A Blood Brain Pharmacokinetic Model

A Blood Brain Pharmacokinetic ModelPharmacokinetics, an emerging firmament in BioPhysics and chemistry is the study of the time variation of do drugs and metabolite take aims in various tissues and fluids of the body. Compartment models are gived to interpret data. In our problem, we consider a simple source-brain compartment model as shown in the figure down the stairsk21 stimulus d(t) k12Kwhere, Compartment 1 = BloodCompartment 2 = BrainThis model is make such that it good deal embolden to help estimate pane specialitys of an orally administered antidepressant drug. The rate of movement of drug from compartment i to compartment j is de noned by the rate constant kji and the rate at which the drug is removed from the countercurrent is represented by the rate constant K. A pharmaceutical confederacy must deal with many factors like battery-acid strengths that exit aid a physician in determining a endurings acid in order to maintain the right tightfistedness levels and as well as minimizing irritation and separate 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 loony toons will be of a periodic square wave given as followsBased on our model and the equations we foot solve the problems1. If we let xj(t) be the amount of drug in milligrams in compartment j, j =1,2. The mass fit law states(i)Using the mass balance law and the figure, we find formation in Blood compartmentSystem in Brain compartmentFrom (i) and the higher up equations, we can find the interest(ii)The systems above are the rates of drug both(prenominal)place 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 dominate(iii)Dividing the above systems by V1 and V2 respectively, we get 3. A ssuming x1(0) =0 and x2(0) =0, and the various parameters listed infrak21k12KV1V2Tb0.29/h0.31/h0.16/h6L0.25L1hand with the numerical theoretical account program Maple , we can perform simulations of the system with given parameters to recommend twain different encapsulated venereal infection strengths A=RTb.= Guidelines to use for recommendation of drug back breaker1) 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.2) Lower frequency of administration (once all(prenominal) 24 mos 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 multiplication per day is unacceptable which makes uttermost allowable dose to be 3, devising 3 doses at 8 hours interval per day the best choice. We can then simul ate from Tp = 8 to Tp = 12, 16 and 24.From the numerical simulations obtained from Maple, we obtain the interest dataTp(h)R (mg/h)Steady-state varianceComments849.04 mg/L to 12.5 mg/LBelow effective cure 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 healing(predicate) concentration12510.9 mg/L to 6.5 mg/LBelow stripped-down therapeutic concentration1268.6 mg/L to 14.1 mg/LBelow stripped 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/L sourish fluctuations Below minimum therapeutic concentration161210.7 mg/L to 23.5 mg/L sharply 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 fluc tuations 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 dot is below minimum therapeutic concentration range.When Tp= 8 and R = 10 , the recommended superman 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 i s 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 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 leap the dose than to try to catch up and double the dose and eventually 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 simulatedScenario missed a dosage and skipped Scenario missing a dosage catching upAnalysis From the scenarios simulations above, we can have a introduce picture of what will go through the patients drug level.In the beginning(a) 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 overly 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 chart seems to kindle 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 drugs 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, colorise = blue)