Application of a two-phenotype color-shift model with heterogeneous growth to a rat hepatocarcinogenesis experiment

Mathematical Biosciences 224 (2010) 95–100

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Application of a two-phenotype color-shift model with heterogeneous growth to a rat hepatocarcinogenesis experiment Jutta Groos, Annette Kopp-Schneider * Department Biostatistics-C060, German Cancer Research Center, Im Neuenheimer Feld 280, D-69120 Heidelberg, Germany

a r t i c l e

i n f o

Article history: Received 10 October 2008 Received in revised form 17 December 2009 Accepted 22 December 2009 Available online 4 January 2010 Keywords: Foci of altered hepatocytes Carcinogenesis model Color-shift model Beta distribution

a b s t r a c t The color-shift model (CSM) was introduced by Kopp-Schneider et al. [1] to describe formation and progression of foci of altered hepatocytes (FAH). It incorporates the field-effect hypothesis which postulates that entire colonies of altered hepatocytes simultaneously alter their phenotype. In the original CSM, FAH grow with deterministic growth rate and change their phenotype after an exponentially distributed waiting time. A modification of the original color-shift model (CSMbeta) is presented here in which the growth rate varies from focus to focus according to a beta distribution. The concept of an exponentially distributed waiting time to phenotype change is modified to the concept of a random radius at which phenotype changes and this radius is modelled as beta distributed. The original and the modified CSM are applied to data from an initiation-promotion rat hepatocarcinogenesis experiment with diethylnitrosomorpholine (DEN) and N-nitrosomorpholine (NNM), in which two phenotypes of FAH were observed in hematoxilin/eosin (H&E) stained liver sections. The Cramer-von-Mises Distance is used as a measure for the discrepancy between empirical and theoretical size distributions. Comparisons of model fit show that considerable improvement is obtained for CSMbeta compared to the original CSM. Ó 2010 Elsevier Inc. All rights reserved.

1. Introduction In the liver the multistage-process of carcinogenesis can be analyzed by the observation of formation and growth of foci of altered hepatocytes (FAH), which represent precursor lesions of carcinoma [2–4]. FAH are studied by morphometric analysis of stained liver sections. Staining with hematoxilin/eosin (H&E) allows for observation of different types of FAH, which represent successive stages on the way from normal liver tissue to the malignant stage [5,2]. Two different biological hypotheses are established to describe formation and progression of FAH. The mutation-hypothesis assumes that FAH change their phenotype through mutation of single FAH cells and that foci grow exclusively through clonal expansion. In contrast the field-effect hypothesis considers foci as entities, which change their phenotype as a whole. In addition, the field-effect hypothesis allows FAH to expand not only through proliferation of FAH cells but also by recruitment of neighboring cells. In contrast to the cellular approach of earlier carcinogenesis models [6] the color-shift model (CSM) describes formation and growth of FAH on the basis of the field-effect hypothesis [1]. In CSM, FAH as a whole are assumed to grow exponentially with deterministic rate and to change their phenotype (‘color’) after an exponentially distributed waiting time.

* Corresponding author. Tel.: +49 6221 42 2391; fax: +49 6221 42 2397. E-mail address: [email protected] (A. Kopp-Schneider). 0025-5564/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.mbs.2009.12.009

CSM has been applied to data from a long-term rat hepatocarcinogenesis experiment in which different types of FAH were observed. Although it predicted the size and number of focal transections well, the model expected more variation in size distribution of FAH than the data exhibited [7]. This rather peculiar observation may be due to the assumption of exponentially distributed waiting times for color change of FAH, which leads to large variations in the time that a focus spends in a phenotype. To reduce this effect the CSM was modified by inserting shift radii for color change of FAH. To take into account that under natural conditions shift radii may differ from focus to focus, the shift radius of a given focus is considered as beta distributed random variable. Since the beta distribution has a compact support, this assumption leads to a bounded support for the size distribution of FAH in one phenotype. Furthermore, the assumption of deterministic growth with a single rate for all FAH of one type seems too simple. Therefore, the second modification of the CSM consists in allowing the growth rate to vary from focus to focus. Specifically, the growth rates of single FAH represent independent realizations of a beta distributed random variable. Based on these two modifications of the CSM the CSM with beta distributed growth rates and beta distributed shift radii (CSMbeta) was developed [8,9]. In the present study, CSM and CSMbeta were applied to data from a medium-term hepatocarcinogenesis experiment. In the course of the experiment the carcinogen N-Nitrosomorpholine (NNM) was administered to male rats following an initiation-promotion-protocol with diethylnitrosomorpholine (DEN) as initiator.

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For the detection of FAH, liver sections were stained with H&E, which allows for the classification of FAH into different types. In this paper the two successive intermediate stages clear-/acidophilic-cell (type 1) and mixed-cell (type 2) FAH [2], are considered. Two-color approaches of the models CSM and CSMbeta were applied to data by maximum likelihood methods. Application of the two models to the same dataset allows for a comparison between the model-predictions concerning the distributions of size and number of the different types of FAH.

distribution of foci in case of one color is given, which constitutes the basis of the derivation of the densities in color 1 and color 2 in the two-color approach. Let the distribution of the growth rate, B, be independent of the color. Furthermore let the shift radius, Rs , be a beta distributed random variable on ðr 0 ; r max . Hence, the density of the shift radius Rs is described by the following expression:

fRs ðr s Þ ¼ 2. Mathematical formulation of models

1 ðr s  r 0 ÞpRs 1 ðr max  r s ÞqRs 1 1ðr0 ;rmax  ðrs Þ; BðpRs ; qRs Þ ðr max  r 0 ÞpRs þqRs 1

where Bðp; qÞ denotes the beta function 2.1. Two-color color-shift model The color-shift model (CSM) describes FAH as entities of spherical shape which change their size by increase of radius [1]. This approach allows for a size increase not only occurring through clonal expansion but also through recruitment of neighboring cells. Centers of spherical foci are generated according to a homogeneous Poisson process in the liver with parameter l. Starting from a fixed size, r 0 , foci grow exponentially with deterministic growth rate, b. FAH are generated in the first preneoplastic stage and may change their phenotype (‘color’) after an exponentially distributed waiting time. As FAH are assumed to be spherical, the size of a focus is completely characterized by its radius. The expressions for the density of the size distribution of FAH-radii in case of m colors were derived in [1]. In the present study the expressions for the density of size distribution for type 1 and type 2 in a two-color approach are of interest. Let RðtÞ and CðtÞ denote radius and color of a focus at time point t and let k be the parameter of the exponentially distributed waiting time for the color-shift from type 1 to type 2. The densities of the size distribution in color 1 and 2 are given by the expressions:

Bðp; qÞ ¼

Z

1

2.2.1. Density of the distribution of size and color of a focus 2.2.1.1. Color 1. For fixed shift radii Rs ¼ rs the density of size and color 1 is the density in case of the one color approach in [9] multiplied with the indicator function 1ðr0 ;rs  , which is:

  pþq1 I lnðrx Þ ðq; ðp  1ÞÞ 1ðr0 ;rs  ðxÞ; xtaðp  1Þ 1 at0

fRðtÞ; CðtÞ ðx; 1jRs ¼ r s Þ ¼

where Ir ðp; qÞ denotes the incomplete beta function, i.e.

Ir ðp; qÞ :¼

1 Bðp; qÞ

Z

r

zðp1Þ ð1  zÞðq1Þ dz

0

with property

Ir ðp; qÞ ¼ 1  I1r ðq; pÞ:

ð2Þ

And hence

fRðtÞ; CðtÞ ðx; 1Þ ¼

Z

rs

fRðtÞ; CðtÞ ðx; 1jRs ¼ r s ÞfRs ðrs Þ dr s rs

2.1.1. color 1

   1 k x fRðtÞ; CðtÞ ðx; 1Þ ¼ exp  ln  1ðr0 ;rmax Þ ðxÞ; tbx b r0

¼

Z

rs rs

  pþq1 I lnðrx Þ ðq; ðp  1ÞÞ xtaðp  1Þ 1 at0

1ðr0 ;rs  ðxÞ  2.1.2. color 2

fRðtÞ; CðtÞ ðx; 2Þ ¼

    1 k x 1  exp  ln  1ðr0 ;rmax Þ ðxÞ; tbx b r0



1 x 2 ðr 0 ; r max  0

1 ðr s  r0 ÞpRs 1 ðr max  rs ÞqRs 1 1ðr0 ;rmax  ðrs Þ dr s : BðpRs ; qRs Þ ðr max  r 0 ÞpRs þqRs 1

Denote by

 ðp þ q  1Þ I

ð1Þ

with rmax :¼ r 0 expðbtÞ the maximum reachable radius at time point t and 1ðr0 ;rmax Þ ðxÞ the indicator function

1ðr0 ;rmax Þ ðxÞ :¼

zðp1Þ ð1  zÞðq1Þ dz:

0

else:

F1 :¼

The assumption of deterministic growth rate was found to be too restrictive and an approach with heterogeneous growth was developed [9,8]. CSMbeta allows the growth rate to differ from focus to focus, so that the growth rate b of each FAH is a realization of a beta distributed random variable B  Betaðp; q; aÞ p; q; a > 0. Additionally to the parameters p and q of the standard beta distribution a parameter a was introduced to vary the upper limit of the support, ½0; a. The shift of a focus from type 1 to type 2 occurs at shift radius Rs , which follows a beta distribution, Rs  BetaðpRs ; qRs ; aRs Þ. As growth is limited by parameter a, aRs is fixed at the maximum reachable radius at time point t, i.e. aRs ¼ r max ¼ r 0 expðatÞ. Hence, given r 0 , the parameter aRs is completely determined by parameter a and, therefore, redundant. In [9] an expression for the density of the size

ðq; ðp  1ÞÞ



xtaðp  1ÞBðpRs ; qRs Þðrmax  r 0 ÞpRs þqRs 1

 1ðr0 ;rmax  ðxÞ;

rs r 0 then by substituting r~s :¼ rmax and the property (2) of the incomr0 plete beta function:

fRðtÞ; CðtÞ ðx; 1Þ ¼ F1  2.2. Color-Shift model with beta distributed growth rate and beta distributed shift radius

lnðrx Þ 1 at0

¼ F1  ¼ F1 

r max r0

Z Z

Z

r max

1ðr0 ;rs  ðxÞðr s  r 0 ÞpRs 1 ðr max  r s ÞqRs 1 dr s

ðr s  r 0 ÞpRs 1 ðr max  r s ÞqRs 1 dr s

x 1 xr0 rmax r0

p 1 q 1 ððrmax  r 0 Þr~s Þ Rs ððr max  r0 Þ  ðr max  r 0 Þr~s Þ Rs

 ðr max  r0 Þ dr~s

  ¼ F1  ðr max  r 0 ÞðpRs þqRs 1Þ BðpRs ; qRs Þ 1  I xr0 ðpRs ; qRs Þ rmax r0    ð2Þ ðp þ q  1Þ ¼ I lnðrx Þ ðq; ðp  1ÞÞ I rmax x ðqRs ; pRs Þ  1ðr0 ;rmax  ðxÞ: r max r0 xtaðp  1Þ 1 at0

2.2.2. Color 2 Given that Rs ¼ rs is fixed and since growth rate B is assumed to be color-independent, the only difference between the expressions

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for the joint density of size and color between color 1 and color 2 consists in the indicator function 1ðrs ;rmax  ðxÞ. Hence:

fRðtÞ; CðtÞ ðx; 2jRs ¼ r s Þ ¼

  pþq1 I lnðrx Þ ðq; ðp  1ÞÞ 1ðrs ;rmax  ðxÞ: xtaðp  1Þ 1 at0

   ðp þ q  1Þ I lnðrx Þ ðq; ðp  1ÞÞ I rmax x ðqRs ; pRs Þ rmax r0 xtaðp  1Þ 1 at0

 1ðr0 ;rmax  ðxÞ;    ðp þ q  1Þ I lnðrx Þ ðq; ðp  1ÞÞ I xr0 ðpRs ; qRs ; Þ fRðtÞ; CðtÞ ðx; 2Þ ¼ rmax r0 xtaðp  1Þ 1 at0  1ðr0 ;rmax  ðxÞ: ð3Þ The models considered here assume that FAH act independently of each other. For details concerning model assumptions and detailed derivation of the density functions (1) see [1]. 3. Methods of application FAH are evaluated on thin stained liver sections, representing two dimensional sections through three dimensional objects. However, the models describe size and number of FAH in 3D. The Wicksell-transformation [10] is used to derive the 2D size distribution of FAH-transections from the 3D FAH-size distribution provided by the model by transforming equations (1) and (3) for the 3D-density of the size distribution fRðtÞ;CðtÞ into the 2D-density of the size distribution fRð2Þ ðtÞ .

fRð2Þ ðtÞ;CðtÞ ðyÞ ¼

y

le

Z y

1

1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fRðtÞ;CðtÞ ðxÞ dx; 2 x  y2

R 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 x2  e fRðtÞ ðxÞ dx denotes the mean size of foci in where le :¼ e 3D and e the minimum radius for detection. As foci are assumed to develop independently of each other, each focus provides an additive contribution to the overall log-likelihood function. The negative log-likelihood was minimized using the MATLAB intrinsic function fmincon where the MEX-Fortran concept was used to perform time-consuming integrations in Fortran77. Goodness of fit Model-predictions by CSM and CSMbeta of FAH number and size are compared graphically to experimental data. Due to the lack of a hierarchical structure of the two models, likelihood ratio tests are not applicable. Cramer-von-Mises Distance (CvMD) is used to confirm the graphical rating of the two models (see [9]). The expression in [9] was modified to eliminate the dependency of the number of observed focal transections in the following way. Let F be the theoretical size distribution of FAHtransections. Let N be the number of FAH-transections with radii r 1 ; . . . ; r N detected in one treatment group, and r½1 ; . . . ; r½N denote the ordered expressions. The value of the empirical distribution function F N at r½i ; i 2 f1; . . . ; Ng is then given by F N ðr ½i Þ ¼ Ni . Modifying the approach given by [11], the CvMD between empirical distribution function and model predicted distribution function is calculated by:

C v MD ¼

Z

1

1

ðFðxÞ  F N ðxÞÞdF N ðxÞ ¼

2 N  1X i Fðr ½i Þ  N i¼1 N

However, CvMD as a measure for the goodness of fit of the size distribution does not assess the number distribution. 4. Application of models to experimental data

The derivation for color 2 is analogous to the derivation for color 1. In summary, the expressions for the density of the distribution of size and color for color 1 and 2 are:

fRðtÞ; CðtÞ ðx; 1Þ ¼

97

ð4Þ

It is equivalent to the Euclidean distance between F and F N at r 1 ; . . . ; rN weighted by the density of the empirical distribution function 1=N. CvMD is used to compare the goodness of fit of the size distribution between the two models within one treatment group.

Data were obtained from a carcinogenesis experiment with 200 juvenile male Wistar rats subdivided into five treatment groups. Three subgroups were initiated by a single dose of DEN at the start of the experiment. Two initiated groups obtained 1 mg NNM ( DEN + Low Dose) and 5 mg NNM ( DEN + High Dose) per kg body weight respectively continuously over 12 weeks. The third initiated group acted as DEN-Control. The fourth group, Low Dose, obtained the low dose of NNM without prior initiation and the fifth group acted as control. FAH were detected in H&E-stained liver sections. Foci with radius smaller than 0.031 mm where excluded. Overall about 16 000 FAH-transections were evaluated, of which 14 500 were characterized as clear/acidophilic-cell foci (type 1) and 1500 as mixed-cell foci (type 2), respectively. This corresponds to a liver area fraction of 0.48% for clear/acidophilic-cell and 0.15% for mixed-cell foci, respectively. Fig. 1 shows empirical cumulative type 1 and type 2 size distributions of focal transections detected in the five treatment groups and the respective predictions by the models CSM and CSMbeta. It is apparent that CSMbeta fits the data much better than CSM. This finding is confirmed by the CvMD-values in Table 1. In each treatment group the CvMD-value associated with the application of CSMbeta is lower than the respective value for CSM for both type 1- and type 2-FAH. The only exception is the CvMD-value for type 2-FAH in the control group, in which only three FAH were detected. In every treatment group, the total CvMD-value (sum of type 1- and type 2-CvMD) from CSMbeta is well below the total CvMD-value from CSM. As CvMD measures the goodness of fit of the FAH-size distributions, predictions about the number of type 1- and type 2-FAH are graphically compared to experimental data in Fig. 2. The number of type 1 and type 2 focal transections is well predicted by both models and there are no remarkable differences between the two model predictions. Therefore the CvMD-values are appropriate to compare the goodness of fit of the two models and show a clear advantage of CSMbeta over CSM. Tables 2 and 3 give maximum likelihood parameter estimates per dose groups for CSMbeta and CSM respectively. The parameter l is the parameter for the Poisson process describing the formation  describe the of FAH in normal tissue and the parameters b and b exponential growth rate in CSM and CSMbeta respectively. In  denotes the expectation of the heterogeneous growth CSMbeta b rate B. Interpretation of the maximum likelihood parameter estimates of both models leads to the conclusion that NNM enhances the rate of formation of FAH with and without previous DEN-initiation. Furthermore the growth of pre-existing FAH is accelerated with increasing dose of NNM with and without initiation. 5. Discussion The application of the two models CSMbeta and CSM shows a clear advantage of CSMbeta over CSM concerning the ability to predict the distribution of size and number of FAH. The fact that the exponential growth rate in CSMbeta varies from focus to focus is obviously a more natural approach than the assumption of a fixed rate for all foci in CSM. To model heterogeneity in growth and color-shift, the distributions of the respective random variables have to satisfy some restrictive conditions. To preclude negative growth and negative shift radii, the supports of the distributions have to be strictly positive. Furthermore both magnitudes have to be bounded above

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Fig. 1. Empirical size distribution (solid line) and predicted size distributions for CSMbeta (black, dashed) and CSM (gray, dashed) separately by treatment group. (A) type 1FAH; (B) type 2-FAH. Radius is displayed on a log-scaled axis. Table 1 CvMD for size distributions of type 1 (T1) and type 2 (T2) FAH for CSMbeta and CSM P (‘LD’ = ‘low dose of NNM’, ‘HD’ = ‘high dose of NNM’, lines in bold: CSMbeta and P CSM denote CvMD (type 1) + CvMD (type 2) for CSM beta and for CSM, respectively). Model

DEN

DEN + LD

DEN + HD

Control

HD

CSMbeta, T1 CSMbeta, T2 P CSMbeta CSM, T1 CSM, T2 P CSM

0.02 0.02

0.03

0.02

0.02

4:76  103 0.03 0.18 0.08 0.26

1:21  103 0.02 0.16 0.05 0.21

0.05 0.03

0.04 0.15 0.08 0.23

0.07 0.15 0.02 0.17

3:19  103 0.02 0.24 0.16 0.40

as growth of FAH is bounded by the organ boundaries. The simplest distribution satisfying the given conditions is the uniform distribution on an interval ð0; a. Former applications of CSM with uniformly distributed growth rates, CSMuni, to experimental data resulted in the conclusion that CSMuni is not flexible enough to describe the growth of real FAH [8,?]. Therefore, due to its positive compact support and its flexible form the beta distribution was selected for the current modification of CSM. In [9], one color approaches of CSM and CSMbeta were applied to experimental data and a small advantage for CSMbeta was detected. However, as FAH were not classified into different phenotypes, the ability

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Type 1, DEN−initiated

2

10

1

0.1

0.01 DEN+0

Type 2, not initiated

100

# Foci per cm

# Foci per cm

2

100

10

1

0.1

DEN+1

0.01

DEN+5

0

Dose NNM in mg pro kg 100

Type 2, DEN−initiated

2

10

# Foci per cm

# Foci per cm

2

100

1

0.1

Type 2, not initiated

10

1

0.1

0.01 DEN+0

5

Dose NNM in mg pro kg

0.01 DEN+1

DEN+5

Dose NNM in mg pro kg

0

5

Dose NNM in mg pro kg

Fig. 2. Empirical number distribution (filled) and predicted number distributions from CSMbeta (black, hollow) and CSM (gray, hollow) for initiated (left panel) and not initiated (right panel) animals. Upper panel shows type 1-FAH; lower panel shows type 2-FAH. The number of foci is displayed on a log-scaled axis.

Table 2 Maximum Likelihood estimates for CSMbeta parameters (‘LD’ = ‘low dose of NNM’, ‘HD’ = ‘high dose of NNM’). Dose group

DEN

DEN + LD

DEN + HD

Control

HD

lð1Þ

0.06 0.023

0.09 0.025

0.13 0.028

0.01 0.017

0.11 0.023

ð2Þ b (1) Rate of FAH-formation [per day and mm3].  ¼ ap . (2) Expected growth rate [per day], b pþq

Table 3 Maximum likelihood estimates CSM parameters (‘LD’ = ‘low dose of NNM’, ‘HD’ = ‘High dose of NNM’). Dose group

DEN

DEN + LD

DEN + HD

Control

HD

lð1Þ

0.03

0.04

0.06

3:91  103 0.022

0.03

ð2Þ 0.032 0.036 0.038 b (1) Rate of FAH-formation [per day and mm3] (2) Growth rate [per day].

0.041

of CSM and CSMbeta to describe the color-shift between different FAH-phenotypes could not be evaluated. Hence the present analysis represents a follow-up of [9] transferring the results to two color approaches. Fig. 1 illustrates the deficiencies of original CSM concerning the prediction of FAH-sizes. Due to the assumption of exponentially distributed waiting times for color- shift, variation in radii of FAH of one color is overestimated by CSM in every treatment group except for the controls. In contrast, the predicted radii of one color in CSMbeta show a similar variation as observed in the data. The change from (random) time for color change to random radius for color change did not essentially change the system dynamics of the CSM but reflects the biological hypothesis that the size more than the ‘age’ of a focus triggers the color-shift. CSM and CSMbeta both include parameters which can be interpreted concerning formation and growth of FAH. Application of these models to data from initiation-promotion experiments

enables us to draw conclusions about the mode of action of the test compound, in this case NNM. It is well-known that NNM is able to generate FAH in normal tissue as well as to accelerate size increase of existing FAH [12]. CSMbeta and CSM had been applied to data from the same experiment [13], but staining with another marker (placental form of gluthadione S-transferase) and, therefore, observation of a single FAH type, resulted in similar parameter interpretations as in the present study. The conclusions drawn from parameter interpretation of the two models coincide and are consistent with earlier experimental findings. In summary, CSMbeta is a mechanistic carcinogenesis model based on the field-effect hypothesis which is able to predict the distribution of size and number of FAH more correctly than the original CSM. Acknowledgments The work of Jutta Groos was supported by DFG-Grant No. KO1886/1–3 and 1–5. We thank Professor Bannasch for letting us use the data. References [1] A. Kopp-Schneider, C. Portier, P. Bannasch, A model for hepatocarcinogenesis treating phenotypical changes in focal hepatocellular lesions as epigenetic events, Math. Biosci. 148 (1998) 181. [2] P. Bannasch, Pathogenesis of hepatocellular carcinoma: Sequential cellular, molecular, and metabolic changes, Prog. Liver Dis. 14 (1996) 161. [3] P. Bannasch, H. Zerban, Predictive value of hepatic preneoplastic lesions as indicators of carcinogenic response, IARC Sci. Publ. 116 (1992) 389. [4] R. Hasegawa, N. Ito, Hepatocarcinogenesis in the rat, in: M.P. Waalkes, J.M. Ward (Eds.), Carinogenesis, Raven, New York, 1994, p. 39. [5] P. Bannasch, The cytoplasm of hepatocytes during carcinogenesis. Electron and light microscopic investigations of the nitrosomorpholine-intoxicated rat liver, Recent Results Cancer Res. 19 (1968) 1. [6] S.H. Moolgavkar, E.G. Luebeck, M. de Gunst, R.E. Port, M. Schwarz, Quantitative analysis of enzyme-altered foci in rat hepatocarcinogenesis experiments-I. Single agent regimen, Carcinogenesis 11 (1990) 1271. [7] I. Burkholder, A. Kopp-Schneider, Incorporating phenotype-dependent growth rates into the color-shift model for preneoplastic hepatocellular lesions, Math. Biosci. 179 (2002) 145.

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[8] J. Groos, A. Kopp-Schneider, Visualization of parametric carcinogenesis models, in: J. Antoch (Ed.), Compstat Proceedings in Computational Statistics, PhysikaVerlag, Heidelberg, 2004, p. 189. [9] J. Groos, A. Kopp-Schneider, Application of a color-shift model with heterogeneous growth to a rat hepatocarcinogenesis experiment, Math. Biosci. 202 (2006) 248. [10] S. Wicksell, The corpuscle problem. A mathematical study of a biometrical problem, Biometrika 17 (1925) 87.

[11] T.W. Anderson, On the distribution of the two-sample Cramer-von Mises criterion, Ann. Math. Stat. 33 (1962) 1148. [12] E. Weber, P. Bannasch, Dose and time dependence of the cellular phenotype in rat hepatic preneoplasia and neoplasia induced by single oral exposures to Nnitrosomorpholine, Carcinogenesis 15 (1994) 1219. [13] J. Groos, P. Bannasch, M. Schwarz, A. Kopp-Schneider, Comparison of mode of action of four hepatocarcinogens: a model based approach, Toxicol. Sci. 99 (2007) 446.