 13mg.
For 16 hour dosage frequencies, sharp fluctuations are observed but the steady-state response is always within the 10mg/L to 30mg/L range. An appropriate rate of uptake is between 10mg/hour and 14mg/hour. An appropriate dosage is consequently between 10mg and 14mg.
For 24 hour dosage frequencies, sharp fluctuations are observed and the steady-state response remained below the minimum therapeutic level for a significant portion of time. An appropriate rate of uptake (and consequential dosage) is not available, seeing as parts of the fluctuations in the steady-state are always either below therapeutic levels or above minimum toxicity levels.
3.4) Final Recommended Dosages
The sharp fluctuations in the 16 and 24 hour frequencies and the drops below therapeutic level with the 24 hour frequency make both suggested frequencies inappropriate. The best frequency is every 12 hours. Recommended dosages are:
The lower dosage corresponds to fluctuations between 10 and 19 mg/L and is intended for less severe cases. The higher dosage corresponds to fluctuations between 15.5 and 28 mg/L and is intended for more severe cases.
4.) Simulation of a Skipped Dosage Followed by a Double Dosage
A common misconception about medication is the idea of doubling the dosage after missing the previous dose. At first glance, it seems like a logical step to take in order to stay on schedule. However, it neglects to take into account the new concentrations within the various volumes of distribution, which could very well be above minimum toxic levels. Below is a numerical simulation of such an event:
The above simulation illustrates the behavior of a 12 hour, 13 mg per dose, perscription regimen. After several days of steady-state response, a dosage is missed, bringing the concentration in the brain to zero mg/L. The next scheduled dosage is then doubled to 26 mg, producing a sharp spike in concentration up to 55 mg/L that remains above the 30 mg/L minimum toxicity for several hours.
Given this simulation, it is clear that doubling the dosage in response to missing a previous dose is dangerous. It would have been far better to simply take the normal dose following the missed one. In that case, there is a time period where there is no drug in the system, but at no point is minimum toxicity reached.
5.0) Modification of the Time-for-Complete-Uptake (Tb) Value
Assuming that the Tb value is changed –perhaps by the development of a delayed release capsule– the question becomes whether dosage recommendations are changed. For this problem, we assume Tb = 8 and that we modify the R values appropriately. The same experimental methodology as in 3.1 is utilized.
5.1) Data Collection
|Tp (h)||R (mg/h) -> dosage||Steady-state variance||notes|
|12||1 -> 8 mg||11mg/L to 17mg/L|
|12||1.25 -> 10 mg||15mg/L to 19mg/L|
|12||1.5 -> 12 mg||18mg/L to 23mg/L|
|12||1.75 -> 14 mg||21mg/L to 27mg/L|
|12||1.875 -> 15 mg||23mg/L to 29mg/L|
|12||2 -> 16 mg||25mg/L to 31mg/L|
|16||1.25 -> 10 mg||10mg/L to 16mg/L|
|16||1.5 -> 12 mg||11mg/L to 19mg/L|
|16||1.75 -> 14 mg||13mg/L to 22mg/L|
|16||2 -> 16 mg||15mg/L to 26mg/L|
|16||2.25 -> 18 mg||18mg/L to 29mg/L|
|16||2.5 -> 20 mg||20mg/L to 32mg/L|
|24||1.25 -> 10 mg||4mg/L to 13mg/L|
|24||1.5 -> 12 mg||5mg/L to 16mg/L|
|24||1.75 -> 14 mg||6mg/L to 19mg/L|
|24||2 -> 16 mg||7mg/L to 22mg/L|
|24||2.25 -> 18 mg||8mg/L to 24mg/L|
|24||2.5 -> 20 mg||9mg/L to 27mg/L|
|24||2.75 -> 22 mg||10mg/L to 30mg/L|
5.2) Data Analysis
For 12 hour dosage frequencies, no sharp fluctuations are observed and the steady-state response is always within the 10mg/L to 30mg/L range. With the 8 hour release, the amplitude is smaller, averaging 6 mg/L. An appropriate dosage is between 8 and 15 mg/L.
For 16 hour dosage frequencies, no sharp fluctuations are observed and the steady-state response is always within the 10mg/L to 30mg/L range. However, the amplitudes are still averaging 9-10 mg/L. An appropriate dosage is between 10 and 18 mg/L.
For 24 hour dosage frequencies, no sharp fluctuations are observed but the steady-state response is still below the minimum therapeutic concentration for nearly all cases. The singular case of a 22mg dose teters between 10mg/L and 30mg/L. An appropriate dosage really isn’t available, since most responses are below steady-state or lack a safety threshold.
5.3) Final Dosage Recommendations
The sharp fluctuations were completely removed in the 12, 16, and 24 hour dosage frequencies with the 8 hour release. The 12 and 16 hour dosage frequencies always stayed within the 10mg/L to 30mg/L concentration, but the 24 hour dosage frequency still had a steady-state response that was below the minimum therapeutic concentration for a significant amount of time. The amplitudes for each dosage frequency were significantly shortened with an 8 hour release when compared to the one hour release.
In conclusion, the 24 hour dosage is still inappropriate. The 16 hour dosage, while within parameters, is an awkward dosage frequency that occurs at different times of the day throughout the week. The 12 hour dosage has the tightest amplitude, permitting the widest range of dosage strengths. The new dosage recommendations are below: