Usually the PK-PD parameters of the TGI model are obtained applying the model to the average tumor growth (TG) curves. For the aim of this study the individual ...
Evaluating the Influence of Different Sources of Variability in the PK/PD Tumor Growth Inhibition Model F. Del Bene(1), M. Germani(1), F. Fiorentini(1), G. De Nicolao(2), P. Magni(2), M. Rocchetti(1) (1) Pharmacokinetics & Modeling, ACCELERA - Nerviano Medical Sciences, Italy (2) Dept. of Computer Engineering and Systems Science, University of Pavia, Italy
ABSTRACT
MATERIALS AND METHODS
Objectives: Generally, the estimate of the in vivo effect of anticancer compounds is based on the evaluation of the changes in the average tumor growth profiles in treated versus untreated group of 8-10 animals. We recently developed a simple and effective pharmacokineticpharmacodynamic model linking the plasma concentrations of anticancer compounds to the effect on the tumor growth [1]. The model has been successfully applied to many different drugs and cell lines, showing also a good predictivity of the expected activity of the tested compounds in the clinics [2]. Essentially, the model describes the system as the interaction of three major processes: the unperturbed tumor growth in untreated animals (representing the biological system), the action of the drug on the tumor growth (representing the pharmacological part of the system) and the pharmacokinetics (PK) of the tested compound. The aim of this communication is to explore and discuss the different sources of variability affecting the response of the system in terms of PD data. Methods: Different sets of experimental data were considered. For each tested compound the inter-animal variability of the PK and the tumor growth PD parameters was estimated through the variance-covariance matrix of parameters derived from individual fittings. The influence of these different sources of variability was investigated analysing series of simulated tumor growth profiles and comparing them with the observed individual data. Results and Conclusions: The Monte Carlo simulations based on the PK/PD model, through its parameters, allowed to identify and assess the contribution of the different processes to the overall behavior of the system. The simulated tumor growth profiles in treated animals indicate the biological process (represented by the parameters modeling unperturbed tumor growth) as the major factor influencing the variability of the system response. These analyses are expected to prove particularly useful for the subsequent development of a comprehensive population approach.
Unperturbed Growth
Pharmacological c(t) Treatment (PK)
W& (t ) = λ 0 ⋅ W (t ) W < Wthreshold W (0 ) = L0 W& (t ) = λ 1 W ≥ Wthreshold Wthreshold = λ 1 λ 0
W& (t ) =
λ 0 ⋅W (t ) ψ ⎡ ⎛λ ⎞ ⎤ ⎢1 + ⎜⎜ 0 ⋅ W (t )⎟⎟ ⎥ λ ⎠ ⎦⎥ ⎣⎢ ⎝ 1
Cycling cells
W (0) = L0
1 ψ
Simeoni et al Cancer Research (2004)
ψ = 20
k 2 · c( t )
Z1
Z2
k1
k1
Z3
Z4
k1
Cell death
Cells damaged by the anticancer agent 6
Overall system of differential equation
Tumor weight Tumor Weight
Usually the PK-PD parameters of the TGI model are obtained applying the model to the average tumor growth (TG) curves. For the 3 aim of this study the individual tumor growth Wthreshold 2 observations in the control groups were 1 considered and corresponding L0i, λ0i and λ1i exponential growth parameters (i = 1,…,8) estimated. Drug effect 0 0 5 10 15 20 W (t ) = Z + Z + Z + Z Time parameters (k1 and k2) were estimated using Time the standard approach. Pharmacological experiments testing the activity of two compounds on A2780 tumor-bearing mice were considered: a known anticancer drug (Paclitaxel) given at 30 mg/Kg i.v. bolus q4dx3 and a candidate drug (Drug X) given at 45 mg/kg 10 i.v. bolus daily. Individual PK parameters were derived from ancillary groups of animals (12 for Paclitaxel, 4 for Drug X) using a two comp. open model. A Global Two Stage (GTS) iterative algorithm was applied to obtain the population parameters vector and the covariance matrix for both PD (L0, λ0, λ1) and PK (V, k10, k12, k21) parameters. Based on these variance-covariance matrices, for each compound, fixing the estimated values of k1 and k2, 200 tumor growth curves were generated assuming multivariate log-normal distribution of the PK parameters and normal distribution (except L0) of the PD control parameters. The generated curves were then compared with the observed tumor weights using the 5th and 95th percentiles. In panel A only the PK parameters are randomly extracted, in panel B only the PD control parameters, in panel C both PK and PD control parameters are extracted. 5
linear growth
4
λ0 ⋅ Z1 (t ) ⎧& − k 2 ⋅ C (t ) ⋅ Z 1 (t ) 1 ⎪ Z1 (t ) = ψ ψ ⎪ ⎡ ⎛λ ⎞ ⎤ ⎪ ⎢1 + ⎜⎜ 0 ⋅ W (t ) ⎟⎟ ⎥ λ ⎪ ⎠ ⎦⎥ ⎣⎢ ⎝ 1 ⎨ ⎪ Z& 2 (t ) = k 2 ⋅ c(t ) ⋅ Z1 (t ) − k1 ⋅ Z 2 (t ) ⎪& Z t k Z t k Z Z1 (0) = L0 ( ) ( ) = ⋅ − ⋅ ⎪ 3 1 2 1 3 (t ) ⎪& Z 2 ,3,4 (0) = 0 ⎩ Z 4 (t ) = k1 ⋅ Z 3 (t ) − k1 ⋅ Z 4 (t ) 1
2
3
4
RESULTS Paclitaxel 6
10
Assuming parameters uncorrelated
-3
12
CV% 32.9 16.4 109.1 231.0 18.9 16.0 100.5
mean * CV% * 695.7 23.9 0.853 9.4 0.022 83.9 0.144 55.5 0.021 16.8 0.024 14.5 0.0049 75.4
240
288
336 384 Time Time (hr) (hr)
432
0 0
480
Individual observed data (untreated animals) Individual observed data ( Paclitaxel 30 mg/kg q4dx3 from 192 hr) median on 200 simulations 5th and 95th percentiles on 200 simulations
12
10
TW (g)
TW (g)
2 0
0
120
240
360 480 Time Time (hr) (hr)
600
240
480
4
240
360 480 Time Time (hr) (hr)
600
0 0
720
120
240
360 480 Time Time (hr) (hr)
600
720
12 10 8 6 4 2 0
5
10
Individual observed data Plasma concentration profile from the average parameters 5th and 95th percentiles on 200 simulations
12
8 TW (g)
3
10
1
10
2
504
12
PD parameters
528
Time (hr) (hr) Time
552
0 0
576
Individual observed data (untreated animals) Individual observed data (Drug X 45 mg/kg daily for 10 days from 312 hr) median on 200 simulations 12 5th and 95th percentiles on 200 simulations
mean 1537.8 1.22 0.223 0.230 0.00970 0.0181 0.0145
(*) Incorporation of the uncertainty of estimation
CV% 37.5 21.3 60.1 35.7 21.2 43.0 53.0
mean * CV% * 1672.6 19.1 1.11 10.6 0.171 35.8 0.213 17.6 0.00962 18.4 0.0172 37.8 0.0143 41.6
960
4
2
2 240
480 720 Time (hr) Time (hr)
960
C
6
4
PAGE, Marseille - France 18-20 June, 2008
480 720 Time (hr) Time (hr)
Individual observed data (untreated animals) Individual observed data (Drug X 45 mg/kg daily for 10 days from 312 hr) median on 200 simulations 5th and 95th percentiles on 200 simulations
8
B
6
0 0
240
10 TW (g)
unit mL 1/h 1/h 1/h 1/h g/h g
8 TW (g)
PK parametes
V k10 k12 k21 λ0 λ1 L0
A
6 4
10
Average and CV% values of the PK and PD control parameters.
Individual observed data (untreated animals) Individual observed data (Drug X 45 mg/kg daily for 10 days from 312 hr) Control from the average parameters median on 200 simulations 5th and 95th percentiles on 200 simulations
10
Conc. (ng/mL)
Observed data, median, 5th and 95th percentiles time points derived from 200 Monte Carlo simulations. In the first figure observed plasma concentration data and simulated PK profiles are presented. In panel A, B and C the expected tumor growth curves in control and treated groups are represented. The pharmacological drug-effect related parameters were fixed at k1 = 0.0284 1/h and k2 = 1.03 e-5 mL/ng/h, estimated from the simultaneous fitting of the average control and treated data.
0 0
240
480 720 Time Time (hr) (hr)
960
1080
CV% 10 10 25
20 20 50
PACLITAXEL
0
240
480
Time (hr)
Drug X
720
Changes in the variability of the TG response considering different CV% values of the PD control parameters (covariance terms were ignored).
C
2
120
360
Time (hr)
Lambda0 Lambda1 L0
4
2
0
720
parameter
6
DRUG X
Time (hr)
720
Individual observed data (untreated animals) Individual observed data ( Paclitaxel 30 mg/kg q4dx3 from 192 hr) median on 200 simulations 5th and 95th percentiles on 200 simulations
8
B
6
0 0
6
10
8
(*) Incorporation of the uncertainty of estimation
8
TW (g)
10 192
12 10 8 6 4 2 0
PACLITAXEL
4
2
TW (g)
mean 699.9 0.910 0.038 0.219 0.022 0.024 0.0049
6 4
10
TW (g)
unit mL 1/h 1/h 1/h 1/h g/h g
TW (g)
0
10
A
TW (g)
PD parameters
V k10 k12 k21 λ0 λ1 L0
Reduction in the variability of the TG response including the covariance terms in the multivariate distribution. TG curves generated including or excluding these terms in the PD parameter distribution. Including correlation among parameters 12
8
3
10
Average and CV% values of the PK and PD control parameters. PK parametes
12
Individual observed data (untreated animals) Individual observed data ( Paclitaxel 30 mg/kg q4dx3 from 192 hr) Control from the average parameters median on 200 simulations 5th and 95th percentiles on 200 simulations
10 Conc. (ng/mL)
Observed data, median, 5th and 95th percentiles time points derived from 200 Monte Carlo simulations. In the first figure observed plasma concentration data and simulated PK profiles are presented. In panel A, B and C the expected tumor growth curves in control and treated groups are represented. The pharmacological drug-effect related parameters were previously estimated and fixed at k1 = 0.0403 1/h and k2 = 2.62 e-5 mL/ng/h as reported in [1,2].
Individual observed data Plasma concentration profile from the average parameters 5th and 95th percentiles on 200 simulations
720
30 30 75
12 10 8 6 4 2 0
DRUG X
0
360
720
1080
Time (hr)
CONCLUSIONS The impact of the variability observed in the PK and TG in control animals on the pharmacological response was explored starting from real cases using Monte Carlo simulations based on the TGI PK/PD model. A GTS method was applied to incorporate the uncertainty of estimation in the model parameters in the variance-covariance matrix. In both the cases presented here (Paclitaxel and Drug X) the major cause of variability observed in the TG in treated animals is due to the biological system. The covariance terms seem to play a role essentially in the first part of the TG curves during the treatment period. The simulations performed with different CV% in the PD parameters show that important reductions could be obtained also with changes of 10-25% in the CV values. The examples reported here are in line with the results observed in other studies with different cell lines and drugs and are indicating that any effort to better standardize the experimental protocol of the biological part of the studies are expected to give clear improvement in the evaluation of the drug efficacy. REFERENCES [1] Simeoni M, Magni P, Cammia C, De Nicolao G, Croci V, Pesenti E, Germani M, Poggesi I, Rocchetti M. Predictive pharmacokinetic-pharmacodynamic modeling of tumor growth kinetics in xenograft models after administrations of anticancer agents. Cancer Research. 2004, 64: 1094-1101. [2] Rocchetti M, Simeoni M, Pesenti E, De Nicolao G, Poggesi I. Predicting the active doses in humans from animal studies: a novel approach in oncology. European Journal of Cancer, 43, 1862,1868, 2007.
