Cancer Res 1988 Norton 7067 71

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CANCER RESEARCH 48 7067 -7071 December 15 1988]

A Gompertzian M odel of Human Breast Cancer Growth1

Larr y No rton2

D ep ar tm en t o fN eo pla st ic D is ea se s, M ou nt S in ai M ed ic al C en te r, N ew Y or k, N ew Y or k 1 00 29

ABSTRACT

T he p attern of g ro wth o f h um an b rea st c an cer is im po rtan t th eo reti

c ally an d c lin ic ally . S pee r et a l. (C an ce r R es., 4 4: 4 12 4-41 .1 0, 1 98 4)

h av e re ce ntly p ro po se d tha t all in div idu al tu mors in itially g ro w w ith

i de nt ic al Gompe rt zi an p ar ame te rs , b ut s ub se qu en tl y d ev el op k in et ic h et

e ro gene it y b y a r an dom t ime -d ep endent p ro ce ss . Th is c on cept h as e li ci te d

i nt er es t b ec au se i t f it s c li ni ca l d at a f or t he s ur vi va l o f u nt re at ed p at ie nt s,

fo r th e p rog re ssion o f sh ad ow s o n serial p aired m ammograms, a nd fo r

t ime -t o- re la ps e f ol low in g ma st ec tomy. Th e s uc ce ss o f t he se c ur ve -f it s i s

c ompe lli ng , a nd t he mod el h as b ee n a pp lie d to c lin ic al tr ia ls . H owev er ,

the a ssu mp tio n o f u nifo rm n asce nt g ro wth is n ot su pp orte d b y th eo ry o r

d ata , a nd in di vi du al c an ce rs h av e n ot b ee n s hown to f ol low th e c ompl ex

g rowth c ur ve s p re di ct ed b y t he Spe er mo de l. A s a n a lt er na tiv e, i f k in eti c

h et er og en eit y is u nd er st oo d t o b e a n i ntr in sic p ro pe rt y o f n eo pl as ia , t he

s am e t hr ee h is to ri ca l d ata s ets a re f it w el l b y a n u na do rn ed Gompe rt zi an

mod el w hic h is p ar simo ni ou s a nd h as m an y o th er i ntu iti ve a nd emp ir ic al

a dv an ta ge s. T he two mo de ls d iff er s ig ni fic an tly i n s uc h c li ni ca l p ro je c

tio ns a s th e e stim ated d uratio n o f s ilen t g ro wth p rio r to d ia gn os is an d

t he a nt ic ip at ed opt ima l c hemothe ra py s ch edul e p os ts ur ge ry .

INTRODUCTION

M any models of tumor growth have been proposed to fit

clinical data and to offer therapeutic guidance (1, 2). Each

model is to some extent dependent on the assumption of a

certain pattern of grow th of the unperturbed tum or. T he land

m ark w ork of Skipper and colleagues, for exam ple, w as based

on exponential kinetics (3). In exponential grow th the cell

number TVis a function of the starting size N(0), the time of

grow th f. and a constant b. by the form ula:

N t

Hence, the tim e fd required for N(t\) to double, i.e., N(t\ +

tÃ¡ /N tÂ¡ 2 , i s a lways cons tan t a t l oge 2 /Â¿Â» .he S kip per m od el,

d ev elo ped p rincipa lly fo r ex perim en tal m urin e leu kem ia, co n

tin ue s to y ield v aluab le in sig hts in to th e relatio nsh ip b etw een

tum or size, g rowth ch aracteristic s, an d th erap eu tic resp onse

(4 ). M ore recen t m ode ls h av e also d ep en ded u po n th e ex po nen

tial curve. A n exam ple is the reconsideration by G oldie and

G oldm an (5) of the D elbruck-L uria m utagenesis m odel (6) as

extended by L aw from bacteria to cancer (7).

T he exponential pattern of grow th can be useful w hen it fits

actual data, but it is now know n not to be universally appropri

ate. In particular, in m any types of grow th â€”¿orm al and neo-

plast ic â€”¿,, i s not constant, but increases as the population size

in cre ase s. T o m an y o f th ese c as es L aird (8 ) su cc es sfu lly a pp lie d

Gompert z' e qu ati on , o rig in all y d ev eloped fo r a ctu ari al a na ly sis ,

but later proposed as a growth curve (9). In Gompertzian

growth /V(') is a function of A'(0), t, and h. but also a lim iting

s ize jV(Â°Â°),y th e e quatio n:

Rec ei ve d 1 2/2 9/ 87 ; r ev is ed 8 /5 /8 8; a cc ep te d 8 /1 1 /8 8.

T he c os ts o f p ub lic atio n o f th is a rtic le w er e d ef ra ye d in p ar t b y th e p ayme nt

o f p ag e c ha rg es. T his a rtic le m us t th er efo re b e h ere by m ark ed a dv er tise me nt in

a cc or da nc e w ith 18 U .S .C . S ec tio n 1 73 4 s ole ly t o i nd ic ate t his f ac t.

1 Supported in part by the C he motherapy F oundation, N ew Y ork, and the T .

J. M arte li M em oria l F ou nd atio n, N ew Y ork .

1 To whom requests for reprints should be addressed, at Departm ent of

N eo pla stic D ise ase s, M ou nt S in ai M ed ic al C en te r, O ne G usta ve L . L ev y P la ce ,

P . O . B ox 1 129 , N ew Y ork, N Y 10029.

N t =

.[l - exp -A f ]|

wher e k = l og ,[N (Â«>)/N( 0)] .

T his equation fits experim ental (10) and clinical data (11),

an d un co vers g rowth-reg ulato ry m ech an ism s in an im al (1 2, 13 )

and human (14) cancers. In addition, when used to relate a

tumo r's size to its rate o f reg ressio n in respo nse to th erap y (1 5),

G ompe rtz 1 e qu atio n h as a id ed in th e d esig n o f su cc es sfu l c lin ic al

tria ls (1 6, 1 7).

R ecently, how ever, the validity of the assum ption of G om

p ertzian g rowth fo r c lin ical breast canc er h as b een ch allen ged

b y th e p ro vo cativ e n ew m ode l o f S peer et al. (1 8). T hese au th ors

p rop ose a sto cha stic pro cess in w hich g rowth is fun dam en tally

G om pertzian, bu t w ith ran dom , sp on tan eo us alteratio ns in the

parameters such that h decreases and A'(Â») increases in a

fu nctio nally related m an ner. T his results in g row th "sp urts."

T hey p ostu late th at b oth b an d N (< *> )re co nstant for all tumo rs

at in cep tio n, an d th e p rob ab ility o f ran dom c han ge is in de pen d

ent o f N (t), i.e., a large tumo r has th e sam e ch an ce o f u nderg oin g

an alteration as a sm all tum or. Tum or heterogeneity is a con

sequence of the randomness of the process. This model was

constructed so as to reconcile the G om pertzian param eters

d escrib ed fo r m ultip le m yeloma an d in v itro b re ast can cer cells

(19) w ith "preconceived notions of the tim ing of the natural

history" of clinical breast cancer. It predicts an average of 8

years of grow th from one cell to clinical recognition at 1 x IO 9

to 5 x 109 cells. O f greatest interest, the m odel fits tw o-point

vo lume data fo r early (m ammo grap hically discern ib le) tumo rs

(20), survival data for untreated cancers (21), and rem ission

d ura tio n d ata p ost-m aste ctomy (2 2). It su gg es ts th at th e s ho rte r

tim e-to-relapse after m astectom y for those patients w ho had

more axillary nodal involvement is due to a positive linear

relationship betw een num ber of involved nodes and num ber of

m eta sta tic s ite s. T his v iew o f th eirs is in c on tra st to a p re vio usly

hy po th esiz ed an d w ide ly acc epted relatio nsh ip , th at patien ts

w ith m ore involved nodes have a greater total-body tum or cell

burden (4). A dditionally, the Speer m odel anticipates m erit in

long-term m aintenance chem otherapy in the postsurgical ad

juvant setting, as opposed to current short-term , intensive ap

proaches.

T he Speer m odel is theoretically intriguing. T he concept of

ran dom sp on tan eo us ch an ge in g rowth rate recalls th e p ow erful

th eo ry of tumo r pro gre ssion (23 , 2 4). T he in verse relatio nsh ip

b etw een b an d N (< *> )h as in de ed b ee n p re vio us ly d esc rib ed (1 2).

N ev erth eless, cu rv es co nsisten t w ith th e S peer m od el h av e n ot

b een fit in in div id ual cases, a s h as b een d on e inn um erab le tim es

for the G om pertzian m odel. A dditionally, the assum ption of a

uniform b and N (Â °Â °)or all prim ary breast cancers at onset of

grow th is not consistent w ith the know n biological heteroge

n eity o f b rea st can cer on th e cellu lar o r ev en m olecu lar lev el (2 ,

2 5, 2 6). In de ed, th e co nstru ctio n o f su ch comp licated cu rve s as

found in the Speer m odel is m ade necessary only if a uniform

initial b is hypothesized. If variability in b is allowed, the

u nmod ifie d G ompe rtz ia n mod el is s uffic ie nt to fit c lin ic al d ata .

This is shown below by the analysis of the sam e three data sets

u se d b y Spe er.

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G OM PER TZIA N B REA ST CA N CER M OD EL

M TER I LS ND M ETHODS

The first data set used by S peer w as provided by B loom et al. w ho

c ol le cte d data f or 2 50 w o me n w i th u ntre ate d b re as t c an ce r f ol lo w ed at

the M iddl es ex H os pi tal, Lo ndo n, duri ng the pe rio d 1 80 5 to 1 93 3 ( 21 ).

A l l pati en ts h ad c on fi rm ati on o f the h is to lo gi cal di ag no si s at n ec ro ps y.

58% of the w omen w ere over the age of 50. The initial symptom for

83% w as a breast lump, and for 71% there w as a delay betw een first

symptom and presentation to hospi tal o f more than 1 year. The duration

o f life fro m onse t o f sy mptom s to de ath is g raphed as square po ints in

Fig. \A : median surv iv al is 2 .7 y ears , and few er than 1 % o f patie nts

surv iv ed fo r mo re than 1 5 y ears. The se re sults are similar to earlier

re po rts ( 27 , 2 8) .

These data are analyzed here as fol lows: , \ ( /) i s an indiv idual patient's

tum or siz e at a time t measured from the o nse t o f sy mptom s at time 0 .

T he tum or s iz e at o ns et o f s ym pto ms i s N ( Q) and / V( Â° Â° )s held constant.

Parameter b is a llowed to vary , which produces a family o f Gompertz ian

c urv es w hi ch i nte rs ec t o nl y at A '( O) . Fo r the g ro up o f 2 50 p ati en ts P i_ (t)

is the pro po rti on o f patie nts w ho hav e die d by o r a t ti me t be caus e the ir

tum or s iz es had re ac he d a l ethal tum or s iz e N L a t s om e ti me l es s than

o r e qu al to t. B y re arran ge me nt o f the G om pe rtz e qu ati on

w he re P \.( ti ) i s th e f rac ti on o f th e 2 50 c anc ers w hi ch h av e G om pe rtz ian

parameters l>less than or equal to />,.This process defines the probabil ity

d is tri bu ti on o f b . U s in g P L( t) f ro m B l oo m an d re as on abl e i ni ti al v al ue s

o f N (0 ), N L, a nd ;V (< Â» ),h e d is tri bu ti on w as f oun d to b e ap pro xi mate ly

l og -n orm al . B y i te rati on , v al ue s o f A ^O ), N Â ¡ _,M 00 ), and the m ean and

s tand ard d ev iati on o f b w e re f ou nd s o as to m ini mi ze th e l eas t- sq uare s

f it o f l og e( A )to the l im ( /> ,)v al ue s c al cu late d ab ov e. T he f it to the ac tu al

d ata o f th e m od el s o g en erate d w as c onf irm ed b y s im ul ati on o f i nd iv id

ual grow th curves. The model is as follow s: A value of Â ¿> Â¡s c ho se n

ran do ml y f ro m th e d eri ve d l og -n orm al di stri bu ti on an d I, i s c al cu late d

s uc h that / V( fi) = N L. T he array o f v al ue s o f f Â¡s us ed to e stim ate P i( t)

b y s tan dard ac tu ari al m eth od s ( 29 ), an d i s g rap hi cal ly c om pare d w i th

P L t f ro m th e B lo om d ata.

The s ec ond data s et us ed by S pe er i s f ro m H eus er e t al. w ho pre se nt

data f or 1 09 pri mary bre as t c anc ers ( in 1 08 w om en) diag no se d am ong

1 0,12 0 sc ree ning m am mo grams (20 ). 4 5 lesio ns w ere dis co vere d o n

i ni tial e xam inati on. O f the 6 4 re maini ng c as es w ith di ag no stic m am

m og ram s, 3 2 h ad h ad p re vi ou s m am mo gram s th at re tro sp ec ti ve ly d em

o ns trate d m eas urable tum ors . The autho rs e stim ate that the se 3 2 e x

am pl es re pre se nt the s lo we st- gro wing 2 3% o f c anc ers in the ir s erie s,

w ith the re mai ning 7 7% g ro wi ng to o rapi dly to re mai n undi ag no se d

c li ni cal ly be tw e en tw o w i de ly s pac ed m am mo gram s. N in e o f th e 3 2 h ad

s uc h s im ilar m eas ure me nts in the tw o c ons ec uti ve e xam inati ons that

g ro wth co uld no t be do cum ented. Fo r the 2 3 canc ers that did g ro w to

a m eas urabl e de gre e the autho rs pro vide m utual ly pe rpe ndic ular di

am ete r v alue s ( L f or l eng th and W f or w idth, s uc h that L~ >W } and the

d ate o f e ac h o f th e f irs t an d s ec ond m eas ure me nts .

The Heuser data are analyzed here as follow s: The time betw een

date s ti is c al cul ate d in m onths ; e ac h tum or v olum e is e sti mate d as the

v olum e o f re vo lutio n o f an e llips e by (ir/6)-/,Â ¡-W h N (Q) is set at the

f irs t v ol um e; N (tu is s et at the s ec ond v ol um e; us ing the v alue o f jV (Â »)

de te rm ine d by the G om pe rtz ian f it to the B lo om data, / >,is c al cul ate d

by the Gompertz equation.

The third data set cited by S peer concerns the grow th of micro-

m etas tati c f oc i o f b re as t c an ce r f ol lo w in g re mo val o f a pri mary tu mo r

b y H al ste d rad ic al m as te cto my . In 1 97 5 F is he r e t al . ( 22 ) p ub li sh ed th e

1 0- ye ar re sul ts o f a N ati onal S urg ic al A dju vant B re as t P ro je ct tri al o f

p os to pe rati ve c he mo th erap y c om pare d w i th p lac eb o f or pati ents w i th

S tag es I o r II di se as e ( re gi onal di se as e no t adv anc ed and no e vi de nc e

o f m Ã© tastase se xce pt to axillary ly mph no des in S tage II). Fo r 3 70

plac ebo -tre ate d w om en at 5 y ears and 3 33 at 1 0 y ears the pe rc entag e

o f free local or reg ional recurrence or mÃ©tastase sw e re e sti mate d by th e

m eth od o f C utl er an d E de re r ( 30 ). T he se are g rap he d i n F ig . 3 f or thre e

s ubg ro uping s by ax il lary ly mph no de po si ti vity at the ti me o f m as te c

to my : al l n ode s n eg ati ve , o ne to thre e p os iti ve , o r f ou r o r m ore p os iti ve .

The model of this paper is applied to the Fisher data by the same

m etho d o f s im ul atio n us ed to c onf irm the f it o f the m ode l to the B lo om

data, w ith thre e e xc epti ons . The l im iti ng e ve nt, re curre nc e, i s s et at a

tum or s iz e .Y K< V , ; a v ari abl e pe rc entag e o f patie nts are as sum ed to

be c ure d [ i.e ., N (0 ) = 0 ] by m as te cto my ; f or the i ndiv iduals w ho are no t

c ure d A (0 ) i s all ow ed to v ary s o as to g iv e the be st l eas t- square s c urv e

fit to the pro bability o f re currenc e I'M ) as a func tio n o f time t after

mastectomy.

RESULTS

By the above methods the Bloom data are found to be

c ons iste nt w ith the G om pe rtz e quatio n w ith N (0 ) â€ ”¿.8 x IO9

cells (almost 5 cc o f densely packed tumor cells), jV (o o)= 3.1 x

IO1 2cells (3 .1 liters), a le thal tumor cell number o f N L = IO'2

cells (one liter), and, expressing t in units of months, a log-

no rm al dis tri buti on o f param ete r Â ¿Â »ith m ean Io n, (/ >) o f â€ ”¿2.9

and a standard deviation of log,(e) of 0.71. The line in Fig. \A

is the fit of this model to the Bloom data. Precise fit is seen at

all data po ints ex cept the 8 6%-alive po int at 1 year, w here the

m ode l pre di cts a 9 1. 6% po int. The S pe er m ode l unde rpre dic ts

thi s po int to a c om parable de gre e. H enc e, the s im ple G om pe rtz

e quatio n fits data fo r unpe rturbe d bre as t c anc er g ro wth fro m

o nse t o f bre as t m ass to le thal tum or s iz e.

Fig. IB presents sample growth curves simulated by the

Go mpe rtz ian m ode l. Cho se n fo r illus trativ e purpo se s are re p

resentatives of the 10th, 30th, 50th, 70th, 90th, and 99th

pe rc entil es , o rde re d by rapidity o f g ro w th. The se c urv es m ay be

compared direc tly w ith the po ints o f 9 0, 70 , 5 0, 3 0, 1 0, and 1 %

s urv iv al in Fig . \A .

The pro bability de ns ity c urv e fo r lo ge (A ) f ro m the G ompe r

tzian fit to the Bloom data is show n in Fig. 2. Graphed for

compariso n are the 2 3 v alues fo r b\ calculated by the method

de sc ribe d abo ve using the H eus er data. The se v alue s fit w ithin

the lo we r 1 5% o f the dis tributio n o f A , w hic h c orre spo nds w ell

w ith H eus er's e stim ate o f 2 3%.

Fig. 3 presents PR(t) postmastectomy. The points are the

ac tual data and the line s are the pre dic tio ns o f the G om pe rtz ian

model using the same probability density of \og,(b) and the

same ./V (oo )as de rived fo r the B lo om data. The tumo r size at

diag no sis o f re laps e is s et at jV R= 1 0 c el ls . F or no de -ne gativ e

patie nts the be st-fit param ete rs are JV (0 ) = IO 2, 7 4.7 % c ure d;

for the one to three positive node group jV(0) = IO4, 33.5%

cured; fo r patients w ith fo ur o r mo re inv olv ed nodes, jV (0) =

1 .7 x IO7 , 1 2 .5 % c ure d. The m ode l no t o nly fits the data po ints

w ell, but does so w ith sig mo id c urv es that simulate the shape

of ac tual c li ni cal obs ervati ons ( 31).

DIS USSION

The abil ity o f the G om pe rtz ian m ode l w ith v ariabl e param e

te r h to s im ulate c linic al data de mo ns trate s that the c om ple xi

ti es o f the Speer mode l, w hi ch are unconfi rmed experi mental ly ,

are unne cessary theo retically as w ell. Ev en w ith fix ed N (Q),

jV (o o), and fixe d tumo r siz e at ev ent (recurrence, death), the

Gompertzian mo de l prov ides a remarkably clo se fit to o bser

v atio ns . The se fix ed v alue s are sim plify ing ass um ptio ns that

may account for the slight deviations of the model from the

data. W hile it has been previously o bserve d in ex perime ntal

animals that b varies more widely than does N (Â °Â°)etween

indiv idual tum ors o f g iv en his to lo gic al ty pe , 7 V (< Â »)o es , ne ve r

the le ss , v ary . Inde ed the re m ay be a no ntriv ial func tio nal re la

tio ns hip be tw ee n b and jV (Â °Â °)12 ). In additio n it is clear from

clinical ex pe rienc e that \(0 ) at first sy mptom is v ariable, and

it is unre as onabl e to as sum e that 7 V (0 )afte r m as te cto my s ho uld

be constant in all individuals. A lso, the tumor size N (t) at

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G OM PER TZ IA N B REA ST C AN CER M OD EL

inv olv ed no de s in the g ro up o f patie nts w ith o ne -to -thre e po si

tiv e no de s, and fiv e inv olv ed no de s in the fo ur-or-mo re no de

c ate go ry . This as sum ptio n m ay be g ratui to us . In additi on, c lin

ic al e xpe rie nc e w ith S tag e IV dis eas e di sc lo se s no m ajo r dif fe r

ence in the spectrum of metastatic sites involved as a conse

quence of the prio r nodal status (31 , 3 4).

The Speer model advises postsurgical adjuvant chemotherapy

of prolonged duration so that chemotherapy exposure may

c oinc ide w ith the hy po the si ze d g ro w th s purts . This re co mm en

datio n has influe nc ed the inte rpre tatio n and de sig n o f c linic al

trials (3 5). Y et prude nc e in this re gard is indic ate d by s ev eral

c ons ide rati ons . Ex po ne nti al and G om pe rtz ian tum ors s ho uld

be tre ate d e arly and inte ns iv ely f or be st re sults ( 15 ). Fo r bre as t

c anc er the pars im onio us Go mpe rtz ian m ode l is he re s ho wn to

s im ul ate c linic al data as w ell o r be tte r than alte rnati ve m ode ls ,

w hic h m ust affo rd it at le as t e qual c re de nc e as re gards c linic al

e xtrapo latio ns . S ec ondly , pe nding the c om ple tio n o f o ng oing

c li ni cal s tudi es ( 36 ), currentl y avai labl e data f ai l to demons trate

s igni fi cant advantage f rom pro longed adjuvant chemotherapy

w ith a sing le drug co mbinatio n, althoug h so me finite limit to

re gim en bre vity w itho ut lo ss o f e ffic ac y m ay w ell be de fine d by

f uture re se arc h (3 7, 3 8) . Las tly , if the g ro w th s purt po stulate d

by Speer w ere to occur, it would more likely be due to the

emergence of an aberrant clone than a change in the grow th

kinetics o f the entire population. W hile it is conceivable that

this abe rrant c lo ne m ig ht be m ore c he mo the rapy s ens itiv e than

the parent cells by v irtue o f a higher g ro wth fraction, it mig ht

als o be m ore ins ens itiv e as a c ons eque nc e o f rando mly ac quire d

bio che mic al drug re sis tanc e ( 2) . H enc e, atte mpts to w ard e rad

ic atio n o f the pare nt po pulatio n prio r to the e me rg enc e o f this

po te ntially re sis tant c lo ne w ould s ee m to be m ore fruitful than

delays in therapy. Indeed, there is no reason to suppose that

the pare nt po pul atio n its elf w o uld be m ore s ens iti ve to the rapy

at a tim e afte r the e me rg enc e o f the m ore rapidly g ro wing c lo ne

than be fo re, so that e ven if the new clo ne w ere chemo therapy

respo nsiv e, the parent po pulatio n w ould be no easier to cure

later rather than so oner. Therefore, ev en if the S pe er mo del

w ere val id phenomenolog ical ly , i ntens ive chemotherapy appl ied

immediately against minimal disease w ould still be mo re rea

s onable than pro lo ng ed lo w-do se o r de lay ed tre atm ent. A rg u

ments i n f avor of i mmediate , i ntens ive i nduc ti on chemotherapy

f ollo w ed by c ro ss -o ve r, inte ns iv e c ons ol idatio n hav e be en pre

s ented e ls ew here ( 2) .

S ince the so le intentio n o f this pape r is to demo nstrate that

an unadorned Gompertzian model is sufficient to simulate

c linic al data, a hig h de gre e o f c onfide nc e in the pre cisio n o f the

e stim ate d param ete rs is ne ithe r m eant no r jus tifie d. S uc h re

f ine me nts as the inc lus io n o f v ari abl e jV (o o) ,jV L, and N R w o uld

s ee m to be re quire d to pro duc e a m ore intuitiv ely satisfac to ry

mo del. The impact o f such modific atio ns o n the pro bability

density of b remains to be determined by further research.

N ev ertheless, it is o f spe culativ e interest to note that settingjV (0) = 1 and N (t{ ) = 4 .8 x IO9 , with N (o o) and the probability

density o f \o g,(b) as abo ve, that the 95 % co nfide nce limits of t,

range fro m about 6 o r 7 mo nths to about 9 y ears, w ith a median

o f about 2 .2 5 y ears. This is a sho rter duratio n o f preclinical

g ro w th than is us ually as sum ed. Epi de mi ol og ic al data re lating

the c arc ino ge nic influe nc e o f sho rt-e xpo sure radiatio n to the

incidenc e o f bre ast cancer w ould sugg est a lo ng er lag perio d

(39 ). This div erg ence is co nsistent w ith the c oncept of a time-

co nsuming prec ance ro us state induc ed by the radiation, fo l

lo we d by a s ec ond, pe rhaps rando m, e ve nt as so ciate d w ith the

initiati on o f the G om pe rtz ian g ro w th o f the ac tual m alig nanc y.

Prim ary c arc ino ge ns o the r than radiatio n c ould also o pe rate

ac cording to this paradig m (2 3). S uch a concept could relate

dire ctly to atte mpts to alte r o r re ve rs e the pre canc ero us s tate

with pharmacological or micronutrient interventions .

The Speer model is innovative and thought provoking. A t

pre se nt, ho w ev er, the re is ins uf fic ie nt e vide nc e to abando n the

G om pe rtz ian m ode l fo r hum an bre as t c anc er. Furthe r re fine

ments o f the mo del presented here may w ell yield better e sti

mates of the volume of the tumor residual after surgery or of

the average durati on of grow th precedi ng c li ni cal presentati on .

N e ve rthe le ss , no c urre nt m ode l fi ts hum an o r anim al data be tte r

than unado rne d G om pe rtz ian g ro wth. The the rape utic im pli

cations of other models in general, and the Speer model in

parti cul ar, s houl d be i nterpre ted and appl ied w ith cauti on .

KNOWLEDGMENTS

I thank Dr. James F. Ho lland for his cri tical rev iew of this manuscript

and Rita B. Mora le s for her secre tarial as s is tance .

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1988;48:7067-7071.Cancer Res

Larry Norton

A Gompertzian Model of Human Breast Cancer Growth

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