Problem-Solving Example #9

PharmacoKinetics

A PharmacoKinetics open-two-compartment model with first order absorption into elimination from central compartment (blood cleared of drug through the liver and/or kidneys) is presented here. The body tissues utilize the drug and therefor an amount is removed by the body's filtering system, i.e. the liver and/or kidneys.

Rate of change in compartments is stated by the following differential equations:

Plasma compartment

Tissue compartment

where Ky represent Rate constants; y = a, 10, 12, and 21;

Ai = Amount of drug at the ith site: 0. Absorption site; 1. Compartment 1; and 2. Compartment 2.

This system of differential equations can be solved analytically using La Place transforms. These solutions are usually expressed in terms of drug concentrations (i.e., parameters A, B, & C). The model equation for compartment 1 is

C = - ( A + B )

where Cp(t) is the plasma concentration at time t;
and 'alpha' & 'beta' are hybrid parameters derived from K12, K21, K10, and Ka.

The half-life of 'beta' is constrained to the range of three to nine years, and thus, adds the two constraints: Half_life >= 3 and <= 9 years where the Half_life = ln(2) / beta

Relative error in curve fitting was chosen due to the huge swing in amplitude over time.

PROSE Program Listing

Global All
Problem Pharmaco	! -Kinetic parameters for open-two-compartment model
  Dimension Time( 12), Plasma( 12), Error( 12), Lows(5), Half(2)
  ! Observed plasma concentrations ... Oral tablet of 10 mg
  Data Time/0, .333, .5, .667, 1, 2, 4, 6, 8, 12, 24, 32/        ! X-Data
  Data Plasma/1.e-4, .657, .727, .763, .695, .51, .307, .161,      ! Y-Data
    .135, .046, .021, .008/	! X-Units=Hr. & Y-Units=Mcg/Ml
  Data Lows/ 5*0.D0/, Half/ 2*0.D0/
  Npoints = 12:   desiredHalflife = 1.2345
  Write(1,*) ' Enter Initial Starting Value ... '
  Read *, X
  A=X:     B=X:    Ka=X:   Alpha=X:   Beta=X      ! Initial Values
  ! The next 'find' statement is many simulations combined into one.
  ! The solver 'jupiter' finds -a solution- that minimizes 'sum'
   Find A, B, Ka, Alpha, Beta; In Concentr; By Jupiter;
     With Lowers Lows; Holding Half; To Minimize Sum

  ! Changed problem to an Inverse Problem ... notice minimal difference.
  ! Takes a run or two to insure model & optimal solution.
  Find A, B, Ka, Alpha, Beta; In Concentr; By Jupiter;
    With Lowers Lows; Holding Half; To Match HalflineError
End
Model Concentr	          ! Concentration In Compartment 1
  Sum = 0
  Do 10 i = 1, Npoints
    T = Time( i)
    C1 = A * Exp( - Alpha * T):   C2 = B * Exp( - Beta * T):   C = -(A + B)
    C3 = C * Exp( - Ka * T):      Cp_t = C1 + C2 + C3
    Error(i) = (Plasma( i) - Cp_t) / Plasma( i)
    Sum = Sum + Error(i)**2
10	Continue
    Halflife = Log( 2) / Beta:     HalflineError = desiredHalflife - Halflife
    Half(1) = Halflife - 3		! 3 Years Minimum
    Half(2) = 9 - Halflife		! 9 Years Maximum
End