Pharmacokinetics is the study of the time variation of drug and metabolite levels in the various...

 

  
 

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Pharmacokinetics is the study of the time variation of drug and metabolite levels in the various fluids and tissues of the body. The discipline frequently makes use of compartment models to interpret data. In this problem, we consider a simple blood-brain compartment model (Figure 3.P.9), that could be used to help estimate dosage strengths of an orally administered antidepressant drug. The rate at which the drug moves from compartment i to compartment j is denoted by the rate constant kj; while the rate at which the drug is removed from the blood is represented by the rate constant K. A pharmaceutical company must weigh many factors in determining drug dosage pa- rameters; of particular importance are dosage strengths that will provide flexibility to a physician in determining individual dosage regimens to conveniently maintain concentra- tion levels at effective therapeutic values while minimizing local irritation and other adverse side effects. R + FIGURE 3.P.9 Tp Compartment 1 = Blood, Compartment 2 = Brain, Input d(t) -Tb Blood x1, V₁, C1 K k21 k12 Brain x2, V₂, C2 A two-compartment model for periodic drug dosages. dxj dt Assuming that the drug is rapidly absorbed into the bloodstream following its introduction into the stomach, a mathematical idealization for the dosage regimen is that of a periodic square wave Project 3 PROBLEMS 1. If x;(t) represents the amount of drug (milligrams) in compartment j, j = 1, 2, use Figure 3.P.9 and the mass balance law dx1 dt dx2 dt to show that x₁ and x2 satisfy the system = compartment j input rate - compartment j output rate, where R is the rate of uptake (milligrams/hour) into the bloodstream, T, is the time period during which the drug is absorbed into the bloodstream following oral administration, and Tp is the length of time between doses. = -(K + k₂1)x1+k12x2 + d(1) d(t) = = K21X1 - K12x2. (i) -18.8 (ii) 2. If ci(t) denotes the concentration of the drug and Vi denotes the apparent volume of distribution in compartment i, use the relation c; = x/V; to show R, 0≤t≤ Tb 0, Tb ≤ t < Tp, that the system (ii) is transformed into dc₁ dt dcz dt k21 0.29/h K12 V2 = -(K + k21)c₁ + -C₂ + V₁ Vik21 V₂ -C₁ - K12c₂. 7/7d(1) 3. Assuming that x₁ (0) = 0 and x2 (0) = 0, use the pa- rameter values listed in the table below to perform numerical simulations of the system (iii) with the goal of recommending two different encapsulated dosage strengths A = RT, for distribution. k12 K V₁ 0.31/h 0.16/h 6L (iii) V2 0.25 L Tb 1 h Use the following guidelines to arrive at your recommendations: It is desirable to keep the target concentra- tion levels in the brain as close as possible to constant levels between 10 mg/L and 30 mg/L, depending on the individual patient. The therapeutic range must be above the minimum effective concentration and below the minimum toxic concentration. For the purpose of this project, we will specify that concentration fluc- tuations should not exceed 25% of the average of the steady-state response. As a matter of convenience, a lower frequency of administration is better than a higher fre- quency of administration; once every 24 hours or once every 12 hours is best. Once every 9.5 hours is unacceptable and more than 4 times per day is unacceptable. Multiple doses are ac- ceptable, that is, "take two capsules every 12 hours." 4. If a dosage is missed, explain through the simula- tions why it is best to skip the dose rather than to try to "catch up" by doubling the next dose, given that it is dangerous and possibly fatal overdose on the drug. Or, does it not really matter in the case of the given parameter values? 5. Suppose the drug can be packaged in a timed- release form so that Tb = 8 h and R is adjusted accordingly. Does this change your recommenda- tions? Pharmacokinetics is the study of the time variation of drug and metabolite levels in the various fluids and tissues of the body. The discipline frequently makes use of compartment models to interpret data. In this problem, we consider a simple blood-brain compartment model (Figure 3.P.9), that could be used to help estimate dosage strengths of an orally administered antidepressant drug. The rate at which the drug moves from compartment i to compartment j is denoted by the rate constant kj; while the rate at which the drug is removed from the blood is represented by the rate constant K. A pharmaceutical company must weigh many factors in determining drug dosage pa- rameters; of particular importance are dosage strengths that will provide flexibility to a physician in determining individual dosage regimens to conveniently maintain concentra- tion levels at effective therapeutic values while minimizing local irritation and other adverse side effects. R + FIGURE 3.P.9 Tp Compartment 1 = Blood, Compartment 2 = Brain, Input d(t) -Tb Blood x1, V₁, C1 K k21 k12 Brain x2, V₂, C2 A two-compartment model for periodic drug dosages. dxj dt Assuming that the drug is rapidly absorbed into the bloodstream following its introduction into the stomach, a mathematical idealization for the dosage regimen is that of a periodic square wave Project 3 PROBLEMS 1. If x;(t) represents the amount of drug (milligrams) in compartment j, j = 1, 2, use Figure 3.P.9 and the mass balance law dx1 dt dx2 dt to show that x₁ and x2 satisfy the system = compartment j input rate - compartment j output rate, where R is the rate of uptake (milligrams/hour) into the bloodstream, T, is the time period during which the drug is absorbed into the bloodstream following oral administration, and Tp is the length of time between doses. = -(K + k₂1)x1+k12x2 + d(1) d(t) = = K21X1 - K12x2. (i) -18.8 (ii) 2. If ci(t) denotes the concentration of the drug and Vi denotes the apparent volume of distribution in compartment i, use the relation c; = x/V; to show R, 0≤t≤ Tb 0, Tb ≤ t < Tp, that the system (ii) is transformed into dc₁ dt dcz dt k21 0.29/h K12 V2 = -(K + k21)c₁ + -C₂ + V₁ Vik21 V₂ -C₁ - K12c₂. 7/7d(1) 3. Assuming that x₁ (0) = 0 and x2 (0) = 0, use the pa- rameter values listed in the table below to perform numerical simulations of the system (iii) with the goal of recommending two different encapsulated dosage strengths A = RT, for distribution. k12 K V₁ 0.31/h 0.16/h 6L (iii) V2 0.25 L Tb 1 h Use the following guidelines to arrive at your recommendations: It is desirable to keep the target concentra- tion levels in the brain as close as possible to constant levels between 10 mg/L and 30 mg/L, depending on the individual patient. The therapeutic range must be above the minimum effective concentration and below the minimum toxic concentration. For the purpose of this project, we will specify that concentration fluc- tuations should not exceed 25% of the average of the steady-state response. As a matter of convenience, a lower frequency of administration is better than a higher fre- quency of administration; once every 24 hours or once every 12 hours is best. Once every 9.5 hours is unacceptable and more than 4 times per day is unacceptable. Multiple doses are ac- ceptable, that is, "take two capsules every 12 hours." 4. If a dosage is missed, explain through the simula- tions why it is best to skip the dose rather than to try to "catch up" by doubling the next dose, given that it is dangerous and possibly fatal overdose on the drug. Or, does it not really matter in the case of the given parameter values? 5. Suppose the drug can be packaged in a timed- release form so that Tb = 8 h and R is adjusted accordingly. Does this change your recommenda- tions?

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Solutions Symbols Explained in the questions are Symbol Use and Units What Symbols Represents x 1 x 2 Function milligrams Represents mass of drug in each compartment at any time t V 1 V 2 Constant lit... View the full answer