A Blood Brain Pharmacokinetic Model

A bloodline Brain Pharmacokinetic ModelPharmacokinetics, an emerging field in BioPhysics and chemistry is the study of the time variation of do do drugss and metabolite aims in various tissues and fluids of the body. Compartment flummoxs are used to interpret data. In our problem, we consider a simple blood-brain compartment model as shown in the figure belowk21Input d(t) k12Kwhere, Compartment 1 = BloodCompartment 2 = BrainThis model is made such that it can attention to help estimate venereal infection 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 support out aid a physician in determining a patient ofs dosage in order to maintain the right concentration take aims and also minimizing bili ousness 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 followsBased on our model and the equations we can solve the problems1. If we permit xj(t) be the nitty-gritty 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 findSystem in Blood compartmentSystem in Brain compartmentFrom (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)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 belowk21k12KV1V2Tb0.29/h0.31/h0.16/h6L0.25L1hand with the numerical simulation program Maple , we can practise simulations of the system with given parameters to recommend two different encapsulated dosage strengths A=RTb.= Guidelines to use for recommendation of drug dosage1) Target concentration level in the brain should be kept as close as possible between levels 10 mg/L and 30 mg/L and concentration hesitation should not exceed 25% of the average of the steady-state response.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)depth psychology 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 foll owing dataTp(h)R (mg/h)Steady-state varianceComments849.04 mg/L to 12.5 mg/L down the stairs effective remediation concentration8511.7 mg/L to 15.5 mg/L8614.4 mg/L to 19.2 mg/L8819.2 mg/L to 25.3 mg/L8921.1 mg/L to 27.9 mg/L81023.2 mg/L to 31.2 mg/LAbove maximum therapeutic concentration12510.9 mg/L to 6.5 mg/LBelow minimum therapeutic concentration1268.6 mg/L to 14.1 mg/LBelow minimum therapeutic concentration1278.32 mg/L to 15.1 mg/LBelow minimum therapeutic concentration12810.6 mg/L to 18.3 mg/L121013.2 mg/L to 22.8 mg/L121317.9 mg/L to 30 mg/L16109.11 mg/L to 19.5 mg/LSharp fluctuations Below minimum therapeutic concentration161210.7 mg/L to 23.5 mg/LSharp fluctuations.161311.5 mg/L to 25.4 mg/LSharp fluctuations.161412.5 mg/L to 27.3 mg/LSharp fluctuations.161614.3mg/L 31.4mg/LSharp fluctuations Above maximum therapeutic concentration24156.19mg/L 24mg/LSharp fluctuations Below minimum therapeutic concentration24208.52mg/L 32mg/LSharp fluctuations Above maximum therapeutic concentrationObtained corresponding Graphs from Maple and their respective Tp and R values are listed belowTp = 8, R = 4 Tp = 8, R = 5Tp = 8, R = 6 Tp = 8, R = 8Tp = 8, R = 9 Tp = 8, R = 10Tp = 12, R = 6 Tp = 12, R = 8Tp = 12, R = 10 Tp = 12, R = 12Tp = 12, R = 13Tp=16, R=10 Tp=16, R=12Tp=16, R=13 Tp=16, R=14Tp=16, R=16Tp=24, R=15 Tp=24, R=20Some CommentsWhen Tp= 8 and R = 4, the recommended dosage is below minimum therapeutic concentration extend.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 concent ration range.=Calculation and Analysis of dosage strength ANow 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 mgA = 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 mgA = RTb = 13 mg/h x 1h= 13 mg4. 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 phoneyScenario missed a dosage and skipped Scenario absent a dosage catching upAnalysis From the scenarios simulations above, we can have a clear picture of what will go through the patients drug level.In the 1st scenario, where the patient missed a dosage and skipped, the concentration level in the brain of the patient stays wi thin 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 MapleTp(h)R(mg/h)Steady-state varianceReasons120.7510.4mg/L 13mg/L12113.9mg/L 17mg/L121.521mg/L 25.5mg/L121.7524.5mg/L 29.8mg/L12228.1mg/L 34mg/LAbove maximum therapeutic concentration1619mg/L 14.3mg/LBelow minimum therapeutic concentration161.2511.2mg/L 17.7mg/L161.513.6mg/L 21.3mg/L16218.3mg/L 28.4mg/L162.2520.5mg/L 31.8mg/LAbove maximum therapeutic concentration162.522.8mg/L 35.4mg/LAbove maximum therapeutic concentration2428.7mg/L 23.3mg/LSharp fluctuation242.259.86mg/L 25.9mg/LSharp fluctuation242.510.9mg/L 29mg/LSharp fluctuationT=12, R=0.75T=12, R=1T=12, R=1.5T=12, R=1.75T=12, R=2T=16, R=1 T=16, R=1.25T=16, R=1.5 T=16, R=2T=16, R=2.25 T=16, R=2.5T=24, R=2 T=24, R=2.5Analysis 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 radical dosage strength AWe can calculate the new strength level of the drugs as(12 hour frequency interval) A=RTb = 0.75 mg/h * 8h = 6mgA=RTb = 1.75 mg/h * 8h = 14mgSame 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 AWe can calculate the new strength level of the d rugs as(16 hour frequency interval) A = RTb=1.25 mg/h * 8h = 10mg A = RTb=2.00 mg/h * 8h = 16mgThus, this changes our recommendation.Simulation Program Maple We used the following code and simulated varying R and P values.g =t piecewise(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)