(CANCER RESEARCH 48, 7067-7071, December 15, 1988]

A Gompertzian Model of Human Breast Cancer Growth1

Larry Norton2

Department ofNeoplastic Diseases, Mount Sinai Medical Center, New York, New York 10029

ABSTRACT

The pattern of growth of human breast cancer is important theoretically and clinically. Speer et al. (Cancer Res., 44: 4124-41.10, 1984)have recently proposed that all individual tumors initially grow withidentical Gompertzian parameters, but subsequently develop kinetic heterogeneity by a random time-dependent process. This concept has elicitedinterest because it fits clinical data for the survival of untreated patients,for the progression of shadows on serial paired mammograms, and fortime-to-relapse following mastectomy. The success of these curve-fits iscompelling, and the model has been applied to clinical trials. However,the assumption of uniform nascent growth is not supported by theory ordata, and individual cancers have not been shown to follow the complexgrowth curves predicted by the Speer model. As an alternative, if kineticheterogeneity is understood to be an intrinsic property of neoplasia, thesame three historical data sets are fit well by an unadorned Gompertzianmodel which is parsimonious and has many other intuitive and empiricaladvantages. The two models differ significantly in such clinical projections as the estimated duration of silent growth prior to diagnosis andthe anticipated optimal chemotherapy schedule postsurgery.

INTRODUCTION

Many models of tumor growth have been proposed to fitclinical data and to offer therapeutic guidance (1, 2). Eachmodel is to some extent dependent on the assumption of acertain pattern of growth of the unperturbed tumor. The landmark work of Skipper and colleagues, for example, was basedon exponential kinetics (3). In exponential growth the cellnumber TVis a function of the starting size N(0), the time ofgrowth f. and a constant b. by the formula:

N(t)

Hence, the time fd required for N(t\) to double, i.e., N(t\ +tá)/N(t¡)= 2, is always constant at loge(2)/¿».The Skipper model,developed principally for experimental murine leukemia, continues to yield valuable insights into the relationship betweentumor size, growth characteristics, and therapeutic response(4). More recent models have also depended upon the exponential curve. An example is the reconsideration by Goldie andGoldman (5) of the Delbruck-Luria mutagenesis model (6) asextended by Law from bacteria to cancer (7).

The exponential pattern of growth can be useful when it fitsactual data, but it is now known not to be universally appropriate. In particular, in many types of growth —¿�normal and neo-plastic —¿�t,, is not constant, but increases as the population sizeincreases. To many of these cases Laird (8) successfully appliedGompertz' equation, originally developed for actuarial analysis,

but later proposed as a growth curve (9). In Gompertziangrowth /V(') is a function of A'(0), t, and h. but also a limiting

size jV(°°),by the equation:

Received 12/29/87; revised 8/5/88; accepted 8/1 1/88.The costs of publication of this article were defrayed in part by the payment

of page charges. This article must therefore be hereby marked advertisement inaccordance with 18 U.S.C. Section 1734 solely to indicate this fact.

1Supported in part by the Chemotherapy Foundation, New York, and the T.

J. Marteli Memorial Foundation, New York.1To whom requests for reprints should be addressed, at Department of

Neoplastic Diseases, Mount Sinai Medical Center, One Gustave L. Levy Place,P. O. Box 1129, New York, NY 10029.

N(t) = .[l - exp(-Af)]|

where k = log,[N(«>)/N(0)].This equation fits experimental (10) and clinical data (11),

and uncovers growth-regulatory mechanisms in animal (12, 13)and human (14) cancers. In addition, when used to relate atumor's size to its rate of regression in response to therapy (15),Gompertz1 equation has aided in the design of successful clinical

trials (16, 17).Recently, however, the validity of the assumption of Gom

pertzian growth for clinical breast cancer has been challengedby the provocative new model of Speer et al. (18). These authorspropose a stochastic process in which growth is fundamentallyGompertzian, but with random, spontaneous alterations in theparameters such that h decreases and A'(»)increases in afunctionally related manner. This results in growth "spurts."

They postulate that both b and N(<*>)are constant for all tumorsat inception, and the probability of random change is independent of N(t), i.e., a large tumor has the same chance of undergoingan alteration as a small tumor. Tumor heterogeneity is a consequence of the randomness of the process. This model wasconstructed so as to reconcile the Gompertzian parametersdescribed for multiple myeloma and in vitro breast cancer cells(19) with "preconceived notions of the timing of the naturalhistory" of clinical breast cancer. It predicts an average of 8years of growth from one cell to clinical recognition at 1 x IO9to 5 x 109 cells. Of greatest interest, the model fits two-point

volume data for early (mammographically discernible) tumors(20), survival data for untreated cancers (21), and remissionduration data post-mastectomy (22). It suggests that the shortertime-to-relapse after mastectomy for those patients who had

more axillary nodal involvement is due to a positive linearrelationship between number of involved nodes and number ofmetastatic sites. This view of theirs is in contrast to a previouslyhypothesized and widely accepted relationship, that patientswith more involved nodes have a greater total-body tumor cell

burden (4). Additionally, the Speer model anticipates merit inlong-term maintenance chemotherapy in the postsurgical adjuvant setting, as opposed to current short-term, intensive ap

proaches.The Speer model is theoretically intriguing. The concept of

random spontaneous change in growth rate recalls the powerfultheory of tumor progression (23, 24). The inverse relationshipbetween b and N(<*>)has indeed been previously described (12).Nevertheless, curves consistent with the Speer model have notbeen fit in individual cases, as has been done innumerable timesfor the Gompertzian model. Additionally, the assumption of auniform b and N(°°)for all primary breast cancers at onset ofgrowth is not consistent with the known biological heterogeneity of breast cancer on the cellular or even molecular level (2,25, 26). Indeed, the construction of such complicated curves asfound in the Speer model is made necessary only if a uniforminitial b is hypothesized. If variability in b is allowed, theunmodified Gompertzian model is sufficient to fit clinical data.This is shown below by the analysis of the same three data setsused by Speer.

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GOMPERTZIAN BREAST CANCER MODEL

MATERIALS AND METHODS

The first data set used by Speer was provided by Bloom et al. whocollected data for 250 women with untreated breast cancer followed atthe Middlesex Hospital, London, during the period 1805 to 1933 (21).All patients had confirmation of the histological diagnosis at necropsy.58% of the women were over the age of 50. The initial symptom for83% was a breast lump, and for 71% there was a delay between firstsymptom and presentation to hospital of more than 1 year. The durationof life from onset of symptoms to death is graphed as square points inFig. \A: median survival is 2.7 years, and fewer than 1% of patientssurvived for more than 15 years. These results are similar to earlierreports (27, 28).

These data are analyzed here as follows: ,\ (/) is an individual patient's

tumor size at a time t measured from the onset of symptoms at time 0.The tumor size at onset of symptoms is N(Q) and /V(°°)is held constant.Parameter b is allowed to vary, which produces a family of Gompertziancurves which intersect only at A'(O).For the group of 250 patients Pi_(t)

is the proportion of patients who have died by or at time t because theirtumor sizes had reached a lethal tumor size NL at some time less thanor equal to t. By rearrangement of the Gompertz equation

where P\.(ti) is the fraction of the 250 cancers which have Gompertzianparameters l>less than or equal to />,.This process defines the probabilitydistribution of b. Using PL(t) from Bloom and reasonable initial valuesof N(0), NL, and ;V(<»),the distribution was found to be approximatelylog-normal. By iteration, values of A^O), N¡_,M00), and the mean andstandard deviation of b were found so as to minimize the least-squaresfit of loge(A)to the lim (/>,)values calculated above. The fit to the actualdata of the model so generated was confirmed by simulation of individual growth curves. The model is as follows: A value of ¿>¡is chosenrandomly from the derived log-normal distribution and I, is calculatedsuch that /V(fi)= NL. The array of values of f¡is used to estimate Pi(t)by standard actuarial methods (29), and is graphically compared withPL(t) from the Bloom data.

The second data set used by Speer is from Heuser et al. who presentdata for 109 primary breast cancers (in 108 women) diagnosed among10,120 screening mammograms (20). 45 lesions were discovered oninitial examination. Of the 64 remaining cases with diagnostic mammograms, 32 had had previous mammograms that retrospectively demonstrated measurable tumors. The authors estimate that these 32 examples represent the slowest-growing 23% of cancers in their series,with the remaining 77% growing too rapidly to remain undiagnosedclinically between two widely spaced mammograms. Nine of the 32 hadsuch similar measurements in the two consecutive examinations thatgrowth could not be documented. For the 23 cancers that did grow toa measurable degree the authors provide mutually perpendicular diameter values (L for length and W for width, such that L~>W} and the

date of each of the first and second measurements.The Heuser data are analyzed here as follows: The time between

dates ti is calculated in months; each tumor volume is estimated as thevolume of revolution of an ellipse by (ir/6)-/,¡-Wh N(Q) is set at thefirst volume; N(tu is set at the second volume; using the value of jV(»)determined by the Gompertzian fit to the Bloom data, />,is calculatedby the Gompertz equation.

The third data set cited by Speer concerns the growth of micro-metastatic foci of breast cancer following removal of a primary tumorby Halsted radical mastectomy. In 1975 Fisher et al. (22) published the10-year results of a National Surgical Adjuvant Breast Project trial ofpostoperative chemotherapy compared with placebo for patients withStages I or II disease (regional disease not advanced and no evidenceof métastasesexcept to axillary lymph nodes in Stage II). For 370placebo-treated women at 5 years and 333 at 10 years the percentageof free local or regional recurrence or métastaseswere estimated by themethod of Cutler and Ederer (30). These are graphed in Fig. 3 for threesubgroupings by axillary lymph node positivity at the time of mastectomy: all nodes negative, one to three positive, or four or more positive.

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

method of simulation used to confirm the fit of the model to the Bloomdata, with three exceptions. The limiting event, recurrence, is set at atumor size .YK< V,; a variable percentage of patients are assumed tobe cured [i.e., N(0) = 0] by mastectomy; for the individuals who are notcured A(0) is allowed to vary so as to give the best least-squares curvefit to the probability of recurrence I'M) as a function of time t after

mastectomy.

RESULTS

By the above methods the Bloom data are found to beconsistent with the Gompertz equation with N(0) —¿�4.8 x IO9cells (almost 5 cc of densely packed tumor cells), jV(oo)= 3.1 xIO12cells (3.1 liters), a lethal tumor cell number of NL = IO'2cells (one liter), and, expressing t in units of months, a log-normal distribution of parameter ¿»with mean Ion,(/>) of —¿�2.9

and a standard deviation of log,(e) of 0.71. The line in Fig. \Ais the fit of this model to the Bloom data. Precise fit is seen atall data points except the 86%-alive point at 1 year, where themodel predicts a 91.6% point. The Speer model underpredictsthis point to a comparable degree. Hence, the simple Gompertzequation fits data for unperturbed breast cancer growth fromonset of breast mass to lethal tumor size.

Fig. IB presents sample growth curves simulated by theGompertzian model. Chosen for illustrative purposes are representatives of the 10th, 30th, 50th, 70th, 90th, and 99thpercentiles, ordered by rapidity of growth. These curves may becompared directly with the points of 90, 70, 50, 30, 10, and 1%survival in Fig. \A.

The probability density curve for loge(A) from the Gompertzian fit to the Bloom data is shown in Fig. 2. Graphed forcomparison are the 23 values for b\ calculated by the methoddescribed above using the Heuser data. These values fit withinthe lower 15% of the distribution of A, which corresponds wellwith Heuser's estimate of 23%.

Fig. 3 presents PR(t) postmastectomy. The points are theactual data and the lines are the predictions of the Gompertzianmodel using the same probability density of \og,(b) and thesame ./V(oo)as derived for the Bloom data. The tumor size atdiagnosis of relapse is set at jVR= 10" cells. For node-negativepatients the best-fit parameters are JV(0) = IO2, 74.7% cured;for the one to three positive node group jV(0) = IO4, 33.5%

cured; for patients with four or more involved nodes, jV(0) =1.7 x IO7, 12.5% cured. The model not only fits the data points

well, but does so with sigmoid curves that simulate the shapeof actual clinical observations (31).

DISCUSSION

The ability of the Gompertzian model with variable parameter h to simulate clinical data demonstrates that the complexities of the Speer model, which are unconfirmed experimentally,are unnecessary theoretically as well. Even with fixed N(Q),jV(oo),and fixed tumor size at event (recurrence, death), theGompertzian model provides a remarkably close fit to observations. These fixed values are simplifying assumptions thatmay account for the slight deviations of the model from thedata. While it has been previously observed in experimentalanimals that b varies more widely than does N(°°)betweenindividual tumors of given histological type, 7V(<»)does, nevertheless, vary. Indeed there may be a nontrivial functional relationship between b and jV(°°)(12). In addition it is clear fromclinical experience that \(0) at first symptom is variable, andit is unreasonable to assume that 7V(0)after mastectomy shouldbe constant in all individuals. Also, the tumor size N(t) at

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I .O»

0.8

0.6

04

0.2

O

i—i—i—r

GOMPERTZIAN BREAST CANCER MODEL

0.4'—i—i—i—i—i—r

Bloom dato (1962)

Fit of model

!- 1- 10

IT"-?—q- ' n ' o—g14 168 10 12

l—i—i—i—i—i—i—i—i—i—i—i—i—i

024 6 8 10 12 14 16YEARS OF GROWTH FROM ONSET OF SYMPTOMS

Fig. 1. Survival of untreated women with breast cancer. A, squares representthe percentage of patients surviving per year after onset of symptoms for 250women with untreated breast cancer (21). The solid line is the Tit of the Gom-pertzian model detailed in the text. Excellent fit is evident. B, solid lines representgrowth curves (tumor cell number as a function of time of growth) for selectedindividual cancers simulated by the Gompertzian model which generated thecurve-fit in A. Ranking the simulated tumors in order of rapidity of growth, theillustrations represent the 10th, 30th. SOth, 70th. 90th. and 99th percentiles. Eachtumor is assumed to be lethal at IO'2 cells (1 liter). Since 90% of the tumors reach

lethal size after the tumor representing the 10th percentile does, 90% of patientswould be expected to survive past the time point at which the 10th percentiletumor reaches !()'•'cells, etc. Using this relationship the growth curves of B

should be directly compared with the survival estimates of the line in I.

relapse or death varies between individuals, and might, in fact,be partially dependent on the tumor growth rate. For example,the slight overprediction by the Gompertzian model of thepercentage of Bloom's patients surviving at least 1 year may be

because rapidly growing tumors are lethal at a smaller size thanindolent cancers. The failure to demonstrate growth on serialmammograms for nine tumors in the Heuser data set mayillustrate the imprecision of the estimation of A^O)from clinicaldata: the diameter measurements from the first mammogrammay have included benign or premalignant tissue, or edema,with only a component of actual neoplastic cells. Since breastcancers frequently arise from areas of carcinoma-/« situ orhyperproliferative fibrocystic disease, this possibility is tenable.Allowance for variability in ./V(O),./V(°°),and N(t) at relapse ordeath could permit the Gompertzian model to simulate clinicalobservations even more accurately than seen here. The inclusionof additional parameters, however, must be based on carefullyanalyzed, actual observations, so as to avoid the dangers ofoverdetermination.

The Gompertzian parameters used here to fit the Bloom data,and shown to be consistent with the Heuser and Fisher datasets, are appropriate for an in vivo model, and must not be

0.3

0.2

O.I

i—i—ii 11

o Heuser data (1979)—¿�Fit of model to

Bloom data

Ql oL<

.001 .01 .1LOG,0(b)

Fig. 2. Probability density of Gompertzian parameter. The solid line representsthe probability density of Gompertzian parameter ft derived from the curve fit todata as shown in Fig. 1. Parameter A is seen to be log-normal with mean log*(ft)of-2.9 and standard deviation of log.(ft) of 0.71 (with time in the Gompertzianequation expressed in months). The small circles are 23 data points for parameterft calculated from two-point mammographie data (20), using the model derivedto fit the survival data of Fig. I. The circles are expected to belong approximatelyin the lower 23% of the cumulative probability distribution and indeed do fall inthe lower 15%.

*,*,• Fisher data (1968!—¿�Fit of model

2468YEARS AFTER MASTECTOMY

Fig. 3. Disease-free survival for women following mastectomy. The data pointsrepresent actuarial disease-free survival following radical mastectomy for womenwith no positive axillary lymph nodes (A), one to three positive nodes (•),andfour or more positive nodes (•)(22). The solid lines are the fits of the Gompertzianmodel detailed in the text. The model fits the data with realistically shaped curves.

compared too strictly with such in vitro estimates as providedby Salmon (19). Intact host mechanisms in vivo may addsignificantly to the cell-loss factor, which serves to slow clinicalgrowth (32). Indeed, the relationship between in vitro measurements and in vivo growth characteristics of the same neoplastictissue may be a novel approach to the estimation of cell lossand perhaps the efficacy of host defense.

The Gompertzian model simulates time-to-recurrence curvesafter mastectomy by assuming that A'(O) and cure rate is a

function of the number of involved nodes at the time of surgery.The analysis presented here thereby agrees with Skipper thatthe residual N(Q) after mastectomy is greater when the numberof positive nodes is greater (33). The Speer model, in contrast,attaches greater importance to the hypothesis of a positivelinear relationship between the number of micrometastatic sitesand the degree of nodal positivity. While a positive relationshipbetween N(0) and the number of different metastatic sites isplausible, the Gompertzian model finds this hypothesis unnecessary to explain the data. It is of note that Speer's linear

relationship was fit by the assumption of an average of two7069

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GOMPERTZIAN BREAST CANCER MODEL

involved nodes in the group of patients with one-to-three positive nodes, and five involved nodes in the four-or-more nodecategory. This assumption may be gratuitous. In addition, clinical experience with Stage IV disease discloses no major difference in the spectrum of metastatic sites involved as a consequence of the prior nodal status (31, 34).

The Speer model advises postsurgical adjuvant chemotherapyof prolonged duration so that chemotherapy exposure maycoincide with the hypothesized growth spurts. This recommendation has influenced the interpretation and design of clinicaltrials (35). Yet prudence in this regard is indicated by severalconsiderations. Exponential and Gompertzian tumors shouldbe treated early and intensively for best results (15). For breastcancer the parsimonious Gompertzian model is here shown tosimulate clinical data as well or better than alternative models,which must afford it at least equal credence as regards clinicalextrapolations. Secondly, pending the completion of ongoingclinical studies (36), currently available data fail to demonstratesignificant advantage from prolonged adjuvant chemotherapywith a single drug combination, although some finite limit toregimen brevity without loss of efficacy may well be defined byfuture research (37, 38). Lastly, if the growth spurt postulatedby Speer were to occur, it would more likely be due to theemergence of an aberrant clone than a change in the growthkinetics of the entire population. While it is conceivable thatthis aberrant clone might be more chemotherapy sensitive thanthe parent cells by virtue of a higher growth fraction, it mightalso be more insensitive as a consequence of randomly acquiredbiochemical drug resistance (2). Hence, attempts toward eradication of the parent population prior to the emergence of thispotentially resistant clone would seem to be more fruitful thandelays in therapy. Indeed, there is no reason to suppose thatthe parent population itself would be more sensitive to therapyat a time after the emergence of the more rapidly growing clonethan before, so that even if the new clone were chemotherapyresponsive, the parent population would be no easier to curelater rather than sooner. Therefore, even if the Speer modelwere valid phenomenologically, intensive chemotherapy appliedimmediately against minimal disease would still be more reasonable than prolonged low-dose or delayed treatment. Arguments in favor of immediate, intensive induction chemotherapyfollowed by cross-over, intensive consolidation have been presented elsewhere (2).

Since the sole intention of this paper is to demonstrate thatan unadorned Gompertzian model is sufficient to simulateclinical data, a high degree of confidence in the precision of theestimated parameters is neither meant nor justified. Such refinements as the inclusion of variable jV(oo),jVL,and NR wouldseem to be required to produce a more intuitively satisfactorymodel. The impact of such modifications on the probabilitydensity of b remains to be determined by further research.Nevertheless, it is of speculative interest to note that settingjV(0) = 1 and N(t{) = 4.8 x IO9, with N(oo) and the probability

density of \og,(b) as above, that the 95% confidence limits of t,range from about 6 or 7 months to about 9 years, with a medianof about 2.25 years. This is a shorter duration of preclinicalgrowth than is usually assumed. Epidemiological data relatingthe carcinogenic influence of short-exposure radiation to theincidence of breast cancer would suggest a longer lag period(39). This divergence is consistent with the concept of a time-

consuming precancerous state induced by the radiation, followed by a second, perhaps random, event associated with theinitiation of the Gompertzian growth of the actual malignancy.Primary carcinogens other than radiation could also operate

according to this paradigm (23). Such a concept could relatedirectly to attempts to alter or reverse the precancerous statewith pharmacological or micronutrient interventions.

The Speer model is innovative and thought provoking. Atpresent, however, there is insufficient evidence to abandon theGompertzian model for human breast cancer. Further refinements of the model presented here may well yield better estimates of the volume of the tumor residual after surgery or ofthe average duration of growth preceding clinical presentation.Nevertheless, no current model fits human or animal data betterthan unadorned Gompertzian growth. The therapeutic implications of other models in general, and the Speer model inparticular, should be interpreted and applied with caution.

ACKNOWLEDGMENTS

I thank Dr. James F. Holland for his critical review of this manuscriptand Rita B. Morales for her secretarial assistance.

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1988;48:7067-7071. Cancer Res   Larry Norton  A Gompertzian Model of Human Breast Cancer Growth

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