Improvement of Parameter Estimations in Tumor Growth Inhibition Models on Xenografted Animals: Handling Sacrifice Censoring and Error Caused by Experimental Measurement on Larger Tumor Sizes

Abstract

The purpose of this study was to explore the impact of censoring due to animal sacrifice on parameter estimates and tumor volume calculated from two diameters in larger tumors during tumor growth experiments in preclinical studies. The type of measurement error that can be expected was also investigated. Different scenarios were challenged using the stochastic simulation and estimation process. One thousand datasets were simulated under the design of a typical tumor growth study in xenografted mice, and then, eight approaches were used for parameter estimation with the simulated datasets. The distribution of estimates and simulation-based diagnostics were computed for comparison. The different approaches were robust regarding the choice of residual error and gave equivalent results. However, by not considering missing data induced by sacrificing the animal, parameter estimates were biased and led to false inferences in terms of compound potency; the threshold concentration for tumor eradication when ignoring censoring was 581 ng.ml⁻¹, but the true value was 240 ng.ml⁻¹.

Appendices

Appendix 1

Determination of the Uncertainty on Tumor Size Issued from Xenograft Experiments

The tumor size calculation can be performed using the formula

$$ \mathrm{Volume}=\frac{p\cdot {q}^2}{2} $$

where p and q are the diameters (length and width) of the ovoid.

If we consider ε the error made on a diameter measure, we have

$$ \mathrm{Volume}=\frac{\left(p+\varepsilon \right)\cdot {\left(q+\varepsilon \right)}^2}{2} $$

which leads after development to

$$ \mathrm{Volume}=\frac{p\cdot {q}^2+\varepsilon \cdot \left(p\cdot \varepsilon +2\cdot p\cdot q+{q}^2+{\varepsilon}^2+2\cdot q\cdot \varepsilon \right)}{2} $$

After removing the high-order negligible terms, we obtain

$$ \mathrm{Volume}=\frac{p\cdot {q}^2+\left(2\cdot p\cdot q+{q}^2\right)\cdot \varepsilon }{2} $$

Regarding the units of the expression, $\frac{p\cdot {q}^2}{2}$ is in a volume unit (mm³) and p · q and q ² have the dimension of an area (mm²) which corresponds to a volume raised to a 2/3 power.

Thereby, we have

$$ \mathrm{Volume}=\frac{p\cdot {q}^2}{2}+g{\left(\frac{p\cdot {q}^2}{2}\right)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\cdot \varepsilon $$

with g representing a function of the volume.

Appendix 2

NONMEM Control Stream for the M3-Like Method Considering the Sacrifice Censoring

$PROBLEM SSE Tumor Growth

$DATA

$INPUT

$SUBROUTINE ADVAN13 TOL = 6

$MODEL NCOMP = 3

COMP = DEP; Depot compartment 1

COMP = CENT; Central compartment 2

COMP = TUMOR_GR; Tumor growth 3

$PK

F1 = 1

KA = 1

K10 = 0.1386

V2 = 5

S2 = V2

LAMB1 = THETA(1)*EXP(ETA(1))

LAMB2 = THETA(2)*EXP(ETA(2))

TG0 = THETA(3)*EXP(ETA(3))

K2 = THETA(4)*EXP(ETA(4))

A_0(1) = 0; Initialization of depot compartment

A_0(2) = 0; Initialization of central compartment

A_0(3) = TG0; Initialization of tumor growth compartment

$DES

;------------------------------------------------------PK model------------------------;

CP = A(2)/S2

DADT(1) = −KA*A(1); Depot compartment first-order absorption

DADT(2) = KA*A(1) − K10*A(2); Central compartment first-order elimination

;-----------------------------------------------Tumor growth model-------------------------;

DADT(3) = ((2*LAMB1*LAMB2*A(3))/((LAMB2 + 2*LAMB1*A(3)))) − K2*CP*A(3)

;Biphasic Koch tumor growth with linear antitumor effect

$ERROR

;----------------------------------------------Residual error model-------------------------;

IPRED = A(3)

W = ((THETA(5)**2)*(IPRED**2) + (THETA(6)**2))**(0.5); Weighting of the combined residual error model

IF (DV.LT.2500) THEN; values below 2500 mm³ treated as continuous data

F_FLAG = 0

Y = IPRED + W*EPS(1)

ENDIF

ULOQ = 2500

IF (DV.GE.ULOQ) THEN; Values above 2500 mm³ treated as categorical data

F_FLAG = 1

DUM = (IPRED-ULOQ)/W

PROB = PHI(DUM); Definition of the probability tumor volume to be >2500 mm³

Y = PROB

ENDIF

IRES = DV-IPRED; Definition of individual residuals

IWRES = IRES/W; Definition of individual weighted residuals

$THETA; Initial estimates for typical values

$SIGMA

1 FIX; EPS 1

$OMEGA; Initial estimates for random effects

$ESTIMATION METHOD = 1 INTER NOABORT MAXEVAL = 9999 LAPLACIAN NUMERICAL SLOW;; Estimation with Laplacian options

$TABLE;; Output tables