Pharmacokinetics: Lecture four

Extravascular

Administration

One compartment Model

Anas Bahnassi PhD RPh

LECTURE’S OBJECTIVES

• Upon completion of this lecture, the student will able to: • Calculate plasma drug concentration at any given time after the

administration of an extravascular dose of a drug, based on known or estimated pharmacokinetic parameters

• Interpret the plasma drug concentration versus time curve of a drug administered extravascularly as the sum of an absorption curve and an elimination curve

• Employ extrapolation techniques to characterize the absorption phase • Calculate the absorption rate constant and explain factors that influence

this constant • Explain possible reasons for the presence of lag time in a drug’s absorption • Calculate peak plasma drug concentration, (Cp)max, and the time, (tmax)at

which this occurs • Explain the factors that influence peak plasma concentration and peak time • Decide when flip-flop kinetics may be a factor in the plasma drug

concentration versus time curve of a drug administered extravascularly.

Anas Bahnassi PhD 2011 2

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Xu 1. Equation for determining the plasma concentration at any time t.

2. Determination of the elimination half life (t½) and rate constant (K or Kel).

3. Determination of the absorption half life (t½)abs and absorption rate constant (Ka).

4. Lag time (t0), if any.

5. Determination of the apparent volume of distribution (V or Vd) and fraction of drug absorbed (F).

6. Determination of the peak time (tmax).

7. Determination of the peak plasma or serum concentration, (Cp)max.

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Amount Remaining in the

Administration Site

−𝑑𝑥𝑎

𝑑𝑡= 𝑘𝑎(𝑥𝑎)𝑡

First Order

Process

(𝑥𝑎)𝑡 = (𝑥𝑎)𝑡=0𝑒−𝑘𝑎𝑡 = 𝐹𝑥𝑜𝑒−𝑘𝑎𝑡

Bioavailability Dose

(𝑥𝑎)𝑡 = 𝑥𝑜𝑒−𝑘𝑎𝑡

100% Absorbed

Monitoring Drug in Site of

Measurement

−𝑑𝑥𝑎

𝑑𝑡= 𝑘𝑎(𝑥𝑎)𝑡 − 𝑘𝑥

First Order

Process

𝑥𝑡 =𝑘𝑎(𝑥𝑎)𝑡=0

𝑘𝑎 − 𝑘𝑒−𝑘𝑡 − 𝑒−𝑘𝑎𝑡

𝑥𝑡 =𝑘𝑎𝐹𝑥0


Elimination Half life (t1/2)

And Elimination Rate Constant(k)

𝐶𝑝𝑡 =𝑘𝑎𝐹𝑥0

𝑉(𝑘𝑎 − 𝑘)𝑒−𝑘𝑡 − 𝑒−𝑘𝑎𝑡

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Anas Bahnassi PhD 2011

8

Calculating Absorption Rate Constant (ka)

Time (h) Observed Plasma

Concentrations

(Cp)observed

Extrapolated Plasma

Concentrations

(Cp)extrapolated

(𝐶𝑝)𝑑𝑖𝑓𝑓

= 𝐶𝑝𝑒𝑥𝑡 − 𝐶𝑝𝑜𝑏𝑠

Time values

corresponding

to observed

plasma

concentrations

for absorption

phase only

Values only from the

absorption phase

(i.e. all values prior to

reaching maximum

or highest plasma

concentration) (units,

e.g. mgmL-1)

Values only from the

extrapolated portion of

the plot of plasma

concentration–time

(units, e.g. (mgmL1)

Differences between

extrapolated and observed

values for each time in the

absorption phase (units,

e.g. mgmL-1)

𝐶𝑝𝑑𝑖𝑓𝑓 =𝑘𝑎𝐹𝑥0

𝑉(𝑘𝑎 − 𝑘)𝑒−𝑘𝑎𝑡

Method of Residuals

Feathering Method


Calculating Absorption Rate Constant (ka)

Slope=−𝑘𝑎

2.303

Method of Residuals

Feathering Method


Lag Time (t0)

Theoretically, intercepts of the terminal linear portion and the feathered line should be the same; however, sometimes, these two lines do not have the same intercepts, Sometimes absorption starts after administration, this delay may be contributed to:

• Slow tablet disintegration • Slow and/or poor dissolution • Incomplete wetting of drug particles • Poor formula • Delayed Release formula


The presence of a negative lag time may be attributed to inadequate data points in the absorption phase as well as in the elimination phase. Another possible reason may be that the absorption rate constant is not much greater than the elimination rate constant.

Negative Lag Time (t0)

Analysis of Absorption

Rate Constant 𝑘𝑎 ≫ 𝑘

Quicker Absorption

Faster Onset of Action

𝒌𝒂 for a given drug can change as a result of:

• Changing the formulation • Changing the dosage form

or the extravascular route of administration.

• Administration of a drug with or without food.


Apparent Volume of

Distribution(Vd) Cannot be calculated from plasma drug concentration data alone WHY? The fraction of drug absorbed (F) is not known. If the drug is 100 percent absorbed; F=1 then

𝐼𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 =𝑘𝑎𝐹𝑥0

𝑉(𝑘𝑎 − 𝑘) If F is not known then it is best to

calculate 𝑉

𝐹

𝑉

𝐹=

𝑘𝑎𝑥0

(𝑘𝑎−𝑘)(

1

𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡)


Calculating Peak Time(tmax)

𝑑𝑥

𝑑𝑡= 𝑘𝑎𝑥𝑎 − 𝑘𝑥

When t=tmax 𝑘𝑎𝑥𝑎 = 𝑘𝑥

𝑑𝑥

𝑑𝑡= 𝑘𝑎(𝑥𝑎)𝑡𝑚𝑎𝑥

− 𝑘(𝑥)𝑡𝑚𝑎𝑥= 0

𝑥𝑡 =𝑘𝑎𝐹𝑥0


𝑥𝑡𝑚𝑎𝑥=

𝑘𝑎𝐹𝑥0

𝑘𝑎 − 𝑘𝑒−𝑘𝑡𝑚𝑎𝑥 − 𝑒−𝑘𝑎𝑡𝑚𝑎𝑥

𝑘𝑎(𝑥𝑎)𝑡𝑚𝑎𝑥= 𝑘(𝑥)𝑡𝑚𝑎𝑥

𝑥𝑎𝑡𝑚𝑎𝑥= 𝐹𝑒−𝑘𝑎𝑡𝑚𝑎𝑥


Calculating Peak Time(tmax)

Taking natural log of both sides:


Significance of Peak Time(tmax)

• To determine comparative bioavailability and/or bioequivalence

• To determine the preferred route of drug administration and the desired dosage form for the patient

• To assess the onset of action.

• Used to determine the comparative bioavailability and/or the bioequivalence between two products.

• Used to determine the superiority between two different dosage forms or two different routes of administration

• Correlates with the pharmacological effect of a drug.

Significance of Peak Concentration(Cmax)


Plot the data and, using the plot, determine the following. a. The elimination half life (t1/2) for each dose. b. The elimination rate constant (K) for each dose. c. The absorption half life, (t1/2)abs, for each dose. d. The absorption rate constant (Ka) for each dose. e. The observed and computed peak time (tmax) for each dose. f. The observed and computed peak plasma concentrations, (Cp)max, for each dose. g. The y-axis intercept for each dose. h. The apparent volume of distribution (V). i. The fraction of drug absorbed (F). j. The characteristics of a plot on rectilinear paper of peak time (tmax) against the administered dose (then make an important observation). k. The characteristics of a plot on rectilinear paper of peak plasma concentrations, (Cp)max, i. Lag time if any.

Question

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Pharmacokinetics

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Pharmacokinetics: Lecture four