ARTICLE IN PRESS

0022-5193/$ - se

doi:10.1016/j.jtb

�CorrespondE-mail addr

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Journal of Theoretical Biology 243 (2006) 407–417

www.elsevier.com/locate/yjtbi

A stochastic model for the sizes of detectable metastases

Leonid Hanina,�, Jason Rosea,b, Marco Zaiderc

aDepartment of Mathematics, Idaho State University, Pocatello, ID 83209-8085, USAbDepartment of Mathematics, College of Southern Idaho, Twin Falls, ID 83303-1238, USA

cDepartment of Medical Physics, Memorial Sloan-Kettering Cancer Center, 1275 York Avenue, New York, NY 10021, USA

Received 18 February 2006; received in revised form 5 July 2006; accepted 10 July 2006

Available online 15 July 2006

Abstract

A stochastic entirely mechanistic model of metastatic progression of cancer is developed. Based on this model the joint conditional

distribution of the ordered sizes of detectable metastases given their number, n, is computed. It is shown that this distribution coincides

with the joint distribution of order statistics for a random sample of size n derived from some probability distribution, and a formula for

the latter is obtained. This formula is specialized for the case of exponentially growing primary and secondary tumors and exponentially

distributed metastasis promotion times, and identifiability of model parameters is ascertained. These results allow for estimation of the

natural history of cancer. As an example, it is estimated for a breast cancer patient with 31 bone metastases of known sizes. The proposed

model for the sizes of detectable metastases provided an excellent fit to these data.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: Cancer natural history; Metastasis; Model identifiability; Poisson process; Primary tumor

0. Introduction

In spite of significant progress in detection and treatmentof primary cancer, its metastatic spread continues to pose aformidable challenge to the improvement of cancer-specificsurvival of patients inflicted with the disease. The greatestunknown faced by an oncologist who designs a curativetreatment plan for a cancer patient is the possibility of thepresence of occult (undetectable) metastases at the start oftreatment for the primary tumor. To increase the chancesof long-term survival of a patient with such metastases,they should be treated concurrently with (or shortly after)the treatment of the primary disease. The advent of modernmethods of metastasis ablation such as conformal stereo-tactic hypofractionated radiosurgery (Schell et al., 1995)and radioimmunotherapy (Bernhardt et al., 2001; Godduet al., 1994; O’Donoghue, 2000) makes this systemicapproach to cancer treatment feasible.

e front matter r 2006 Elsevier Ltd. All rights reserved.

i.2006.07.005

ing author. Tel.: +1208 282 3293; fax: +1 208 282 2636.

esses: hanin@isu.edu (L. Hanin), jrose@csi.edu (J. Rose),

c.org (M. Zaider).

However, a curative therapy developed for an individualpatient can only be as good as the information about thenatural history of his/her disease. Unfortunately, manyimportant parameters descriptive of the course of thedisease (such as the age at onset, growth rates for primarytumor and metastases, intensity of metastasizing, andpromotion time, that is, the time between shedding of ametastasis by the primary tumor and inception of thismetastasis in the host organ or tissue) are unobservable. Toestimate them, one has to develop a comprehensivemathematical model of the disease natural history and fitit to the individual data observable in a given clinicalsetting. These observations include clinical characteristicsof the disease that become available at the time of primarydiagnosis (such as age, tumor size, stage, localization,histological grade, and various biochemical markers atdetection of the primary tumor) as well as variablesdescriptive of the metastatic progression of the disease(most notably, the number and sizes of detectablemetastases). A detailed mechanistic model of individualcancer natural history allows one to estimate the distribu-tional characteristics of the number and sizes of metastasesthat remain occult at the time of diagnosis or the start of

ARTICLE IN PRESSL. Hanin et al. / Journal of Theoretical Biology 243 (2006) 407–417408

treatment. With this information at hand, an oncologistcan design a therapeutic intervention plan (including mode,dosage, and time course of treatment) that fits a givenpatient best and maximizes the probability of cure or theresidual lifetime.

In the present paper, which is predominantly methodo-logical, we develop a mathematical model of metastasisformation and growth in a given host site. These processesare very complex, heterogeneous, and selective (Barbourand Gotley, 2003; Chambers et al., 1995; Evans, 1991;Fidler, 1990, 1991, 1997, 2003). To form a micrometastasis,a tumor cell has to separate itself from the primary tumor,penetrate a blood or lymph vessel, traverse the circulationsystem, evade attacks of the immune system, extravasate,invade a host site, proliferate, and induce angiogenesis. Asa result, only a small fraction of cells shed by the primarytumor gives rise to viable metastases.

The formation of metastases involves substantial varia-bility and random fluctuations in the characteristics ofmetastatic cells and the host microenvironment. Therefore,it can be best described using stochastic models. Thisapproach to modeling metastatic progression of cancer wasapplied, mostly in the form of Monte Carlo simulationstudies, in Bernhardt et al. (2001) and Kendal (2006). Astatistical estimation of the empirical distribution of thesizes of metastases resulting from autopsy studies was doneby Kendal (2001). A semi-stochastic description of thekinetics of the number and sizes of metastases based on thevon Forster equation was obtained by Iwata et al. (2000).An attempt at developing a mechanistic, biologicallymotivated, stochastic model of the metastasis was madein Bartoszynski et al. (2001), where the probability that atthe time of cancer detection the primary tumor has not yetmetastasized was computed. One of the basic ideasentertained in Bartoszynski et al. (2001) was to relate therate of metastases formation to the size of the primarytumor within the framework of quantal response models(Puri, 1967, 1971; Puri and Senturia, 1972). This approachwill be followed in the present work as well.

In order to assess the extent of the disease, a patientdiagnosed with primary cancer is given an imagingprocedure which, after reading of the images, allows thenumber of detected metastases and their sizes (or volumes)to be determined. Therefore, the first step in developing acomprehensive model of the natural history of metastasisconsists in deriving a formula for the conditional jointdistribution of the sizes of detectable metastases given theirnumber. Such a formula is a key to statistical estimation ofsome of the unobservable parameters of the natural historyof the disease. This leads to an important question as towhat characteristics of occult metastases (such as thedistributions of their number, sizes, and total volume) canbe estimated on the basis of data available for detectablemetastases.

When dealing with the sizes of metastases observed at agiven time t one faces a considerable mathematical andmethodological challenge. The problem is that the sizes of

metastases cannot in general be thought of as resultingfrom a sequence of independently repeated trials, and thusthey do not form a random sample from a probabilitydistribution. The reasons for this are two-fold. First,metastases that were shed later tend to have smaller sizes attime t. Second, the rate of metastasis shedding may dependon the primary tumor size which increase typically causesacceleration of the process of metastasis formation.Therefore, although one can always construct a frequencydistribution (histogram) of the sizes of detectable metas-tases at any time t post-diagnosis, it is generally not truethat it represents an empirical counterpart of the distribu-tion of the ‘‘size of detectable metastasis at time t.’’ In fact,the latter random variable is not well-defined! Labeling (ornumbering) metastases represents yet another concomitantproblem. The most natural way to label detectablemetastases is through ordering their sizes taken at a giventime from the smallest to the largest (or vice versa).However, with such a labeling the sizes of metastases arerepresented by random variables that are neither indepen-dent nor identically distributed.The most important findings of this work consist of

showing that under certain biologically plausible assump-tions the joint distribution of the sizes of detectablemetastases conditional on their number coincides with thatof the vector of order statistics derived from someprobability distribution and, furthermore, obtaining aformula for the latter. Although the mechanism ofsampling from this distribution is elusive, for manypurposes, including parameter estimation in the maximumlikelihood setting, this distribution may serve as asurrogate of the distribution of the ‘‘size of detectablemetastasis.’’ In particular, fitting this distribution to theobserved sizes of metastases makes it possible to estimatemany parameters descriptive of the natural history of thedisease and gain an insight into the process of metastasisformation and growth.The structure of the paper is as follows. In Section 1 we

formulate a model of the natural history of cancer. Section2 deals with the derivation of a formula for the distributionunderlying the sizes of a given number of detectablemetastases. Specialization of this formula for the case ofexponentially growing tumors and exponentially distribu-ted promotion times is treated at length in Section 3.Additionally, in this section we discuss various issuesrelated to model identifiability and statistical parameterestimation. Section 4 deals with data analysis based on ourtheoretical results from Sections 2 and 3 and estimation ofmodel parameters. Finally, our conclusions are formulatedin Section 5.

1. The model

The natural history of invasive cancer is commonlydivided into the periods of tumor latency, primary tumorgrowth, and metastatic progression. These periods andrelevant model assumptions are described below.

ARTICLE IN PRESSL. Hanin et al. / Journal of Theoretical Biology 243 (2006) 407–417 409

1.1. Tumor latency

The latent period starts with the birth of an individualand ends with the appearance of the first malignantclonogenic cell. This event is termed the onset of the

disease.

1.2. Primary tumor growth

The size of the primary tumor (that is, the number ofcells comprising the tumor) at any time w counted from theonset of the disease will be denoted by FðwÞ. It is assumedthat F is a strictly increasing continuous function such thatFð0Þ ¼ 1. The function F may depend on one or severalparameters that can be deterministic or random. Wedenote by j the function inverse to F.

1.3. Metastasis formation

Following Bartoszynski et al. (2001), we will assume thatmetastasis shedding is governed by a non-homogeneousPoisson process with intensity m proportional to somepower of the current size of the primary tumor. Thus,

mðwÞ ¼ aFyðwÞ, (1)

where a40 and yX0 are constants, and time w is countedfrom the onset time t of the disease. Note that model (1)with y ¼ 0 describes stationary metastasis shedding gov-erned by a homogeneous Poisson process with constantrate a. It is further assumed that metastases shed by theprimary tumor reach a given host site and get establishedthere independently of each other with some probability q.Therefore, (see e.g. Ross, 1997, pp. 257–259), inception ofmetastases in the host site is governed by a Poisson processwith the intensity n ¼ qm. Each established metastasisspends some random time between detachment from theprimary tumor and inception in the host site (which mayinclude a period of dormancy, see Barbour and Gotley,2003 and references therein) after which it starts toproliferate. We assume that these promotion times fordifferent metastases are independent and identicallydistributed with some probability density function (p.d.f.)f and the corresponding cumulative distribution function(c.d.f.) F . It is well-known (see e.g. Hanin and Yakovlev,1996) that the resulting delayed Poisson process is againPoisson with the rate

lðwÞ ¼Z w

0

nðsÞf ðw� sÞds. (2)

1.4. Secondary metastasis

To retain mathematical tractability of the model, weassume that secondary metastasizing (that is, formation of‘‘metastasis of metastasis’’) to the given site both fromother sites and from within is negligible.

1.5. Metastasis growth

After inception in the host site the growth of a metastasisis irreversible and its size at time w measured from theinception is equal to CðwÞ, where C is a strictly increasingdifferentiable function such that Cð0Þ ¼ 1. The function Cmay depend on some deterministic or random parameters.The inverse function for C will be denoted by c.

1.6. Metastasis detection

The volume of a metastasis becomes measurable whenthe size of the metastasis reaches some threshold value m.The value of m and the accuracy of volume measurementare determined by the sensitivity of the imaging technol-ogy. In the case of PET/CT imaging involved in the presentstudy the threshold volume was 0:5 cm3, and the accuracyof volume determination was one pixel, which is approxi-mately 0:065 cm3.

1.7. Effects of treatment

After surgical removal of the primary tumor its size is setto zero. Then, in accordance with formula (2), if theprimary tumor was resected at age v then mðwÞ ¼ 0 forw4v� t. Because the rate of secondary metastasizing isassumed to be negligible, at the time of primary tumorextirpation the process of new metastasis formation isstopped. Finally, chemotherapeutic or hormonal treatmentof metastases is assumed to affect them only through therate of their growth and the distribution of their promotiontimes.

2. Joint distribution of the sizes of detectable metastases

Suppose that at age u a patient was diagnosed withprimary cancer and that the primary tumor size atdiagnosis was S. It follows from our assumptions inSection 1.2 that the age t at the disease onset is given by

t ¼ u� jðSÞ. (3)

Suppose also that at age v; vXu, the primary tumor wasresected, and that at age t; tXu, the patient developed n

detectable metastases localized in the same host site withthe observed sizes x1; x2; . . . ; xn, where mpx1ox2o � � �oxnpCðt� tÞ. We intend to compute the joint conditionalp.d.f. of the observed sizes of metastases given that theirnumber, N, is equal to n. Note that tumor resection aftertime t has no bearing on the sizes of metastases measuredat time t so that in this case (as well as in the case of anuntreated primary tumor) we can set, for the purpose ofour computation, v ¼ t. Thus, we will assume without lossof generality that 0ptoupvpt, see Fig. 1.Let X 1;X 2; . . . ;X n be the sizes of detectable metastases

at time t ordered from the smallest to the largest. Let alsoT ¼ ðT1;T2; . . . ;TnÞ be the vector of the correspondinginception times counted from the onset of the disease. We

ARTICLE IN PRESS

0 t u v τ

birth of theindividual

of metastasesof the volumesmeasurement

primary tumorresection of thedetection of the

primary tumordiseaseonset of the

Fig. 1. Timeline of the natural history of cancer.

L. Hanin et al. / Journal of Theoretical Biology 243 (2006) 407–417410

have X i ¼ Cðt� t� TiÞ, hence Ti ¼ t� t� cðX iÞ; 1pi

pn. Clearly, these inception times form a decreasingsequence. Note also that a metastasis with size X i isdetectable if and only if 0pTipt� t� cðmÞ; 1pipn.

In accordance with our assumptions and due to formula(1) the inception times of metastases follow a Poissonprocess with the intensity nðsÞ ¼ qaFyðsÞ for 0pspv� t

and nðsÞ ¼ 0 for v� tospt� t. Therefore, in view of (2)

lðwÞ ¼ qaZ minfw;v�tg

0

FyðsÞf ðw� sÞds; 0pwpt� t. (4)

We are interested in the joint distribution of inceptiontimes on the interval ½0; t� t� cðmÞ� given that n inceptionevents occurred on that interval. It follows from a well-known basic theorem about Poisson processes (see e.g.Ross, 1997, pp. 264–265) that the p.d.f. of this conditionaldistribution has the form

pT ðt1; t2; . . . ; tnjN ¼ nÞ ¼ n! oðt1Þ � � �oðtnÞ

if t� t� cðmÞXt14t24 � � �4tnX0, ð5Þ

and equals 0 otherwise, where

oðwÞ ¼lðwÞR t�t�cðmÞ

0 lðsÞds; 0pwpt� t� cðmÞ. (6)

Substituting (4) into (6) and changing the order ofintegration in the denominator of the resulting formulawe represent the function o as follows:

oðwÞ ¼

Rminfw; v�tg

0 FyðsÞf ðw� sÞdsRminft�t�cðmÞ; v�tg

0FyðsÞF ðt� t� cðmÞ � sÞds

,

0pwpt� t� cðmÞ. ð7Þ

Also, if the duration of the promotion time is negligiblethen formula (7) is reduced to

oðwÞ ¼FyðwÞRminft�t�cðmÞ; v�tg

0 FyðsÞds,

0pwpminft� t� cðmÞ; v� tg. ð8Þ

Because the random vector X ¼ ðX 1;X 2; . . . ;X nÞ is relatedto the absolutely continuous random vector T ¼

ðT1;T2; . . . ;TnÞ through the transformation X i ¼ Cðt�t� TiÞ; 1pipn, the distribution of X is also absolutelycontinuous. To evaluate the conditional p.d.f. of therandom vector X given that N ¼ n at a pointðx1; x2; . . . ; xnÞ, where mpx1ox2o � � �oxnpCðt� tÞ,pick a number d40 small enough so that dom and the

intervals ½xi � d;xi þ d�; 1pipn, are disjoint. Then ac-cording to (5)

PrfX i 2 ½xi � d; xi þ d�; 1pipnjN ¼ ng

¼ PrfTi 2 ½t� t� cðxi þ dÞ; t� t� cðxi � dÞ�,

1pipnjN ¼ ng

¼ n!Yn

i¼1

Z t�t�cðxi�dÞ

t�t�cðxiþdÞoðwÞdw. ð9Þ

Dividing both sides of (9) by ð2dÞn and taking the limit asd! 0 we find that

pX ðx1; x2; . . . ; xnjN ¼ nÞ ¼ n!Yn

i¼1

oðt� t� cðxiÞÞc0ðxiÞ

(10)

if mpx1ox2o � � �oxnpCðt� tÞ, and vanishes elsewhere.This proves the following result.

Theorem. The sizes X 1oX 2o � � �oX n of metastases in a

certain host site that are detectable at age t are equidis-

tributed, given their number n, with the vector of order

statistics for a random sample of size n drawn from the

distribution with the p.d.f. defined by

pðxÞ ¼ oðt� t� cðxÞÞc0ðxÞ; mpxpCðt� tÞ, (11)

and pðxÞ ¼ 0 for xe½m;Cðt� tÞ�, where t is given by (3) and

function o is specified in (7).

Remark 1. The p.d.f. p given in (11) is independent of thenumber n of metastases detected at time t and is free of theparameter qa that characterizes the intensity of metastasisseeding. The latter parameter, however, is indispensable fordetermining the distribution of the number of metastases inthe site in question at any time post-diagnosis. In fact, thedistribution of the number of metastases in a given site thatare detectable at time t is Poisson with parameter

qaZ minft�t�cðmÞ; v�tg

0

FyðsÞF ðt� t� cðmÞ � sÞds.

The same formula with m ¼ 1 gives the distribution of thetotal number of metastases in the site at time t.

Remark 2. If the primary tumor size S at presentation isunknown then the onset time t should be treated as arandom variable. In this case an additional integration in(11) with respect to the distribution of the onset time isrequired. This distribution can be obtained by utilizing oneof the established mechanistic models of tumor latency

ARTICLE IN PRESSL. Hanin et al. / Journal of Theoretical Biology 243 (2006) 407–417 411

such as the two-stage clonal expansion model (also termedMoolgavkar–Venzon–Knudson model) (Moolgavkar et al.,1988; Moolgavkar and Knudson, 1981; Moolgavkar andLuebeck, 1990; Moolgavkar and Venzon, 1979) orYakovlev–Polig model (Yakovlev and Polig, 1996). Alter-natively, one can assume that the tumor latency timefollows a distribution from a flexible parametric family(e.g. of gamma or Weibull distributions). For methodolo-gical approaches to parameter estimation for such modelsfrom population data on cancer incidence, see Hanin et al.(2006), Luebeck and Moolgavkar (2002), Zorin et al.(2005).

Remark 3. If the laws of the primary tumor and/ormetastasis growth contain random parameters then for-mula (11) should be additionally integrated with respect totheir distribution. More generally, if the growth of theprimary tumor is governed by a stochastic process thenformula (11) can be applied to any of its sample paths Fand then integrated with respect to the correspondingprobability measure on the space of sample paths of theprocess (which is typically extremely hard to obtain). In thecase where growth of metastases is governed by a stochasticprocess this approach is feasible only if sample paths of theprocess are all increasing and never cross each other, theassumptions used in a very essential way in our derivationof formula (11). The main difficulty that arises forstochastic processes whose sample paths are not necessarilyincreasing or may intersect is that metastases that wereshed earlier may have smaller sizes at the time of detectiont than those shed later or even remain occult at or becomeextinct by time t.

Remark 4. Formula (11) with m ¼ 1 describes the dis-tribution underlying the sizes of all (detectable and occult)metastases. In this case function o in (7) takes on a simplerform

oðwÞ ¼

Rminfw;v�tg

0 FyðsÞf ðw� sÞdsR v�t

0FyðsÞF ðt� t� sÞds

; 0pwpt� t.

In the next section, we will explicate formulas (7) and(11) in the special case of non-random exponential growthof both the primary tumor and its metastases combinedwith an exponentially distributed metastasis promotiontime. In particular, we will consider the most commonscenario with regard to the primary tumor, namely, whenthe latter is surgically removed at the time of diagnosis.

3. The distribution of the sizes of detectable metastases for

exponentially growing tumors

3.1. Theoretical distribution

Suppose that the primary tumor and metastases growexponentially with constant rates b and g, respectively:FðwÞ ¼ ebw and CðwÞ ¼ egw. Then jðyÞ ¼ ln y=b andcðyÞ ¼ ln y=g. Also, we assume that metastasis promotion

times follow an exponential distribution with the expectedvalue r40. The condition tX0 implies in view of (3) that

bXlnS

u. (12)

Denote by M ¼ Cðt� tÞ the maximum possible size of ametastasis. Then it follows from (3) that

M ¼ Sg=begðt�uÞ. (13)

3.1.1. Full model

A straightforward computation based on formula (7)yields the following expression for the p.d.f. (11) under-lying the conditional joint distribution of the sizes ofdetectable metastases given their number:

pðxÞ ¼ðCxÞ�1½ðM=AÞa � ðA=MÞb�ðx=AÞb if mpxoA;

ðCxÞ�1½ðM=xÞa � ðx=MÞb� if ApxpM;

(

(14)

where M is given by (13),

a ¼ybg; b ¼

1

gr; A ¼ maxfm; egðt�vÞg, (15)

and

C ¼ b�1½ðM=AÞa � ðA=MÞb�½1� ðm=AÞb�

þ a�1½ðM=AÞa � 1� � b�1½1� ðA=MÞb�

is a normalization constant that makes (14) a properprobability distribution. Observe that if egðt�vÞpm thendistribution (14) takes on a simpler form

pðxÞ ¼ ðC1xÞ�1½ðM=xÞa � ðx=MÞb�; mpxpM, (16)

with

C1 ¼ a�1½ðM=mÞa � 1� � b�1½1� ðm=MÞb�.

In particular, this is the case for v ¼ t, that is, when theprimary tumor remained untreated by the time when thesizes of metastases were measured.Note that the function p in (16) is decreasing for all

values of parameters a; b40 and M4m. The same is truefor function p given by the more general model (14)provided that bo1. In the case b41, function (14)increases on the interval ½m;A� (or remains constant ifb ¼ 1) and decreases on the interval ½A;M�. Also, function(14) is continuous at the point A, and both functions (14)and (16) vanish at the point M.Some of the limiting forms of the full model (14) and its

particular case (16) are discussed below.

3.1.2. Homogeneous model

This model arises when y ¼ 0 so that by setting a ¼ 0 in(14) we obtain

pðxÞ ¼ðC2xÞ�1½1� ðA=MÞb�ðx=AÞb if mpxoA;

ðC2xÞ�1½1� ðx=MÞb� if ApxpM ;

(

(17)

ARTICLE IN PRESSL. Hanin et al. / Journal of Theoretical Biology 243 (2006) 407–417412

where parameters M ;A; b are given in (13) and (15), and

C2 ¼ lnM

A� b�1½ðm=AÞb � ðm=MÞb�.

If Apm then distribution (17) reduces to

pðxÞ ¼ ðC3xÞ�1½1� ðx=MÞb�; mpxpM, (18)

with

C3 ¼ lnM

m� b�1½1� ðm=MÞb�.

3.1.3. Instantaneous seeding model

If promotion time is very short (in other words, if r! 0)then the p.d.f. p takes on an especially simple form.Proceeding from (8) and (11) we obtain the powerdistribution

pðxÞ ¼ C4x�a�1; ApxpM, (19)

where

C4 ¼a

A�a �M�a .

A homogeneous version of this model ðy ¼ 0Þ is obtainedby setting a ¼ 0. In this case

pðxÞ ¼1

lnðM=AÞx; ApxpM. (20)

3.2. Identification of model parameters

To reconstruct the natural history of the disease, one hasto estimate kinetic parameters b; y; g; r from the observedsample of sizes of detectable metastases. Such estimation ispossible only if the corresponding model is identifiable(Hanin, 2002). We first establish identifiability of theparameters M;A; a; b of the full model assuming thata; b40 and mpAoM.

Proposition. Parameters M;A; a; b and M; a; b are uniquely

determined by the models (14) and (16), respectively.

The proof of the proposition is given in the Appendix.Similar but simpler arguments would show that respectiveparameters of the models (17)–(20) are jointly identifiableas well.

We now address the reconstruction of the kineticparameters b; y; g; r of the full model based on the knownparameters M;A; a; b (or assuming they were alreadyestimated from the observed sizes of detectable metastases)and given quantities t; u; v;S. Consider the following cases:

(1) Suppose that t4u and A4m (the latter implies inparticular that t4v). Then parameters b; y; g;r areuniquely determined by the parameters M;A; a; b providedthe latter satisfy certain conditions. In fact, using formulas

(13) and (15) we find easily that

b ¼lnA lnS

ðt� vÞ lnM � ðt� uÞ lnA; g ¼

lnA

t� v,

r ¼t� v

b lnAð21Þ

and

y ¼ aðt� vÞ lnM � ðt� uÞ lnA

ðt� vÞ lnS. (22)

Clearly, conditions y40 and (12) are satisfied if and only if

t� u

t� vo

lnM

lnAp

tt� v

. (23)

In particular, in the case v ¼ u (that is, when the primarytumor is resected at the time of diagnosis) we have

b ¼lnA lnS

ðt� uÞ lnðM=AÞ; g ¼

lnA

t� u,

r ¼t� u

b lnAand y ¼ a

lnðM=AÞ

lnSð24Þ

under the condition that

lnM

lnAp

tt� u

. (25)

(2) Keeping the assumption t4u we now suppose thatA ¼ m. In this case only three out of the four parametersb; y; g; r of model (16) can be determined from (13) and thefirst two equations in (15).(3) We are now left with the case t ¼ u ¼ v where the

sizes of metastases were measured concurrently with thesize of the primary tumor. Here A ¼ m, and the relationsbetween kinetic parameters b; y; g;r and parameters a; b;Mof model (16) are given by

ybg¼ a; gr ¼ b�1 and

gb¼

lnM

lnS.

In this case, surprisingly, parameter y is identifiable:y ¼ a lnM= lnS. However, it is only the combinations grand b=g that can be estimated from the given observations.Turning to the homogeneous model we find that in the

case where t4u and A4m, parameters b; g;r of the model(17) are given by formulas (21) under condition (25) whilein all other cases these three parameters are not determineduniquely by parameters M ;A; b of model (18). Finally,parameters b; y; g of the instantaneous seeding model (19)can be obtained from M ;A; a only in the case t4u; A4m;they are given by the relevant formulas in (21) and (22)under conditions (23). The same is true regarding therecovery of parameters b; g of the model (20) from itsnatural parameters M and A.

3.3. Statistical estimation of model parameters

Parameters M ;A; a; b involved in models (17)–(20) canbe estimated by maximizing the joint likelihood ofobservations. Due to formula (10), the likelihood function(with the factor n! omitted) for either of these models is

ARTICLE IN PRESSL. Hanin et al. / Journal of Theoretical Biology 243 (2006) 407–417 413

given by

Lðx1;x2; . . . ;xnÞ ¼Yn

i¼1

pðxiÞ. (26)

Notice that it has exactly the same form it would takeshould the observations be independent.

We first discuss the instantaneous seeding model (19).The positivity of likelihood requires that Apx1 andMXxn. Also, because the p.d.f. (19) is monotonicallyincreasing in A and decreasing in M we conclude that A ¼

x1 and M ¼ xn, where ^ stands for the maximum likelihoodestimator. Thus, the likelihood (26) becomes

Lðajx1; x2; . . . ; xnÞ ¼a

x�a1 � x�a

n

� �n Yn

i¼1

xi

!�a�1

.

By taking the logarithm, changing the sign, and dividing byn, we reduce the problem to minimizing the function

LðaÞ:¼�1

nlnLðaÞ ¼ lnðx�a

1 � x�an Þ þ ðaþ 1Þx� ln a; a40,

where

x:¼1

n

Xn

i¼1

ln xi. (27)

We have

L0ðaÞ ¼lnðxn=x1Þ

ðxn=x1Þa� 1þ x� ln x1 �

1

a

¼ � lnðxn=x1Þ1

lnðyþ 1Þ�

1

y� c

� �, ð28Þ

where y:¼ðxn=x1Þa� 1 and

c:¼x� ln x1

lnðxn=x1Þ. (29)

Notice that y40 and 0oco1. The function

gðyÞ:¼1

lnðyþ 1Þ�

1

y; y40,

is decreasing, limy!0þ gðyÞ ¼ 12and limy!1 gðyÞ ¼ 0. There-

fore, it follows from (28) that in the case co12the function

L has a unique minimizer a40 while for cX12

theminimizer is a ¼ 0 in which case the best fitting modelbecomes

pðxÞ ¼1

lnðxn=x1Þx; x1pxpxn.

The full and homogeneous models do not seem to allow fora similar explicit computation of the maximum likelihoodestimates of their parameters. In the data analysis examplediscussed in the next section these estimates were obtainednumerically.

4. Data analysis

To ascertain applicability of the above model for makinginference about the natural history of disseminated cancer,

we used a data base of breast cancer patients who werediagnosed and treated at the Memorial Sloan-KetteringCancer Center (MSKCC) and for whom the whole bodyPET/CT scans are available. The treatments includedvarious combinations and time courses of radical surgery,radiation-, chemo-, and hormonal therapies. Reading PET/CT scans to detect metastases and measure their volumes isa laborious procedure that requires significant effort andexpertise. As of now such analysis has been done for 40patients from the data base. To be useful for our analysis,the patients had to satisfy several requirements: (1) thenumber of metastases in a single site is large enough; (2)primary tumor volume at presentation is available; (3) thevolumes of the primary tumor and metastases weremeasured at different times; and (4) the time course oftreatment allows application of the above parametricmodel of metastatic cancer natural history with a singleset of kinetic parameters b; g; y; r.Only one patient among the 40 satisfied these conditions.

At the age of 74 (specifically, on 4/1/96), this patient wasdiagnosed with stage III estrogen receptor positive primarybreast cancer, and the volume of the primary tumor wasfound to be 10:3 cm3. Shortly after the diagnosis sheunderwent radical surgery and was put on an adjuvanthormonal therapy with tamoxifen. At the age of 82 (moreexactly, on 4/6/04), 37 detectable bone, lung, lymph andsoft tissue metastases were discovered, and their sizes wereidentified through PET/CT images. The prevalent meta-static site was the skeletal system that was found to contain31 bone metastases. The sizes of these metastases were

26; 31; 31; 31; 33; 34; 38; 47; 49; 51; 52; 54; 54; 55; 65; 67; 73,

78; 78; 81; 84; 87; 98; 101; 114; 139; 142; 172; 196; 213; 354

pixels, the volume of one pixel being approximately0:065 cm3. To convert these readings into volumes, weresolved equal metastasis sizes in pixels by spreading themuniformly over the corresponding pixel bins. The resultingvolumes rounded to two decimal places are

1:69; 1:98; 2:01; 2:04; 2:14; 2:20; 2:46; 3:05; 3:18; 3:31; 3:37,

3:48; 3:52; 3:57; 4:22; 4:34; 4:73; 5:04; 5:08; 5:25; 5:45; 5:64,

6:36; 6:55; 7:39; 9:01; 9:21; 11:15; 12:71; 13:81; 22:96 cm3.

In addition to 31 bone metastases, three lung metastaseswith the volumes of 1.30, 2.01 and 7:26 cm3, two lymphnode metastases with the volumes of 2.85 and 9:66 cm3, andone soft tissue metastasis with the volume of 11:41 cm3

were detected. The threshold of measurable volumes forthe PET/CT scanner at hand was m ¼ 5� 108 cells thatcorresponds to 0:5 cm3 based on the volume of one tumorcell of 10�9 cm3 that was assumed in this study.Parameters of the full, homogeneous, and instantaneous

seeding models were estimated using the maximum like-lihood methodology. Our computation was based on theabove volumes of n ¼ 31 detectable metastases, the knownquantities u ¼ v ¼ 74 years, t ¼ 82 years, S ¼ 10:3 cm3 and

ARTICLE IN PRESSL. Hanin et al. / Journal of Theoretical Biology 243 (2006) 407–417414

m ¼ 5� 108 cells. The quantity c defined in (29) and (27)turned out to be about 0.382 which suggests the use of theinstantaneous seeding model (19) with a40, as opposed toits degenerate version (20). The maximum likelihoodestimate of parameter b in the full model turned out tobe very large which implies, in accordance with (24), thatthe mean promotion time r is very small. Thus, the bestfitting full model degenerated into the instantaneousseeding model. This is also corroborated by a decreasingpattern in the histogram of the observed volumes ofdetectable metastases, see Fig. 2. Maximum likelihoodestimates of the parameters M;A; a; b of the instantaneousseeding and homogeneous models are compared in Table 1.Clearly, conditions t4u; A4m and (25) for both modelsare satisfied, and hence their kinetic parameters areidentifiable (see Section 3.2). The values of these para-meters and the estimated date of the disease onsetcomputed using formulas (24) and (3) are presented inTable 2.

To assess the adequacy of the instantaneous seeding andhomogeneous models, we compared the empirical c.d.f.and the model-based c.d.f. with the optimum parameters,

0

0.05

0.1

0.15

0.2

0.25

5 10 15 20 25Metastasis Volume (in cm3)

Fig. 2. Equal areas histogram of the volumes of n ¼ 31 detectable bone

metastases.

Table 1

Maximum likelihood estimates of model parameters with 95% confidence bou

Model M ðcm3Þ A ðcm3Þ

Instantaneous seeding 22:96� 8:81 1:69� 0:24Homogeneous 24.58 2.01

Table 2

Parameters of the disease natural history

Model b ðyears�1Þ g ðyears�1Þ

Instantaneous seeding 23.47 2.66

Homogeneous 24.66 2.68

see Fig. 3(a), (b). A visual comparison shows that, withthese parameters, the instantaneous seeding model pro-vides an excellent fit to the empirical distribution of thesizes of detectable metastases, whereas the fit of thehomogeneous model is considerably worse. This conclu-sion is confirmed by the values of L2 distance between theempirical and theoretical c.d.f.’s as well as the values of thelog-likelihood L ¼ �ð1=nÞ lnL for the two models re-ported in Table 1. Finally, 95% confidence bounds for theparameters of the best fitting instantaneous seeding modelare also given in Table 1.

5. Discussion

Mathematical modeling of complex biomedical pro-cesses is always a balancing act that attempts to achieve adelicate compromise between model adequacy, its mathe-matical tractability, and identifiability of model para-meters. The complexity of the model should match theavailable data from which model parameters are estimatedand against which model adequacy is tested. To be useful,the model should provide a reasonably good fit to the dataand capture the most salient features of the process ofinterest while disregarding its less important aspects.In this work we developed a detailed mechanistic model

of metastatic progression of cancer in a given site based onthe assumption that metastasis formation is governed by anon-homogeneous Poisson process with the intensityproportional to some power of the primary tumor size.This model allows for arbitrary laws of growth of primarytumor and metastases and any distribution of metastasispromotion time. We also formulated and thoroughlystudied a parametric model (termed the full model), thatassumes non-random exponential growth of primarytumor and metastases and exponentially distributedmetastasis promotion time, as well as its two limiting cases(called instantaneous seeding and homogeneous models).The full and instantaneous seeding models with themaximum likelihood parameters estimated from thevolumes of 31 detectable bone metastases observed in abreast cancer patient turned out to be almost indistinguish-able and provided a remarkably good fit to the empirical

nds for instantaneous seeding model

a b L L2 distance

0:56� 0:49 1 2.396 0.200

0 5.87 2.514 0.549

y r (years) Date of onset

0.063 0 4/7/95

0 0.064 4/25/95

ARTICLE IN PRESS

0

0.2

0.4

0.6

0.8

1

cdf

5 10 15 20 25 30Metastasis Volume (in cm3)

0

0.2

0.4

0.6

0.8

1

cdf

5 10 15 20 25 30Metastasis Volume (in cm3)

(a)

(b)

Fig. 3. Comparison between the empirical and model-based c.d.f.’s of the

volumes of n ¼ 31 detectable bone metastases: (a) instantaneous seeding

model ðr ¼ 0Þ; (b) homogeneous model ðy ¼ 0Þ.

L. Hanin et al. / Journal of Theoretical Biology 243 (2006) 407–417 415

distribution of the volumes of metastases. This supportsthe validity of our methodological approach.

One of the main limitations of our general non-parametric model of disseminated cancer natural historyis that it neglects secondary metastasis. The parametricversion of this model is of necessity even more simplistic inseveral respects. First, it proceeds from the exponentialgrowth laws for primary tumor and metastases that maynot be completely adequate. Second, it assumes anexponential distribution for the metastasis promotion timethat may prove to be too rigid. Third, it postulates thesame metastatic growth rates and mean promotion timesfor the entire period from the onset of the disease to thetime of metastasis surveying. However, taking into accountsecondary metastases, assuming more flexible multipara-metric growth laws and promotion time distributions (forexample, Gompertz growth and gamma distributed pro-motion times), and changing model parameters at the timeof primary tumor resection and/or start of treatment formetastases would make the number of model parametersprohibitively large, given the data available in our study.

Our analyses led us to the following conclusions:1. Within the adopted model of cancer development the

sizes of site-specific detectable metastases at a given timepost-treatment rearranged in an increasing order areequidistributed, conditional on their number n, with theorder statistics for a sample of size n drawn from aprobability distribution p given by formula (11). Thisdistribution is free of n and independent of the rate ofmetastasis inception. However, the latter parameter iscritical for determination of the distribution of the numberof both detectable and occult metastases. Therefore,estimation of the extent and dynamics of metastasis in agiven patient requires a series of measurements of thenumber and volumes of detectable metastases taken atdifferent time points. Furthermore, formula (11) with m ¼

1 allows the computation of the distribution of the sizes ofall metastases at any time post-diagnosis and hence of thesizes of occult metastases alone.2. The four-parameter full model (14) for the p.d.f. p,

that governs the distribution of sizes of detectablemetastases, and its particular case (16) and degenerateversions (17)–(20) are represented by various combinationsof power functions and display a variety of patternsincluding decreasing and peak-shaped patterns. Theinstantaneous seeding model (19) and its degenerateversion (20) are reduced to a single power function. Allthese models are identifiable in that the set of their naturalparameters M ;A; a; b (or a subset thereof) is uniquelydetermined by the p.d.f. p. However, the set of biologicallymeaningful kinetic parameters b; g; y; r descriptive of thenatural history of the disease or its respective subsets areidentifiable from the models (14), (17), (19) and (20) only iftime points u and t at which the sizes of primary tumor andmetastases were surveyed are distinct ðt4uÞ and A4m.However, even when t ¼ u parameter y can still berecovered from models (16) and (19). If the size S of theprimary tumor at diagnosis is unknown then in the caset4u; A4m the rate of growth of metastases g and theirmean promotion time r can nevertheless be found throughformulas (21).3. Knowledge of the distribution of the sizes of

detectable metastases at a certain time post-diagnosismakes it possible to estimate, under the conditions specifiedin Section 3.2, all parameters of the individual naturalhistory of the disease except for the intensity of metastasesshedding by the primary tumor. These parameters includethe time of onset t, the rates of growth b and g of theprimary tumor and metastases, respectively, and the meanmetastasis promotion time r. In particular, we found thatthe patient under analysis was inflicted with an extremelyaggressively growing primary tumor (b ¼ 23:47 years�1

that corresponds to the tumor doubling time of about10.8 days). By contrast, the metastases were growing muchslower as suggested by the growth rate g ¼ 2:66 years�1

which corresponds to the doubling time of approximately95 days. This difference is most likely due to estrogendeprivation caused by hormonal therapy. It is worth noting

ARTICLE IN PRESSL. Hanin et al. / Journal of Theoretical Biology 243 (2006) 407–417416

that, as follows from Table 2, the onset of the diseaseoccurred just about 1 year prior to the detection of theprimary tumor. Because this time is much smaller than thetime of about 8 years between the primary diagnosis anddetection of metastases, our extension of the rates ofgrowth of metastases and their inception in the host siteunder hormonal therapy to the entire period betweenmetastasis shedding and detection is justified. We observealso that, although parameters y and r differed substan-tially between the instantaneous seeding and homogeneousmodels, the growth rates b and g estimated from thesemodels remained remarkably stable (see Table 2). Finally,the estimated promotion time for bone metastases provedto be negligibly small.

4. Interestingly enough, the full model applied to theentire set of volumes of 37 detectable metastases did notdegenerate into the instantaneous seeding model andprovided an excellent fit to the observed distribution ofvolumes of detectable metastases with the log-likelihood L ¼ 2:486 and L2 distance between the theoreticaland empirical c.d.f.’s of 0.150, compare with Table 1.The following estimates of the kinetic parameters ofthe full model were obtained: b ¼ 24:26 years�1; g ¼2:68 years�1; r ¼ 0:059 years and y ¼ 0:064. The meanmetastasis promotion time r was the only kineticparameter that changed significantly when six additionalnon-skeletal metastases were added to the data set. In fact,it increased from almost zero for bone metastases to 21.5days for multiple metastatic sites. This observation canserve as an indirect evidence that the main differencebetween various metastatic sites is manifested in themetastasis promotion times rather than growth rates, andthat promotion times for non-skeletal metastatic sites maybe quite long.

5. Within our model the metastasis that was formed firstis the one that has the largest volume ð22:96 cm3Þ at thetime of metastasis surveying (4/6/04). The inception time ofthis metastasis estimated through the instantaneous seed-ing model is 4/17/95, that is, as early as 10 days after theonset of the primary tumor. Although direct extrapolationof the laws of growth of primary tumor and metastases tosuch early times would be misleading, the primary tumorwas clearly undetectable at the time of inception of the firstmetastasis. This very preliminary observation may shedsome light on the limited success with which treatment ofmetastatic breast cancer has met so far. It may also suggestthat certain categories of patients should be treated foroccult metastases concurrently with or shortly afterextirpation of the primary tumor.

6. Parameter y exerts its influence on the sizes ofdetectable metastases through the rate m of metastasisshedding by the primary tumor (see formula (1)). Becausehomogeneous model in which y ¼ 0 is clearly inferior tothe instantaneous seeding model in fitting the sizes ofdetectable metastases, the impact of parameter y and henceof the size of the primary tumor is significant. However, thevalues of parameter y produced by the instantaneous

seeding model applied to 31 bone metastases and fullmodel applied to 37 multiple site metastases (0.063 and0.064, respectively) are surprisingly low. One could expectthat m should be proportional to the tumor surface area, inwhich case y should be about 2/3. Yet another plausibleconsideration is that y should be close to the fractaldimension of blood vessels that feed the primary tumor,which was also estimated to be about 2/3, see Iwata et al.(2000). The observation that y is relatively small—ifconfirmed in further analyses—may have important con-sequences for the still debated question of whether theoutcome of local treatment has a significant effect ondistant metastatic failure. Indeed, a negligible value of ywould show that the rate of metastasis shedding dependsonly weakly on the primary tumor volume and would thussupport the notion that failure to eradicate the primarytumor may not contribute in any important way to cause-specific mortality.

Acknowledgements

The authors are grateful to the anonymous reviewers fortheir constructive criticism and helpful suggestions.

Appendix

To prove the proposition in Section 3.2 we note that thesupport of the function p given in (14) is the interval½m;M�. Therefore, parameter M is uniquely determined bythe distribution (14). Next, denote qðxÞ:¼xpðxÞ; mpxpM.Assuming that A4m and using formulas (14) we computethe left- and right-sided derivatives of the function q at thepoint A:

q0ðA�Þ ¼b

CA½ðM=AÞa � ðA=MÞb�40

and

q0ðAþÞ ¼ �1

CM½aðM=AÞaþ1 þ bðA=MÞb�1�o0.

Therefore, A is the only point on the interval ðm;MÞ wherethe derivative of the function q is discontinuous. Thus,parameter A is identifiable from the distribution (14). Also,because function (16) is continuously differentiable onðm;MÞ, the same property of the point A enablesdiscrimination between the cases A ¼ m and A4m.Finally, we will show that parameters a; b are also

uniquely determined by the p.d.f. (14) or its particular case(16). A straightforward computation shows that

q0ðM�Þ ¼ �aþ b

CM. (A.1)

Also, for the function rðxÞ:¼q0ðxÞx2 we have

r0ðM�Þ ¼ �aða� 1Þ � bðbþ 1Þ

C(A.2)

ARTICLE IN PRESSL. Hanin et al. / Journal of Theoretical Biology 243 (2006) 407–417 417

and

r00ðM�Þ ¼ �a2ða� 1Þ þ b2

ðbþ 1Þ

CM. (A.3)

Dividing Eqs. (A.3) and (A.2) by (A.1) yields

r0ðM�Þ

Mq0ðM�Þ¼ b� aþ 1

and

r00ðM�Þ

q0ðM�Þ¼ a2 � abþ b2

þ b� a ¼ ðb� aÞðb� aþ 1Þ þ ab.

Therefore, the combinations b� a and ab of the para-meters a and b are uniquely determined by the distributions(14) or (16). Hence, so are the parameters a and b, whichcompletes the proof.

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