Mathematical Material. Calculus: Objectives. Calculus. 17 January Calculus

Transcription

1 Mathematical Material Calculus Differentiation Integration Laplace Transforms Calculus: Objectives To understand and use differential processes and equations (de s) To understand Laplace transforms and their use in the integration of de s (analytical integration) To understand Euler s method of numerical integration To understand integration as a summation process Calculus Pharmacokinetics is the study of the rate of drug movement and transformation Studied using differential calculus Drug performance is related to total amount of drug absorbed Studied using integral calculus

2 Differential Equations Most (but not all) rate processes are first or zero order Write differential equations from diagram Convert differential equation into integrated form Numerical methods - Euler s method and others nalytical method - Laplace transform Draw the diagram Zero order rate constant: First order rate constants: and Write the de - Component 2

3 Write the de - Component Positive Rates - arrow into component Write the de - Component Negative Rates - arrow away from component Write the de - Component dx =+ X Zero order just enter rate constant First order multiply rate constant by component at the END of the arrow 3

4 Write the de - Component 2 dx 2 =+ X X 2 Zero order just enter rate constant First order multiply rate constant by component at the END of the arrow Write the de - Component 3 dx 3 =+ X 2 Zero order just enter rate constant First order multiply rate constant by component at the END of the arrow nother example Write the de - Component where is a first order rate constant L 4

5 nother example Write the de - Component dx = X Rate of Change of mount dx = X Initial Condition olus Dose = X (0) mount Remaining - X X (0) dx = X mount X Slope = - X Time 5

6 Numerical Integration - Euler Point-Slope Method Point Initial Value - X (0) Slope Differential Equation - X The Equation dx = X n Example - Euler s Method Choose stepsize (ss) = X (new) = X (old) + slope ss = X (old) + ( X (old)) ss =00 + ( ) 0. X = = 97.5 (new) 0. X (0) = 00 = 0.25 n Example - Euler s Method Time X X X (0) = 00 = 0.25 L 6

7 nalytical Integration using Laplace Transform Laplace transform converts Differential Equation into the Laplace s domain Equation can be rearranged algebraically ack transform provide the solution Similar to how logarithms convert multiplication and division into addition and subtraction Laplace Transform - General Method Write the differential equation(s) Transform to the Laplace, s, domain Rearrange to solve for variable(s) of interest Take the back transform to the time domain to get the integrated equation Laplace Integral Laplace transform is based on the Laplace Integral Lf(t) = e s t 0 f(t) where f(t) is the function to be transformed from the time (t) domain to the Laplace (s) domain - for the mathematicians in the class 7

8 Three Useful Transforms Transform of the constant L() = s Transform of variable such as drug amount L( X) = X Transform of a differential equation L dx = s X X 0 Laplace Transform Table Function: f(t) Laplace: f(s) s s e a t ( s + a) a ( e a t ) s ( s +a) ( b a) ( e a t e b t ) ( s + a) ( s + b) a t a ( ) 2 e a t s 2 ( s + a) From Table I, Mayersohn and Gibaldi, mer. J. Pharm. Ed., 34(4) (970) s a PDF file Example - I.V. olus One compartment Model - I.V. olus Transform d.e. and variable Solve for X dx = kel X s X X 0 = kel X s X +kel X = X 0 = Dose 8

9 Example - I.V. olus Rearrange and solve for X X = Dose s + kel ack transform using Laplace Table X = Dose e kel t Divide by V to Determine Cp Cp = X V = Dose V e kel t Example - I.V. Infusion One compartment Model - I.V. Infusion dx = kel X Transform d.e. and variable s X X 0 = s kel X Solve for X s X +kel X = s Since X 0 = 0 Example - I.V. Infusion Rearrange and solve for X X = s ( s+ kel) ack transform using Laplace Table ( ) X = e kel t kel Divide by V to Determine Cp Cp = X V = V kel e kel t ( ) 9

10 Integration of Cp versus Time Integration of dcp/ give Cp versus time Integration of Cp gives rea under the Time versus Cp curve (UC) Integration dcp/ Cp UC Differentiation cceleration Speed Distance Integration of Cp versus Time UC is used a measure the dosage form performance If the model or equation is known then an analytical solution is possible One model independent, numerical method is the trapezoidal rule rea under the Curve (UC) Concentration (mg/ml) dd time slices to calculate the total area 00 UC = 80 t=0 Cp Time (hr) Cp 0

11 UC - nalytical Solution UC = t=0 Cp if Cp = Dose V e kel t Dose UC = V e kel t t=0 taking the constants (Dose/V) outside the integral UC = Dose V e kel t t=0 UC - nalytical Solution... UC = Dose V e kel t t=0 now taking the integral using Math Tables gives UC = Dose V e kel t kel t=0 that is integration from 0 to expanding using both limits gives UC = Dose V kel e kel 0 kel e kel UC - nalytical Solution... UC = Dose V kel e kel 0 kel e kel since e -kel = 0 and e -kel 0 = UC = Dose V 0 kel kel and UC = Dose V kel = Dose V kel = Cp(0) kel

12 UC - Numerical Solution Concentration (mg/ml) UC Segment 60 UC Time (hr) One segment - Shape of a Trapezoid UC - One Segment Concentration (mg/ml) t(3) t =t(4) UC t =t(3) = Cp(3) Cp(3) + Cp(4) ( t(4) t(3) ) 2 Cp(4) Time (hr) t(4) UC - 0 to Last Data Point t =t(last) Cp(0) + Cp() UC t =0 = 2 Cp(6) + Cp(last) Cp(0) t() + ( t(last) t(6) ) Cp()+ Cp(2) ( t(2) t() ) + 2 Concentration (mg/ml) Time (hr)? 2

13 UC - Zero to Infinity t UC = UC = t= t =0 = UC t(last) t= 0 + UC t =t(last) with t = UC t =t(last) = Cp t=t(last) = Cp(last) kel UC - Example Calculation Time (hr) Cp (µg/ml) UC UC (µg.hr/ml) 0 00 * Total 8.9 # 29.9 * estimated by back-extrapolated # using kel = hr - 3