Application of a color-shift model with heterogeneous growth to a rat hepatocarcinogenesis...

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Mathematical Biosciences 202 (2006) 248–268

Application of a color-shift model with heterogeneousgrowth to a rat hepatocarcinogenesis experiment

Jutta Groos, Annette Kopp-Schneider *

Biostatistics-C060, German Cancer Research Center, Im Neuenheimer Feld 280, D-69120 Heidelberg, Germany

Received 2 February 2005; received in revised form 20 October 2005; accepted 21 February 2006Available online 17 April 2006

Abstract

Several hypotheses are established to describe the formation and progression of foci of altered hepato-cytes (FAH). A common model of hepatocarcinogenesis is the mutation model (MM), which is based onthe assumption that cells have to undergo multiple successive changes on their way from the normal tothe malignant stage. This model describes growth and phenotype change of foci on the cellular level andis based on the assumption that single cells change their phenotype through mutation into the next stageand proliferate according to a linear stochastic birth–death process.

In contrast, the color-shift model (CSM) was introduced by Kopp-Schneider et al. to describe that wholecolonies of altered hepatocytes simultaneously alter their phenotype. In this paper two modifications of thecolor-shift model are considered which allow the growth rate to vary from focus to focus. All four modelsare compared with respect to their ability to predict number and radii of foci in a rat hepatocarcinogenesisexperiment, in which rats were treated with the carcinogens N-nitrosomorpholine, 2-acetylaminofluorenand Phenobarbital. Maximum likelihood parameter estimates are given, and predicted and empiricalFAH size distributions are visualized. The Cramer–von-Mises distance is used as a measure for the discrep-ancy between empirical and theoretical size distributions.� 2006 Elsevier Inc. All rights reserved.

Keywords: Mutation model; Color-shift model; Hepatocarcinogenesis; Cramer–von-Mises distance; Foci of alteredhepatocytes; GSTP staining

0025-5564/$ - see front matter � 2006 Elsevier Inc. All rights reserved.doi:10.1016/j.mbs.2006.02.002

* Corresponding author. Tel.: +49 6221 42 2391; fax: +49 6221 42 2397.E-mail address: [email protected] (A. Kopp-Schneider).

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J. Groos, A. Kopp-Schneider / Mathematical Biosciences 202 (2006) 248–268 249

1. Introduction

Hepatocarcinogenesis experiments identify focal lesions consisting of intermediate cells at dif-ferent preneoplastic stages. Foci of altered hepatocytes (FAH) are studied by the morphometricanalysis of stained liver sections. Depending on the staining method either different types or onlyone type of FAH can be observed. Several hypotheses are established to describe the formationand progression of FAH.

A common model of hepatocarcinogenesis is the mutation model (MM) also known as twostage clonal expansion model [1], which is based on the assumption that cells have to undergomultiple successive heritable transformations on their way from the normal to the malignantstage. This model describes formation, growth and phenotype change of foci on the cellularlevel and is based on the assumption that single cells change their phenotype through mutationinto the next stage and proliferate according to a linear stochastic birth–death process. There issome evidence that the MM cannot adequately describe the formation and progression of pre-neoplastic lesions in the rat liver when several successive types of preneoplastic lesion are ob-served [2].

In contrast, the color-shift model (CSM) was introduced by Kopp-Schneider et al. [3] to de-scribe that whole colonies of altered hepatocytes more or less simultaneously alter their pheno-type. The CSM formalizes the biological hypothesis that the change of FAH is due to a fieldeffect in which the modification of the biochemical environment induces a gradual metabolic shiftduring progression from early to advanced FAH. In the CSM, FAH as a whole are assumed togrow exponentially with deterministic rate and to change their phenotype (‘color’) after an expo-nentially distributed waiting time. The CSM has been applied to data from a long-term rathepatocarcinogenesis experiment in which different types of FAH were observed. Although it pre-dicted the size and number of focal transections more appropriately than the MM, the model ex-pected more variation in size distribution of FAH than the data exhibited [4]. This rather peculiarobservation may be due to the assumption of exponentially distributed waiting times for colorchange 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 deterministic radii for color change ofFAH, which leads to a bounded support for the size distribution of FAH in one phenotype. Espe-cially in this context, the assumption of deterministic growth with a single rate for all FAH of onetype seems too simple. Therefore the second modification of the color-shift model consists ofallowing the growth rate to vary from focus to focus. Specifically the growth rate of every singleFAH represents an independent realization of either a uniformly (CSMuni) or a beta distributed(CSMbeta) random variable [5]. Variations in growth rate are effective from the time point of for-mation onward. Two reasons may lead to different growth rates. Either differences in growth rateare due to biological variation of FAH of the same phenotype, or variations in growth rate can beinterpreted as reflecting the fact that the FAH population consists of a mixture of different sub-populations representing early stages of the hepatocarcinogenic process with different speeds ofgrowth, but indistinguishable by the staining method.

Morphometrical evaluation of hepatocarcinogenesis experiments results in datasets consistingof sizes of focal transections. Biologically based models, however, are developed to describe thethree-dimensional situation in the liver. Moolgavkar et al. [1] have shown that the appropriate

250 J. Groos, A. Kopp-Schneider / Mathematical Biosciences 202 (2006) 248–268

evaluation method of FAH data consists in transformation of the model-derived three-dimen-sional number and size distributions of FAH into two-dimensional number and size distribu-tions of focal transections by the Wicksell transformation [6]. The transformed models areapplied to the observed data about number and sizes of focal transections by maximum likeli-hood methods.

Comparison of hepatocarcinogenesis models is often hampered by the fact that models are notnested hierarchically and therefore the likelihood ratio test cannot be used for comparison of themodel fit. However, the value of the likelihood at the maximum can be taken as heuristic indica-tion for comparing the model fit as long as the number of parameters involved in the models isidentical [7,8]. In the present study, all models predict the total number of FAH in every groupcorrectly and the difference in model fit shows in the predicted size distribution of FAH. In thissituation, the Cramer–von-Mises distance (CvMD) between data and model is an adequate mea-sure for the goodness-of-fit of the model [9] and comparison of the CvMD values allows to iden-tify the best fitting model in a descriptive sense.

The present study shows the model-based evaluation of data from a rat liver foci experiment inwhich the liver sections obtained at a single observation time point were stained with glutathione-S-transferase placental form (GSTP). The experiment involved several experimental groups ofwhich we evaluated those treated with the carcinogens N-nitrosomorpholine (NNM), 2-acetylami-nofluoren (2-AAF) and Phenobarbital (PB) [10]. The staining method marks FAH which exhibitan increased expression of GSTP. It has proven a reliable and easily detectable cytochemical mar-ker for FAH [11]. Other staining methods, such as hematoxylin and eosin (H&E), reveal a largevariety of different FAH types [12] which represent lesions in different stages towards malignancy.Detection and evaluation of FAH in H&E-stained liver sections has to be performed manually,whereas GSTP staining allows for the application of image analysis programs for evaluation ofFAH. It is known that GSTP is negative in preneoplastic and neoplastic cell populations with in-creased basophilic components, lesions which are known to be more advanced in the carcinogenicprocess [12,13].

The model-based analysis of GSTP-marked FAH data can be used to investigate whether FAHloose GSTP staining when they advance towards malignancy. The implementation of this hypo-thesis involves the change of FAH phenotype into a type which is not detected by GSTP, i.e. two-color models in which only FAH in the first color are observed.

Due to the fact that in the present experiment liver sections were evaluated at an early timepoint (after 10 weeks of treatment) most foci are of earlier types, which was confirmed byH&E staining [10]. Since a single observation time point was available in the experiment, temporalpatterns of growth rates are not identifiable from the data.

The manuscript is organized in the following way. Section 2 briefly shows the mathematicaldescription of hepatocarcinogenesis models MM, CSM, CSMuni and CSMbeta. The Wickselltransformation is applied and the resulting number and size distributions for focal transectionsare given. CvMD is introduced as a descriptive measure of goodness-of-fit. The implementationof maximum likelihood methods in MATLAB [14] is presented. Section 3 shows the applicationof the different models to experimental data. Maximum likelihood parameter estimates are gi-ven, and predicted and empirical distributions are visualized. A discussion of results is given inSection 4.


2. Methods

2.1. Model development

2.1.1. M-stage color-shift modelThe color-shift model (CSM) introduced by Kopp-Schneider et al. [3] describes the formation

and growth of FAH which are distinguished from normal tissue by staining.Centers of spherical foci are generated according to a homogeneous Poisson process in the liver.

Starting from a fixed size, foci grow exponentially with deterministic growth rate. FAH are gen-erated in the first preneoplastic stage and may change their phenotype (‘color’) after an exponen-tially distributed waiting time. The color shifts sequentially and irreversibly over time until colorm is reached (see Fig. 1). As FAH are assumed to be spherical, the size of a focus is completelycharacterized by its radius.

Let the color C(t) and the radius R(t) of a focus at time point t both be stochastic processes intime. Let the location X of a focus be independent of the time point of formation T, the color C(t)and the radius R(t), and let C(t) and R(t) be conditionally independent given T. The followingparameters characterize the distribution of size and number of FAH:

• Radius at formation r0.• Poisson parameter l of the formation process.• Growth rate b.• Parameter k of the exponentially distributed waiting time for color shift.

At time point t > s, the radius of a focus generated at time point T = s and growing with rate bis given by

Fig. 1

RðtÞ ¼ r0 expðbðt � sÞÞ.
Since centers of FAH are generated by a Poisson process, the time point of formation T is uni-formly distributed on [0, t]. The following expressions for the joint density of radius R(t) and colorC(t) at time point t were derived in [3] (R(t) 2 [r0,1),C(t) 2 {1, . . . ,m}).
Color 1

fRðtÞ;CðtÞðr; 1Þ ¼1

tbrexp � k

bln

rr0

� �� 1½r0;r0 expðbtÞ�ðrÞ;

where 1[a,b] denotes the indicator function

1½a;b�ðxÞ ¼1 x 2 ½a; b�;0 otherwise:

�

. Cube of liver tissue at four sequenced time points. Light spheres: FAH of type 1. Dark spheres: FAH of type 2.


Color 1 < c < m

fRðtÞ;CðtÞðr; cÞ ¼1

tbrðc� 1Þ! exp � kb

lnrr0

� �� kb

lnrr0

� �� c�1

1½r0;r0 expðbtÞ�ðrÞ.

Color m

fRðtÞ;CðtÞðr;mÞ ¼1

tbr1�

Xm�2

k¼0

1

k!exp � k

bln

rr0

� �� kb

lnrr0

� �� k( )

1½r0;r0 expðbtÞ�ðrÞ.

2.1.2. Color-shift models with heterogeneous growthThe assumption of deterministic growth rates for FAH in CSM seems to oversimplify the real

process, wherefore a CSM with heterogeneous growth rate is introduced [5]. As in CSM the for-mation of spherical foci with initial radius r0 is described by a homogeneous Poisson process withrate l. Every single FAH grows exponentially with rate b which is a realization of a positive ran-dom variable B with density fB. B is assumed to be independent of the time point of FAH forma-tion T.

FAH are assumed to change their color from C(t) = c to C(t) = c + 1 whenever they reach thesize R(t) = rc, where c 2 {1, . . . ,m}, r0 6 r1 6 � � � 6 rm 61. The color of a focus is, therefore,completely determined by its size. The joint density of radius R(t) and color C(t) at time t is givenby

fRðtÞ;CðtÞðx; cÞ ¼1

xt

Z 1

lnð xr0Þ

t

fBðbÞb

db � 1½ðrc�1;rcÞðxÞ.

This expression is derived by integration of the density of the one-color CSM (CSM1), setting theparameter of the exponential waiting time for color change, k, to 0, so that the waiting time forFAH color-shift is infinity.

In this paper two special cases of CSM with heterogeneous growth are considered. In the pres-ent study, only one type of FAH was evaluated experimentally, therefore, shift radii could not beestimated. Hence the expressions for densities and model parameters are given in a one-stageapproach.

CSMuni assumes uniformly distributed growth rates, i.e.

fBðbÞ ¼1

jIj 1IðbÞ; I ¼ ½b; b� � Rþ.

CSMbeta assumes beta distributed growth rates, where an additional parameter a is introduced tovary the support of the beta distribution, i.e.

fBðbÞ ¼1

Bðp; qÞbðp�1Þða� bÞðq�1Þ

aðpþq�1Þ � 1½0;a�ðbÞ p; q; a > 0; ð1Þ

where B(p,q) is the Beta-function

Bðp; qÞ ¼Z 1

0

zðp�1Þð1� zÞðq�1Þ dz.


The densities of radius R(t) at time t for CSMuni and CSMbeta are given in Appendixes A.1and A.2.

The CSMs with heterogeneous growth contain the following model parameters:Model parameters of CSMuni:

• Radius r0 of a focus at formation.• Poisson parameter l of the formation process.• Upper and lower limit b and b for growth rate.

Model parameters of CSMbeta:

• Radius r0 of a focus at formation.• Poisson parameter l of the formation process.• Parameters p, q and a for growth rate.

2.1.3. Mutation modelIn contrast to CSM, the mutation model (MM) describes formation, growth and phenotype

change of FAH on the basis of individual hepatocytes rather than entire FAH. This model as-sumes that single cells alter their phenotype through mutation. After a normal cell is transformedinto a preneoplastic stage with initiation rate l it divides into two preneoplastic cells or dies andleaves the process with rates b and d, respectively. Under the assumption that cells are denselypacked in space and that FAH are spherical, a focus containing n cells has a radius ofr ¼ rcell �

ffiffiffin3p

, where rcell is the radius of a single cell. The distribution of size and number of fociat time t is characterized by the following parameters:

• Radius of a cell rcell.• Initiation rate l.• Cell division rate b.• Cell death rate d.

Moolgavkar et al. [1] derived the following expression for the density of size distribution ofFAH:

fRðtÞðxÞ ¼3

bd

� �pðtÞ

� �ð xr0Þ3

r lnd

d� bpðtÞ

� �

with

pðtÞ ¼

d� d expð�ðb� dÞtÞb� d expð�ðb� dÞtÞ b 6¼ d;

bt1þ bt

b ¼ d:

8>><>>: ð2Þ

All models considered here assume that FAH act independently of each other.


2.2. Stereological considerations

In hepatocarcinogenesis experiments, measurements are made in two-dimensional liver sectionsand inference about the reality in three-dimensional liver is limited by the stereological problem.This problem is described briefly by the fact that the probability of a focus to be cut increases withits size. All models introduced here describe the three-dimensional situation. Moolgavkar et al. [1]suggested to transform the expressions for the distributions of size and number of foci into thecorresponding two-dimensional expressions by the Wicksell transformation [6] and to apply themodel to two-dimensional measurements by maximum likelihood methods (see Fig. 2). This pro-cedure was found to be optimal for the model based analysis of FAH data [15].

The morphometrical evaluation of stained liver sections records focal transections with transec-tion radius larger than e. Kopp-Schneider et al. [3] derived the following expressions for the ex-pected number of observable focal transections of color 1 at time point t in CSM

~n2 ¼ 2ltZ 1

e

ffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 � e2p

fRðtÞ;CðtÞðx; 1Þdx. ð3Þ

The density of the size distribution of observable focal transections of color 1 at time point t intwo dimensions for CSM is

fRð2ÞðtÞjCðtÞðyj1Þ ¼yZ 1

e


fRðtÞ;CðtÞðx; 1Þdx

Z 1

y

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 � y2

p fRðtÞ;CðtÞðx; 1Þdx. ð4Þ

In MM the expected three-dimensional number of non-extinct FAH at time point t is Poissondistributed with mean KðtÞ ¼ l

b ½lnð dd�bpðtÞÞ�, where p(t) is given by Eq. (2) [1].

In this case the expected number of focal transections at time point t is given by

~n2 ¼ 2KðtÞZ 1

e


fRðtÞðxÞdx. ð5Þ

The density of the two-dimensional size distribution of focal transections at time point t in MM is

fRð2ÞðtÞðyÞ ¼yZ 1

e


fRðtÞðxÞdx

Z 1

y

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 � y2

p fRðtÞðxÞdx. ð6Þ

Fig. 2. Cube of liver tissue with foci and transections in 2D.


2.3. Likelihood construction

Let n2 denote the number of focal transections observed in a liver section of area A and let r2,j

denote the radius of the jth focal transection of color 1. In CSM this liver section contributes theloglikelihood

ðn2 lnðA~n2Þ � A~n2Þ þXn2

j¼1

lnðfRð2ÞðtÞjCðtÞðr2;jj1ÞÞ" #

þ C; ð7Þ

and in MM

ðn2 lnðA~n2Þ � A~n2Þ þXn2

j¼1

lnðfRð2ÞðtÞðr2;jÞÞ" #

þ C; ð8Þ

where C :¼ �ln(n2!).As liver sections of one experiment are independent of each other, the loglikelihood of the com-

plete dataset is the sum of the contributions of every section. In the present application liver sec-tions of animals in each treatment group were pooled. Therefore, the predicted foci densities areidentical for all animals in one experimental group and the actually predicted transection numberfor every animal depends on its total liver section area. Due to the Poisson nature of the FAHformation process the maximum likelihood parameter estimates are not affected by the poolingprocedure.

2.4. Cramer–von-Mises distance

Let N be the number of focal transections detected in one experiment and let the radii be de-noted by r1, . . . , rN. Let r[1], . . . , r[N] denote the ordered observations and F(r([1]), . . . ,F(r[N]) the cor-responding values of the theoretical distribution function F predicted by the model.

The expression for the Cramer–von-Mises distance in a form given by Durbin and Knott in1972 [16]:

CvMD :¼ N þ 1

N

XN

i¼1

F ðr½i�Þ �i

N þ 1

� �2

is a measure for the discrepancy between empirical and theoretical distribution function of thefocal transection sizes.

Therefore CvMD can be used as a descriptive measure of goodness-of-fit of the models, and bycomparison of the CvMD values the best fitting model can be identified from a set of non-nestedmodels. However, CvMD is a non-parametric measure of goodness-of-fit and the number ofparameters does not enter the calculation of CvMD. The expression ‘best fitting model’ is usedonly in a descriptive sense.

2.5. Implementation

The MATLAB environment is used to compute loglikelihood functions, to compute the max-imum likelihood parameter estimates and to visualize the results. In CSMbeta the density of the


size distribution in two dimensions contains double integrals which are not solvable analytically.Therefore, the Fortran–Mex Interface is used to include Fortran NAg routines for numericaldouble integration into the MATLAB environment [14,17]. The negative loglikelihood is mini-mized by the MATLAB function fmincon. Theoretical results are compared to the empiricaldata by visualization of the size distributions and the CvMD is used to asses the goodness-of-fit.

3. Application to rat liver foci data

3.1. Data

Data from a rat hepatocarcinogenesis experiment published by Ittrich et al. [10] are chosen toillustrate the method and to compare different carcinogenesis models. From the large dataset weconcentrate on the subset generated in Professor Bannasch’s laboratory at the German CancerResearch Center. In this study male Wistar rats were continuously treated with one of threechemicals:

• Eight animals were treated with 5 mg of the chemical carcinogen N-nitrosomorpholine (NNM)per kg body weight dissolved in distilled water and administered by gavage.

• Eight animals were treated with 10 mg 2-acetylaminoflouren (2-AAF) per kg body weight sus-pended in olive oil and administered by gavage.

• Eight animals received 500 pm Phenobarbital (PB) in the feed.

After 10 weeks two independent liver sections of each rat were treated immunohistochemicallyfor the demonstration of GSTP and GSTP-positive focal transections were observed. The mor-phometrical evaluation of the stained liver sections generated datasets consisting of the area ofevery liver section and the area of every focal transection detected in this liver section. The mor-phometrical evaluation accounted for focal transections with area exceeding 0.003 mm2.

The total liver section area and number of foci evaluated in the three groups were N = 1270 inA = 1325 mm2 for 2-AAF, N = 627 in A = 1260 mm2 for NNM and N = 12 in A = 1701 mm2 forPB.

The foci densities in the three groups ranged from 0.13 to 2.05 per mm2 (median 0.94) for 2-AAF, from 0.26 to 0.65 per mm2 (median 0.54) for NNM and from 0.00 to 0.02 per mm2 (median0.005) for PB.

3.2. Application of hepatocarcinogenesis models to data

All four models are applied to the data by maximum likelihood methods. In this applicationwe assume that in color-shift models FAH originate from a single cell, i.e. the radius of forma-tion corresponds to the size of one hepatocyte. Biological studies suggest that the radius of ahepatocyte can be approximated by r0 = rcell = 0.014 mm [18]. The morphological detectionlimit of 0.003 mm2 translates to a minimum detection radius of e = 0.031 mm for the focaltransection.


The assumption that FAH are of spherical shape may seem unrealistic. Histopathologicalexamination of liver sections shows that focal transections of medium size are more or less circu-lar, whereas very large focal transections sometimes seem to have less circular shape. It is knownthat very small GSTP-positive FAH may have very irregular shapes [19]. In the current experi-mental study, no large lesions developed due to the short duration of the study. Very small tran-sections were not included, neither in the statistical nor in the model-based analysis and, therefore,non-spherical shapes do not pose a serious problem for our model-based evaluation.

Table 1 summarizes the maximum likelihoood parameter estimates with Wald-confidence inter-vals for all four models. The goodness-of-fit, measured by CvMD and the likelihood at the max-imum, is reported in Table 2. Figs. 4–6 visualize the theoretical and empirical size distributions forthe three treatment groups. Since only a single observation time point was available, the expectedtotal number of detected focal transections of one treatment group is completely determined bythe Poisson parameter l for FAH formation. Hence the total number of focal transections forevery treatment group expected by every model matches exactly the correct value.

3.2.1. Application of one-color CSM to dataFigs. 4(a), 5(a) and 6(a) show the application of the one-color CSM (CSM1) to the data for all

three chemicals. The comparison of the theoretical and empirical size distributions shows thatCSM1 is not adequate to predict the size distribution of FAH. The parameter estimates (see Table1) for l predict that during 10 weeks of 2-AAF, NNM and PB treatment 2940, 5376 and 101FAH, respectively, are generated in 1 cm3 of liver tissue. Inspection of the parameter estimatefor the growth rate of FAH reveals that b is determined by the size of the largest transection ob-served in the dataset. Consider exemplarily the 2-AAF treatment group in which the growth rateis estimated as b = 0.039 mm per day. Assuming that a focus was generated instantly at the startof the treatment period, it would reach a three-dimensional radius of 0.68 mm at termination ofthe study. When cut centrally this corresponds exactly to the largest observed transection radius.

3.2.2. Application of two-color CSM to dataAs mentioned before, FAH have to go through more than one successive stage until the malig-

nant stage is reached. The marker GSTP used in the present study stains early FAH, but it isknown that GSTP fails to stain progressed FAH. To address the question whether FAH mayloose their potential to be marked by GSTP, a two-color CSM (CSM2) is applied to the datain which foci of the second type are treated as ‘invisible’. Figs. 4(b), 5(b) and 6(b) show the appli-cation of CSM2. It is apparent that CSM2 with ‘invisible’ foci of the second type is more appro-priate than CSM1. CSM1 can be interpreted as a CSM2 with infinite waiting time for phenotypechange, i.e. k = 0. Hence a likelihood ratio test can be performed to compare the fit of CSM2

against CSM1. For 2-AAF and NNM there is a highly significant advantage for the two-color ap-proach (p-value = 0). The result for PB is less clear (p-value = 0.21), which is probably due to thesmall number of observed FAH in this group. The mean waiting time to phenotype change is esti-mated to be about 2 weeks for 2-AAF and NNM and about 4 weeks for PB. At the end of theexperiment, 83%, 82% and 70% of FAH are expected to be ‘invisible’ for 2-AAF, NNM andPB, respectively. As to the focal transections, CSM2 expects that 98% for 2-AAF and NNMand 90% for PB cannot be detected by GSTP staining. Both of these values seem unrealistic.To compensate for the loss of visible FAH the Poisson parameter l has to reach larger values


in the two-color approach. The growth parameter estimates are virtually identical to those ob-tained in CSM1.

3.2.3. Application of CSMuni to dataIn a first step CSM1 is modified by incorporating a growth rate which is assumed to be uni-

formly distributed on ½b; b� 2 Rþ. Maximum likelihood estimation for a full CSMuni with variablelower limit b and a CSMuni with fixed b = 0 result in the same likelihood at the maximum. Henceb = 0 can be assumed without loss of generality.

Figs. 4(c), 5(c) and 6(c) show the application of CSMuni. CSMuni is apparently more appro-priate than CSM1. Visual inspection reveals that the goodness-of-fit is comparable to the resultsfor CSM2. As there is no hierarchical structure between CSM1 and CSMuni the CvMD is used tocompare the goodness-of-fit. The values for CvMD demonstrate that there is a clear advantage ofCSMuni over CSM1 (see Table 2). Also the value for the likelihood at the maximum is larger in allthree treatment groups. The parameter estimates (see Table 1) for l predict that during 2-AAF,NNM and PB treatment 10920, 18480 and 1008 FAH, respectively are generated in 1 cm3 of livertissue. These values are relatively large compared to the values for CSM1. The estimates for themaximum growth rate b are comparable to the estimates for growth rate b in CSM1 and CSM2.Hence in CSMuni some FAH grow with smaller rate than in CSM. This leads to a smaller meansize of FAH in CSMuni. As the probability for a focus to be detected increases with its size, alarger three-dimensional number of small FAH in CSMuni leads to the same number of focaltransections.

In an additional investigation, a two-color CSMuni with deterministic shift radius r1 is applied.For the application to GSTP-marked foci data, FAH of the second type are assumed to be invis-ible which means that they vanish after reaching radius r1. In all three treatment groups the max-imum likelihood parameter estimate for the shift radius r1 exceeds the maximal reachable radiusin color one. Therefore, the one- and two-color approach leads to identical maximum likelihoodparameter estimates and identical model fits.

3.2.4. Application of CSMbeta to dataIn a second step CSM1 is modified by incorporating a growth rate with a more flexible distri-

bution than in CSMuni. The growth rate is assumed to be beta distributed on the interval [0,a].The uniform distribution on [0,a] is a special case of the beta distribution on [0,a] settingp = q = 1 in Eq. (1). In the application of CSMbeta the parameter a for the support of the betadistribution is fixed to the same value as b determined from maximum likelihood estimation inCSMuni for the corresponding dataset. The three remaining parameters in CSMbeta l, p and qare estimated by maximum likelihood methods. This approach allows to perform a likelihoodratio test for the two CSMs with stochastic growth rates on basis of the hierarchical structureof the uniform and the beta distribution (cp. Fig. 3).

Figs. 4(d), 5(d) and 6(d) show the results for CSMbeta. The comparison of theoretical andempirical size distribution of foci after 2-AAF and NNM treatment shows a clear advantage ofCSMbeta over CSMuni.

The computation of the CvMD confirms the graphical impression that CSMbeta generatesbetter model fits than CSMuni. The likelihood ratio test shows that the improvement is significant(p-value = 0 for 2-AAF and p-value = 2.3 · 10�6 for NNM). In the case of Phenobarbital

0 0.01 0.02 0.03 0.0390

5

10

15

20

25

30

35

Growth rate [mm/day]

Pro

babi

lity

dens

ity fu

nctio

n

Beta

Uniform

Fig. 3. Comparison of the estimated distribution of the growth rate in CSMbeta and CSMuni in case of 2-AAFtreatment.


treatment only 12 foci were detected and only marginal differences between the model fits can beobserved. Fig. 3 shows the estimated density function of beta and uniformly distributed growthrates in case of 2-AAF treatment. In this case the beta distribution is almost symmetric. Hencethe mean growth rate is the same for the beta and the uniform approach. For 2-AAF andNNM the estimated values for p and q are dissimilar to 1. Therefore the beta and the uniformdensity function have clearly different shapes, which explains the difference in the predictions ofCSMbeta and CSMuni. For PB treatment the estimates for p and q are comparatively close to1, so that there is not much difference between the two approaches.

Analog to the application of CSMuni, application of a two-color CSMbeta with deterministicshift radius r1 and invisible second type leads to the conclusion that the shift radius r1 is notreached during treatment. Therefore, parameter estimates and model fits are identical in theone- and two-stage approach for all three treatment groups.

3.2.5. Application of MM to dataApplication of the mutation model (MM) is shown in Figs. 4(e), 5(e) and 6(e). Since only a sin-

gle evaluation time point was available, not all of the three parameters b, d and l, are identifiable.Therefore, d

b is kept constant. The model fit and the likelihood value at the maximum are unaf-fected by the choice of d

b, but the parameter estimates for b and l vary depending on the choiceof d

b, their ratio, however, remains constant. Although typically the ratio of death to birth rate db

exceeds 0.9 [1,2], the present data show that large values of db lead to values of the birth rate which

are much too large to reflect the duration of cell cycle. Therefore the ratio of cell death rate d tocell division rate b is fixed to d

b ¼ 0:75. Application to data from NNM and PB treatment leads tomodel fits comparable to those obtained for CSMbeta. Data from 2-AAF treatment, however, arenot adequately predicted, which is confirmed by the relatively large CvMD.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1CSM1 applied to 2–AAF

Radius [mm]

F (

Rad

ius)

DataCSM1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1CSM2 applied to 2–AAF

Radius [mm]

F (

Rad

ius)

DataCSM2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Radius [mm]

F (

Rad

ius)

CSMuni applied to 2–AAF

DataCSMuni

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Radius [mm]

F (

Rad

ius)

CSMbeta applied to 2–AAF

DataCSMbeta

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1MM applied to 2–AAF

Radius [mm]

F (

Rad

ius)

DataMM

(a) (b)

(c) (d)

(e)

Fig. 4. Empirical (solid) and theoretical (dotted) size distributions after 2-AAF treatment for one-color CSM, two-color CSM, CSMuni, CSMbeta and MM.


0 0.05 0.1 0.150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1CSM1 applied to NNM

Radius [mm]

F (

Rad

ius)

DataCSM1

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

CSM2 applied to NNM

Radius [mm]

F (

Rad

ius)

DataCSM2

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1CSMuni applied to NNM

Radius [mm]

F (

Rad

ius)

DataCSMuni

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1CSMbeta applied to NNM

Radius [mm]

F (

Rad

ius)

DataCSMbeta

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1MM applied to NNM

Radius [mm]

F (

Rad

ius)

DataMM

(a) (b)

(c) (d)

(e)

Fig. 5. Empirical (solid) and theoretical (dotted) size distributions after NNM treatment for one-color CSM, two-colorCSM, CSMuni, CSMbeta and MM.


0 0.02 0.04 0.06 0.08 0.1 0.120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1CSM1 applied to PB

Radius [mm]

F (

Rad

ius)

DataCSM1

0 0.02 0.04 0.06 0.08 0.1 0.120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1CSM2 applied to PB

F (

Rad

ius)

Radius [mm]

DataCSM2

0.02 0.04 0.06 0.08 0.1 0.120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1CSMuni applied to PB

Radius [mm]

F (

Rad

ius)

DataCSMuni

0 0.02 0.04 0.06 0.08 0.1 0.120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1CSMbeta applied to PB

Radius [mm]

F (

Rad

ius)

DataCSMbeta

0 0.02 0.04 0.06 0.08 0.1 0.120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1MM applied to PB

Radius [mm]

F (

Rad

ius)

DataMM

(a) (b)

(c) (d)

(e)

Fig. 6. Empirical (solid) and theoretical (dotted) size distributions after PB treatment for one-color CSM, two-colorCSM, CSMuni, CSMbeta and MM.


Table 1Maximum likelihood estimates for model parameters with Wald-confidence intervals depending on model andtreatment group (CSM1 = one-color CSM, CSM2 = two-color CSM)

2-AAF NNM PB

CSM1

la 0.029 [0.028,0.030] 0.053 [0.049,0.057] 0.001 [0.000,0.002]bb 0.039 [0.039,0.039] 0.024 [0.024,0.024] 0.021 [0.020,0.022]

CSM2

la 1.67 [1.58,1.76] 2.63 [2.61,2.66] 0.01 [0.00,0.02]kc 0.060 [0.059,0.061] 0.057 [0.056,0.057] 0.032 [0.024,0.039]bb 0.039 [0.039,0.039] 0.024 [0.024,0.024] 0.022 [0.020,0.024]

CSMuni

la 0.11 [0.10,0.12] 0.18 [0.17,0.20] 0.003 [0.002,0.005]bb 0 0 0bb 0.039 [0.039,0.039] 0.024 [0.024,0.024] 0.023 [0.021,0.027]

CSMbeta

la 0.13 [0.12,0.14] 0.2 [0.18,0.21] 0.003 [0.002,0.004]p 1.78 [1.68,1.89] 1.50 [1.37,1.63] 1.00 [0.33,1.68]q 1.82 [1.75,1.89] 1.37 [1.28,1.45] 0.83 [0.46,1.21]ad 0.039 0.024 0.023

MM

la 0.35 [0.32,0.38] 0.55 [0.49,0.62] 0.01 [0.00,0.02]be 0.43 [0.41,0.43] 0.23 [0.23,0.23] 0.25 [0.18,0.34]db 0.75 0.75 0.75

a Number of foci per day and mm3.b Millimeter per day.c Per day.d a in mm per day is fixed to �b from the corresponding CSMuni.e Per cell per day.


The parameter estimates (see Table 1) for l predict that during 10 weeks of 2-AAF, NNM andPB treatment 24886, 38597 and 487 FAH, respectively are generated in 1 cm3 of liver tissue ofwhich 75% are expected to die out asymptotically. The predicted number of FAH generated dur-ing treatment depends on the choice of d

b. For every choice of db, the rate of cell division during 2-

AAF treatment is faster than during NNM treatment. PB and NNM show similar rates of celldivision. Comparing the parameter estimates of cell division rates for the three chemicals, it isapparent that treatment with PB and NNM leads to comparable rates of cell division, whereascell division is increased for 2-AAF. Formation of FAH induced by 2-AAF and NNM is of sim-ilar size, whereas much fewer FAH are induced by PB. The ratio of initiation to cell division rate l

bis 0.84 for 2-AAF, 2.41 for NNM and 0.03 for PB. The estimates for cell division rate and theratio l

b suggest that NNM increases both initiation and cell division rate, but the emphasis liesmore on enhancement of the initiation rate. The effect of PB is mostly the increase of cell divisionrate. 2-AAF on the other hand increases both initiation and cell division rate in a balanced way.These considerations remain valid for other choices of d

b because the relationship of parametersbetween chemicals and the ratio l

b is independent of the constant chosen for db.

Table 2Loglikelihood at the maximum and CvMD as measures for goodness-of-fit depending on model and treatment group(CSM1 = one-color CSM, CSM2 = two-color CSM)

2-AAF NNM PB

CSM1

Loglikelihooda 8809.6 4889.0 48.6CvMD 279.62 45.22 0.26

CSM2

Loglikelihood 9652.7 5112.3 49.5CvMD 2.92 1.19 0.03

CSMuni


CSMbeta


MM


a Omitting a data dependent constant C.


4. Discussion

The present study shows the application of different models to the same set of GSTP-stainedfocal liver transections induced by three carcinogenic substances. Whenever models are hierarchi-cally nested, the goodness-of-fit can be compared by likelihood ratio test. For non-nested models,comparison of the CvMD-value for the size distribution of focal transections can be used to iden-tify the most appropriate model whenever the transection number distribution coincides amongdifferent models. Due to the Poisson nature of the FAH formation process and since a single eval-uation time point was considered, the rate of FAH formation in all models can be adjusted suchthat the total transection number per group predicted by the model matches the observation.

Application of the one-color CSM showed unsatisfactory model fits for the three substances.Two potential reasons for this finding were analyzed:

1. Limitation of the CSM postulating that all FAH grow deterministically with the same rate.2. Progression of FAH to a second type which is undetectable in the staining method chosen for

morphometric evaluation of the liver sections.

Two modified CSMs were introduced and applied to the experimental data to allow FAH togrow with different rates. Growth rates in CSMuni are uniformly distributed on ½0; b�, whereasgrowth rates in CSMbeta follow a beta distribution on the same interval. Visual comparison ofthe one-color CSM, CSMuni and CSMbeta model fits shows that CSMbeta is obviously by farthe most flexible model to predict FAH size distributions. Inspection of the parameter estimates


reveals that the estimate for the growth rate of FAH in the one-color CSM is completely deter-mined by the size of the largest transection observed in the data set. The same is true for the upperbound b of the growth rates in CSMuni. The beta distribution allows for more flexibility in thechoice of growth rates and approximates the true size distribution better.

Different biological interpretations can be drawn from the finding that variable growth rates areneeded to appropriately predict the size distribution of FAH. Either FAH of the same type mayvary in growth rate or FAH detected by GSTP staining may represent a mixture of different sub-populations which are in different early stages of the hepatocarcinogenic process. With a singleobservation time point available in the experiment, there is no possibility to discriminate betweenthese two hypotheses.

The existence of a second ‘invisible’ stage was implemented by applying a two-color CSM andformulating the likelihood just for FAH in the first color. The model fit is dramatically improvedby this approach but not as good as the fit obtained from the (one-color) CSMbeta. Applicationof two-color CSMbeta and CSMuni shows that for all treatment groups the maximum likelihoodestimate for r1 is larger than the maximum reachable radius, i.e. the radius of a FAH generated atthe start of the experiment and growing with the largest growth rate b. Hence this model modi-fication leads to identical values of the likelihood at the maximum and CvMD. The two-colormodels therefore do not predict that any FAH reach the invisible stage and therefore vanish dur-ing the observation period. Summarizing the results of the application of the two-color CSM,two-color CSMuni and two-color CSMbeta, no definite conclusion can be drawn concerningthe existence of a second ‘invisible’ stage. Although the fit of the deterministic CSM is improvedby incorporating a second ‘invisible’ stage, this may merely be due to the additional parameter kintroduced into the model. There is biological evidence for the existence of more progressed FAHwhich are negative for GSTP staining, but these FAH usually develop at later time points.

The mutation model MM has been used for evaluation of FAH data in earlier studies [1]. It isconceptually different from the CSMs since it describes FAH cells rather than entire FAHs. Theaim of the deterministic CSM was to test hypotheses about the phenotype change of FAH. In thepresent hepatocarcinogenesis experiment however, a single phenotype of FAH was detected. It is,therefore, not surprising that application of the CSM involving a rather crude growth processgives a much less appropriate model fit than the MM which describes FAH growth by a stochasticbirth–death process. Application of the improved color-shift models CSMuni and especiallyCSMbeta, however, yields a comparable model fit to the one obtained by MM. In the case of2-AAF treatment both the likelihood value at the maximum and the CvMD are better for CSM-beta than for MM. This suggests that CSMbeta is at least as adequate as MM to describe GSTP-marked FAH data.

So far, the comparison of model fits does not take the number of model parameters into ac-count. CSMbeta involves three parameters estimated from the data, whereas in CSM, CSMuniand MM only two parameters are estimated. Therefore, CSMbeta seems to be the most flexiblemodel. Allowing heterogeneous cell division rates in MM by assuming that b comes from a ran-dom distribution would increase the number of model parameters and possibly improve the modelfit to the 2-AAF data and outperform CSMbeta. As FAH growth in MM is stochastic already,there appears to be less necessity for heterogeneous cell division rate in MM.

The salient feature of biologically based modeling is that model parameters can be interpretedbiologically. Inspection of the parameter estimates allows to draw conclusions about the mode of


action of the chemicals involved in the experimental study. Table 1 shows that irrespective of themodel we observe that PB has much less potential to generate FAH than either NNM or 2-AAFand that NNM generates more FAH than 2-AAF. Considering the growth rates in all models, PBand NNM seem to induce similar FAH growth behavior, whereas 2-AAF appears to be a strongerpromoter of FAH growth.

CSMbeta can be applied to FAH data with more than two phenotypes of FAH. Future appli-cations of CSMs with heterogeneous growth rates will, therefore, focus on the evaluation of dataobtained from staining methods which allow to discriminate between different phenotypes ofFAH.

Acknowledgments

The work of Jutta Groos was supported by DFG-Grant No. KO1886/1-3. We thank ProfessorBannasch for letting us use the data. We thank two anonymous reviewers for their valuablecomments.

Appendix A. Mathematical Appendix

A.1. Density of size distribution for color-shift model with uniformly distributed growth rates

Let the growth rate B be uniformly distributed and positive, that is

fBðbÞ ¼1

jIj 1IðbÞ; I ¼ ½b; b� � Rþ.

Since there is an upper limit for the growth rates, b, the maximum radius of foci is

rmax :¼ r0 expðbtÞ;

fRðtÞðxÞ ¼1

xt

Z 1

ln xr0ð Þt

fBðbÞb

db � 1ðr0;rmax�ðxÞ ¼1

xtjIj

Z 1

ln xr0ð Þt

1

b1IðbÞdb � 1ðr0;rmax�ðxÞ

¼

lnðbÞ�lnln x

r0ð Þt

� xtjIj � 1ðr0;rmax�ðxÞ b 6

ln xr0

� t ;

lnðbÞ�lnðbÞxtjIj � 1ðr0;rmax�ðxÞ b >

ln xr0

� t :

8>>>><>>>>:

A.2. Density of 1-stage CSM with beta distributed growth rates

Let the random variable B which describes the exponential growth be beta distributed withparameters p, q and a. The parameter a was introduced additionally to the parameters of the stan-dard beta distribution, p and q, to modify the support of the distribution function (p,q,a > 0).


Hence the growth rate B is a positive random variable with the following density:

fBðbÞ ¼1

Bðp; qÞbðp�1Þða� bÞðq�1Þ

aðpþq�1Þ � 1½0;a�ðbÞ p; q; a > 0;

where B(p,q) is the Beta-function

Bðp; qÞ ¼Z 1

0

zðp�1Þð1� zÞðq�1Þdz.

Since there is an upper limit for the growth rates, a, the maximum radius of foci is

rmax :¼ r0 expðatÞ.
So the density function of the size distribution of foci is
fRðtÞðtÞ ¼1

xtBðp; qÞaðpþq�1Þ

Z 1

ln xr0ð Þt

bðp�1Þða� bÞðq�1Þ

b1½0;a�ðbÞdb � 1½0;rmax�ðxÞ

¼ 1


Z a

ln xr0ð Þt

bðp�1Þða� bÞðq�1Þ

bdb � 1½0;rmax�ðxÞ

¼ 1


Z a

ln xr0ð Þt

bðp�2Þða� bÞðq�1Þ db � 1½0;rmax�ðxÞ

¼ 1


Z 1

ln xr0ð Þ

a�t

ða~bÞðp�2Þða� a~bÞðq�1Þ � ad~b � 1½0;rmax�ðxÞ

¼ 1

xtBðp; qÞa

Z 1

ln xr0ð Þ

a�t

~bðp�2Þð1� ~bÞðq�1Þ d~b � 1½0;rmax�ðxÞ;

with substitution b ¼ a � ~b.

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