Simplified Mathematical Approach for Back Calculation in Wagner-Nelson Method

Applications in In Vitro and In Vivo Correlation and Formulation Development Work

The objective of this work is to present a simplified mathematical approach
for back calculation in the Wagner-Nelson method.

The literature reported values of in vitro study, obtained using
a discriminative and biorelevant test methodology, were used to generate the
calculated plasma concentration of a drug. The volume of distribution, elimination
rate constant and the dose of the drug are needed to predict the calculated
plasma concentration. For validating the proposed equation, Wagner-Nelson
method was adopted to generate the values of elimination rate constant and
volume of distribution in a literature reported in vivo data set.
The applications of the rearranged form of the Wagner-Nelson equation are
shown in judging in vitro in vivo correlation and also in dosage
form design. The steps for obtaining the rearranged form of the Wagner-Nelson
equation are given in an appendix. The proposed method can be used if the
custom made computer programs are not available.

An in vitro in vivo correlation (IVIVC) deals with a relationship
(preferably linear) between an in vitro characteristic (e.g. in vitro
drug dissolution) and a biological parameter (maximum plasma drug concentration
(C_max), time at which C_max reach (T_max) or area
under the curve). The FDA guidance document states that the main objective of
developing and evaluating an IVIVC is to enable the dissolution test to serve
as a surrogate for in vivo bioavailability study¹. The IVIVC is used
in the formulation development work and also in scale up and post approval changes
(SUPAC). The four categories of in vitro in vivo correlation described
in the FDA guidance document are Level A, Level B, Level C and multiple Level
C. Out of these four categories, level A correlation is the most common type of
correlation observed in new drug application (NDA), since it represents a point-to-point
relationship between in vitro drug dissolution and in vivo bioavailability
of the drug from a dosage form.

Mojaverian et al. stated that it is possible to obtain IVIVC by deconvoluting the plasma concentration-time curve using a model independent method such as Wagner-Nelson method or direct mathematical deconvolution and time correction factor². Even though there are numerous examples of IVIVCs in the literature, many of the correlation have not been rigorously tested through a systematic evaluation of their predictability and majority of them are written by keeping well experienced pharmacokineticians in mind, who are having an access to sophisticated custom made computer programs. Balan et al., used "Kinetica" Software (version 2.0.2, Innaphase, France) for the establishment of Level A correlation.³ Rossi et al., carried out simulations using a computer program to generate plasma concentration-time profile for diltiazem HCl, whose disposition obey to an open two-compartment model⁴.

Wagner and Nelson developed an equation (see Appendix - I) for calculating absorption rate constant (Ka) and fraction of dose absorbed from plasma drug concentration time profile (in vivo data) for open-compartment model⁵. The Wagner-Nelson method does not require a model assumption concerning the absorption process⁶.

Mathematical calculation of fraction of dose absorbed at various time points, from given plasma concentration versus time profile, has been demonstrated very well in literature^6-8. The in vitro in vivo correlation is generated using pooled mean fraction of dose dissolved (FRD) and pooled mean fraction of dose absorbed (FRA) from two or more formulations. The objective of the present study is to present a rearranged form of the Wagner-Nelson equation for evaluating IVIVC.

Discussion

Pharmacist will appreciate that calculation of the plasma drug concentration
from the data of fraction of dose absorbed at different time points requires
thorough understanding of Wagner-Nelson method. An effort is made in this
work to compute the calculated plasma drug concentration using an EXCEL worksheet.

Appendix I shows the steps to evolve the re-arranged form of the Wagner-Nelson
equation. It is assumed that a perfect Level A correlation exist between
the vitro and the vivo performance of a drug. ΔF is the difference
between fractions of drug dissolved at two successive sampling time points.
Δt is the difference between two time points used for calculating
ΔF. When the drug is given by an extravascular route, the concentration
of drug at zero time is taken as zero. The proposed equation can be used
to compute plasma drug concentration using the in vitro dissolution data
set as an input. It is desirable the in vitro dissolution test is discriminative
and biorelevant. The dose of the drug (D), elimination rate constant (K_e),
and volume of distribution (V_d) are needed for computing plasma
drug concentration. The volume of distribution of a drug is a constant.⁹

 

Validation of Model

The actual in vivo data (time and plasma drug concentrations) are necessary to validate the proposed equation (Eq. 1.5). The plasma drug concentration and fraction of drug absorbed at zero time are considered as zero. The elimination rate constant (K_e=0.04211 h^-1) was calculated by performing regression analysis between time and natural log plasma drug concentration for the terminal data points. A perfect Level A IVIVC yields a straight line with slope equal to one and intercept equal to zero. This can happen if there is a perfect match between FRD and FRA. For the purpose of validating the rearranged form of the Wagner - Nelson equation (Appendix - I), FRA is considered as the results obtained in a biorelevant dissolution test (FRD). The reported results of vivo studies (the first two columns in Table 1) were used as the input values¹⁰. The fraction of dose absorbed was calculated by the reported method^6-8. The amount of drug absorbed/V_d at the last sampling time (24 h) was considered as A_max/V_d.⁷ The calculated value of A_max/V_d was 4.234 (Table 1). The volume of distribution was calculated by using the relationship; V_d = (1*200000)/4.234. Fraction of dose absorbed is considered as 1 and the dose is 200,000 mg. The volume of distribution was calculated as 47.23 liters.

Equation 1.5, reported in Appendix I, was used to generate the calculated values of plasma drug concentration at different time points (see Table 1). A sample calculation is shown in Table 1A. The observed and the calculated values of plasma drug concentration at different time points are identical, indicating the suitability of eq. 1.5 for back calculation in the Wagner-Nelson method.

Application in IVIVC

In order to show the application of the proposed equation in IVIVC, the actual in vivo and in vitro data reported by Shishoo et al., were used.¹⁰ It is worthwhile to note that in this part of study FRA is not considered as equal to FRD, since the objective of this part of the study was to show application of eq.1.5 in IVIVC. The values of K_e (0.04211h ^-1) and V_d (47.233 L) were taken as that obtained in the in vivo studies. Table 2 shows the results of calculated plasma drug concentration and percentage prediction error. The percentage prediction error was calculated as ((observed Cp-predicted Cp)/observed Cp)*100. Good agreement can be seen between the observed and the predicted values.

A pharmacist often fails in establishing level A IVIVC in the first attempt, i.e. after conducting the in vivo studies in human and in vitro study under one set of conditions. The in vitro test is carried out by varying different parameters in statistically designed experiments under such circumstances. A set of in vitro dissolution conditions that show good correlation with in vivo results is then used for quality assurance.

Application in dosage form design

For showing the application of the proposed equation in
formulation and development work, volume of distribution of theophylline was
obtained from a reference book¹¹. The reported value of V_d
for theophylline is 0.5 l/kg. The average weight of human being was taken
as 70 kg¹². The V_d was calculated as 35 l. The elimination
rate constant was calculated from the reported value of elimination half-life
value (3-13 h^-1)¹². The average value of half-life (8
h) was used to calculate elimination rate constant (0.0863 h^-1).
Generally, K_e is estimated by administering a drug in solution
or rapidly dissolving dosage form. The actual in vitro data was used
as input¹⁰ (Table 2). The calculated plasma drug concentration
is shown in Table 3. Good agreement can be seen between the observed and the
predicted values. The accuracy of prediction will depend up on the correctness
of the pharmacokinetic parameters.

The applications of the proposed equation can be extended to the selection of suitable dissolution conditions (type of media, agitation rate and type of dissolution equipment) and also in investigating the influence of minor changes in the formulation of dosage form during scale-up. While seeking a bio-waiver, such a treatment may prove to be useful. The USP XXVI has included special guidance for bioequivalence (BE) study <1090> and the BE study results are also reported in ANDA applications¹³. The failure rate in BE studies is very high and this results in delay in submission of ANDA. The usual practice at industry is to make changes in formulation and again repeat the expensive BE study. Different formulations may be screened at this time by using the proposed equation for the selection of a bio-batch, i.e. a batch to be used in BE studies. A copy of EXCEL worksheet can be obtained from the authors upon request.

In conclusion, the rearranged from of the Wagner-Nelson equation can be used for establishing IVIVC and also in dosage form design. All the computations can be quickly performed in EXCEL. The Loo-Riegelman equation can also be given similar treatment to deal with the drugs that do not satisfy the requirements of Wagner-Nelson equation.

Appendix - I

Derivation of Equation for Back Calculation of Wagner - Nelson
Equation

According to the Wagner - Nelson equation,. ⁵

........................(1.1)

Where,

A_t = Amount of drug absorbed at time ‘t'.

A_∞ = Amount of drug absorbed at time ‘infinite'.

K_e = Elimination rate constant of the drug.

= Area under the curve of the plasma concentration versus time profile of drug, for time period between t = 0 to t = t.

= Area under the curve of the plasma concentration versus time profile of drug, for time period between t = 0 to t = ∞.

 

Now, F_t = A_t/A_∞ = fraction of drug absorbed at time‘t',

However,

(1.2)

Where,

D = Dose of drug administered

V_d = Apparent volume of distribution

Assuming that at infinite time, the administered dose is completely absorbed, i.e. F_∞=1 in equation (1.2)

The rearranged forms of Wagner - Nelson for time t and t+1 are given below:

.................................(1.3)

..........................(1.4)

Subtracting equation (1.3) from equation (1.4)

 

Taking F_t+1-F_t = ?F and, t_t+1-t_t = ?t,

....................(1.5)

TABLE 1: RESULTS OF VALIDATION STUDY OF REARRANGED FORM OF WAGNER-NELSON
EQUATION.

Time (h)	Observed Plasma Concentration+ (µg/ml)	At/Vd	Fraction of dose absorbed	Calculated Plasma Concentration (µg/ml)*
1	0.891	0.90	0.214	0.891
2	1.997	2.07	0.490	1.997
3	2.616	2.79	0.659	2.616
4	3.411	3.71	0.877	3.411
5	3.33	3.77	0.891	3.33
6	3.868	4.46	1.054	3.868
7	3.371	4.12	0.973	3.371
8	3.433	4.32	1.021	3.433
10	2.916	4.07	0.962	2.916
12	2.877	4.28	1.011	2.877
24	1.679	4.23	1	1.679

+ Data source: reference 10, * See Equation 1.5 in Appendix I

A perfect Level A correlation between in vitro and in vivo studies

is assumed (i.e. FRA=FRD), Elimination Rate Constant (K_e) = 0.04211 h ^-1,

A_t/V_d = Cp + K_e*AUC⁶

Volume of Distribution (V_d) = 47.234 L, Dose = 200000 µg

TABLE 1A: SAMPLE CALCULATION FOR VALIDATION OF THE PROPOSED MODEL

	A	B	C
	Time (h)	FRA = FRD	Calculated plasma drug concentration
1
2	1	0.2148	0.891 (See equation 1.5)

?F = (0.2148-0) = 0.2148, Dose (D) = 200000 mg,

V_d = 47234 ml, C_t = 0 (Plasma drug concentration at 0 time is Zero),

K_e = 0.04211 h^-1, ?t = (1-0) = 1

The above mentioned values are inserted in equation 1.5 to calculate

C_t+1 (Plasma drug concentration at 1 h = 0.891)

TABLE 2: RESULTS OF BACK CALCULATION IN WAGNER-NELSON EQUATION- APPLICATION IN IVIVC

Time (h)	Fraction of drug dissolved*	Predicted Cpx	Observed Cp+	% Prediction Error
				0
1	0.187	0.778	0.891	12.61
2	0.453	1.848	1.997	7.41
3	0.624	2.479	2.616	5.22
4	0.773	2.997	3.411	12.11
5	0.849	3.186	3.33	4.30
6	0.915	3.328	3.868	13.93
7	0.927	3.241	3.371	3.83
8	0.958	3.236	3.433	5.73

*Source: Reference 10 (actual in vitro data)

The values of pharmacokinetics parameters are shown in Table 1

Cp = Plasma drug concentration, ^x Calculated using Eq.1.5,

⁺ Refer Table 1

TABLE 3: RESULTS OF BACK CALCULATION IN WAGNER-NELSON EQUATION - APPLICATION IN DOSAGE FORM DESIGN

Time (h)	Fraction of drug dissolved*	Predicted Cp	Observed Cp	Absolute % Prediction Error
				00.00
1	0.187	1.028	0.891	15.41
2	0.453	2.399	1.997	20.13
3	0.624	3.133	2.616	19.77
4	0.773	3.692	3.411	08.26
5	0.849	3.799	3.33	14.08
6	0.915	3.845	3.868	00.59
7	0.927	3.591	3.371	06.55
8	0.958	3.462	3.433	00.87

* Source: Reference 10 (actual in vitro data), literature reported values of

V_d (35 L) and K_e (0.0863h ^-1) were used for calculating predicted Cp¹¹

References