the multistage model of cancer development: some...

The multistage model of cancer development some implications

Gordon Rittera Richard Wilsona Francesco Pompeiab and Dimitriy Burmistrovac

aHarvard University Department of Physics Cambridge USAbExergen Corp Watertown MA USAcMenzie-Cura and Associates Winchester MA USA

The multistage model introduced by Armitage and Doll was very successful at describing manyfeatures of cancer development Doll and Peto noted a significant departure below the prediction of

the model and suggested that this could be due to undercounting of cases at older ages or to the

lsquobiology of extreme old agersquo Moolgavkar pointed out that it could also be due to the approximation

used The recent observation that cancer incidence falls rapidly above age 80 has stimulated new

modelling investigations such as the PompeiWilson beta model (which does reproduce the rapid

fall) In the present paper we argue that Moolgavkarrsquos criticisms while mathematically correct do

not affect the conclusions particularly the constancy of the number of stages across different cancer

registries (Cook Doll and Fellingham 1969 A mathematical model for the age distribution ofcancer in man International Journal of Cancer 4 93112) We discuss several exact solutions

compare them with the most recent data and prove rigorously that the standard ArmitageDoll

multistage model can never reproduce the sharp turnaround in cancer incidence at old age seen in the

data We discuss in detail multistage processes which have a property observed in many laboratory

studies namely that some stages progress much faster than the others We verify mathematically the

intuition that sufficiently fast stages do not appreciably affect the incidence rate of cancer and

discuss implications of this fact for cancer treatment strategies We also show that the simplest

possible modification of the ArmitageDoll model to incorporate cellular senescence just leads tothe PompeiWilson beta model Toxicology and Industrial Health 2003 19 125145

Key words beta model cancer incidence carcinogenesis cellular senescence multistage model old

age

Introduction

The idea that cancer proceeds by a number of stages

was developed by Armitage and Doll (1954) soon

after cancer registries became available in many

countries They noted that cancer incidence was not

very reliable whereas mortality is much more

objective Therefore they plotted cancer mortality

against age rather than incidence However evenmortality was not often well described above age70 since many death certificates had the nebulousentry lsquoold agersquo They therefore only used themultistage model to describe cancer mortality upto age 70 Cancer incidence I(t) is defined byepidemiologists as the rate of diagnosing new casesdivided by the number of persons at risk Similarlynormalized cancer mortality is the rate of cancerdeaths divided by the number of persons at riskArmitage and Doll found that cancer mortalityfitted a function

Address all correspondence to Gordon Ritter Harvard UniversityDepartment of Physics 17 Oxford Street Cambridge MA 02138 USAE-mail ritterfasharvardedu

Toxicology and Industrial Health 2003 19 125145

wwwtihjournalcom

Arnold 2003 1011910748233703th195oa

I a tk1 (1)

where k was interpreted as the number of stages inthe progression of cancer and a is proportional tothe product of the probabilities at each stage Thismodel successfully described several importantknown features of cancer

1 There is usually a latency period between theinsult which appears to cause the cancer and theexpression of the cancer

2 If cancer-causing agents act at different stages ofthe cancer progression and are present at highdoses then a multiplicative synergy is predictedas seen for example between smoking andasbestos exposure

3 Cook et al (1969) found that while k varied fordifferent tumor sites it is constant betweencountries The constant a varied between coun-tries as might be expected from the differentenvironmental situations

These successes led to widespread use of themodel It was soon noted that above age 60 theage-specific mortality rate flattened and fell belowthe curve predicted by Armitage and Doll Twomajor reasons for such a flattening have beensuggested the first is a variation in susceptibility(variation in a) between different members of thesame group being considered and the second is afailure of the mathematical approximations used byDoll and Armitage

Cook Doll and Fellingham considered varia-tions of susceptibility in the population Theyplotted incidence versus age on a logarithmicscale for varying fractions C of sensitive persons(Cook et al 2004 Figure 4 reproduced here asFigure 1) They concluded that if only 1 of thepopulation is susceptible to cancer then at olderages the normalized incidence curves must fallwhen all the 1 have developed cancer

Even if the number of susceptibles is 10 then aflattening occurs Figure 1 reproduced from Cooket al (1969) indicates that by suitable choice of theparameter C one might be able to describe a rapidfall off in incidence a suggestion we discuss laterCook Doll and Fellingham ultimately rejected thispossibility because data showed a constant agefor the peak of cancer incidence while theirparticular assumption of insensitive individuals

indicates that incidence peaks at a younger age

for rarer cancersThe variations in cure rates for cancer in the last

50 years suggest that it is now more important to

discuss incidence rather than mortality and in

addition that incidence may now be much better

determined than 50 years ago In this paper there-

fore we reopen the discussion with an emphasis on

cancer incidence rather than mortalityLet f(t) denote the event rate for the kth change

at time t in a single cell According to our

definition cancer incidence is related to f(t) by

I(t)Nf(t) where N is the number of affected cells

The theoretical model calculates f(t) we will per-

form this calculation exactly for a variety of

interesting situationsIn the case of the ArmitageDoll model

IAD(t)h(t) is an infinitely growing hazard func-

tion and satisfies the criterion that the cumulative

probability of cancer given by

Figure 1 Cook Doll and Fellingham (1969) considered variablesusceptibility in the population This plot shows incidence versus ageon a logarithmic scale for varying fractions C of sensitive personsModified from Cook et al (1969) Figure 4

Multistage model of cancer developmentG Ritter et al

126

F (t)1exp

g

t

0

h(t)dt

(2)

approaches 1 in the limit as t 0 According tothe definition of the hazard function we considerthe differential equation

IAD(t)atk1F (t)

1 F(t) (3)

Solving this equation with the initial conditionF(0)o yields

F (t)1(o1) exp

atk

k

The condition o0 corresponds to fetal cancerwhich while a real condition is ignored in thispaper Hence we are concerned with the limit o00

F(t)1exp(atk=k) (4)

This satisfies limt0F (t)1 which describes thefact that according to this model everyone willdevelop cancer if they do not die of something elseIt is obvious from these considerations thatthe ArmitageDoll approximation (Equation (1))increases without bound and given sufficienttime any susceptible cell eventually becomes ma-lignant

As noted above the ArmitageDoll model(Equation (1)) must be viewed as an approximationThe correct equations were written down byMoolgavkar (1991) and Armitage (1985) and itwas seen that (Equation (1)) is only valid for smallvalues of t It was seen also that (Equation (1))overestimates incidence Moolgavkar (1991) haspublished a simple example supporting the ideathat ArmitageDoll approximation can be inade-quate if the transition rates for cancer stages are notsmall enough His example presupposes a smallernumber of cells N109 in the tissue of concern thanis usually taken In contrast to the ArmitageDollapproximation the hazard function in the exactmodel has a finite asymptote limt0h(t)Nmminwhere mmin is the minimal transition rate Thesuggestion that the failure of the ArmitageDollapproximation may be responsible for the observedflattening of incidence for ages above 60 followsfrom this

Pompei and Wilson (2001) reopened the issue

They noted that since 1955 data on cancer

incidence has improved and data on both incidence

and mortality above age 70 has improved For

discussions of cancer development incidence may

now be more reliable (for many tumor sites) than

mortality due to improvements in cancer treat-

ment Making the assumption that the data and

particularly the SEER data in the United States are

reliable Pompei and Wilson plotted the normalized

cancer incidence data against age finding a rapid

drop in normalized incidence above age 80 which is

not explained by simple parameter adjustments

of the ArmitageDoll model This is illustrated

in Figure 2 for a common cancer (colon) an

Figure 2 SEER data from Pompei et al (2004) and SEER (2002)shown together with the best possible fit of the ArmitageDoll model


127

intermediate-incidence cancer (non-Hodgkins lym-phoma) and a rare cancer (brain)

Pompei and Wilson added an extra term toEquation (1) leading to a beta function I

atk1(1bt) which fits the data for most cancersremarkably well and speculated on possible biolo-gical explanations for the extra term PompeiPolkanov and Wilson (Pompei et al 2001) exam-ined the limited data on laboratory mice that havebeen followed for their natural life of 1000 days ormore rather than being sacrificed at 750 days Theage of cancer incidence is not well determined formice but cancer mortality suffers neither fromcomplications of improved treatment nor frommisreporting In the absence of these systematicerrors it was observed that the cancer mortality fellto zero before the end of life This is shown inFigure 3

In the present paper we re-examine the criticismsof Moolgavkar and improve upon the speculationsof Pompei and Wilson This is done byexact calculations in the multistage model In theprocess we derive several closed-form solutionsto the model Following previous studies (Pompeiand Wilson 2001) we continue to make theassumption that the SEER data in the UnitedStates is reliable above age 80 We prove that underthis model as one would expect the probabilitythat a single cell becomes cancerous given infinitetime is 1

Crucial parameters for the multistage model arethe transition rates which we denote mi We give anexact solution for a model with a number ns of slowstages with rate ms and one fast stage with rate mfand show that if mfms the fast stage only slightlyperturbs the final solution and may be neglectedWe show that if the multistage model is assumed tocalculate the probability of cancer for a single cell

then this model can only be consistent with therapid fall-off observed in the data if some unusualbiological assumptions are made

Methods

Assumptions of the multistage model

In this section we repeat starting with the funda-mental assumptions the derivation of the exactmultistage model and suggest various extensionsthereof

In the ArmitageDoll approximation cancerincidence has a simple power law growth This isan approximation to the exact multistage model(Moolgavkar 1978 Moolgavkar et al 1999 Pom-pei and Wilson 2001) described by the Batemanequations (Bateman 1910) The multistage modelmakes several basic assumptions

1 Malignant tumors arise from a series of mod-ifications of a single progenitor cell

2 The process of developing a malignancy isequally likely for all cells in the same tissue

3 The process of developing a malignancy in onecell is independent of the process in any othercell

4 After a malignancy has developed in a cellproliferation to an overt cancer is rapid andinvolves many cells in the same tissue and mayeven involve metastasis to another tissue

Under these assumptions cancer is the last of aseries of k sudden and irreversible changes whichmust take place in a specific order For a cell whichhas experienced (i1) changes the event rate forthe next change is mi We choose the conventionthat a single transition from a state p0 to a state p1

p0 0m1

p1 (5)

corresponds to one lsquostagersquo and has k1 Thisimplies that the number of transition rate constantsm1 mk always equals the number of stages

Cancer in a cell

Cancer is the last of a series of k sudden andirreversible changes which must take place in a cellin a specific order As is standard for Poisson

Figure 3 Mice data from Pompei et al (2001) shown together withthe best possible fit of the ArmitageDoll model


128

random processes we neglect the probability of twoor more events taking place in (t tdt) as dt 0 0This means that if a cell is in state pi at time t theprobability of transformation to state pi1 in asmall time interval Dt is given by mi1Dto(Dt)where o(Dt) is very small compared to Dt ando(Dt)=Dt 0 0 as Dt 0 0 Also the probability oftransformation i 0 i j with j1 in time Dt isassumed to be o(Dt) This implies that 1=mi1 is theaverage time required for the cell to go from state ito state i1

The probability to find a cell in the ith stage bythe end of time interval (t tdt) is then given by

pi(tdt) (1mi1dt) pi(t)mi1dt 0

midt pi1(t) (6)

Taking the limit dt 0 0 Equation (6) becomes

p0(t)m1p0(t)

pi(t)mi1pi(t)mipi1(t) (7)

pk(t)mkpk1(t)

The first and the last equations in this system aredifferent because there are no stages before thestage i0 (normal cell) and no stages after the kth

stage which corresponds to cancer The system(Equation (7)) should be completed with appro-priate initial conditions

p0(0)1 pi(0)0 (i1 2 k) (8)

which mean that the cell was normal at time t0The last equation in Equation (7) also gives theprobability to find the cell at its last cancerousstage Following Armitage and Doll we reserve thenotation f(t) for this quantity

f (t)pk(t)mkpk1(t) (9)

Thus f(t) denotes the event rate for the kth

change (corresponding to cancer) at time t in asingle cell

Cancer in a tissue and in humans

The event rate f(t) is a probability distributionfunction (PDF) for appearance of cancerous cellswhich means that f

0f (t)dt1 which means that

given enough time each cell will become cancerousEven though the PDF is convenient for mathema-

tical modelling it is not directly observable Onlydeveloped cancer in a tissue rather than a singlecancer cell could be recognized as a cause of deathor even diagnosed at all In practical studies cancerincidence is commonly observed Following thenotation of Armitage and Doll cancer incidenceI(t) (which is the same quantity as the hazard

function used in biostatistics and denoted there ash(t)) is defined by epidemiologists as the rate ofdiagnosing new cases divided by the number ofpersons at risk

Let us define a cumulative distribution function(CDF) for cancer per cell as F(t)f

t

0f (t)dt that is

the probability for a cell to become a cancer bymoment t Then the probability of at least onecancer cell in a tissue of N cells by the moment t isG(t)1[1F(t)]N This definition assumes theindependence of cells Cancer incidence can bedefined in terms of G(t) as

I(t)G(t)=(1G(t))NF (t)=(1F (t))

Cancer is a rare event even on the tissue leveland therefore F (t)1 and hence the approxi-mation

I(t)Nf (t) (10)

is quite good since the number of cells in a tissue Nis greater than 109 Equation (10) allows fitting ofthe theoretically derived cancer event rate per cellf(t) and the observed incidence of cases in popula-tion I(t) This result was derived by Armitage andby Moolgavkar

In the above the function I(t)Nf (t) is inci-dence of cancer in a tissue or equivalently in aperson When multiplied by the size of a popula-tion this becomes the true number of cancers inthat population at risk or the incidence Howeverthe number of cells N may not be large enough toensure that the ArmitageDoll approximation isaccurate so it is important to discuss this para-meter There are about 1014 cells in a human bodyand in any given tissue they would be proportionalto weight Thus as an order of magnitude approx-imation the colon has roughly 1012 cells Thepostulate that only cells on the surface are im-portant may reduce this to 1011 Therefore in whatfollows we take the number used by Moolgavkar(109 cells) as a lower limit


129

The Bateman equations

The exact solution to the multistage model can be

derived by analogy from Batemanrsquos solution of

successive radioactive decays (Bateman 1910)The simplest form of Batemanrsquos solution de-

scribes a linear decay chain leading ultimately to a

stable nucleus More complex forms have also been

used to describe for example the decay of radium

The isotope radium C decays into radium C and

radium C which both decay into lead so the

diagram contains a loop It is highly likely that

these more complex forms are also relevant to

cancer incidence although in the present work we

restrict attention to linear decay chains A k-stage

radioactive decay process depicted schematically

by the diagram

is described by the Bateman equations a set ofcoupled linear first-order differential equations inwhich pi(t) denotes the quantity of objects in state i

at time t

dp0

dtm01p0

dpj

dtmj1 jpj1mj1pj (1B iBk) (11)

dpk

dtmk1kpk1

where mj j1 is the partial rate constant for thetransition from state j to state j1 and mj is thetotal decay rate1 of state j1 The partial decayrates are related to the total decay rates byconstants bj j1 called branching ratios Thisrelationship takes the form

mj j1bj j1mj1

In the case of a linear decay chain the branchingratios equal unity and the partial rate constants aregiven by

mj j1mj1

In this simple case Equation (11) is equivalent tothe equations derived earlier in our discussion ofcancer in a cell Most discussions of the modelmake this assumption However we note that notall lsquoinitiatedrsquo cells proceed to cancer but the vastmajority are either repaired or excreted Somemay undergo clonal expansion and multiply Thiscould be modelled by taking values of thebranching ratio bj j1 less than unity (in the firstcase) or greater than 1 for clonal expansionHowever this is a very simplistic view of clonalexpansion which treats the expansion as a defini-tive process rather than a stochastic one We areattempting to understand this further The reali-zation that the multistage model only takes intoaccount the slow late limiting steps in the cancerprocess allows a theory in which clonal expansionoccurs rapidly but is not noticed as a separatestage

However if these branching ratios are the samefor each cell they may be subsumed in an overallconstant factor in the formula for incidence andthe fitted values of m adjusted accordingly Thenthe algebra is the same as if each and every celleventually becomes cancerous

We will work with linear decay chains in whatfollows In that case the diagram of the processsimplifies to

p0 0m1

p1 0m2 0mk

pk (12)

In the earlier section lsquoCancer in a cellrsquo it wasshown that the same differential equations describethe multistage model for the formation of malig-nant cells Every cell eventually transforms givensufficient time so in a realistic model

pk()g

0

f (t)dt1 (13)

In Lemma 3 we show that Equation (13) is a truemathematical consequence of the Bateman equa-tions with appropriate initial conditions

1In a model describing radioactivity mi1ln(2)=t1=2 where t1=2 is thehalf-life of that species Our notation differs slightly from that ofMoolgavkar who identifies the rate constant with the state being leftrather than with the state to which the transition is being made


130

The most general initial conditions for the

system Equation (11) are to allow pi(0) to be arbit-

rary nonzero constants for all i The solution

to the system Equation (11) with general initial

conditions is

for m1 kA simplifying assumption consistent with the

use of Equation (11) to analyse the probability of asingle cell becoming cancerous (or the radioactivedecay of a single nuclide) is that the concentrationsof the later stages (or daughter nuclides) are allinitially zero This assumption is equivalent to theinitial conditions

p0(0)a

pi(0)0 (for all i1) (15)

With these initial conditions the solution (Equa-tion (14)) reduces immediately to the simpler formvalid for m f1 kg

m1(t)cm

Xm

j1

xjm emj t (16)

where cmaYm1

j1

mj1j

Here c1 is an empty product which is 1 byconvention c2m12 etc and we have defined time-independent constants

xjmY

l1m

lj

(mlmj)1 (17)

Equations (16) and (17) were shown for thelinear chain by Moolgavkar (1991) [(A1) p 217] inhis work on the multistage theory of carcinogenesisWhen evaluating the sum in Equation (14) the usershould be warned that the terms tend to be largeand of opposite sign Indeed when the m are equalanother formula has to be used The cancerincidence in a tissue with N cells (which is the

approximation for a person) is Nf(t) to a very highdegree of accuracy and

f (t)pk (t)mkpk1(t)

We state our most important mathematical asser-tions as a series of Lemmas for which we provideproofs in an appendix

Lemma 1 (Solution of the Bateman equations)Assume that mi mj for all i j Then the functionspm(t) defined by Equation (16) and Equation (17)

solve the Bateman equations Equation (11) withinitial conditions Equation (15)

Generally we take the initial condition a1 Itis easy to see that the first nonvanishing term in aTaylor series of pm1(t) is of the ArmitageDollform where m is an index for an intermediate stageIndeed pm1(t) is the constant cm times a linearcombination of exponentials am

j1xjm emj t the nth

term in the Taylor series of this linear combinationis clearly

Xm

j1

xjm

(mjt)n

n (18)

This vanishes by algebraic identity if nBm1 andfor nm1 one may check that am

j1xjmmm1j 1

so to lowest nonvanishing orderpm1(t)cmtm1=(m1)

The next nonvanishing term (nm) also has asimple form Equation (18) with nm becomes

tm

m

Xm

j1

mj tm

(m 1)m

where m1

kak

i1mi denotes the average m We may

then write

pm1(t)cm

tm1

(m 1)(1m t )

Substituting mk for the last stage we find

I(t)atk1(1m t )

where

a

Qk

j1 mj

(k 1) (19)

pm1(t)Xm1

i0

Ym2

j0

mj j1

Xm

j1

pi(0)emj tQl (i 1) m

l j

(ml mj)

26664

37775 (14)


131

This is the Taylor expansion discussed by Mool-gavkar It was referred to (Moolgavkar 1978) as theMacLaurin series but that is a special case of themore general Taylor series We use the more generalterm in this paper

As one might expect the solution (Equation(16)) has a simple series expansion which can beexpressed in closed form This is given in thefollowing Lemma

Lemma 2 (Taylor expansion)Let fmi i1 kg be any set of nonzero constants

with mi mj if i j Then

Xm

i1

ximemi tX

j]m1

ajtj

with am11=(m1) and

aj1(1) jm

j

Xi155ijm

mi1mi2

mijm(20)

if jm This implies that

pm1(t)cm

tm1

(m 1)

1

m

Xm

i1

mi

tmO(tm1)

Remark 1 In various applications it is mostconvenient to use the lowest nonvanishing order(the ArmitageDoll approximation) in this seriesexpansion and itrsquos necessary to have conditionswhich guarantee that this truncation is valid Wecompare the lowest nonvanishing order to the(assumably better) approximation obtained byconsidering higher-order terms In particularlet us compare the lowest nonvanishing order(jm1 in the above formulae) to the followingterm In this paper as with the papers ofMoolgavkar we are concerned with the last stageonly and this corresponds to mk in the aboveequations Let Dn denote the nth nonzero term inthe Taylor expansion which has coefficientakn2

By a simple calculation

D2

D1

t m (21)

This formula shows that D2=D1 can be arbi-trarily large by inclusion of a fast stage or

stages each with large m When attempting as

we are doing to find a model to fit the data

from ages 20 to 70 years the constant a in the

first term of Equation (19) is heavily constrained

by the data If k is not varied the constant a

is proportional to the product m1 mk One

can only make m arbitrarily large if this pro-

duct is fixed to the correct value This is

discussed further in the section on determination

of k We will show later that if nf of the stages are

much faster than the remaining nsknf stages

then the exact solution for the k-stage model is

approximately equal to the exact solution for the

remaining ns stages which we call slow stages

Equation (21) indicates that the Taylor series for

k stages converges more slowly than the Taylor

series for ns stages because inclusion of fast

stages increases the average transition constant

In the presence of fast stages a large number of

terms must be included before the Taylor series

becomes accurate and consideration of only the

first and second terms in the expansion is

incorrect However the exact solution for the

model without the fast stages has a different

Taylor series which converges quite rapidly This

realization leads to important practical conclu-

sions that are discussed in more detail later

Specifically if inclusion of a few fast stages set

m103 then Equation (21) shows that D2 is 10

percent of D1 at t100 If the stages are fast

enough to set m102 per year then D2=D1 is

of order one at 100 years It is clear that by

including fast stages D2=D1 can be made arbi-

trarily largeMoolgavkar deliberately chose a set of para-

meters which included some slow and some fast

(but not very fast) stages (see for example

Moolgavkar 1978 Table 1) Although the above

estimate for D2=D1 does hold for Moolgavkarrsquos

example a discussion of his chosen parameters

needs more careful numerical analysis Consider

a tissue of 109 cells affected by a hypothetical

six-stage2 malignant tumor with transition rates

(per year) given by

2Moolgavkar (1991) refers to this as a seven-stage process but onlygives values for six transition constants We assume here that six stagesare meant


132

104 2104 34104 7103

8103 9103

In this example

m00046 per year

So our formula Equation (21) predicts D2046D1

at t100 years This is precisely what is observed inFigure 4 which shows the exact solution togetherwith its first five Taylor approximants The numer-ical factor relating the second approximationwhich lies below the exact curve to the firstapproximation lying above the curve at t100years is seen to be about 146

As noted in the introduction 109 cells may be toofew Bearing in mind that the model is used to fitincidence data each value of the six constants m

would have to be reduced by a factor of 2 if N wereincreased to 64109 and the exact formula at aget100 years would then be only 128 times theArmitageDoll approximation

The implications for determination of the valueof k using Moolgavkarrsquos parameters is discussedlater More generally

Dn1

Dn

akn1

akn2

t

t

k n

Pnk

Pn1k

where Pnk is the symmetric degree n polynomial ink variables given by ai155in

mi1 min

These poly-nomials satisfy

Pnk mkPn1k(terms involving m1 mk1)

It follows that for n]2

Dn1

Dn

t

k n

mk

ratio approaching

zero as mk 0

13(22)

The surprising conclusion of Equation (22) isthat if we add one very fast (mm104) stage at theend it may be necessary to consider hundreds orthousands of terms in the Taylor expansion in orderto attain accurate results The rate of convergenceof this expansion is affected strongly by theappearance of fast stages while the exact solutionis not This is a very important paradoxical resultthe solution for which may be found in the earliercomments If there are ns slow stages and nf veryfast stages the exact solution is close to the

ArmitageDoll approximation for the ns slowstages alone the ArmitageDoll approximationfor the full exact model with nsnf stages is notaccurate For equal transition probabilities givenby the single transition constant m the aboveimplies D2=D1mt

Lemma 3 (Normalization)Let p0(t) pk(t) satisfy the k-stage Batemanequations (Equation (11)) with any collection(m1 mk) of transition constants Then

g

0

pk(t)dtYk

i1mi

Xk

j1

xjk

mj

1 (23)

This equation shows that given sufficient timethe cancer process will eventually reach the laststage

General properties of the solutions

The general solution for reasonable values of k has far too many constants to be determined byepidemiological data alone However there arevarious cases in which the solution of the Batemanequations Equations (16) and (17) can be simplifiedenough to depend on only a few independentconstants even if the number of stages is large

Equation (21) implies that the ratio of the secondterm to the first in the Taylor series for I(t)

D2

D1

is proportional to the average m But the product ofthe ms is constrained by the value of the observedincidence With this constraint the average m will

Figure 4 Exact solution for Moolgavkarrsquos example shown with itsfirst five Taylor approximants


133

be a minimum when all the values of m are equalFor this important special case D2=D1 is muchsmaller than for example in Moolgavkarrsquos case

Another interesting result is that permuting thestages makes no difference to the final result pk(t)or more generally to mmpm1(t) for any m5k Thisis self-evident physically for radioactive decaywhich is also described by the Bateman equationsThis result used implicitly in the ArmitageDollapproximation is a fundamental property of theexact solution It may be proved as follows Itsuffices to consider a transposition of the ith stageand the jth stage as any permutation maybe represented as a product of transpositionsNow xim is a product of (mlmi)

1 where l iApplying the transposition these terms become(mlmj)

1except for the l j term which be-

comes (mimj)1 The result is thenQ

lj(mlmj)1

xjmA third general result is that if there are say 4

stages which proceed slowly (small equal values ofm) and many stages which act much faster (values ofm that are 10 times larger) then the final resultdepends only on the 4 slow stages This is self-evident physically and was probably implicit in thethinking of earlier users of the multistage modelThis is proven rigorously in Lemma 6 and dis-cussed with a specific example in the section onmeasurement of k

We pay special attention to a simple case (onenot covered by Lemma 1) in which all stages haveequal transition probabilities This simple case isdescribed by Lemma 4

Equal transition ratesLemma 4 (Equal transition rates)Consider the Bateman equations pj (t)m(pj1pj)arising from equal transition rates m1m2

mkm For any natural number j5k1 the solutionis

pj(t)1

jemt(mt) j (24)

In particular for the penultimate stage

pk1(t)1

(k 1)emt(mt)k1

The global maximum of pj(t) occurs at the agetpeak j=m Further as before we have

g

0

pk (t)dtg

0

mpk1(t)dt1

Remark 2 Since f(t) applies to a single cell m isvery small (on the order of 103) and hence for anynonzero number of stages tpeak j=m always occursat a much greater age than those contained in ahuman lifespan This important point is discussedin detail in the earlier section lsquoCan the multistagemodel ever fit the datarsquo

The algebraic simplicity of equation Equation(24) makes it attractive as a starting point for ourdiscussions It has two adjustable constants k andm A useful numerical example is the followingSuppose we would like to know how the exactsolution Equation (24) differs from the first term inits series expansion

pj(t)Xn0

(mt) jn

jn (25)

In general the first term in the series expansion ofexxn (n0 an integer) is xn and jexxnxn j

xn(1ex) In a numerical example with G03 G001 per year and k7 we have mt502for all t in a human lifespan Then frac121emtfrac12502and so for the first Taylor remainder R1 of pj(t) wehave

jR1j51

5k1256106 (26)

Equally separated transition rates

Another case which is amenable to an exactanalytical solution is when the transition constantsmi for the various stages increase by a fixed amountOf course it is not necessary to assume that in theactual cancer process the transition rates occur inincreasing order thanks to invariance of the processunder general permutations as remarked above sothis solution applies to the general situation whenthere exists some arbitrary reordering of thetransition constants so that they increase linearly

Lemma 5 (Equal separation)Suppose that the transition constants mm1m2 mk are uniformly separated so that mi1mi d for all i and for some constant d0 Then for anyn5k the solution is given by


134

pn1(t)Cdmn emt(1etd)n1 (27)

where CdmnGm

d n 1

G(m=d)G(n)

is a constant

Remark 3 To calculate the penultimate stagepk1(t) simply replace n by k in the abovepk1(t)Cdmk emt(1etd)k1

Remark 4 This model Equation (27) althoughan exact model with unequal transition constantsallows us to find an analytic expression for theposition of the peak Setting p k1(t)0 in Equa-tion (27) leads to

tpeakd1ln

1

d

m(k1)

(28)

This is an exact expression If dm whichcorresponds to the model with equal transitionrates the log can be expanded leading to

tpeakk 1

m (29)

This agrees with what was found in Lemma 4Remark 5 For consistency Lemma 5 should

reduce to the exact solution for equal transitionrates given in Lemma 4 This is true since for anym5k cm clearly approaches mm1 and

pm1

cm

limd00

Xm

j1

xjmemj t

emtlimd00

d1m

Xm

j1

(1)j1

(j 1)(m j)e(j1)dt

emt

G(m)limd00

d1metd (1 etd)m

etd 1

emt tm1

(m 1)

This is consistent with Lemma 4Remark 6 The solution obtained in Lemma 5 is

an analytic expression containing no sums orproducts which would force the number of stagesk to be an integer therefore it is amenable tocomputer simulations which allow k to be acontinuous variable In particular one can

perform nonlinear regression to optimize thevalue of k together with d and m which are theother free parameters in the model given byEquation (27)

Remark 7 The analytic solution for tpeak allowsus to also find the height of the function at its peak

pk1(tpeak)Cdmk

(k 1) d

(k 1) d m

k1

1

(k 1) d

m

m

d

From this one obtains the qualitative observa-tion that large n shifts the peak down To seethis note that the asymptotic expansion for largek is

pk1(tpeak) 0large k

const1

kO

1

k

2

where the constant is em

dd

m

m

d

Gm

d

1

This is

important because it explains why when weattempt to fit real-world data the curve-fittingroutine finds unrealistically high values of k thenumber of stages

Fast and slow stages

In the discussion of Equation (21) we outlinedthe effect of including a stage much faster thanthe others on the Taylor expansion In thissection we analyse this in a more precise way(Figure 5) To illustrate the general effect whichoccurs when one or a few stages are much fasterthan the rest we first consider a simple examplewhich entails the convergence of the two-stagesolution to the single-stage solution as thetransition rate for the second stage becomeslarge We exhibit this convergence in the functionpk(t) which corresponds to the last stage inEquations (30) and (31) If k1 we have (forI(t)pk(t))

I(t)m1em1t (30)

and if k2


135

I(t)m1m2

m2 m1

em1tm1m2

m1 m2

em2t (31)

One may now observe directly that Equation (31)converges to Equation (30) in the limit that m2m1In fact this phenomenon is quite general asfollows from Lemma 6

Lemma 6 (Slow and fast stages)Suppose that the first ns stages have transitionconstant ms followed by one stage with transitionconstant mf Then the exact solution is given by

pns1(t)(ms)

ns1

nsemf tg

t

0

e(mfms)xxnx dx

emf t

ms

ms mf

ns11

G(1 ns t(ms mf ))

ns

13(32)

Further we assert that

limmf 0(mf pns1(t))mspns(t) (33)

The statement of Lemma 6 places the fast stageat the end However previously we have shown that

permuting the stages has no effect on the finalsolution ie on the incidence function ThereforeLemma 6 and in particular Equation (33) appliesto a multistage progression in which any one of thestages is very fast compared to the others Byinduction the same is then true of the case withmultiple fast stages

This is an exact solution no approximations havebeen made Our intuition is that if the fast stage isfast enough then it will have no effect on theproperly normalized solution and Equation (32)will reduce to pns

(t) This intuition is rigorouslyexpressed as Equation (33) We note that both sidesof Equation (33) integrate to one This generalizesthe simple observation following Equation (31) Weillustrate this general phenomenon with a specificexample Consider the Bateman equations with ns

stages at rate n and the remaining kns stages atrate m Let g(tm n) denote the solution pk(t)mpk1(t) and let

S(t n)n

nsetn(tn)ns

denote the exact solution for ns slow stages ofrate n To illustrate that the effect of the fast stagesbecomes negligible for mn we plot g(t nn n) andS(t n) on the same graph for increasing values of n with k5 total stages and ns3 slow stages For theslow transition constant we take a reasonable valueof 5103 which is comparable to the valueschosen by Moolgavkar (1991) The relevant Bate-man equations are

p0(t)m p0(t)

p1(t)m p0n p1

p2(t)n p1n p2 p5(t)n p4

We conclude in this numerical example that form50n the effect of the faster stages is in the orderof a few per cent In the limit as m=n 0 weobserve that g(tm n) converges to S(t n) for all t

Results and discussion

On the number of stages

In the introduction we pointed out that a funda-mental success of the multistage model (1) was by

Figure 5 Exact solutions with three slow stages having rates n5103 followed by two lsquofastrsquo stages at rate m Graphs are shown form2n and m50n The solution S(t n) with ns3 slow stages and nofast stages is shown as a dashed line for comparison


136

showing that whereas normalization constant a inthe incidence formula varied from country tocountry number of stages k stayed constant Thisthey did by plotting lnI against lnt The slope was(k1)

How does this change when we go to the exact

model The theorem that one very fast stage makesno difference to the result suggests that Armitageand Doll were measuring the number of slow stagesHowever Moolgavkar showed a plausible set ofparameters where the ArmitageDoll approxima-tion was 146 times the exact model at age t100We show that a straight-line fit in loglog scale isstill good approximation in this example and thevalue of k that Armitage and Doll would havederived in such a case is only slightly less than thetrue number of stages We investigate this asfollows

We simulate data from the exact model using theMoolgavkar parameters (Moolgavkar 1978) Thenwe analyse these simulated data in the Armitage

Doll manner to derive k by fitting these data usinga least squares fit

In Figure 6 line 1 we show the simulated datausing Moolgavkarrsquos parameters (squares) and thebest linear fit of lnI versus lnt From the slope ofthis line 48068 we see that Armitage and Dollwould have derived 58 stages instead of the correct

60 This is greater than the small number 2 of

slowest stages but less than the total number of

stages If the number of cells is 64 times greater than

Moolgavkar assumed then the values of m must all

be reduced a factor of 2 to give similar incidence

(Figure 6 line 4) and the derived number of stages

59 is even closerWe can go further and ask whether such a

derived value of k is stable under changing an

individual transition rate m corresponding to a

difference between cancer registries in different

countries Line 2 of Figure 6 shows a similar plot

with one of the stages m1 increased tenfold to

0001 The value of k that Armitage and Doll

would have derived is the same 58 Increasing the

transition rate tenfold for the fastest stage m6

resulted in k54 which deviates from the exact

k by less than 10 We conclude that although

using an approximation can reduce the derived

number of stages this reduction is not dramatic

and it is stable across registries so that one of the

principal successes of the multistage model is

unalteredThat many more mutations are present in cancers

than the 48 slow stages predicted from the

epidemiological data is well supported For exam-

ple a test for mutated DNA for colon cancer

includes 21 specific mutations (Tagore et al

2003) Far larger numbers of mutations are known

to exist (Lengauer et al 1998 Duensing and

Munger 2002) as a consequence of genetic instabil-

ities caused by early stage alterations and 11 000

are reported by Stoler et al (1999) for colon

cancers Clearly the vast majority of these altera-

tions must occur very rapidly and thus do not affect

the age distribution of cancer which is determined

by the much slower (and therefore rarer) rate

limiting stagesAs a strategy for reducing cancer incidence it

appears that it is much more productive to develop

environmental diet or behavioral strategies which

would further slow (making them less probable) the

slow stages to reduce cancers rather than strategies

which make fast stages less probable which would

not reduce cancers appreciably It is a challenge to

cancer biologists to classify the stages as fast or

slow in order to identify the most effective preven-

tion strategies

Figure 6 ArmitageDoll approximation (line fit in loglog scale) ofthe exact multistage model in the example proposed by Moolgavkar(1991) 1 Squares exact model N1e9 k6 m100001 m200002m300034 m40007 m50008 m60009 Line (solid) least squarefit lnI48068lnt2146 2 Triangles exact model m10001(increased 10 times) Line least square fit lnI48001lnt19139 3Circles exact model m6009 (increased 10 times) Line least squarefit lnI44395lnt18261 4 Dashed line exact model m1000005m2000001 m300017 m400035 m50004 m600045 (all transi-tion rates decreased 2 times) N64e10 (increased 64 times)


137

There are arguments that numerous stagesare unnecessary for modelling which entail a two-stage clonal expansion (Moolgavkar and Luebeck2003) However it is important to recognize thatthe additional stages are part of the carcino-genesis process and it is clearly of interest topossess models which reflect in as much detail aspossible the correct structure of the carcinogenesisprocess

Can the multistage model ever fit the data

A salient feature of the data is that cancer incidenceappears to fall off fast above age 80 to nearlyzero at age 100 (Pompei et al 2004) Consider anexact model for I(t)Nf(t) with equal transitionrates

Nm (t m)k

et m k

which as discussed above has its peak at tpeakk=m Since k is the number of stages it is at least oforder one and on the other hand if the model isapplied to a single cell then epidemiological datarequires that even for common cancers the transi-tion rate m must be O(103) Therefore tpeak

O(103) and the turnaround can never occur within ahuman lifespan

We now give a second argument that the turn-around is not relevant within a human lifespanNote that pk(t) is a linear combination of exponen-tials so for any set of positive real transitionconstants it increases from zero to a peak turnsover and asymptotically approaches zero as t 0 Qualitatively this has the same feature as thedata and a more careful analysis is necessary to seethe problem

As noted by Heidenreich et al (2002) theincidence function

I(t)Npk(t)

1 pk(t)(34)

increases to a finite asymptote as t 0 Withstages and transition constants based on fittingreal-world data pk(t) is around 109 or smaller for0BtB100 years so in this range Npk(t) is wellapproximated by Equation (34) which is monotoneincreasing However the time scale at which pk(t)

starts to turn over is precisely the scale at whichpk(t) becomes of order one and the approximationpk(t)1 is no longer valid In conclusion we havethe following assertion

Lemma 7 (Turnover)The turnover in pk(t) can never explain the turnoverin the data

It is of course possible to fit an exact solutionwith Ncells1010 to the (monotone increasing) partof the data set between 20 years and 70 yearsWe consider both equal transition constants(Figure 7) and equally separated transition con-stants (Figure 8) In the region shown in Figure 7which corresponds to a human lifespan the lowest-order Taylor approximation to the exact solutionwith equal transition rates is an extremely goodapproximation (see Remark 1 above) and thus theexact curve is indistinguishable to the Armitage

Doll approximation Iatk1Interestingly even though we only fitted the

equal separation model to the data up to age 70one can see in Figure 8 that the equal separationmodel correctly predicted the next few data points

Susceptibility differences

An early attempted explanation of the decrease inincidence after age 80 is that the cancer-susceptiblepeople are being depleted Thus the incidence levelsoff and drops to zero when all the senistive peoplehave developed cancer This was described by Cooket al (1969) who used the ArmitageDoll approx-imation and made the simple assumption thatonly a fixed fraction of population is susceptible(sensitive) eg able to develop cancer We derive a

Figure 7 Fit of exact solution with equal transition constants to malecolon cancer data with Ncells1010


138

more general expression for the age-specific inci-dence Let S(t) be the number of sensitive personsat time t Then

d

dtS(t)S(t)I1(t)

where I1(t) is the incidence per sensitive person LetP be the total number of persons at t0 This leadsto

S(t)PCexp

g

t

0

I1(t)dt

where C is the fraction of sensitive persons Theincidence in the population which includes bothsensitive and insensitive persons is

I(t)new cases per unit time

persons at risk at time t

dS(t)=dt

(1 C)P CPexpg

t

0

I1(t)dt

CPeg

t

0I1(t)dtI1(t)

(1 C)P CPegt

0I1(t)dt

I1(t)

1

1 C

C

expg

t

0

I1(t)dt(35)

The choice C1 corresponds to the assumptionthat all people are sensitive and implies I(t)I1(t)With C0 which corresponds to a completelyinsensitive population we have I(t)0 Further forany 0BCB1 we have I(t) 0 0 as t 0 Thus if

there are any immune persons in the populationI(t) will eventually turn over and tend to zerowhich might enable such an equation to fit thedata The challenge is whether with reasonableassumptions the incidence will approach zero forall cancers within a human lifetime as SEER data

suggests If we define G 1 C

C

expf

t

0I1(t)dt then

Equation (35) implies that the equation determiningthe peak I (t)0 takes the form

I 1I2

1

G

G 1

In the general case one would not expect to solvethis except numerically but for the ArmitageDoll

case I1(t)atk1 which implies I 1=(I1)2 k1

atk

and we have

k 1

a

tk(1 C)

Ceatk=k 1 C

Taylor expanding the right hand side gives theapproximate solution

tpeak

k 1

ak(1 C)

1=k

(36)

For typical laboratory numbers k6 a1013C01 the approximation Equation (36) givestpeak65 However this depends sensitively on k

With I1(t) given by the exact model with equaltransition rates (ETR) in a tissue with N cells wefind

IETR(t)

N

G(k)emt(mt)k1

1 1 C

Cexp

N

m

G(k) G(k mt)

G(k)

(37)

The fact that the function IETR(t) presented inEquation (37) is extremely sensitive to its funda-mental parameters presents a problem from amodelling perspective With typical biological para-

meters of N1013 and m103 the factor N=mappearing under the exponential in Equation (37) isof order 1016 To obtain a reasonable function the

Figure 8 Fit of exact equal separation model to male colon cancerdata with Ncells1010 Although the fit was done using data up to age70 the model remains accurate to age 90


139

factor

G(k) G(k mt)

G(k)must cancel this divergence

which implies fine-tuning of k and mCook et al (1969) rejected this approach because

it appeared that the peak shifted with tumorprobability In contrast we find that we can directlyfit the SEER data for our three example cancers byadjusting the parameter values a k and C andobtain the results in Figure 9 Also shown forcomparison is the fit obtained with the senescencemodel of (Pompei and Wilson (2001 2004) I(t)atk1(1bt) The fitted values of a and k aresimilar for both the susceptibility and senescencefits Figure 10 shows the three cancers plotted onthe same axes each with a corresponding suscept-ibility model fit

Both these fits to the data use the ADapproximation but we believe that use of the exactmodel will not make a change in the generalconclusions The senescence model modifies theAD assumption of the inevitability of cancer atthe cellular level by including the idea that due tosenescence every cell has an increasing probabilitywith age that it never reaches the cancerous finalstage

The susceptibility and senescence fits are verysimilar except for age t105 years The suscept-ibility fit has a small tail and incidence neverreaches zero while incidence does reach zero inthe senescence fit The SEER data (and the ED01mouse data) actually reach zero incidence but thenumber of men alive at this age (700) is small andage reporting is unverified thus there is someuncertainty that the true incidence number is zeroThe susceptibility model requires that the fractionof susceptible persons C be very different for eachcancer to fit the data (0185 0046 00095 for the

cancers shown) Although it is widely believed thatthere are differences up to a factor of 3 or so inindividual susceptibility there is no indication inthe study of chemical produced cancers that there issuch a large difference in susceptibility

The senescence conjecture has important possibleimplications on the relationship between cancerand longevity as well as on the age distributionof cancer as discussed by Pompei and Wilson(2004) The basic idea is that as each cell ages ittends to progress to a senescent stage a statecharacterized by normal functioning but inabilityto divide and repair genomic damage There isevidence although not conclusive that as tissueages a larger fraction of cells senesce and thustissue is slow to repair The mathematical modellingmakes the simple assumption that at the end of ahuman lifespan (110 years) all cells are senescentand lack of repair capacity is related to the end oflife It is the same lack of repair capacity by celldivision that stops the carcinogenesis process andthus produces the sharp turnover in cancer inci-dence when a significant fraction of cells are unableto reproduce

Senescence and the beta model

Interestingly the beta model introduced by Pompeiand Wilson (2001) may also be derived from asimplified model of senescence which we calldeterministic senescence For this derivation weassume that cancer cells are mortal in the sense ofHayflick (1965) which means that a finite length isimposed on each chain of cell divisions At eachpoint in the process the number of divisionsremaining is called the residual doubling potential Assume that senescence does not affect the stage

Figure 9 Comparison between SEER data (m) susceptibility model fit ( a631013 221013 961012 k668 698 622 C

0185 0046 00095) and senescence model fit ( - a1901013 1361013 471012 k677 544 b00093 00096 001)


140

transitions or transition rates in the ArmitageDoll

process Assume a constant (deterministic) cell

division rate then the residual doubling potential

for a cell decreases linearly with ageIf a malignant cell is completely senescent so

that the residual number of doublings is zero then

this cell does not produce observable cancer More

generally if the residual doubling potential is small

then the malignant cell is limited to a small number

of descendants and this produces an unobservable

tumor or one which is effectively destroyed by the

immune system A larger tumor has a proportion-

ally greater chance of survivalArmitageDoll theory assumes that each malig-

nant cell leads to a tumor but naturally this cell

must undergo many divisions to produce an

observable tumor Accordingly the Armitage

Doll model Iatk1 must be modified to reflect

the possibility that the cell as it progresses to

cancer may be simultaneously approaching mor-

tality in the sense of Hayflick The appropriate

modification which takes account of this effect is to

introduce a linear multiplicative factor which must

be of the form (1bt) and which now represents

the conditional probability for an observable tu-

mor given one cell which is malignant (or in the kth

stage of the multistage process) at time t The

incidence function now takes the form of the

PompeiWilson beta model As before the para-

meter b may be determined through curve-fitting

As a caveat to this approach it must be noted that

the existence of in vivo senescence in the human

body has not been established

If the explanation is accepted that cellularsenescence is responsible for the rapid fall off incancers above age 80 there are several conse-quences some of which are discussed in Pompeiand Wilson (2004) For example it is not sufficientfor a drug or environmental agent (or lack thereof)to be shown to reduce cancer If the action of thedrug or agent is to increase senescence to reducecancer then it will be accompanied by the seriousside effect of reduction in longevity Alterations inthe p53 gene have been shown to do this

Melatonin a known antioxidant that reducesDNA damage has been shown in mice experimentsto both increase cancers and increase longevitysuggesting that antioxidants might require morecareful consideration If a drug or environmentalagent could be targeted to one or more specificstages in the multistage cancer interpretation thenperhaps reduction in cancer might be accomplishedwithout reduction in longevity This is seems to bethe case when known strong carcinogens areprevented from acting on cells such as stoppingsmoking or stopping b-naphthylamine inhalation

Another consequence is that drugs such ascortisone which are carcinogenic and dangerousfor use on the young may well be relatively safe foranyone over age 80 for treatment for example ofarthritis Further study of this point is veryimportant If modeling and tests bear this out itwould be welcome news for the aged

At the very least anyone testing or regulating ananticancer drug must be aware of the possibilitythat the drug might be acting as an accelerator ofsenescence which would reduce life expectancy anddemand the appropriate test regimen

Conclusions

This study set out to investigate whether it ispossible to describe cancer incidence above age 80using the exact multistage model but no otherassumptions We prove rigorously that it is notpossible In the process we have proved severaltheorems about the multistage model that may beuseful in further investigations For example weconclude that the multistage model considers onlythe slow stages and that the incidence function isinvariant under permutation of the rate constantsCommon cancers are thought to have 48 slow

Figure 10 Comparison between SEER data for male colon cancer(k) non-Hodgkins lymphoma (^) and brain cancer (I) each withsusceptibility model fit


141

stages our conclusion is that there could be manymore fast stages without an appreciable effect onthe age distribution of cancer The possible numberof these lsquofastrsquo stages depends on how much fasterthey are If fast stages are present they maydramatically affect the convergence of the Taylorexpansion

We found that with additional biological as-sumptions it is possible to fit the data and wepresented two possibilities One is the addition ofcellular senescence the other is an assumption thatmost people are not susceptible to cancer and infact for rare cancers very few people are suscep-tible

The incidence data alone cannot decide betweenthe model that there are large differences insusceptibility and the model that cellular senes-cence is important or whether either one is likelyWe merely present the data fits and the problems intheir understanding for others to consider

Acknowledgements

The authors would like to thank Suresh Moolgav-kar for helpful discussions

References

Armitage P 1985 The multistage models of carcinogenesis

Environmental Health Perspectives 53 195201

Armitage P and Doll R 1954 The age distribution of cancer

and a multi-stage theory of carcinogenesis British Journal

of Cancer VIII 112

Bateman H 1910 Solutions of a system of differential

equations occurring in the theory of radioactive transfor-

mation Proceedings of the Cambridge Philosophical

Society 15 42327

Cook PJ Doll R and Fellingham SA 1969 A mathema-

tical model for the age distribution of cancer in man

International Journal of Cancer 4 93112

Duensing S and Munger K 2002 Human papillomaviruses

and centrosome duplication errors modeling the origins

of genomic instability Oncogene 21 624148

Hayflick L 1965 The limited in vitro lifetime of human

diploid cell strains Experimental Cell Research 37 614 36

Heidenreich WF Luebeck EG and Hazelton WD 2002

Multistage models and the incidence of cancer in the

cohort of A-bomb survivors Radiation Research 158

60714

Lengauer C Kinzler KW and Vogelstein B 1998 Genetic

instabilities in human cancers Nature 396 64349

Moolgavkar SH 1978 The multistage theory of carcinogen-

esis and the age distribution of cancer in man Journal of

the National Cancer Institute 61 4952

Moolgavkar SH 1991 Carcinogenesis models an overview

Basic Life Science 58 38796

Moolgavkar SH Krewski D and Schwarz M 1999

Mechanisms of carcinogenesis and biologically-based

models for quantitative estimation and prediction of

cancer risk In Moolgavkar SH Krewski D and Zeise

L et al editors Quantitative Estimation and Prediction of

Cancer Risk IARC Scientific Publications Lyon France

Oxford University Press 131

Moolgavkar SH and Luebeck EG 2003 Multistage carci-

nogenesis and the incidence of human cancer Genes

Chromosomes and Cancer 38 302306

Pompei F and Wilson R 2001 The age distribution of

cancer the turnover at old age Health and Environmental

Risk Assessment 7 161950

Pompei F and Wilson R 2004 A quantitative model of

cellular senescence influence on cancer and longevity

Toxicology and Industrial Health 18 36576

Pompei F Polkanov M and Wilson R 2001 Age distribu-

tion of cancer in mice the incidence turnover at old age


Pompei F Lee EE and Wilson R 2004 Cancer turnover

(Letter) httpwwwnaturecomnrc (accessed January

2004)

Pompei F Lee EE and Wilson R Cancer incidence

turnover to age 110+ Gerontological Society of America

Annual Meeting Nov 2004 Washington DC

SEER Public-Use 19951999 when Using SEERStat Sur-

veillance Epidemiology and End Results (SEER) Pro-

gram (wwwseercancergov) SEERStat Database

Incidence-SEER 11 Regs Nov 2001 Sub (19732000)

National Cancer Institute DCCPS Surveillance Research

Program Cancer Statistics Branch released April 2002

based on the November 2001 submission

Stoler DL Chen N and Basik M et al 1999 The onset and

extent of genomic instability in sporadic colorectal tumor

progression Proceedings of the National Academy of

Science 96 1512126

Tagore KS Lawson MJ and Yucaitis JA et al 2003

Sensitivity and specificity of a stool DNA multitarget

assay panel for the detection of advanced colorectal

neoplasia Clinical Colorectal Cancer 3 4753


142

Appendix A

Proofs of assertions

Proof of Lemma 1

cm1mm1mcm and xjm1 (mm1mj)1

xjm

for all jm1

d

dtpmcm1

Xm1

j1

xjm1(mj)emj t

cm1

Xm1

j1

xjm1(mm1mj)emj t

cm1

Xm1

j1

mm1xjm1emj t

mm1mcm

Xm1

j1

xjmemj tmm1pm

mm1mpm1mm1pm

as desiredProof of Lemma 2 Equation (20) is proved by

double induction on m and j The series aaj tj

converges absolutely for all t C since Equation(16) is a linear combination of exponentials

Proof of Lemma 3 pk(t)mkpk1(t) so thelemma is equivalent to the statement that for0BiBk

g

0

pi(t)dt1=mi1

Generally speaking pi(t) represents the numberof objects in state i of the process (normalized bythe initial condition) Considering the process forone object forces the initial condition to be p0(0)

1 pi(0)0 i1 and it follows that pi(t)1 if theobject is in state i and zero otherwise It is thenobvious that f

0pi(t)dt does not depend on the rates

to get into state i and is inversely proportional tothe rate to get out of state i Therefore

g

0

pi(t)dt1=mi1

(this can also be proved by a tedious directcomputation) Setting ik1 gives the result that

g

0

mkpk1(t)dt1

as desiredProof of Lemma 4

pj (t)m

1

(j 1)emt(mt)j1

emt(mt)j

j

13

m(pj1pj)

We infer that the global maximum occurs at thesolution of pjpj1 which immediately gives tj=m The value of the integral follows by directcomputation as well as by Lemma 3 in the limit ofequal transition rates

Proof of Lemma 5 mj m(j1)d j]1 sothat mpmj (p j)d This implies that

xjnd1nY

p1n

pj

1

p jd1n

Yj1

i1

1

(i)

Yj1

i1

1

i

d1n(1)j1

(j 1)(n j)

cnYn1

j1

mj Yn2

i0

(m id)dn1

m

d

n1

dn1

Gm

d n 1

G(m=d)

where in the last line we have made use of thePochhammer symbol or lsquoforward factorialrsquo and itsidentity in terms of gamma functions

(a)na(a1) (an1)G(an)=G(a)

We find by direct calculation that

Xn

j1

(1) j1

(j 1)(n j)e(j1)dt

etd(1 etd)n

(etd 1)G(n)


143

hence

pn1(t)cn

Xn

j1

xjnemj t

Gm

d n 1

dn1G(m=d)

Xn

j1

(1)j1e(m(j1)d)t

(j 1)(n j)

Gm

d n 1

G(m=d)

emt etd(1 etd)n

(etd 1)G(n)

Gm

d n 1

Gm

d

G(n)

emt(1etd)n1

This completes the proof

Proof of Lemma 6 Let A be any constantIn general the solution of the differential equa-tion

f (t)Af (t)g(t) f (0)0

where g is an arbitrary function and f is to besolved for is given by

f (t)eAt

gt

0

eAxg(x)dx

Suppose that stages zero through ns are describedby the transition constant ms and the stages ns

through k are described by transition constant mf Later we will consider the limit mf 0 whilekeeping ms fixed This means that

d

dtpns1mspns

mf pns1

By the above reasoning with Amf gmspns

this implies that

pns1(t)emf t

gt

0

emf xmspns(x)dx

However from our solution of the model withequal transition rates we know that

pns(x)

1

nsemsx(msx)ns

so we have

pns1(t)(ms)

ns1

nsemf t

gt

0

e(mf ms)xxns dx

This is as far as we can go without the use ofspecial functions To proceed note that

gt

0

eaxxbdxbG(b) G(1 b ta)

ab1

Apply this with amsmf and bns Thisgives

gt

0

e(mfms)xxnsdxnsG(ns) G(1 ns t(ms mf ))

(ms mf )ns1

We infer that

pns1(t)(ms)

ns1(ns G(1 ns tms tmf ))

nsetmf (ms mf )

ns1

emf t

ms

ms mf

ns1

1G(1 ns t(ms mf ))

ns

13

We note that the last line is precisely Equation(32) and completes the proof of that assertion Itremains to examine the limit as mf 0 Ele-mentary methods of asymptotic analysis show thatfor fixed a and frac12zfrac12 0

G(a z)za1ez(1O(z1))

Thus neglecting terms proportional to inversepowers of mf we find


144

mf pns1(t)mf emf t

ms

msmf

ns1

1(ms mf )

nset(msmf )tns

ns

13

emstmf

ms

ms mf

mns

s

tns

ns

emst

ms

ms=mf 1

(mst)

ns

ns

0 ms

1

nsemst(mst)

ns mspns(t)

This completes the proof We remark in passing

that

pns1(t)

ms

ms mf

ns1

etmf g(ns

1 t(msmf ))=ns

where g is the lower incomplete gamma function


145

I a tk1 (1)

where k was interpreted as the number of stages inthe progression of cancer and a is proportional tothe product of the probabilities at each stage Thismodel successfully described several importantknown features of cancer

1 There is usually a latency period between theinsult which appears to cause the cancer and theexpression of the cancer

2 If cancer-causing agents act at different stages ofthe cancer progression and are present at highdoses then a multiplicative synergy is predictedas seen for example between smoking andasbestos exposure

3 Cook et al (1969) found that while k varied fordifferent tumor sites it is constant betweencountries The constant a varied between coun-tries as might be expected from the differentenvironmental situations

These successes led to widespread use of themodel It was soon noted that above age 60 theage-specific mortality rate flattened and fell belowthe curve predicted by Armitage and Doll Twomajor reasons for such a flattening have beensuggested the first is a variation in susceptibility(variation in a) between different members of thesame group being considered and the second is afailure of the mathematical approximations used byDoll and Armitage

Cook Doll and Fellingham considered varia-tions of susceptibility in the population Theyplotted incidence versus age on a logarithmicscale for varying fractions C of sensitive persons(Cook et al 2004 Figure 4 reproduced here asFigure 1) They concluded that if only 1 of thepopulation is susceptible to cancer then at olderages the normalized incidence curves must fallwhen all the 1 have developed cancer

Even if the number of susceptibles is 10 then aflattening occurs Figure 1 reproduced from Cooket al (1969) indicates that by suitable choice of theparameter C one might be able to describe a rapidfall off in incidence a suggestion we discuss laterCook Doll and Fellingham ultimately rejected thispossibility because data showed a constant agefor the peak of cancer incidence while theirparticular assumption of insensitive individuals

indicates that incidence peaks at a younger age

for rarer cancersThe variations in cure rates for cancer in the last

50 years suggest that it is now more important to

discuss incidence rather than mortality and in

addition that incidence may now be much better

determined than 50 years ago In this paper there-

fore we reopen the discussion with an emphasis on

cancer incidence rather than mortalityLet f(t) denote the event rate for the kth change

at time t in a single cell According to our

definition cancer incidence is related to f(t) by

I(t)Nf(t) where N is the number of affected cells

The theoretical model calculates f(t) we will per-

form this calculation exactly for a variety of

interesting situationsIn the case of the ArmitageDoll model

IAD(t)h(t) is an infinitely growing hazard func-

tion and satisfies the criterion that the cumulative

probability of cancer given by

Figure 1 Cook Doll and Fellingham (1969) considered variablesusceptibility in the population This plot shows incidence versus ageon a logarithmic scale for varying fractions C of sensitive personsModified from Cook et al (1969) Figure 4


126

F (t)1exp

g

t

0

h(t)dt

(2)


IAD(t)atk1F (t)

1 F(t) (3)


F (t)1(o1) exp

atk

k


F(t)1exp(atk=k) (4)




















127







Methods









p0 0m1

p1 (5)


Cancer in a cell




128




midt pi1(t) (6)


p0(t)m1p0(t)


pk(t)mkpk1(t)



p0(0)1 pi(0)0 (i1 2 k) (8)












t

0f (t)dt that is


I(t)G(t)=(1G(t))NF (t)=(1F (t))


I(t)Nf (t) (10)




129















by the diagram


at time t

dp0

dtm01p0

dpj


dpk

dtmk1kpk1


mj j1bj j1mj1


mj j1mj1




p0 0m1

p1 0m2 0mk

pk (12)


pk()g

0

f (t)dt1 (13)




130





conditions is



p0(0)a

pi(0)0 (for all i1) (15)


m1(t)cm

Xm

j1

xjm emj t (16)

where cmaYm1

j1

mj1j


xjmY

l1m

lj

(mlmj)1 (17)



f (t)pk (t)mkpk1(t)





j1xjm emj t the nth


Xm

j1

xjm

(mjt)n

n (18)


j1xjmmm1j 1



tm

m

Xm

j1

mj tm

(m 1)m

where m1

kak


then write

pm1(t)cm

tm1

(m 1)(1m t )


I(t)atk1(1m t )

where

a

Qk

j1 mj

(k 1) (19)

pm1(t)Xm1

i0

Ym2

j0

mj j1

Xm

j1


l j

(ml mj)

26664

37775 (14)


131





Xm

i1

ximemi tX

j]m1

ajtj

with am11=(m1) and

aj1(1) jm

j

Xi155ijm

mi1mi2

mijm(20)


pm1(t)cm

tm1

(m 1)

1

m

Xm

i1

mi

tmO(tm1)



D2

D1

t m (21)












































(per year) given by



132

104 2104 34104 7103

8103 9103

In this example

m00046 per year






Dn1

Dn

akn1

akn2

t

t

k n

Pnk

Pn1k


mi1 min




Dn1

Dn

t

k n

mk

ratio approaching

zero as mk 0

13(22)




g

0

pk(t)dtYk

i1mi

Xk

j1

xjk

mj

1 (23)





D2

D1




133






lj(mlmj)1






pj(t)1

jemt(mt) j (24)


pk1(t)1

(k 1)emt(mt)k1


g

0

pk (t)dtg

0

mpk1(t)dt1



pj(t)Xn0

(mt) jn

jn (25)



jR1j51

5k1256106 (26)





134


where CdmnGm

d n 1

G(m=d)G(n)

is a constant



tpeakd1ln

1

d

m(k1)

(28)


tpeakk 1

m (29)



pm1

cm

limd00

Xm

j1

xjmemj t

emtlimd00

d1m

Xm

j1

(1)j1

(j 1)(m j)e(j1)dt

emt

G(m)limd00

d1metd (1 etd)m

etd 1

emt tm1

(m 1)





pk1(tpeak)Cdmk

(k 1) d

(k 1) d m

k1

1

(k 1) d

m

m

d


pk1(tpeak) 0large k

const1

kO

1

k

2


dd

m

m

d

Gm

d

1

This is




I(t)m1em1t (30)

and if k2


135

I(t)m1m2

m2 m1

em1tm1m2

m1 m2

em2t (31)



pns1(t)(ms)

ns1

nsemf tg

t

0

e(mfms)xxnx dx

emf t

ms

ms mf

ns11

G(1 ns t(ms mf ))

ns

13(32)








S(t n)n

nsetn(tn)ns


p0(t)m p0(t)

p1(t)m p0n p1








136



















































tion strategies



137




Nm (t m)k

et m k





I(t)Npk(t)

1 pk(t)(34)











138


d

dtS(t)S(t)I1(t)


S(t)PCexp

g

t

0

I1(t)dt




dS(t)=dt

(1 C)P CPexpg

t

0

I1(t)dt

CPeg

t

0I1(t)dtI1(t)

(1 C)P CPegt

0I1(t)dt

I1(t)

1

1 C

C

expg

t

0

I1(t)dt(35)




C

expf

t

0I1(t)dt then


I 1I2

1

G

G 1



atk

and we have

k 1

a

tk(1 C)

Ceatk=k 1 C


tpeak

k 1

ak(1 C)

1=k

(36)



IETR(t)

N

G(k)emt(mt)k1

1 1 C

Cexp

N

m

G(k) G(k mt)

G(k)

(37)





139

factor

G(k) G(k mt)













140




































Conclusions




141




Acknowledgements


References





of Cancer VIII 112




Society 15 42327












60714





























2004)














Science 96 1512126






142

Appendix A


Proof of Lemma 1


xjm

for all jm1

d

dtpmcm1

Xm1

j1

xjm1(mj)emj t

cm1

Xm1

j1

xjm1(mm1mj)emj t

cm1

Xm1

j1

mm1xjm1emj t

mm1mcm

Xm1

j1

xjmemj tmm1pm

mm1mpm1mm1pm





g

0

pi(t)dt1=mi1





g

0

pi(t)dt1=mi1


g

0

mkpk1(t)dt1


pj (t)m

1

(j 1)emt(mt)j1

emt(mt)j

j

13

m(pj1pj)



xjnd1nY

p1n

pj

1

p jd1n

Yj1

i1

1

(i)

Yj1

i1

1

i

d1n(1)j1

(j 1)(n j)

cnYn1

j1

mj Yn2

i0

(m id)dn1

m

d

n1

dn1

Gm

d n 1

G(m=d)




Xn

j1

(1) j1

(j 1)(n j)e(j1)dt

etd(1 etd)n

(etd 1)G(n)


143

hence

pn1(t)cn

Xn

j1

xjnemj t

Gm

d n 1

dn1G(m=d)

Xn

j1

(1)j1e(m(j1)d)t

(j 1)(n j)

Gm

d n 1

G(m=d)

emt etd(1 etd)n

(etd 1)G(n)

Gm

d n 1

Gm

d

G(n)

emt(1etd)n1



f (t)Af (t)g(t) f (0)0


f (t)eAt

gt

0

eAxg(x)dx



d

dtpns1mspns

mf pns1


this implies that

pns1(t)emf t

gt

0

emf xmspns(x)dx


pns(x)

1

nsemsx(msx)ns

so we have

pns1(t)(ms)

ns1

nsemf t

gt

0

e(mf ms)xxns dx


gt

0


ab1


gt

0


(ms mf )ns1

We infer that

pns1(t)(ms)


nsetmf (ms mf )

ns1

emf t

ms

ms mf

ns1

1G(1 ns t(ms mf ))

ns

13


G(a z)za1ez(1O(z1))



144

mf pns1(t)mf emf t

ms

msmf

ns1

1(ms mf )

nset(msmf )tns

ns

13

emstmf

ms

ms mf

mns

s

tns

ns

emst

ms

ms=mf 1

(mst)

ns

ns

0 ms

1

nsemst(mst)

ns mspns(t)


that

pns1(t)

ms

ms mf

ns1

etmf g(ns

1 t(msmf ))=ns



145

F (t)1exp

g

t

0

h(t)dt

(2)


IAD(t)atk1F (t)

1 F(t) (3)


F (t)1(o1) exp

atk

k


F(t)1exp(atk=k) (4)




















127







Methods









p0 0m1

p1 (5)


Cancer in a cell




128




midt pi1(t) (6)


p0(t)m1p0(t)


pk(t)mkpk1(t)



p0(0)1 pi(0)0 (i1 2 k) (8)












t

0f (t)dt that is


I(t)G(t)=(1G(t))NF (t)=(1F (t))


I(t)Nf (t) (10)




129















by the diagram


at time t

dp0

dtm01p0

dpj


dpk

dtmk1kpk1


mj j1bj j1mj1


mj j1mj1




p0 0m1

p1 0m2 0mk

pk (12)


pk()g

0

f (t)dt1 (13)




130





conditions is



p0(0)a

pi(0)0 (for all i1) (15)


m1(t)cm

Xm

j1

xjm emj t (16)

where cmaYm1

j1

mj1j


xjmY

l1m

lj

(mlmj)1 (17)



f (t)pk (t)mkpk1(t)





j1xjm emj t the nth


Xm

j1

xjm

(mjt)n

n (18)


j1xjmmm1j 1



tm

m

Xm

j1

mj tm

(m 1)m

where m1

kak


then write

pm1(t)cm

tm1

(m 1)(1m t )


I(t)atk1(1m t )

where

a

Qk

j1 mj

(k 1) (19)

pm1(t)Xm1

i0

Ym2

j0

mj j1

Xm

j1


l j

(ml mj)

26664

37775 (14)


131





Xm

i1

ximemi tX

j]m1

ajtj

with am11=(m1) and

aj1(1) jm

j

Xi155ijm

mi1mi2

mijm(20)


pm1(t)cm

tm1

(m 1)

1

m

Xm

i1

mi

tmO(tm1)



D2

D1

t m (21)












































(per year) given by



132

104 2104 34104 7103

8103 9103

In this example

m00046 per year






Dn1

Dn

akn1

akn2

t

t

k n

Pnk

Pn1k


mi1 min




Dn1

Dn

t

k n

mk

ratio approaching

zero as mk 0

13(22)




g

0

pk(t)dtYk

i1mi

Xk

j1

xjk

mj

1 (23)





D2

D1




133






lj(mlmj)1






pj(t)1

jemt(mt) j (24)


pk1(t)1

(k 1)emt(mt)k1


g

0

pk (t)dtg

0

mpk1(t)dt1



pj(t)Xn0

(mt) jn

jn (25)



jR1j51

5k1256106 (26)





134


where CdmnGm

d n 1

G(m=d)G(n)

is a constant



tpeakd1ln

1

d

m(k1)

(28)


tpeakk 1

m (29)



pm1

cm

limd00

Xm

j1

xjmemj t

emtlimd00

d1m

Xm

j1

(1)j1

(j 1)(m j)e(j1)dt

emt

G(m)limd00

d1metd (1 etd)m

etd 1

emt tm1

(m 1)





pk1(tpeak)Cdmk

(k 1) d

(k 1) d m

k1

1

(k 1) d

m

m

d


pk1(tpeak) 0large k

const1

kO

1

k

2


dd

m

m

d

Gm

d

1

This is




I(t)m1em1t (30)

and if k2


135

I(t)m1m2

m2 m1

em1tm1m2

m1 m2

em2t (31)



pns1(t)(ms)

ns1

nsemf tg

t

0

e(mfms)xxnx dx

emf t

ms

ms mf

ns11

G(1 ns t(ms mf ))

ns

13(32)








S(t n)n

nsetn(tn)ns


p0(t)m p0(t)

p1(t)m p0n p1








136



















































tion strategies



137




Nm (t m)k

et m k





I(t)Npk(t)

1 pk(t)(34)











138


d

dtS(t)S(t)I1(t)


S(t)PCexp

g

t

0

I1(t)dt




dS(t)=dt

(1 C)P CPexpg

t

0

I1(t)dt

CPeg

t

0I1(t)dtI1(t)

(1 C)P CPegt

0I1(t)dt

I1(t)

1

1 C

C

expg

t

0

I1(t)dt(35)




C

expf

t

0I1(t)dt then


I 1I2

1

G

G 1



atk

and we have

k 1

a

tk(1 C)

Ceatk=k 1 C


tpeak

k 1

ak(1 C)

1=k

(36)



IETR(t)

N

G(k)emt(mt)k1

1 1 C

Cexp

N

m

G(k) G(k mt)

G(k)

(37)





139

factor

G(k) G(k mt)













140




































Conclusions




141




Acknowledgements


References





of Cancer VIII 112




Society 15 42327












60714





























2004)














Science 96 1512126






142

Appendix A


Proof of Lemma 1


xjm

for all jm1

d

dtpmcm1

Xm1

j1

xjm1(mj)emj t

cm1

Xm1

j1

xjm1(mm1mj)emj t

cm1

Xm1

j1

mm1xjm1emj t

mm1mcm

Xm1

j1

xjmemj tmm1pm

mm1mpm1mm1pm





g

0

pi(t)dt1=mi1





g

0

pi(t)dt1=mi1


g

0

mkpk1(t)dt1


pj (t)m

1

(j 1)emt(mt)j1

emt(mt)j

j

13

m(pj1pj)



xjnd1nY

p1n

pj

1

p jd1n

Yj1

i1

1

(i)

Yj1

i1

1

i

d1n(1)j1

(j 1)(n j)

cnYn1

j1

mj Yn2

i0

(m id)dn1

m

d

n1

dn1

Gm

d n 1

G(m=d)




Xn

j1

(1) j1

(j 1)(n j)e(j1)dt

etd(1 etd)n

(etd 1)G(n)


143

hence

pn1(t)cn

Xn

j1

xjnemj t

Gm

d n 1

dn1G(m=d)

Xn

j1

(1)j1e(m(j1)d)t

(j 1)(n j)

Gm

d n 1

G(m=d)

emt etd(1 etd)n

(etd 1)G(n)

Gm

d n 1

Gm

d

G(n)

emt(1etd)n1



f (t)Af (t)g(t) f (0)0


f (t)eAt

gt

0

eAxg(x)dx



d

dtpns1mspns

mf pns1


this implies that

pns1(t)emf t

gt

0

emf xmspns(x)dx


pns(x)

1

nsemsx(msx)ns

so we have

pns1(t)(ms)

ns1

nsemf t

gt

0

e(mf ms)xxns dx


gt

0


ab1


gt

0


(ms mf )ns1

We infer that

pns1(t)(ms)


nsetmf (ms mf )

ns1

emf t

ms

ms mf

ns1

1G(1 ns t(ms mf ))

ns

13


G(a z)za1ez(1O(z1))



144

mf pns1(t)mf emf t

ms

msmf

ns1

1(ms mf )

nset(msmf )tns

ns

13

emstmf

ms

ms mf

mns

s

tns

ns

emst

ms

ms=mf 1

(mst)

ns

ns

0 ms

1

nsemst(mst)

ns mspns(t)


that

pns1(t)

ms

ms mf

ns1

etmf g(ns

1 t(msmf ))=ns



145







Methods









p0 0m1

p1 (5)


Cancer in a cell




128




midt pi1(t) (6)


p0(t)m1p0(t)


pk(t)mkpk1(t)



p0(0)1 pi(0)0 (i1 2 k) (8)












t

0f (t)dt that is


I(t)G(t)=(1G(t))NF (t)=(1F (t))


I(t)Nf (t) (10)




129















by the diagram


at time t

dp0

dtm01p0

dpj


dpk

dtmk1kpk1


mj j1bj j1mj1


mj j1mj1




p0 0m1

p1 0m2 0mk

pk (12)


pk()g

0

f (t)dt1 (13)




130





conditions is



p0(0)a

pi(0)0 (for all i1) (15)


m1(t)cm

Xm

j1

xjm emj t (16)

where cmaYm1

j1

mj1j


xjmY

l1m

lj

(mlmj)1 (17)



f (t)pk (t)mkpk1(t)





j1xjm emj t the nth


Xm

j1

xjm

(mjt)n

n (18)


j1xjmmm1j 1



tm

m

Xm

j1

mj tm

(m 1)m

where m1

kak


then write

pm1(t)cm

tm1

(m 1)(1m t )


I(t)atk1(1m t )

where

a

Qk

j1 mj

(k 1) (19)

pm1(t)Xm1

i0

Ym2

j0

mj j1

Xm

j1


l j

(ml mj)

26664

37775 (14)


131





Xm

i1

ximemi tX

j]m1

ajtj

with am11=(m1) and

aj1(1) jm

j

Xi155ijm

mi1mi2

mijm(20)


pm1(t)cm

tm1

(m 1)

1

m

Xm

i1

mi

tmO(tm1)



D2

D1

t m (21)












































(per year) given by



132

104 2104 34104 7103

8103 9103

In this example

m00046 per year






Dn1

Dn

akn1

akn2

t

t

k n

Pnk

Pn1k


mi1 min




Dn1

Dn

t

k n

mk

ratio approaching

zero as mk 0

13(22)




g

0

pk(t)dtYk

i1mi

Xk

j1

xjk

mj

1 (23)





D2

D1




133






lj(mlmj)1






pj(t)1

jemt(mt) j (24)


pk1(t)1

(k 1)emt(mt)k1


g

0

pk (t)dtg

0

mpk1(t)dt1



pj(t)Xn0

(mt) jn

jn (25)



jR1j51

5k1256106 (26)





134


where CdmnGm

d n 1

G(m=d)G(n)

is a constant



tpeakd1ln

1

d

m(k1)

(28)


tpeakk 1

m (29)



pm1

cm

limd00

Xm

j1

xjmemj t

emtlimd00

d1m

Xm

j1

(1)j1

(j 1)(m j)e(j1)dt

emt

G(m)limd00

d1metd (1 etd)m

etd 1

emt tm1

(m 1)





pk1(tpeak)Cdmk

(k 1) d

(k 1) d m

k1

1

(k 1) d

m

m

d


pk1(tpeak) 0large k

const1

kO

1

k

2


dd

m

m

d

Gm

d

1

This is




I(t)m1em1t (30)

and if k2


135

I(t)m1m2

m2 m1

em1tm1m2

m1 m2

em2t (31)



pns1(t)(ms)

ns1

nsemf tg

t

0

e(mfms)xxnx dx

emf t

ms

ms mf

ns11

G(1 ns t(ms mf ))

ns

13(32)








S(t n)n

nsetn(tn)ns


p0(t)m p0(t)

p1(t)m p0n p1








136



















































tion strategies



137




Nm (t m)k

et m k





I(t)Npk(t)

1 pk(t)(34)











138


d

dtS(t)S(t)I1(t)


S(t)PCexp

g

t

0

I1(t)dt




dS(t)=dt

(1 C)P CPexpg

t

0

I1(t)dt

CPeg

t

0I1(t)dtI1(t)

(1 C)P CPegt

0I1(t)dt

I1(t)

1

1 C

C

expg

t

0

I1(t)dt(35)




C

expf

t

0I1(t)dt then


I 1I2

1

G

G 1



atk

and we have

k 1

a

tk(1 C)

Ceatk=k 1 C


tpeak

k 1

ak(1 C)

1=k

(36)



IETR(t)

N

G(k)emt(mt)k1

1 1 C

Cexp

N

m

G(k) G(k mt)

G(k)

(37)





139

factor

G(k) G(k mt)













140




































Conclusions




141




Acknowledgements


References





of Cancer VIII 112




Society 15 42327












60714





























2004)














Science 96 1512126






142

Appendix A


Proof of Lemma 1


xjm

for all jm1

d

dtpmcm1

Xm1

j1

xjm1(mj)emj t

cm1

Xm1

j1

xjm1(mm1mj)emj t

cm1

Xm1

j1

mm1xjm1emj t

mm1mcm

Xm1

j1

xjmemj tmm1pm

mm1mpm1mm1pm





g

0

pi(t)dt1=mi1





g

0

pi(t)dt1=mi1


g

0

mkpk1(t)dt1


pj (t)m

1

(j 1)emt(mt)j1

emt(mt)j

j

13

m(pj1pj)



xjnd1nY

p1n

pj

1

p jd1n

Yj1

i1

1

(i)

Yj1

i1

1

i

d1n(1)j1

(j 1)(n j)

cnYn1

j1

mj Yn2

i0

(m id)dn1

m

d

n1

dn1

Gm

d n 1

G(m=d)




Xn

j1

(1) j1

(j 1)(n j)e(j1)dt

etd(1 etd)n

(etd 1)G(n)


143

hence

pn1(t)cn

Xn

j1

xjnemj t

Gm

d n 1

dn1G(m=d)

Xn

j1

(1)j1e(m(j1)d)t

(j 1)(n j)

Gm

d n 1

G(m=d)

emt etd(1 etd)n

(etd 1)G(n)

Gm

d n 1

Gm

d

G(n)

emt(1etd)n1



f (t)Af (t)g(t) f (0)0


f (t)eAt

gt

0

eAxg(x)dx



d

dtpns1mspns

mf pns1


this implies that

pns1(t)emf t

gt

0

emf xmspns(x)dx


pns(x)

1

nsemsx(msx)ns

so we have

pns1(t)(ms)

ns1

nsemf t

gt

0

e(mf ms)xxns dx


gt

0


ab1


gt

0


(ms mf )ns1

We infer that

pns1(t)(ms)


nsetmf (ms mf )

ns1

emf t

ms

ms mf

ns1

1G(1 ns t(ms mf ))

ns

13


G(a z)za1ez(1O(z1))



144

mf pns1(t)mf emf t

ms

msmf

ns1

1(ms mf )

nset(msmf )tns

ns

13

emstmf

ms

ms mf

mns

s

tns

ns

emst

ms

ms=mf 1

(mst)

ns

ns

0 ms

1

nsemst(mst)

ns mspns(t)


that

pns1(t)

ms

ms mf

ns1

etmf g(ns

1 t(msmf ))=ns



145




midt pi1(t) (6)


p0(t)m1p0(t)


pk(t)mkpk1(t)



p0(0)1 pi(0)0 (i1 2 k) (8)












t

0f (t)dt that is


I(t)G(t)=(1G(t))NF (t)=(1F (t))


I(t)Nf (t) (10)




129















by the diagram


at time t

dp0

dtm01p0

dpj


dpk

dtmk1kpk1


mj j1bj j1mj1


mj j1mj1




p0 0m1

p1 0m2 0mk

pk (12)


pk()g

0

f (t)dt1 (13)




130





conditions is



p0(0)a

pi(0)0 (for all i1) (15)


m1(t)cm

Xm

j1

xjm emj t (16)

where cmaYm1

j1

mj1j


xjmY

l1m

lj

(mlmj)1 (17)



f (t)pk (t)mkpk1(t)





j1xjm emj t the nth


Xm

j1

xjm

(mjt)n

n (18)


j1xjmmm1j 1



tm

m

Xm

j1

mj tm

(m 1)m

where m1

kak


then write

pm1(t)cm

tm1

(m 1)(1m t )


I(t)atk1(1m t )

where

a

Qk

j1 mj

(k 1) (19)

pm1(t)Xm1

i0

Ym2

j0

mj j1

Xm

j1


l j

(ml mj)

26664

37775 (14)


131





Xm

i1

ximemi tX

j]m1

ajtj

with am11=(m1) and

aj1(1) jm

j

Xi155ijm

mi1mi2

mijm(20)


pm1(t)cm

tm1

(m 1)

1

m

Xm

i1

mi

tmO(tm1)



D2

D1

t m (21)












































(per year) given by



132

104 2104 34104 7103

8103 9103

In this example

m00046 per year






Dn1

Dn

akn1

akn2

t

t

k n

Pnk

Pn1k


mi1 min




Dn1

Dn

t

k n

mk

ratio approaching

zero as mk 0

13(22)




g

0

pk(t)dtYk

i1mi

Xk

j1

xjk

mj

1 (23)





D2

D1




133






lj(mlmj)1






pj(t)1

jemt(mt) j (24)


pk1(t)1

(k 1)emt(mt)k1


g

0

pk (t)dtg

0

mpk1(t)dt1



pj(t)Xn0

(mt) jn

jn (25)



jR1j51

5k1256106 (26)





134


where CdmnGm

d n 1

G(m=d)G(n)

is a constant



tpeakd1ln

1

d

m(k1)

(28)


tpeakk 1

m (29)



pm1

cm

limd00

Xm

j1

xjmemj t

emtlimd00

d1m

Xm

j1

(1)j1

(j 1)(m j)e(j1)dt

emt

G(m)limd00

d1metd (1 etd)m

etd 1

emt tm1

(m 1)





pk1(tpeak)Cdmk

(k 1) d

(k 1) d m

k1

1

(k 1) d

m

m

d


pk1(tpeak) 0large k

const1

kO

1

k

2


dd

m

m

d

Gm

d

1

This is




I(t)m1em1t (30)

and if k2


135

I(t)m1m2

m2 m1

em1tm1m2

m1 m2

em2t (31)



pns1(t)(ms)

ns1

nsemf tg

t

0

e(mfms)xxnx dx

emf t

ms

ms mf

ns11

G(1 ns t(ms mf ))

ns

13(32)








S(t n)n

nsetn(tn)ns


p0(t)m p0(t)

p1(t)m p0n p1








136



















































tion strategies



137




Nm (t m)k

et m k





I(t)Npk(t)

1 pk(t)(34)











138


d

dtS(t)S(t)I1(t)


S(t)PCexp

g

t

0

I1(t)dt




dS(t)=dt

(1 C)P CPexpg

t

0

I1(t)dt

CPeg

t

0I1(t)dtI1(t)

(1 C)P CPegt

0I1(t)dt

I1(t)

1

1 C

C

expg

t

0

I1(t)dt(35)




C

expf

t

0I1(t)dt then


I 1I2

1

G

G 1



atk

and we have

k 1

a

tk(1 C)

Ceatk=k 1 C


tpeak

k 1

ak(1 C)

1=k

(36)



IETR(t)

N

G(k)emt(mt)k1

1 1 C

Cexp

N

m

G(k) G(k mt)

G(k)

(37)





139

factor

G(k) G(k mt)













140




































Conclusions




141




Acknowledgements


References





of Cancer VIII 112




Society 15 42327












60714





























2004)














Science 96 1512126






142

Appendix A


Proof of Lemma 1


xjm

for all jm1

d

dtpmcm1

Xm1

j1

xjm1(mj)emj t

cm1

Xm1

j1

xjm1(mm1mj)emj t

cm1

Xm1

j1

mm1xjm1emj t

mm1mcm

Xm1

j1

xjmemj tmm1pm

mm1mpm1mm1pm





g

0

pi(t)dt1=mi1





g

0

pi(t)dt1=mi1


g

0

mkpk1(t)dt1


pj (t)m

1

(j 1)emt(mt)j1

emt(mt)j

j

13

m(pj1pj)



xjnd1nY

p1n

pj

1

p jd1n

Yj1

i1

1

(i)

Yj1

i1

1

i

d1n(1)j1

(j 1)(n j)

cnYn1

j1

mj Yn2

i0

(m id)dn1

m

d

n1

dn1

Gm

d n 1

G(m=d)




Xn

j1

(1) j1

(j 1)(n j)e(j1)dt

etd(1 etd)n

(etd 1)G(n)


143

hence

pn1(t)cn

Xn

j1

xjnemj t

Gm

d n 1

dn1G(m=d)

Xn

j1

(1)j1e(m(j1)d)t

(j 1)(n j)

Gm

d n 1

G(m=d)

emt etd(1 etd)n

(etd 1)G(n)

Gm

d n 1

Gm

d

G(n)

emt(1etd)n1



f (t)Af (t)g(t) f (0)0


f (t)eAt

gt

0

eAxg(x)dx



d

dtpns1mspns

mf pns1


this implies that

pns1(t)emf t

gt

0

emf xmspns(x)dx


pns(x)

1

nsemsx(msx)ns

so we have

pns1(t)(ms)

ns1

nsemf t

gt

0

e(mf ms)xxns dx


gt

0


ab1


gt

0


(ms mf )ns1

We infer that

pns1(t)(ms)


nsetmf (ms mf )

ns1

emf t

ms

ms mf

ns1

1G(1 ns t(ms mf ))

ns

13


G(a z)za1ez(1O(z1))



144

mf pns1(t)mf emf t

ms

msmf

ns1

1(ms mf )

nset(msmf )tns

ns

13

emstmf

ms

ms mf

mns

s

tns

ns

emst

ms

ms=mf 1

(mst)

ns

ns

0 ms

1

nsemst(mst)

ns mspns(t)


that

pns1(t)

ms

ms mf

ns1

etmf g(ns

1 t(msmf ))=ns



145















by the diagram


at time t

dp0

dtm01p0

dpj


dpk

dtmk1kpk1


mj j1bj j1mj1


mj j1mj1




p0 0m1

p1 0m2 0mk

pk (12)


pk()g

0

f (t)dt1 (13)




130





conditions is



p0(0)a

pi(0)0 (for all i1) (15)


m1(t)cm

Xm

j1

xjm emj t (16)

where cmaYm1

j1

mj1j


xjmY

l1m

lj

(mlmj)1 (17)



f (t)pk (t)mkpk1(t)





j1xjm emj t the nth


Xm

j1

xjm

(mjt)n

n (18)


j1xjmmm1j 1



tm

m

Xm

j1

mj tm

(m 1)m

where m1

kak


then write

pm1(t)cm

tm1

(m 1)(1m t )


I(t)atk1(1m t )

where

a

Qk

j1 mj

(k 1) (19)

pm1(t)Xm1

i0

Ym2

j0

mj j1

Xm

j1


l j

(ml mj)

26664

37775 (14)


131





Xm

i1

ximemi tX

j]m1

ajtj

with am11=(m1) and

aj1(1) jm

j

Xi155ijm

mi1mi2

mijm(20)


pm1(t)cm

tm1

(m 1)

1

m

Xm

i1

mi

tmO(tm1)



D2

D1

t m (21)












































(per year) given by



132

104 2104 34104 7103

8103 9103

In this example

m00046 per year






Dn1

Dn

akn1

akn2

t

t

k n

Pnk

Pn1k


mi1 min




Dn1

Dn

t

k n

mk

ratio approaching

zero as mk 0

13(22)




g

0

pk(t)dtYk

i1mi

Xk

j1

xjk

mj

1 (23)





D2

D1




133






lj(mlmj)1






pj(t)1

jemt(mt) j (24)


pk1(t)1

(k 1)emt(mt)k1


g

0

pk (t)dtg

0

mpk1(t)dt1



pj(t)Xn0

(mt) jn

jn (25)



jR1j51

5k1256106 (26)





134


where CdmnGm

d n 1

G(m=d)G(n)

is a constant



tpeakd1ln

1

d

m(k1)

(28)


tpeakk 1

m (29)



pm1

cm

limd00

Xm

j1

xjmemj t

emtlimd00

d1m

Xm

j1

(1)j1

(j 1)(m j)e(j1)dt

emt

G(m)limd00

d1metd (1 etd)m

etd 1

emt tm1

(m 1)





pk1(tpeak)Cdmk

(k 1) d

(k 1) d m

k1

1

(k 1) d

m

m

d


pk1(tpeak) 0large k

const1

kO

1

k

2


dd

m

m

d

Gm

d

1

This is




I(t)m1em1t (30)

and if k2


135

I(t)m1m2

m2 m1

em1tm1m2

m1 m2

em2t (31)



pns1(t)(ms)

ns1

nsemf tg

t

0

e(mfms)xxnx dx

emf t

ms

ms mf

ns11

G(1 ns t(ms mf ))

ns

13(32)








S(t n)n

nsetn(tn)ns


p0(t)m p0(t)

p1(t)m p0n p1








136



















































tion strategies



137




Nm (t m)k

et m k





I(t)Npk(t)

1 pk(t)(34)











138


d

dtS(t)S(t)I1(t)


S(t)PCexp

g

t

0

I1(t)dt




dS(t)=dt

(1 C)P CPexpg

t

0

I1(t)dt

CPeg

t

0I1(t)dtI1(t)

(1 C)P CPegt

0I1(t)dt

I1(t)

1

1 C

C

expg

t

0

I1(t)dt(35)




C

expf

t

0I1(t)dt then


I 1I2

1

G

G 1



atk

and we have

k 1

a

tk(1 C)

Ceatk=k 1 C


tpeak

k 1

ak(1 C)

1=k

(36)



IETR(t)

N

G(k)emt(mt)k1

1 1 C

Cexp

N

m

G(k) G(k mt)

G(k)

(37)





139

factor

G(k) G(k mt)













140




































Conclusions




141




Acknowledgements


References





of Cancer VIII 112




Society 15 42327












60714





























2004)














Science 96 1512126






142

Appendix A


Proof of Lemma 1


xjm

for all jm1

d

dtpmcm1

Xm1

j1

xjm1(mj)emj t

cm1

Xm1

j1

xjm1(mm1mj)emj t

cm1

Xm1

j1

mm1xjm1emj t

mm1mcm

Xm1

j1

xjmemj tmm1pm

mm1mpm1mm1pm





g

0

pi(t)dt1=mi1





g

0

pi(t)dt1=mi1


g

0

mkpk1(t)dt1


pj (t)m

1

(j 1)emt(mt)j1

emt(mt)j

j

13

m(pj1pj)



xjnd1nY

p1n

pj

1

p jd1n

Yj1

i1

1

(i)

Yj1

i1

1

i

d1n(1)j1

(j 1)(n j)

cnYn1

j1

mj Yn2

i0

(m id)dn1

m

d

n1

dn1

Gm

d n 1

G(m=d)




Xn

j1

(1) j1

(j 1)(n j)e(j1)dt

etd(1 etd)n

(etd 1)G(n)


143

hence

pn1(t)cn

Xn

j1

xjnemj t

Gm

d n 1

dn1G(m=d)

Xn

j1

(1)j1e(m(j1)d)t

(j 1)(n j)

Gm

d n 1

G(m=d)

emt etd(1 etd)n

(etd 1)G(n)

Gm

d n 1

Gm

d

G(n)

emt(1etd)n1



f (t)Af (t)g(t) f (0)0


f (t)eAt

gt

0

eAxg(x)dx



d

dtpns1mspns

mf pns1


this implies that

pns1(t)emf t

gt

0

emf xmspns(x)dx


pns(x)

1

nsemsx(msx)ns

so we have

pns1(t)(ms)

ns1

nsemf t

gt

0

e(mf ms)xxns dx


gt

0


ab1


gt

0


(ms mf )ns1

We infer that

pns1(t)(ms)


nsetmf (ms mf )

ns1

emf t

ms

ms mf

ns1

1G(1 ns t(ms mf ))

ns

13


G(a z)za1ez(1O(z1))



144

mf pns1(t)mf emf t

ms

msmf

ns1

1(ms mf )

nset(msmf )tns

ns

13

emstmf

ms

ms mf

mns

s

tns

ns

emst

ms

ms=mf 1

(mst)

ns

ns

0 ms

1

nsemst(mst)

ns mspns(t)


that

pns1(t)

ms

ms mf

ns1

etmf g(ns

1 t(msmf ))=ns



145





conditions is



p0(0)a

pi(0)0 (for all i1) (15)


m1(t)cm

Xm

j1

xjm emj t (16)

where cmaYm1

j1

mj1j


xjmY

l1m

lj

(mlmj)1 (17)



f (t)pk (t)mkpk1(t)





j1xjm emj t the nth


Xm

j1

xjm

(mjt)n

n (18)


j1xjmmm1j 1



tm

m

Xm

j1

mj tm

(m 1)m

where m1

kak


then write

pm1(t)cm

tm1

(m 1)(1m t )


I(t)atk1(1m t )

where

a

Qk

j1 mj

(k 1) (19)

pm1(t)Xm1

i0

Ym2

j0

mj j1

Xm

j1


l j

(ml mj)

26664

37775 (14)


131





Xm

i1

ximemi tX

j]m1

ajtj

with am11=(m1) and

aj1(1) jm

j

Xi155ijm

mi1mi2

mijm(20)


pm1(t)cm

tm1

(m 1)

1

m

Xm

i1

mi

tmO(tm1)



D2

D1

t m (21)












































(per year) given by



132

104 2104 34104 7103

8103 9103

In this example

m00046 per year






Dn1

Dn

akn1

akn2

t

t

k n

Pnk

Pn1k


mi1 min




Dn1

Dn

t

k n

mk

ratio approaching

zero as mk 0

13(22)




g

0

pk(t)dtYk

i1mi

Xk

j1

xjk

mj

1 (23)





D2

D1




133






lj(mlmj)1






pj(t)1

jemt(mt) j (24)


pk1(t)1

(k 1)emt(mt)k1


g

0

pk (t)dtg

0

mpk1(t)dt1



pj(t)Xn0

(mt) jn

jn (25)



jR1j51

5k1256106 (26)





134


where CdmnGm

d n 1

G(m=d)G(n)

is a constant



tpeakd1ln

1

d

m(k1)

(28)


tpeakk 1

m (29)



pm1

cm

limd00

Xm

j1

xjmemj t

emtlimd00

d1m

Xm

j1

(1)j1

(j 1)(m j)e(j1)dt

emt

G(m)limd00

d1metd (1 etd)m

etd 1

emt tm1

(m 1)





pk1(tpeak)Cdmk

(k 1) d

(k 1) d m

k1

1

(k 1) d

m

m

d


pk1(tpeak) 0large k

const1

kO

1

k

2


dd

m

m

d

Gm

d

1

This is




I(t)m1em1t (30)

and if k2


135

I(t)m1m2

m2 m1

em1tm1m2

m1 m2

em2t (31)



pns1(t)(ms)

ns1

nsemf tg

t

0

e(mfms)xxnx dx

emf t

ms

ms mf

ns11

G(1 ns t(ms mf ))

ns

13(32)








S(t n)n

nsetn(tn)ns


p0(t)m p0(t)

p1(t)m p0n p1








136



















































tion strategies



137




Nm (t m)k

et m k





I(t)Npk(t)

1 pk(t)(34)











138


d

dtS(t)S(t)I1(t)


S(t)PCexp

g

t

0

I1(t)dt




dS(t)=dt

(1 C)P CPexpg

t

0

I1(t)dt

CPeg

t

0I1(t)dtI1(t)

(1 C)P CPegt

0I1(t)dt

I1(t)

1

1 C

C

expg

t

0

I1(t)dt(35)




C

expf

t

0I1(t)dt then


I 1I2

1

G

G 1



atk

and we have

k 1

a

tk(1 C)

Ceatk=k 1 C


tpeak

k 1

ak(1 C)

1=k

(36)



IETR(t)

N

G(k)emt(mt)k1

1 1 C

Cexp

N

m

G(k) G(k mt)

G(k)

(37)





139

factor

G(k) G(k mt)













140




































Conclusions




141




Acknowledgements


References





of Cancer VIII 112




Society 15 42327












60714





























2004)














Science 96 1512126






142

Appendix A


Proof of Lemma 1


xjm

for all jm1

d

dtpmcm1

Xm1

j1

xjm1(mj)emj t

cm1

Xm1

j1

xjm1(mm1mj)emj t

cm1

Xm1

j1

mm1xjm1emj t

mm1mcm

Xm1

j1

xjmemj tmm1pm

mm1mpm1mm1pm





g

0

pi(t)dt1=mi1





g

0

pi(t)dt1=mi1


g

0

mkpk1(t)dt1


pj (t)m

1

(j 1)emt(mt)j1

emt(mt)j

j

13

m(pj1pj)



xjnd1nY

p1n

pj

1

p jd1n

Yj1

i1

1

(i)

Yj1

i1

1

i

d1n(1)j1

(j 1)(n j)

cnYn1

j1

mj Yn2

i0

(m id)dn1

m

d

n1

dn1

Gm

d n 1

G(m=d)




Xn

j1

(1) j1

(j 1)(n j)e(j1)dt

etd(1 etd)n

(etd 1)G(n)


143

hence

pn1(t)cn

Xn

j1

xjnemj t

Gm

d n 1

dn1G(m=d)

Xn

j1

(1)j1e(m(j1)d)t

(j 1)(n j)

Gm

d n 1

G(m=d)

emt etd(1 etd)n

(etd 1)G(n)

Gm

d n 1

Gm

d

G(n)

emt(1etd)n1



f (t)Af (t)g(t) f (0)0


f (t)eAt

gt

0

eAxg(x)dx



d

dtpns1mspns

mf pns1


this implies that

pns1(t)emf t

gt

0

emf xmspns(x)dx


pns(x)

1

nsemsx(msx)ns

so we have

pns1(t)(ms)

ns1

nsemf t

gt

0

e(mf ms)xxns dx


gt

0


ab1


gt

0


(ms mf )ns1

We infer that

pns1(t)(ms)


nsetmf (ms mf )

ns1

emf t

ms

ms mf

ns1

1G(1 ns t(ms mf ))

ns

13


G(a z)za1ez(1O(z1))



144

mf pns1(t)mf emf t

ms

msmf

ns1

1(ms mf )

nset(msmf )tns

ns

13

emstmf

ms

ms mf

mns

s

tns

ns

emst

ms

ms=mf 1

(mst)

ns

ns

0 ms

1

nsemst(mst)

ns mspns(t)


that

pns1(t)

ms

ms mf

ns1

etmf g(ns

1 t(msmf ))=ns



145





Xm

i1

ximemi tX

j]m1

ajtj

with am11=(m1) and

aj1(1) jm

j

Xi155ijm

mi1mi2

mijm(20)


pm1(t)cm

tm1

(m 1)

1

m

Xm

i1

mi

tmO(tm1)



D2

D1

t m (21)












































(per year) given by



132

104 2104 34104 7103

8103 9103

In this example

m00046 per year






Dn1

Dn

akn1

akn2

t

t

k n

Pnk

Pn1k


mi1 min




Dn1

Dn

t

k n

mk

ratio approaching

zero as mk 0

13(22)




g

0

pk(t)dtYk

i1mi

Xk

j1

xjk

mj

1 (23)





D2

D1




133






lj(mlmj)1






pj(t)1

jemt(mt) j (24)


pk1(t)1

(k 1)emt(mt)k1


g

0

pk (t)dtg

0

mpk1(t)dt1



pj(t)Xn0

(mt) jn

jn (25)



jR1j51

5k1256106 (26)





134


where CdmnGm

d n 1

G(m=d)G(n)

is a constant



tpeakd1ln

1

d

m(k1)

(28)


tpeakk 1

m (29)



pm1

cm

limd00

Xm

j1

xjmemj t

emtlimd00

d1m

Xm

j1

(1)j1

(j 1)(m j)e(j1)dt

emt

G(m)limd00

d1metd (1 etd)m

etd 1

emt tm1

(m 1)





pk1(tpeak)Cdmk

(k 1) d

(k 1) d m

k1

1

(k 1) d

m

m

d


pk1(tpeak) 0large k

const1

kO

1

k

2


dd

m

m

d

Gm

d

1

This is




I(t)m1em1t (30)

and if k2


135

I(t)m1m2

m2 m1

em1tm1m2

m1 m2

em2t (31)



pns1(t)(ms)

ns1

nsemf tg

t

0

e(mfms)xxnx dx

emf t

ms

ms mf

ns11

G(1 ns t(ms mf ))

ns

13(32)








S(t n)n

nsetn(tn)ns


p0(t)m p0(t)

p1(t)m p0n p1








136



















































tion strategies



137




Nm (t m)k

et m k





I(t)Npk(t)

1 pk(t)(34)











138


d

dtS(t)S(t)I1(t)


S(t)PCexp

g

t

0

I1(t)dt




dS(t)=dt

(1 C)P CPexpg

t

0

I1(t)dt

CPeg

t

0I1(t)dtI1(t)

(1 C)P CPegt

0I1(t)dt

I1(t)

1

1 C

C

expg

t

0

I1(t)dt(35)




C

expf

t

0I1(t)dt then


I 1I2

1

G

G 1



atk

and we have

k 1

a

tk(1 C)

Ceatk=k 1 C


tpeak

k 1

ak(1 C)

1=k

(36)



IETR(t)

N

G(k)emt(mt)k1

1 1 C

Cexp

N

m

G(k) G(k mt)

G(k)

(37)





139

factor

G(k) G(k mt)













140




































Conclusions




141




Acknowledgements


References





of Cancer VIII 112




Society 15 42327












60714





























2004)














Science 96 1512126






142

Appendix A


Proof of Lemma 1


xjm

for all jm1

d

dtpmcm1

Xm1

j1

xjm1(mj)emj t

cm1

Xm1

j1

xjm1(mm1mj)emj t

cm1

Xm1

j1

mm1xjm1emj t

mm1mcm

Xm1

j1

xjmemj tmm1pm

mm1mpm1mm1pm





g

0

pi(t)dt1=mi1





g

0

pi(t)dt1=mi1


g

0

mkpk1(t)dt1


pj (t)m

1

(j 1)emt(mt)j1

emt(mt)j

j

13

m(pj1pj)



xjnd1nY

p1n

pj

1

p jd1n

Yj1

i1

1

(i)

Yj1

i1

1

i

d1n(1)j1

(j 1)(n j)

cnYn1

j1

mj Yn2

i0

(m id)dn1

m

d

n1

dn1

Gm

d n 1

G(m=d)




Xn

j1

(1) j1

(j 1)(n j)e(j1)dt

etd(1 etd)n

(etd 1)G(n)


143

hence

pn1(t)cn

Xn

j1

xjnemj t

Gm

d n 1

dn1G(m=d)

Xn

j1

(1)j1e(m(j1)d)t

(j 1)(n j)

Gm

d n 1

G(m=d)

emt etd(1 etd)n

(etd 1)G(n)

Gm

d n 1

Gm

d

G(n)

emt(1etd)n1



f (t)Af (t)g(t) f (0)0


f (t)eAt

gt

0

eAxg(x)dx



d

dtpns1mspns

mf pns1


this implies that

pns1(t)emf t

gt

0

emf xmspns(x)dx


pns(x)

1

nsemsx(msx)ns

so we have

pns1(t)(ms)

ns1

nsemf t

gt

0

e(mf ms)xxns dx


gt

0


ab1


gt

0


(ms mf )ns1

We infer that

pns1(t)(ms)


nsetmf (ms mf )

ns1

emf t

ms

ms mf

ns1

1G(1 ns t(ms mf ))

ns

13


G(a z)za1ez(1O(z1))



144

mf pns1(t)mf emf t

ms

msmf

ns1

1(ms mf )

nset(msmf )tns

ns

13

emstmf

ms

ms mf

mns

s

tns

ns

emst

ms

ms=mf 1

(mst)

ns

ns

0 ms

1

nsemst(mst)

ns mspns(t)


that

pns1(t)

ms

ms mf

ns1

etmf g(ns

1 t(msmf ))=ns



145

104 2104 34104 7103

8103 9103

In this example

m00046 per year






Dn1

Dn

akn1

akn2

t

t

k n

Pnk

Pn1k


mi1 min




Dn1

Dn

t

k n

mk

ratio approaching

zero as mk 0

13(22)




g

0

pk(t)dtYk

i1mi

Xk

j1

xjk

mj

1 (23)





D2

D1




133






lj(mlmj)1






pj(t)1

jemt(mt) j (24)


pk1(t)1

(k 1)emt(mt)k1


g

0

pk (t)dtg

0

mpk1(t)dt1



pj(t)Xn0

(mt) jn

jn (25)



jR1j51

5k1256106 (26)





134


where CdmnGm

d n 1

G(m=d)G(n)

is a constant



tpeakd1ln

1

d

m(k1)

(28)


tpeakk 1

m (29)



pm1

cm

limd00

Xm

j1

xjmemj t

emtlimd00

d1m

Xm

j1

(1)j1

(j 1)(m j)e(j1)dt

emt

G(m)limd00

d1metd (1 etd)m

etd 1

emt tm1

(m 1)





pk1(tpeak)Cdmk

(k 1) d

(k 1) d m

k1

1

(k 1) d

m

m

d


pk1(tpeak) 0large k

const1

kO

1

k

2


dd

m

m

d

Gm

d

1

This is




I(t)m1em1t (30)

and if k2


135

I(t)m1m2

m2 m1

em1tm1m2

m1 m2

em2t (31)



pns1(t)(ms)

ns1

nsemf tg

t

0

e(mfms)xxnx dx

emf t

ms

ms mf

ns11

G(1 ns t(ms mf ))

ns

13(32)








S(t n)n

nsetn(tn)ns


p0(t)m p0(t)

p1(t)m p0n p1








136



















































tion strategies



137




Nm (t m)k

et m k





I(t)Npk(t)

1 pk(t)(34)











138


d

dtS(t)S(t)I1(t)


S(t)PCexp

g

t

0

I1(t)dt




dS(t)=dt

(1 C)P CPexpg

t

0

I1(t)dt

CPeg

t

0I1(t)dtI1(t)

(1 C)P CPegt

0I1(t)dt

I1(t)

1

1 C

C

expg

t

0

I1(t)dt(35)




C

expf

t

0I1(t)dt then


I 1I2

1

G

G 1



atk

and we have

k 1

a

tk(1 C)

Ceatk=k 1 C


tpeak

k 1

ak(1 C)

1=k

(36)



IETR(t)

N

G(k)emt(mt)k1

1 1 C

Cexp

N

m

G(k) G(k mt)

G(k)

(37)





139

factor

G(k) G(k mt)













140




































Conclusions




141




Acknowledgements


References





of Cancer VIII 112




Society 15 42327












60714





























2004)














Science 96 1512126






142

Appendix A


Proof of Lemma 1


xjm

for all jm1

d

dtpmcm1

Xm1

j1

xjm1(mj)emj t

cm1

Xm1

j1

xjm1(mm1mj)emj t

cm1

Xm1

j1

mm1xjm1emj t

mm1mcm

Xm1

j1

xjmemj tmm1pm

mm1mpm1mm1pm





g

0

pi(t)dt1=mi1





g

0

pi(t)dt1=mi1


g

0

mkpk1(t)dt1


pj (t)m

1

(j 1)emt(mt)j1

emt(mt)j

j

13

m(pj1pj)



xjnd1nY

p1n

pj

1

p jd1n

Yj1

i1

1

(i)

Yj1

i1

1

i

d1n(1)j1

(j 1)(n j)

cnYn1

j1

mj Yn2

i0

(m id)dn1

m

d

n1

dn1

Gm

d n 1

G(m=d)




Xn

j1

(1) j1

(j 1)(n j)e(j1)dt

etd(1 etd)n

(etd 1)G(n)


143

hence

pn1(t)cn

Xn

j1

xjnemj t

Gm

d n 1

dn1G(m=d)

Xn

j1

(1)j1e(m(j1)d)t

(j 1)(n j)

Gm

d n 1

G(m=d)

emt etd(1 etd)n

(etd 1)G(n)

Gm

d n 1

Gm

d

G(n)

emt(1etd)n1



f (t)Af (t)g(t) f (0)0


f (t)eAt

gt

0

eAxg(x)dx



d

dtpns1mspns

mf pns1


this implies that

pns1(t)emf t

gt

0

emf xmspns(x)dx


pns(x)

1

nsemsx(msx)ns

so we have

pns1(t)(ms)

ns1

nsemf t

gt

0

e(mf ms)xxns dx


gt

0


ab1


gt

0


(ms mf )ns1

We infer that

pns1(t)(ms)


nsetmf (ms mf )

ns1

emf t

ms

ms mf

ns1

1G(1 ns t(ms mf ))

ns

13


G(a z)za1ez(1O(z1))



144

mf pns1(t)mf emf t

ms

msmf

ns1

1(ms mf )

nset(msmf )tns

ns

13

emstmf

ms

ms mf

mns

s

tns

ns

emst

ms

ms=mf 1

(mst)

ns

ns

0 ms

1

nsemst(mst)

ns mspns(t)


that

pns1(t)

ms

ms mf

ns1

etmf g(ns

1 t(msmf ))=ns



145






lj(mlmj)1






pj(t)1

jemt(mt) j (24)


pk1(t)1

(k 1)emt(mt)k1


g

0

pk (t)dtg

0

mpk1(t)dt1



pj(t)Xn0

(mt) jn

jn (25)



jR1j51

5k1256106 (26)





134


where CdmnGm

d n 1

G(m=d)G(n)

is a constant



tpeakd1ln

1

d

m(k1)

(28)


tpeakk 1

m (29)



pm1

cm

limd00

Xm

j1

xjmemj t

emtlimd00

d1m

Xm

j1

(1)j1

(j 1)(m j)e(j1)dt

emt

G(m)limd00

d1metd (1 etd)m

etd 1

emt tm1

(m 1)





pk1(tpeak)Cdmk

(k 1) d

(k 1) d m

k1

1

(k 1) d

m

m

d


pk1(tpeak) 0large k

const1

kO

1

k

2


dd

m

m

d

Gm

d

1

This is




I(t)m1em1t (30)

and if k2


135

I(t)m1m2

m2 m1

em1tm1m2

m1 m2

em2t (31)



pns1(t)(ms)

ns1

nsemf tg

t

0

e(mfms)xxnx dx

emf t

ms

ms mf

ns11

G(1 ns t(ms mf ))

ns

13(32)








S(t n)n

nsetn(tn)ns


p0(t)m p0(t)

p1(t)m p0n p1








136



















































tion strategies



137




Nm (t m)k

et m k





I(t)Npk(t)

1 pk(t)(34)











138


d

dtS(t)S(t)I1(t)


S(t)PCexp

g

t

0

I1(t)dt




dS(t)=dt

(1 C)P CPexpg

t

0

I1(t)dt

CPeg

t

0I1(t)dtI1(t)

(1 C)P CPegt

0I1(t)dt

I1(t)

1

1 C

C

expg

t

0

I1(t)dt(35)




C

expf

t

0I1(t)dt then


I 1I2

1

G

G 1



atk

and we have

k 1

a

tk(1 C)

Ceatk=k 1 C


tpeak

k 1

ak(1 C)

1=k

(36)



IETR(t)

N

G(k)emt(mt)k1

1 1 C

Cexp

N

m

G(k) G(k mt)

G(k)

(37)





139

factor

G(k) G(k mt)













140




































Conclusions




141




Acknowledgements


References





of Cancer VIII 112




Society 15 42327












60714





























2004)














Science 96 1512126






142

Appendix A


Proof of Lemma 1


xjm

for all jm1

d

dtpmcm1

Xm1

j1

xjm1(mj)emj t

cm1

Xm1

j1

xjm1(mm1mj)emj t

cm1

Xm1

j1

mm1xjm1emj t

mm1mcm

Xm1

j1

xjmemj tmm1pm

mm1mpm1mm1pm





g

0

pi(t)dt1=mi1





g

0

pi(t)dt1=mi1


g

0

mkpk1(t)dt1


pj (t)m

1

(j 1)emt(mt)j1

emt(mt)j

j

13

m(pj1pj)



xjnd1nY

p1n

pj

1

p jd1n

Yj1

i1

1

(i)

Yj1

i1

1

i

d1n(1)j1

(j 1)(n j)

cnYn1

j1

mj Yn2

i0

(m id)dn1

m

d

n1

dn1

Gm

d n 1

G(m=d)




Xn

j1

(1) j1

(j 1)(n j)e(j1)dt

etd(1 etd)n

(etd 1)G(n)


143

hence

pn1(t)cn

Xn

j1

xjnemj t

Gm

d n 1

dn1G(m=d)

Xn

j1

(1)j1e(m(j1)d)t

(j 1)(n j)

Gm

d n 1

G(m=d)

emt etd(1 etd)n

(etd 1)G(n)

Gm

d n 1

Gm

d

G(n)

emt(1etd)n1



f (t)Af (t)g(t) f (0)0


f (t)eAt

gt

0

eAxg(x)dx



d

dtpns1mspns

mf pns1


this implies that

pns1(t)emf t

gt

0

emf xmspns(x)dx


pns(x)

1

nsemsx(msx)ns

so we have

pns1(t)(ms)

ns1

nsemf t

gt

0

e(mf ms)xxns dx


gt

0


ab1


gt

0


(ms mf )ns1

We infer that

pns1(t)(ms)


nsetmf (ms mf )

ns1

emf t

ms

ms mf

ns1

1G(1 ns t(ms mf ))

ns

13


G(a z)za1ez(1O(z1))



144

mf pns1(t)mf emf t

ms

msmf

ns1

1(ms mf )

nset(msmf )tns

ns

13

emstmf

ms

ms mf

mns

s

tns

ns

emst

ms

ms=mf 1

(mst)

ns

ns

0 ms

1

nsemst(mst)

ns mspns(t)


that

pns1(t)

ms

ms mf

ns1

etmf g(ns

1 t(msmf ))=ns



145


where CdmnGm

d n 1

G(m=d)G(n)

is a constant



tpeakd1ln

1

d

m(k1)

(28)


tpeakk 1

m (29)



pm1

cm

limd00

Xm

j1

xjmemj t

emtlimd00

d1m

Xm

j1

(1)j1

(j 1)(m j)e(j1)dt

emt

G(m)limd00

d1metd (1 etd)m

etd 1

emt tm1

(m 1)





pk1(tpeak)Cdmk

(k 1) d

(k 1) d m

k1

1

(k 1) d

m

m

d


pk1(tpeak) 0large k

const1

kO

1

k

2


dd

m

m

d

Gm

d

1

This is




I(t)m1em1t (30)

and if k2


135

I(t)m1m2

m2 m1

em1tm1m2

m1 m2

em2t (31)



pns1(t)(ms)

ns1

nsemf tg

t

0

e(mfms)xxnx dx

emf t

ms

ms mf

ns11

G(1 ns t(ms mf ))

ns

13(32)








S(t n)n

nsetn(tn)ns


p0(t)m p0(t)

p1(t)m p0n p1








136



















































tion strategies



137




Nm (t m)k

et m k





I(t)Npk(t)

1 pk(t)(34)











138


d

dtS(t)S(t)I1(t)


S(t)PCexp

g

t

0

I1(t)dt




dS(t)=dt

(1 C)P CPexpg

t

0

I1(t)dt

CPeg

t

0I1(t)dtI1(t)

(1 C)P CPegt

0I1(t)dt

I1(t)

1

1 C

C

expg

t

0

I1(t)dt(35)




C

expf

t

0I1(t)dt then


I 1I2

1

G

G 1



atk

and we have

k 1

a

tk(1 C)

Ceatk=k 1 C


tpeak

k 1

ak(1 C)

1=k

(36)



IETR(t)

N

G(k)emt(mt)k1

1 1 C

Cexp

N

m

G(k) G(k mt)

G(k)

(37)





139

factor

G(k) G(k mt)













140




































Conclusions




141




Acknowledgements


References





of Cancer VIII 112




Society 15 42327












60714





























2004)














Science 96 1512126






142

Appendix A


Proof of Lemma 1


xjm

for all jm1

d

dtpmcm1

Xm1

j1

xjm1(mj)emj t

cm1

Xm1

j1

xjm1(mm1mj)emj t

cm1

Xm1

j1

mm1xjm1emj t

mm1mcm

Xm1

j1

xjmemj tmm1pm

mm1mpm1mm1pm





g

0

pi(t)dt1=mi1





g

0

pi(t)dt1=mi1


g

0

mkpk1(t)dt1


pj (t)m

1

(j 1)emt(mt)j1

emt(mt)j

j

13

m(pj1pj)



xjnd1nY

p1n

pj

1

p jd1n

Yj1

i1

1

(i)

Yj1

i1

1

i

d1n(1)j1

(j 1)(n j)

cnYn1

j1

mj Yn2

i0

(m id)dn1

m

d

n1

dn1

Gm

d n 1

G(m=d)




Xn

j1

(1) j1

(j 1)(n j)e(j1)dt

etd(1 etd)n

(etd 1)G(n)


143

hence

pn1(t)cn

Xn

j1

xjnemj t

Gm

d n 1

dn1G(m=d)

Xn

j1

(1)j1e(m(j1)d)t

(j 1)(n j)

Gm

d n 1

G(m=d)

emt etd(1 etd)n

(etd 1)G(n)

Gm

d n 1

Gm

d

G(n)

emt(1etd)n1



f (t)Af (t)g(t) f (0)0


f (t)eAt

gt

0

eAxg(x)dx



d

dtpns1mspns

mf pns1


this implies that

pns1(t)emf t

gt

0

emf xmspns(x)dx


pns(x)

1

nsemsx(msx)ns

so we have

pns1(t)(ms)

ns1

nsemf t

gt

0

e(mf ms)xxns dx


gt

0


ab1


gt

0


(ms mf )ns1

We infer that

pns1(t)(ms)


nsetmf (ms mf )

ns1

emf t

ms

ms mf

ns1

1G(1 ns t(ms mf ))

ns

13


G(a z)za1ez(1O(z1))



144

mf pns1(t)mf emf t

ms

msmf

ns1

1(ms mf )

nset(msmf )tns

ns

13

emstmf

ms

ms mf

mns

s

tns

ns

emst

ms

ms=mf 1

(mst)

ns

ns

0 ms

1

nsemst(mst)

ns mspns(t)


that

pns1(t)

ms

ms mf

ns1

etmf g(ns

1 t(msmf ))=ns



145

I(t)m1m2

m2 m1

em1tm1m2

m1 m2

em2t (31)



pns1(t)(ms)

ns1

nsemf tg

t

0

e(mfms)xxnx dx

emf t

ms

ms mf

ns11

G(1 ns t(ms mf ))

ns

13(32)








S(t n)n

nsetn(tn)ns


p0(t)m p0(t)

p1(t)m p0n p1








136



















































tion strategies



137




Nm (t m)k

et m k





I(t)Npk(t)

1 pk(t)(34)











138


d

dtS(t)S(t)I1(t)


S(t)PCexp

g

t

0

I1(t)dt




dS(t)=dt

(1 C)P CPexpg

t

0

I1(t)dt

CPeg

t

0I1(t)dtI1(t)

(1 C)P CPegt

0I1(t)dt

I1(t)

1

1 C

C

expg

t

0

I1(t)dt(35)




C

expf

t

0I1(t)dt then


I 1I2

1

G

G 1



atk

and we have

k 1

a

tk(1 C)

Ceatk=k 1 C


tpeak

k 1

ak(1 C)

1=k

(36)



IETR(t)

N

G(k)emt(mt)k1

1 1 C

Cexp

N

m

G(k) G(k mt)

G(k)

(37)





139

factor

G(k) G(k mt)













140




































Conclusions




141




Acknowledgements


References





of Cancer VIII 112




Society 15 42327












60714





























2004)














Science 96 1512126






142

Appendix A


Proof of Lemma 1


xjm

for all jm1

d

dtpmcm1

Xm1

j1

xjm1(mj)emj t

cm1

Xm1

j1

xjm1(mm1mj)emj t

cm1

Xm1

j1

mm1xjm1emj t

mm1mcm

Xm1

j1

xjmemj tmm1pm

mm1mpm1mm1pm





g

0

pi(t)dt1=mi1





g

0

pi(t)dt1=mi1


g

0

mkpk1(t)dt1


pj (t)m

1

(j 1)emt(mt)j1

emt(mt)j

j

13

m(pj1pj)



xjnd1nY

p1n

pj

1

p jd1n

Yj1

i1

1

(i)

Yj1

i1

1

i

d1n(1)j1

(j 1)(n j)

cnYn1

j1

mj Yn2

i0

(m id)dn1

m

d

n1

dn1

Gm

d n 1

G(m=d)




Xn

j1

(1) j1

(j 1)(n j)e(j1)dt

etd(1 etd)n

(etd 1)G(n)


143

hence

pn1(t)cn

Xn

j1

xjnemj t

Gm

d n 1

dn1G(m=d)

Xn

j1

(1)j1e(m(j1)d)t

(j 1)(n j)

Gm

d n 1

G(m=d)

emt etd(1 etd)n

(etd 1)G(n)

Gm

d n 1

Gm

d

G(n)

emt(1etd)n1



f (t)Af (t)g(t) f (0)0


f (t)eAt

gt

0

eAxg(x)dx



d

dtpns1mspns

mf pns1


this implies that

pns1(t)emf t

gt

0

emf xmspns(x)dx


pns(x)

1

nsemsx(msx)ns

so we have

pns1(t)(ms)

ns1

nsemf t

gt

0

e(mf ms)xxns dx


gt

0


ab1


gt

0


(ms mf )ns1

We infer that

pns1(t)(ms)


nsetmf (ms mf )

ns1

emf t

ms

ms mf

ns1

1G(1 ns t(ms mf ))

ns

13


G(a z)za1ez(1O(z1))



144

mf pns1(t)mf emf t

ms

msmf

ns1

1(ms mf )

nset(msmf )tns

ns

13

emstmf

ms

ms mf

mns

s

tns

ns

emst

ms

ms=mf 1

(mst)

ns

ns

0 ms

1

nsemst(mst)

ns mspns(t)


that

pns1(t)

ms

ms mf

ns1

etmf g(ns

1 t(msmf ))=ns



145



















































tion strategies



137




Nm (t m)k

et m k





I(t)Npk(t)

1 pk(t)(34)











138


d

dtS(t)S(t)I1(t)


S(t)PCexp

g

t

0

I1(t)dt




dS(t)=dt

(1 C)P CPexpg

t

0

I1(t)dt

CPeg

t

0I1(t)dtI1(t)

(1 C)P CPegt

0I1(t)dt

I1(t)

1

1 C

C

expg

t

0

I1(t)dt(35)




C

expf

t

0I1(t)dt then


I 1I2

1

G

G 1



atk

and we have

k 1

a

tk(1 C)

Ceatk=k 1 C


tpeak

k 1

ak(1 C)

1=k

(36)



IETR(t)

N

G(k)emt(mt)k1

1 1 C

Cexp

N

m

G(k) G(k mt)

G(k)

(37)





139

factor

G(k) G(k mt)













140




































Conclusions




141




Acknowledgements


References





of Cancer VIII 112




Society 15 42327












60714





























2004)














Science 96 1512126






142

Appendix A


Proof of Lemma 1


xjm

for all jm1

d

dtpmcm1

Xm1

j1

xjm1(mj)emj t

cm1

Xm1

j1

xjm1(mm1mj)emj t

cm1

Xm1

j1

mm1xjm1emj t

mm1mcm

Xm1

j1

xjmemj tmm1pm

mm1mpm1mm1pm





g

0

pi(t)dt1=mi1





g

0

pi(t)dt1=mi1


g

0

mkpk1(t)dt1


pj (t)m

1

(j 1)emt(mt)j1

emt(mt)j

j

13

m(pj1pj)



xjnd1nY

p1n

pj

1

p jd1n

Yj1

i1

1

(i)

Yj1

i1

1

i

d1n(1)j1

(j 1)(n j)

cnYn1

j1

mj Yn2

i0

(m id)dn1

m

d

n1

dn1

Gm

d n 1

G(m=d)




Xn

j1

(1) j1

(j 1)(n j)e(j1)dt

etd(1 etd)n

(etd 1)G(n)


143

hence

pn1(t)cn

Xn

j1

xjnemj t

Gm

d n 1

dn1G(m=d)

Xn

j1

(1)j1e(m(j1)d)t

(j 1)(n j)

Gm

d n 1

G(m=d)

emt etd(1 etd)n

(etd 1)G(n)

Gm

d n 1

Gm

d

G(n)

emt(1etd)n1



f (t)Af (t)g(t) f (0)0


f (t)eAt

gt

0

eAxg(x)dx



d

dtpns1mspns

mf pns1


this implies that

pns1(t)emf t

gt

0

emf xmspns(x)dx


pns(x)

1

nsemsx(msx)ns

so we have

pns1(t)(ms)

ns1

nsemf t

gt

0

e(mf ms)xxns dx


gt

0


ab1


gt

0


(ms mf )ns1

We infer that

pns1(t)(ms)


nsetmf (ms mf )

ns1

emf t

ms

ms mf

ns1

1G(1 ns t(ms mf ))

ns

13


G(a z)za1ez(1O(z1))



144

mf pns1(t)mf emf t

ms

msmf

ns1

1(ms mf )

nset(msmf )tns

ns

13

emstmf

ms

ms mf

mns

s

tns

ns

emst

ms

ms=mf 1

(mst)

ns

ns

0 ms

1

nsemst(mst)

ns mspns(t)


that

pns1(t)

ms

ms mf

ns1

etmf g(ns

1 t(msmf ))=ns



145




Nm (t m)k

et m k





I(t)Npk(t)

1 pk(t)(34)











138


d

dtS(t)S(t)I1(t)


S(t)PCexp

g

t

0

I1(t)dt




dS(t)=dt

(1 C)P CPexpg

t

0

I1(t)dt

CPeg

t

0I1(t)dtI1(t)

(1 C)P CPegt

0I1(t)dt

I1(t)

1

1 C

C

expg

t

0

I1(t)dt(35)




C

expf

t

0I1(t)dt then


I 1I2

1

G

G 1



atk

and we have

k 1

a

tk(1 C)

Ceatk=k 1 C


tpeak

k 1

ak(1 C)

1=k

(36)



IETR(t)

N

G(k)emt(mt)k1

1 1 C

Cexp

N

m

G(k) G(k mt)

G(k)

(37)





139

factor

G(k) G(k mt)













140




































Conclusions




141




Acknowledgements


References





of Cancer VIII 112




Society 15 42327












60714





























2004)














Science 96 1512126






142

Appendix A


Proof of Lemma 1


xjm

for all jm1

d

dtpmcm1

Xm1

j1

xjm1(mj)emj t

cm1

Xm1

j1

xjm1(mm1mj)emj t

cm1

Xm1

j1

mm1xjm1emj t

mm1mcm

Xm1

j1

xjmemj tmm1pm

mm1mpm1mm1pm





g

0

pi(t)dt1=mi1





g

0

pi(t)dt1=mi1


g

0

mkpk1(t)dt1


pj (t)m

1

(j 1)emt(mt)j1

emt(mt)j

j

13

m(pj1pj)



xjnd1nY

p1n

pj

1

p jd1n

Yj1

i1

1

(i)

Yj1

i1

1

i

d1n(1)j1

(j 1)(n j)

cnYn1

j1

mj Yn2

i0

(m id)dn1

m

d

n1

dn1

Gm

d n 1

G(m=d)




Xn

j1

(1) j1

(j 1)(n j)e(j1)dt

etd(1 etd)n

(etd 1)G(n)


143

hence

pn1(t)cn

Xn

j1

xjnemj t

Gm

d n 1

dn1G(m=d)

Xn

j1

(1)j1e(m(j1)d)t

(j 1)(n j)

Gm

d n 1

G(m=d)

emt etd(1 etd)n

(etd 1)G(n)

Gm

d n 1

Gm

d

G(n)

emt(1etd)n1



f (t)Af (t)g(t) f (0)0


f (t)eAt

gt

0

eAxg(x)dx



d

dtpns1mspns

mf pns1


this implies that

pns1(t)emf t

gt

0

emf xmspns(x)dx


pns(x)

1

nsemsx(msx)ns

so we have

pns1(t)(ms)

ns1

nsemf t

gt

0

e(mf ms)xxns dx


gt

0


ab1


gt

0


(ms mf )ns1

We infer that

pns1(t)(ms)


nsetmf (ms mf )

ns1

emf t

ms

ms mf

ns1

1G(1 ns t(ms mf ))

ns

13


G(a z)za1ez(1O(z1))



144

mf pns1(t)mf emf t

ms

msmf

ns1

1(ms mf )

nset(msmf )tns

ns

13

emstmf

ms

ms mf

mns

s

tns

ns

emst

ms

ms=mf 1

(mst)

ns

ns

0 ms

1

nsemst(mst)

ns mspns(t)


that

pns1(t)

ms

ms mf

ns1

etmf g(ns

1 t(msmf ))=ns



145


d

dtS(t)S(t)I1(t)


S(t)PCexp

g

t

0

I1(t)dt




dS(t)=dt

(1 C)P CPexpg

t

0

I1(t)dt

CPeg

t

0I1(t)dtI1(t)

(1 C)P CPegt

0I1(t)dt

I1(t)

1

1 C

C

expg

t

0

I1(t)dt(35)




C

expf

t

0I1(t)dt then


I 1I2

1

G

G 1



atk

and we have

k 1

a

tk(1 C)

Ceatk=k 1 C


tpeak

k 1

ak(1 C)

1=k

(36)



IETR(t)

N

G(k)emt(mt)k1

1 1 C

Cexp

N

m

G(k) G(k mt)

G(k)

(37)





139

factor

G(k) G(k mt)













140




































Conclusions




141




Acknowledgements


References





of Cancer VIII 112




Society 15 42327












60714





























2004)














Science 96 1512126






142

Appendix A


Proof of Lemma 1


xjm

for all jm1

d

dtpmcm1

Xm1

j1

xjm1(mj)emj t

cm1

Xm1

j1

xjm1(mm1mj)emj t

cm1

Xm1

j1

mm1xjm1emj t

mm1mcm

Xm1

j1

xjmemj tmm1pm

mm1mpm1mm1pm





g

0

pi(t)dt1=mi1





g

0

pi(t)dt1=mi1


g

0

mkpk1(t)dt1


pj (t)m

1

(j 1)emt(mt)j1

emt(mt)j

j

13

m(pj1pj)



xjnd1nY

p1n

pj

1

p jd1n

Yj1

i1

1

(i)

Yj1

i1

1

i

d1n(1)j1

(j 1)(n j)

cnYn1

j1

mj Yn2

i0

(m id)dn1

m

d

n1

dn1

Gm

d n 1

G(m=d)




Xn

j1

(1) j1

(j 1)(n j)e(j1)dt

etd(1 etd)n

(etd 1)G(n)


143

hence

pn1(t)cn

Xn

j1

xjnemj t

Gm

d n 1

dn1G(m=d)

Xn

j1

(1)j1e(m(j1)d)t

(j 1)(n j)

Gm

d n 1

G(m=d)

emt etd(1 etd)n

(etd 1)G(n)

Gm

d n 1

Gm

d

G(n)

emt(1etd)n1



f (t)Af (t)g(t) f (0)0


f (t)eAt

gt

0

eAxg(x)dx



d

dtpns1mspns

mf pns1


this implies that

pns1(t)emf t

gt

0

emf xmspns(x)dx


pns(x)

1

nsemsx(msx)ns

so we have

pns1(t)(ms)

ns1

nsemf t

gt

0

e(mf ms)xxns dx


gt

0


ab1


gt

0


(ms mf )ns1

We infer that

pns1(t)(ms)


nsetmf (ms mf )

ns1

emf t

ms

ms mf

ns1

1G(1 ns t(ms mf ))

ns

13


G(a z)za1ez(1O(z1))



144

mf pns1(t)mf emf t

ms

msmf

ns1

1(ms mf )

nset(msmf )tns

ns

13

emstmf

ms

ms mf

mns

s

tns

ns

emst

ms

ms=mf 1

(mst)

ns

ns

0 ms

1

nsemst(mst)

ns mspns(t)


that

pns1(t)

ms

ms mf

ns1

etmf g(ns

1 t(msmf ))=ns



145

factor

G(k) G(k mt)













140




































Conclusions




141




Acknowledgements


References





of Cancer VIII 112




Society 15 42327












60714





























2004)














Science 96 1512126






142

Appendix A


Proof of Lemma 1


xjm

for all jm1

d

dtpmcm1

Xm1

j1

xjm1(mj)emj t

cm1

Xm1

j1

xjm1(mm1mj)emj t

cm1

Xm1

j1

mm1xjm1emj t

mm1mcm

Xm1

j1

xjmemj tmm1pm

mm1mpm1mm1pm





g

0

pi(t)dt1=mi1





g

0

pi(t)dt1=mi1


g

0

mkpk1(t)dt1


pj (t)m

1

(j 1)emt(mt)j1

emt(mt)j

j

13

m(pj1pj)



xjnd1nY

p1n

pj

1

p jd1n

Yj1

i1

1

(i)

Yj1

i1

1

i

d1n(1)j1

(j 1)(n j)

cnYn1

j1

mj Yn2

i0

(m id)dn1

m

d

n1

dn1

Gm

d n 1

G(m=d)




Xn

j1

(1) j1

(j 1)(n j)e(j1)dt

etd(1 etd)n

(etd 1)G(n)


143

hence

pn1(t)cn

Xn

j1

xjnemj t

Gm

d n 1

dn1G(m=d)

Xn

j1

(1)j1e(m(j1)d)t

(j 1)(n j)

Gm

d n 1

G(m=d)

emt etd(1 etd)n

(etd 1)G(n)

Gm

d n 1

Gm

d

G(n)

emt(1etd)n1



f (t)Af (t)g(t) f (0)0


f (t)eAt

gt

0

eAxg(x)dx



d

dtpns1mspns

mf pns1


this implies that

pns1(t)emf t

gt

0

emf xmspns(x)dx


pns(x)

1

nsemsx(msx)ns

so we have

pns1(t)(ms)

ns1

nsemf t

gt

0

e(mf ms)xxns dx


gt

0


ab1


gt

0


(ms mf )ns1

We infer that

pns1(t)(ms)


nsetmf (ms mf )

ns1

emf t

ms

ms mf

ns1

1G(1 ns t(ms mf ))

ns

13


G(a z)za1ez(1O(z1))



144

mf pns1(t)mf emf t

ms

msmf

ns1

1(ms mf )

nset(msmf )tns

ns

13

emstmf

ms

ms mf

mns

s

tns

ns

emst

ms

ms=mf 1

(mst)

ns

ns

0 ms

1

nsemst(mst)

ns mspns(t)


that

pns1(t)

ms

ms mf

ns1

etmf g(ns

1 t(msmf ))=ns



145




































Conclusions




141




Acknowledgements


References





of Cancer VIII 112




Society 15 42327












60714





























2004)














Science 96 1512126






142

Appendix A


Proof of Lemma 1


xjm

for all jm1

d

dtpmcm1

Xm1

j1

xjm1(mj)emj t

cm1

Xm1

j1

xjm1(mm1mj)emj t

cm1

Xm1

j1

mm1xjm1emj t

mm1mcm

Xm1

j1

xjmemj tmm1pm

mm1mpm1mm1pm





g

0

pi(t)dt1=mi1





g

0

pi(t)dt1=mi1


g

0

mkpk1(t)dt1


pj (t)m

1

(j 1)emt(mt)j1

emt(mt)j

j

13

m(pj1pj)



xjnd1nY

p1n

pj

1

p jd1n

Yj1

i1

1

(i)

Yj1

i1

1

i

d1n(1)j1

(j 1)(n j)

cnYn1

j1

mj Yn2

i0

(m id)dn1

m

d

n1

dn1

Gm

d n 1

G(m=d)




Xn

j1

(1) j1

(j 1)(n j)e(j1)dt

etd(1 etd)n

(etd 1)G(n)


143

hence

pn1(t)cn

Xn

j1

xjnemj t

Gm

d n 1

dn1G(m=d)

Xn

j1

(1)j1e(m(j1)d)t

(j 1)(n j)

Gm

d n 1

G(m=d)

emt etd(1 etd)n

(etd 1)G(n)

Gm

d n 1

Gm

d

G(n)

emt(1etd)n1



f (t)Af (t)g(t) f (0)0


f (t)eAt

gt

0

eAxg(x)dx



d

dtpns1mspns

mf pns1


this implies that

pns1(t)emf t

gt

0

emf xmspns(x)dx


pns(x)

1

nsemsx(msx)ns

so we have

pns1(t)(ms)

ns1

nsemf t

gt

0

e(mf ms)xxns dx


gt

0


ab1


gt

0


(ms mf )ns1

We infer that

pns1(t)(ms)


nsetmf (ms mf )

ns1

emf t

ms

ms mf

ns1

1G(1 ns t(ms mf ))

ns

13


G(a z)za1ez(1O(z1))



144

mf pns1(t)mf emf t

ms

msmf

ns1

1(ms mf )

nset(msmf )tns

ns

13

emstmf

ms

ms mf

mns

s

tns

ns

emst

ms

ms=mf 1

(mst)

ns

ns

0 ms

1

nsemst(mst)

ns mspns(t)


that

pns1(t)

ms

ms mf

ns1

etmf g(ns

1 t(msmf ))=ns



145




Acknowledgements


References





of Cancer VIII 112




Society 15 42327












60714





























2004)














Science 96 1512126






142

Appendix A


Proof of Lemma 1


xjm

for all jm1

d

dtpmcm1

Xm1

j1

xjm1(mj)emj t

cm1

Xm1

j1

xjm1(mm1mj)emj t

cm1

Xm1

j1

mm1xjm1emj t

mm1mcm

Xm1

j1

xjmemj tmm1pm

mm1mpm1mm1pm





g

0

pi(t)dt1=mi1





g

0

pi(t)dt1=mi1


g

0

mkpk1(t)dt1


pj (t)m

1

(j 1)emt(mt)j1

emt(mt)j

j

13

m(pj1pj)



xjnd1nY

p1n

pj

1

p jd1n

Yj1

i1

1

(i)

Yj1

i1

1

i

d1n(1)j1

(j 1)(n j)

cnYn1

j1

mj Yn2

i0

(m id)dn1

m

d

n1

dn1

Gm

d n 1

G(m=d)




Xn

j1

(1) j1

(j 1)(n j)e(j1)dt

etd(1 etd)n

(etd 1)G(n)


143

hence

pn1(t)cn

Xn

j1

xjnemj t

Gm

d n 1

dn1G(m=d)

Xn

j1

(1)j1e(m(j1)d)t

(j 1)(n j)

Gm

d n 1

G(m=d)

emt etd(1 etd)n

(etd 1)G(n)

Gm

d n 1

Gm

d

G(n)

emt(1etd)n1



f (t)Af (t)g(t) f (0)0


f (t)eAt

gt

0

eAxg(x)dx



d

dtpns1mspns

mf pns1


this implies that

pns1(t)emf t

gt

0

emf xmspns(x)dx


pns(x)

1

nsemsx(msx)ns

so we have

pns1(t)(ms)

ns1

nsemf t

gt

0

e(mf ms)xxns dx


gt

0


ab1


gt

0


(ms mf )ns1

We infer that

pns1(t)(ms)


nsetmf (ms mf )

ns1

emf t

ms

ms mf

ns1

1G(1 ns t(ms mf ))

ns

13


G(a z)za1ez(1O(z1))



144

mf pns1(t)mf emf t

ms

msmf

ns1

1(ms mf )

nset(msmf )tns

ns

13

emstmf

ms

ms mf

mns

s

tns

ns

emst

ms

ms=mf 1

(mst)

ns

ns

0 ms

1

nsemst(mst)

ns mspns(t)


that

pns1(t)

ms

ms mf

ns1

etmf g(ns

1 t(msmf ))=ns



145

Appendix A


Proof of Lemma 1


xjm

for all jm1

d

dtpmcm1

Xm1

j1

xjm1(mj)emj t

cm1

Xm1

j1

xjm1(mm1mj)emj t

cm1

Xm1

j1

mm1xjm1emj t

mm1mcm

Xm1

j1

xjmemj tmm1pm

mm1mpm1mm1pm





g

0

pi(t)dt1=mi1





g

0

pi(t)dt1=mi1


g

0

mkpk1(t)dt1


pj (t)m

1

(j 1)emt(mt)j1

emt(mt)j

j

13

m(pj1pj)



xjnd1nY

p1n

pj

1

p jd1n

Yj1

i1

1

(i)

Yj1

i1

1

i

d1n(1)j1

(j 1)(n j)

cnYn1

j1

mj Yn2

i0

(m id)dn1

m

d

n1

dn1

Gm

d n 1

G(m=d)




Xn

j1

(1) j1

(j 1)(n j)e(j1)dt

etd(1 etd)n

(etd 1)G(n)


143

hence

pn1(t)cn

Xn

j1

xjnemj t

Gm

d n 1

dn1G(m=d)

Xn

j1

(1)j1e(m(j1)d)t

(j 1)(n j)

Gm

d n 1

G(m=d)

emt etd(1 etd)n

(etd 1)G(n)

Gm

d n 1

Gm

d

G(n)

emt(1etd)n1



f (t)Af (t)g(t) f (0)0


f (t)eAt

gt

0

eAxg(x)dx



d

dtpns1mspns

mf pns1


this implies that

pns1(t)emf t

gt

0

emf xmspns(x)dx


pns(x)

1

nsemsx(msx)ns

so we have

pns1(t)(ms)

ns1

nsemf t

gt

0

e(mf ms)xxns dx


gt

0


ab1


gt

0


(ms mf )ns1

We infer that

pns1(t)(ms)


nsetmf (ms mf )

ns1

emf t

ms

ms mf

ns1

1G(1 ns t(ms mf ))

ns

13


G(a z)za1ez(1O(z1))



144

mf pns1(t)mf emf t

ms

msmf

ns1

1(ms mf )

nset(msmf )tns

ns

13

emstmf

ms

ms mf

mns

s

tns

ns

emst

ms

ms=mf 1

(mst)

ns

ns

0 ms

1

nsemst(mst)

ns mspns(t)


that

pns1(t)

ms

ms mf

ns1

etmf g(ns

1 t(msmf ))=ns



145

hence

pn1(t)cn

Xn

j1

xjnemj t

Gm

d n 1

dn1G(m=d)

Xn

j1

(1)j1e(m(j1)d)t

(j 1)(n j)

Gm

d n 1

G(m=d)

emt etd(1 etd)n

(etd 1)G(n)

Gm

d n 1

Gm

d

G(n)

emt(1etd)n1



f (t)Af (t)g(t) f (0)0


f (t)eAt

gt

0

eAxg(x)dx



d

dtpns1mspns

mf pns1


this implies that

pns1(t)emf t

gt

0

emf xmspns(x)dx


pns(x)

1

nsemsx(msx)ns

so we have

pns1(t)(ms)

ns1

nsemf t

gt

0

e(mf ms)xxns dx


gt

0


ab1


gt

0


(ms mf )ns1

We infer that

pns1(t)(ms)


nsetmf (ms mf )

ns1

emf t

ms

ms mf

ns1

1G(1 ns t(ms mf ))

ns

13


G(a z)za1ez(1O(z1))



144

mf pns1(t)mf emf t

ms

msmf

ns1

1(ms mf )

nset(msmf )tns

ns

13

emstmf

ms

ms mf

mns

s

tns

ns

emst

ms

ms=mf 1

(mst)

ns

ns

0 ms

1

nsemst(mst)

ns mspns(t)


that

pns1(t)

ms

ms mf

ns1

etmf g(ns

1 t(msmf ))=ns



145

mf pns1(t)mf emf t

ms

msmf

ns1

1(ms mf )

nset(msmf )tns

ns

13

emstmf

ms

ms mf

mns

s

tns

ns

emst

ms

ms=mf 1

(mst)

ns

ns

0 ms

1

nsemst(mst)

ns mspns(t)


that

pns1(t)

ms

ms mf

ns1

etmf g(ns

1 t(msmf ))=ns



145

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