Methods Of Determining Absorption Rate Constant

```A SEMINAR ON
by
V. Sandeep Kumar
M.Pharmacy, I Sem.
Department of Pharmaceutics
University College Of Pharmaceutical Sciences
Kakatiya University
Warangal
CONTENTS
 Introduction
 Methods To Detect Absorption Rate Constant





Method of Residuals
Wagner-Nelson Method
Loo - Riegelman Method
Deconvolution Method
Estimation of ka from Urinary Data
• Significance of Absorption Rate Constants
• Conclusion
• References
INTRODUCTION
Absorption can be defined as the process of movement of
unchanged drug from site of administration to site of
measurement i.e plasma.
The actual drug absorption process may be zero-order, firstorder, or a combination of rate processes that is not easily
quantitated.
METHOD OF RESIDUALS
The technique is also known as feathering, peeling and stripping.
 For a drug that follows one-compartment kinetics and
administered extra vascularlly, the time course of drug
concentration in plasma is expressed by a bi exponential
equation 1.
→Equation-1
Equation 1 can be written as
Cp = A.e-kel.t –A.e-ka.t
→Equation- 2
where
Cp
True plasma
concentration
values
Time(hours)
Figure 1. Semi-log
plot of Cp versus
Time after Oral
During the elimination phase, when absorption is almost over,( Ka
>> Kel ) and the value of second exponential approaches zero (ekat) whereas the first exponentional (e-ket ) retains some finite
value.
At this time, the equation 2 reduced to
Cp = A.e-kel.t
→equation -3
where cp represents the back extrapolated plasma concentration
values.
A plot of log cp verses t gives terminal linear phase having slope =
-kel /2.303.
Back extrapolation of this straight line to time zero yields
y-intercept equals to=log A
KaFX 0
log C=  log[
]  Kt / 2.303
Intercept
Vd ( Ka  K )
Back extrapolation terminal portion of
curve logCp
Cp
Slope =- kel /2.303
Time(hours)
Figure 2. Semi-log Plot of Cp versus Time after oral
Substracting of true plasma concentration values i.e.
equation 2 from extrapolated plasma concentration values i.e.
equation 3 yields a series of residual concentration values .
Cr = cp- cp
Cr = A.e –ka t
→equation 4
Plotting the Cr versus time should give another straight
line graph with a slope equal to – ka/2.303 and the same
intercept as before, i.e. log A
log C  log[
Intercept=
KaFX 0
]  Kt / 2.303
Vd ( Ka  K )
Back extrapolated terminal portion of curve
Residual curve
Cp
True plasma concentration values
Slope=- ka/2.303
Time(hours)
Figure 3. Semi-log Plot of Cp versus Time
Slope= -kel/2.303
From the slope, the absorption rate constant Ka
estimated..
can be
In this method of calculation it is important to remember
that the following assumptions are made:
1. It is assumed that ka is at least five times larger than k el, if not
neither constant can be determined accurately.
2. It is assumed that the absorption and elimination processes
both follow the first order, if not the residual line and, perhaps,
the elimination line will not be straight.
LAG TIME
 The time delay prior to the commencement of the first order
drug absorption is known as Lag time( t 0 ).
 In some instances absorption of drug a single oral dose not
started immediately due to such physiological factors as
stomach-emptying time and intestinal mobility or due to
formulation itself.
where Fk aD 0/V D(k a–k) is the y
value at the point of intersection of
the residual lines in .
where A and B represents the
intercepts after extrapolation of the
residual lines for absorption and
elimination, respectively.
Plasma drug level
Back extrapolated terminal portion of curve
Residual curve
Time(hours)
Lag time t0
Figure 4. Determination of lag time by graphically
Flip-Flop of ka and kel
 The estimation of the rate constant for absorption and
elimination by method of residuals is based on the assumption
that ka>>kel.
 If kel >> ka, then the values of ka from the terminal phase and
kel from the residual line are obtained .
 This phenomenon is called flip-flop of the absorption and
elimination rate constant.
 The only way to be sure of estimates is to compare kel
calculated from oral administration with kel from intravenous
data
METHOD OF RESIDUALS FOR TWO
COMPARTMENT MODEL
There are three first order processes occurring simultaneously
i.e. absorption, distribution and elimination
Plasma concentration of the drug depends initially on three
process (three exponents), then on two processes of distribution
and elimination (two exponentials) and finally on elimination
process only ( mono exponential).
C = C0 e-kat + A e-αt + B e-βt
Log CO
True plasma concentration curve
Log A
Back extrapolated distribution curve
Log B
Back extrapolated elimination curve
First residual curve
Second residual curve
Slope=-β/2.303
Log C
Slope=-ka/2.303
Time(hours)
Figure 5
Slope=-α/2.303
APPLICATIONS
 To
calculate absorption rate constant for a drug administered
orally ,absorbed by first order kinetics and confer the
characteristics of one and two compartment open model .
For a drug following intravenous administration and confer
multy compartmental characteristics .
LIMITATIONS
When the absorption is complex rather than a simple first order
process .
WAGNER-NELSON METHOD
The Wagner-Nelson method of calculation does not require a model
assumption concerning the absorption process
The assumptions are
(1) The body behaves as a single homogeneous compartment,
(2) Drug elimination obeys the first order kinetics.
The amount administered = The amount absorbed (A)+ The
Amount unabsorbed (U)
The amount absorbed (A) to any time t = the amount of the drug
in the body (X) + the amount of the drug eliminated from the
body to any time, t (Xe)
A = X + Xe
6
Taking the derivative with respective time
dA/dt = dX/dt + dXe/dt
7
but X = Vd. C ,hence
dX/dt = Vd. dC/dt
and
therefore,
dXe/dt = KX therefore,
dXe/dt = K.Vd.C
dA/dt = Vd. dC/dt + K.Vd.C
dA = Vd.dC + K.Vd.C.dt
8
integrating equation 8 between limits of t = 0 to t = t gives,
t
t
t
0
0
0
 dA  Vd  dC  K.Vd .  C. dt
t
At  A0  Vd Ct  C0  K.Vd  C. dt
0
A0 = amount of drug absorbed at t = 0 is zero ,& C0 =0. so,
t
At  Vd . Ct  KVd
.  C. dt
0
t
Rearranging the above equation
At
 Ct  K .  C. dt
Vd
0
9
Where At/Vd is the amount of drug absorbed up to time t divided
by the volume of distribution
Ct = plasma concentration at time t
t
Ct  K .  C. dt
= AUC up to time t.
Integrating equation 8 between the limits of t = 0 to t = 
And rearranging the equation, give the following
0



0
0
0
 dA  Vd  dC  K.Vd .  C. dt

A  Vd . C  C0  K.Vd .  C. dt
but C = 0, C 0= 0
0
10
Where, A/Vd = the total amount of drug absorbed from the dosage
form up to infinity time divided by the volume of the distribution of
the drug.

 C. dt
= AUC up to 
0
The fraction of absorbed at any time is obtained when equation 9 is
divided by equation 10
t


 Ct  K  C. dt 

0
At / A 


 K  Cdt 


0
11
the fraction of unabsorbed at any time t is
12
figure 7
Slope=absorption
rate constant
Time(hours)
Percent of unabsorbed
drug versus time plot-Zero
order
Log Percent of unabsorbed
Percent of unabsorbed
figure 6
Slope=-ka/2.303
Time(hours)
Logarithm Percent of
unabsorbed drug versus
time plot-First order
Wagner Nelson Method Procedure
1.
2.
3.
4.
5.
6.
7.
Plot log concentration of drug versus time.
Find K from the terminal part of slope when the slope is –
K/2.303.
Find AUCt0 by plotting Cp versus time.
Find K.AUCtO by multiplying each AUCtO by K.
Find AUC0 by adding up all the AUC pieces, from t = 0 to t
= .
Determine the 1-(Ab/Ab) value corresponding to each time
point using by the table.
Plot 1-(Ab/Ab) versus time on semi log paper, with 1(Ab/Ab)on the logarithmic axis.
For Example
k = 0.1 hr– 1
APPLICATIONS
 To understand the absorption kinetics without prior assumption .
 Two formulations of a drug that differ substantially in terms of
how much of drug is eventually absorbed which is reflected in
t
At
 Ct  K .  C. dt
Vd
0
Vs time plots
LIMITATIONS
 It applies rigorously only to the drugs with one compartmental
characteristics .
 However , when conc vs time curve after oral administration
shows multi compartmental characteristics and on IV
administration shows one compartmental model, analysis by
this method gives incorrect result
LOO-RIEGELMAN METHOD
 Loo -Riegelman method is useful in determining the absorption
rate constant for a drug follows a two compartment model.
 It requires the plasma concentration time data after i.v. bolus
and oral administration to obtain all necessary kinetic
constants.
 This method can be applied to drug that can distributed by any
number of compartments
Ab = Xc +Xt +X3
→
Equation 3.1
Xc = Vc.Cp
Xt = Vt.Ct
X3 = Vc.k13 ∫ C.dt = Vc.k13 .[AUC]0t
Substituting of Xc and X3 into equation 3.1
Ab = Vc.Cp + Xt + Vc.k13 .[AUC]0t
→Equation 3.2
 Dividing the equation 3.2 by Vc, we get
Ab/ Vc= Cp + Xt/Vc + K13 [AUC]0t →Equation 3.3
Setting the value of t = ∞, this equation becomes
Ab∞ /Vc = 0+ 0 + K13 [AUC]0∞
Ab∞ /Vc = K13 [AUC]0∞
→Equation 3.4
 Where, Ab∞ is the amount of the drug that will be ultimately
absorbed from the dosage form.
F = Ab∞ /X0
→Equation 3.5
 The fraction of the dose absorbed at any time in comparison with
Ab∞ can be obtained by dividing the equation 3.3 by equation 3.4.
Slope= -Ka/2.303
→equation3.6
Where, Xt /Vc = Ct = tissue
concentration
figure 8
Absorption rate constant by
Loo- Riegelman method
→Equation 3.7
Where
 Ct= Apparent tissue concentration
 tn= time of sampling for sample n
 tn-1 = time of sampling for the sampling point preceding sample n
 (Cp) tn-1 = concentration of drug at central compartment for
sample n-1
 ΔCp = concentration difference at central compartment between
two sampling times.
 t = Time difference between two sampling times.
Example To Calculate Ct values
K=0.16 hr– 1 ,k 12 = 0.29 hr– 1, k 21 = 0.31hr– 1.
APPLICATIONS
 Loo Riegelman method is applicable for the drugs that confers
multi compartmental characteristics .
LIMITATIONS
 It requires the concentration vs time data of both oral and IV
 An inherent limitation of this method is intra subject variability
between oral and IV administration studies . The assumption be
made that kinetics of drug distribution and elimination remain
unchanged in interval between doses.
DECONVOLUTION METHOD
It is a model independent method for determining the
absorption rate and its use has been limited.
It requires no assumptions regarding the no of compartments
or kinetics of absorption .
Linear distribution and elimination are assumed.
 It require both the data after oral and IV administration in
same subject .
ESTIMATION Ka FROM URINARY DATA
Using a plot of percent of drug unabsorbed versus time.
For a one-compartment model
Ab =DB+DE
5.1
5.2
Assuming first-order elimination kinetics with renal elimination
constant k e
5.3
5.4
Rearranging Equation 5.3
Assuming a one-compartment model,
Substituting V D C p into Equation 5.2
5.5
Substituting for dC p/ dt into Equation 5.5 and kDu/ k e for D E,
5.6
When the above expression is integrated from zero to time t,
At t = ∞. The total amount of drug absorbed is Ab ∞and dD u/ dt = 0
D ∞u - total amount of
unchanged drug excreted in the
urine.
The fraction of drug absorbed at
any time t
Slope= -Ka/2.303
figure 9
LIMITATIONS
If the drug is rapidly absorbed, it may be difficult to obtain
multiple early urine samples to describe the absorption phase
accurately.
 Drugs with very slow absorption will have low
concentrations.
SIGNIFICANCE OF ABSORPTION RATE CONSTANT
 The calculation of k a is useful in designing a multiple-dosage
regimen. Knowledge of the k a and k allows for the prediction
of peak and trough plasma drug concentrations following
multiple dosing
 The peak time (t max) in the plasma conc. versus time curve
provides a convenient measure of absorption rate.
 With the increase in absorption rate constant, C max also
increases.
Effect of a change in the absorption rate constant, k a, on the
plasma drug concentration-versus-time curve.
CONCLUSION
To compare different formulations of same drug.
The method of residual is used for the drugs which follow one
or multi compartmental characteristics but the absorption process
should not be complex .
Wagner nelson method is used for the drug confers one
compartmental characteristics by orally.
Loo Riegelman method is applicable for the drug that confers
multi compartmental characteristics .
Deconvolution method has limited use due to its complexity.
When there is lack of sufficiently sensitive analytic techniques
to measure concentration of drugs in plasma, urinary excretion
data is used.
REFERENCES
1. Leon Shargel, Susanna, Wu Pong, Andrew B.C.Yu, Applied Biopharmaceutics and
Pharmacokinetics, Fifth Edition, Mc Graw Hill., pp.161-182.
2. Malcolm Rowland, Thomas N.Tozer, Clinical Pharmacokinetics,concepts and
Applications, third edition,Waverly.,pp.119-130,478-484.
3. Milo Gibaldi and Donald Perrier, Pharmacokinetics; Second edition volume. 15,
Marcel dekker., pp33-36,145-167,433-444.
4. V. Venkateshwarlu., Biopharmaceutics and pharmacokinetics, Pharma Book
syndicate.,pp.221-224,259-263,385-387.
5. D.M.Brahmankar, SunilB.Jaiswal, Biopharmaceutics and pharmacokinetics ,a
Treatise,pp.222-224,244-268,
6. www.australianprescriber.com
7. www.ncbi.nlm.nih.gov
8. www.boomer.org
9. www.medscape.com
10. www.pharmainfo.net
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