Print Medical Pharmacokinetics #3

Topics in this printout

• Pharmacokinetic models
• Formulae that must be recalled from memory

Pharmacokinetic Models

Models

• Model Independent Pharmacokinetics
• One Compartment Pharmacokinetic Model
• Two Compartment Pharmacokinetic Model

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Last modified: 03 Sep 1996 22:49 glc

Model Independent Pharmacokinetics

• Introduction
• Calculations

Introduction

• Have been using this approach so far in this course
• Treats body as black box, making no assumptions
• Monitor Inputs, Outputs, and Concentration in Plasma
• Basic equation of pharmacokinetic dose calculations

• Comments on the formula:
  • ... fine for every-day, simple dose adjustments where one has "measured" peaks and/or troughs
  • ... cannot predict peaks or troughs
  • ... cannot use it to extrapolate concentrations to estimate "withdrawal times"

Calculations

Estimating dosing rate

• Use formula

• Get estimate of CLEARANCE from literature for this drug
• Find desired STEADY STATE CONCENTRATION from literature
  • Aim for middle of therapeutic range
  • Assume Css(ave) corresponds to Css
    Css is knowable only for intravenously infused drugs
• Multiply to find DOSING RATE
• Determine optimal DOSING INTERVAL
  (usually from literature as first approximation)
• Multiply DOSING RATE * DOSING INTERVAL to get DOSE
• Are PEAKS and TROUGHS in the therapeutic window?
  If not adjust regimen by decreasing interval
  (Note: Need information in "compartmental models" to estimate peaks and troughs

Questions

• Will giving the drug at the computed rate immediately produce a therapeutic concentration?
• Do you need to know the Vd to use this formula?
• Do you need to know the size of the animal to use this formula?
  Indirectly, YES. The CLEARANCE must be the whole animal clearance for the animal in question OR
  Clearance must be normalized to a relative mass, e.g., L/hr/kg of body weight.
  Many times you will have to do conversions since the literature often (usually) expresses the CL in a similar form.

One Compartment Model

• Definition and assumptions
• 1-COM formula (Ke & A)
• Accumulation with repeated dosing
• Oscillation on repeated dosing
  • Peak at steady state with repeated doses
  • Trough at steady state repeated doses
  • Degree of oscillation

Definition and assumptions for 1-COM

• Definition: The One compartment open model treats the body as one homogeneous volume in which mixing is instantaneous. Input and output are from this one volume.
  • [full] [icon] Figure: Diagram of 1-COM with terms
• Assumptions:
  • Body is one homogeneous compartment -- Problems? Practicalities?
  • Instantaneous mixing -- Must be instantaneous "relative to the time scale of measurement"
  • Applicable only to first order processes

1-COM Formula

• [full] [icon] Figure: Plot of 1-COM plus formula
• [do plot][how?] Plot 1-COM formula and enter various values for A and Ke (actually, elimination half-life) using a spreadsheet. (NOTE: You must have Microsoft Excel installed on your machine and your browser must be appropriately setup for this to work)

Questions and exercises -- 1-COM

• Get the semi-log plot you made for miraclemycin in the 70 kg patient in the volume of distribution discussion.
• Identify the various parts of the 1-COM formula on the graph.
  

  Cp(t) = A * e(-Ke * t)

• Focus on the "A" term
  • At what time after a drug is given can one make the best "guestimate" of the value of "A" if the drug is given IV?
  • What does the term Cp₍₀₎ mean?
  • How could one estimate Cp₍₀₎ given values for the following and assuming instantaneous absorption and distribution? Vd, Dose, F
  • Understanding the "A" is a "scaling" factor and that it can be any place on the time axis where one has a reasonable ability to know its value will give one a big boost in understanding simple clinical pharmacokinetics.
  • [do_plot] Exercises involving "A" -- Enter different numbers for Dose, F, and Vd into spreadsheet to see the effect on plot of 1-COM. (NOTE:Your browser must be appropriately setup for this to work)
  • Does "A" have to be at the zero-time point on the graph? What if one knows from measuring drug concentration, the value at 5 hours? Could one estimate the concentration at, e.g., 10 hours after a dose, given a measured 5-hour concentration and a Ke?

Accumulation with repeated doses

The plasma concentration of drugs given by infusion at constant rate or by repeated dosing at a constant rate will rise until the concentration high enough that elimination is equal to input. This is termed "accumulation".

Retrieve or remake graph of Miraclemycin from data in the lecture on volume of distribution (Vd). [70 Kg data] The data to make the graph are repeated here.

• What is the concentration at 8 hr after administering the drug?
• What was the "zero-time" concentration in that graph?
• Assuming IV administration and instantaneous distribution, what would the concentration be if an identical dose (2800 mg) were given at 4 hours? (Note: rise in concentration with each dose is 20 mg/L)
• What is the new peak?
• What is the concentration 4 hr after 2nd dose?
• Do this exercise for at least 2 more doses.
• Does the curve keep rising at the same rate?
• Concentrations in this scenario are:

 Calculation

• Calculation of accumulation factor for repeated doses

• When T < 5 half-lives accumulation will occur
• Examples of relationship between T & Ke on accumulation

  Accumulation Factor: Effect of Half-life and Dose Interval
  Rates		Accumulation Factor
  half-life (h)	Ke (frcn/h)	Dose Interval (h)
  		1	2	12	24
  2	0.3465	3.4	2.0	1.0	1.0
  50	0.0139	72.71	36.6	6.5	3.5

   

   

• Note from the table that given equal dose intervals, e.g., 24 h, the drug with the longer half-life will accumulate more. See multiplier 1.0 versus 3.5.
  • Changes in half-life: It should be apparent from this that changes in half-life during therapy can have tremendous effects on steady state drug concentrations. Such changes can occur during disease
  • Interindividual variation in half-life: . There is also a large interindividual variation in elimination half-life so accumulation may be more pronounced in one patient than another.
• Examples of some actual concentrations

Oscillation with repeated doses

On repeated "bolus" administration of drug, the concentration in the plasma oscillates between the peak and the trough. The importance of the degree of oscillation is drug dependent and depends on the dose, dose interval, and elimination half-life.

Degree of oscillation

• Oscillation can be viewed as a (an) --
  • ratio -- determined by Dose interval and half-life
  • absolute amount in mg/L (determined by dose interval, half-life, and (F* dose/Vd))
  • Ratio of peak to trough depends on --
    • Dose interval
    • Elimination half-life
  • [Excel] [how?] Use spreadsheet to see influence of varying T and half-life on the ratio of peak to trough

• • Oscillation ratio = 1/e-(Ke* T)

    Ke = elimination rate constant T = dose interval

  • Absolute oscillation also includes factor: F*D/Vd
• Ratio: Peak to trough
  • At T = half-life, Ratio is 2!, i.e., Peak is twice the trough
  • At T<half-life, Ratio is <2
  • At T>half-life, Ratio is >2
  • As T increases, Ratio approaches infinity
  • As T decreases, Ratio approaches infinitesimal

• Importance of degree of oscillation
  • Importance is drug dependent
  •  

Height and shape of peak after one dose

• The height and shape of the peak on a plot of drug concentration versus time depends on
  • Ka = absorption rate
  • Ke = elimination rate
  • F*D/Vd
  • IV administration gives the sharpest peaks and is the standard for Ka>>Ke.
    Ka >> Ke means absorption is much faster than elimination so nearly all of the drug is absorbed before a significant amount is eliminated.
  • PO (per os) and other routes of adminstration and use of slowly absorbing dose forms may cause peaks to be much flatter.
  • Formulae used below to calculate Peak and Trough concentrations at steady state and with repeated doses, assume Ka>>Ke. When this condition is NOT true, actual peaks will be less and actual troughs will be higher than predicted by the formulae.
• [EXCEL] [how?] Try different values of absorption half-life given a constant elimination half-life to see the influence of varying absorption rate on the shape of the curve. Note the ordinate is in terms of "multiples of the elimination half-life."

Peak concentration with repeated dosing

To calculate Peak concentration at steady state (Css(max))

• Use of (F*D/Vd) to estimate peak concentration of the first dose
• Qualifier that Ka >> Ke, i.e., absorption rate must be much faster than elimination rate. This is true for IV injections an some intramuscular injections.
• When Ka >> Ke is NOT TRUE, the true peak will be less than that estimated by this formula. This is a safety factor in many situations where we use this formula to estimate the peak concentration.

Trough Concentration

To calculate Trough concentration at steady state (Css(min))

• Note the obvious similarity of this formula to the standard 1-COM formula!

Two Compartment Model

• Definition and assumptions
• 2-COM formula

Definition and assumptions

• Definition: The two compartment open model treats the body as two compartments. Input and output are from the central compartment. Mixing occurs between the two compartments.
  • [full] Figure: Box diagram
  • [full] Figure: Plot of 2-COM with terms
• Assumptions
  • Mixing is instantaneous in within each compartment
  • Input and output are from the "central" compartment
  • Mixing between the compartments is slow relative to mixing within the compartments

2-COM formula (alpha, beta, A, B)

• [full] Figure: Plot of 2-COM with terms
• [do_plot] Plot 2-COM formula and enter various values for parameters. (NOTE: You must have Microsoft Excel installed on your machine and your browser must be appropriately setup for this to work)

Key Formulas

The following formulas and definitions should be understood so well that they are essentially memorized. You should be able to use them in solving problems. Consult other portions of these notes for details and qualifications of these terms and formulas.

Note that the preceding formula can be rearranged to provide an estimate of the amount of drug in the body, a loading dose (DL), or a volume of distribution (Vd). You should be comfortable in using it in all of its forms.