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Statistics in Clinical Cancer Research

CureModelsasaUsefulStatisticalToolforAnalyzingSurvival

Megan Othus1, Bart Barlogie3, Michael L. LeBlanc1, and John J. Crowley2

Abstract

Cure models are a popular topic within statistical literature but are not as widely known in the clinical

literature. Many patients with cancer can be long-term survivors of their disease, and cure models can be a

useful tool to analyze and describe cancer survival data. The goal of this article is to review what a cure

model is, explain when cure models can be used, and use cure models to describe multiple myeloma

survival trends. Multiple myeloma is generally considered an incurable disease, and this article shows that

by using cure models, rather than the standard Cox proportional hazards model, we can evaluate whether

there is evidence that therapies at the University of Arkansas for Medical Sciences induce a proportion of

patients to be long-term survivors. Clin Cancer Res; 18(14); 3731–6. ?2012 AACR.

Introduction

Progress in the treatment of cancer has led to a spate of

statisticalresearchtodevelopcuremodels.Manyanalysesof

cancer survival data are based on overall survival or pro-

gression-free survival (PFS). No patient can be "cured" of

death, so in these situations cure models can be used to

model long-term survivors rather than cured patients.

Cure models can be used to investigate the heterogeneity

between patients with cancer who are long-term survivors

and those who are not. A straightforward way to identify

whether a particular data set might have a subset of long-

term survivors is to look at the survival curve. If the survival

curvehasaplateauattheendofthestudy,acuremodelmay

be an appropriate and useful way to analyze the data.

Anexampleofdataforwhichcuremodelscouldbeuseful

is provided in Fig. 1, the PFS curve for patients treated on

the University of Arkansas for Medical Sciences (UAMS;

LittleRock,AR)firstTotalTherapystudy(TT1),whichtested

a tandem autotransplant approach for patients with mul-

tiple myeloma (1, 2). Here, PFS is defined from the time of

response to the first of death or progression, with patients

last known to be alive without progression censored at the

date of last contact. With the current amount of follow-up,

thereisaflatplateauafter15years.Onewaytointerpretthis

curve is that there are 2 groups of patients in this trial. One

group of patients are long-term survivors and will not fail

during the follow-up of the study, whereas the rest of the

patients will fail during the first 15 years of the study.

Cure models can be a useful alternative to the standard

Cox proportional hazards models (3) for data with

survivaltrendslikethoseshowninFig.1forseveralreasons.

First, the assumption of proportional hazards can fail

when survival curves have plateaus at their tails. Second,

survival plots with long plateaus may indicate hetero-

geneity within a patient population that can be useful to

describeexplicitly.Curemodelsallowustoinvestigatewhat

covariatesareassociatedwitheithershort-termorlong-term

effects. For example, cure models can allow us to evaluate

whether a new therapy is associated with an increase or

decrease in the probability of being a long-term survivor

or an improvement or detriment in survival for those who

are not long-term survivors.

While cure models have been a popular component of

statistical literature for the past 20 years or more, they have

not been implemented in some areas of the clinical liter-

ature. The purpose of this article is to review cure models

with the hope that some researchers will find the models a

useful alternative to standard survival models when ana-

lyzing some types of cancer survival data. To this end, we

first describe in a fairly nontechnical manner what cure

models are and how they differ from more widely used

survival models. Then, we present a cure model analysis

of multiple myeloma data from the UAMS. Multiple mye-

loma is generally considered an incurable disease (4), but

researchers at UAMS have developed an approach called

Total Therapy that may allow some patients with multiple

myeloma to be long-term survivors. The analysis will high-

lightwhatadditionalinformationcanbegainedfromusing

a cure model analysis beyond a standard Cox analysis.

Cure Models

There are 2 major classes of cure models, mixture and

nonmixture models. Mixture cure models, as the name

suggests, explicitly model survival as a mixture of 2 types

of patients: those who are cured and those who are not

Authors'Affiliations:1FredHutchinsonCancerResearchCenter;2Cancer

Research And Biostatistics, Seattle, Washington; and3Myeloma Institute

for Research and Therapy, University of Arkansas for Medical Sciences,

Little Rock, Arkansas

Note: Supplementary data for this article are available at Clinical Cancer

Research Online (http://clincancerres.aacrjournals.org/).

Corresponding Author:Megan Othus,FredHutchinsonCancerResearch

Center 1100 Fairview Ave NW, M3-C102, Seattle, WA 98117. Phone: 206-

667-5749; Fax: 206-667-4408; E-mail: mothus@fhcrc.org

doi: 10.1158/1078-0432.CCR-11-2859

?2012 American Association for Cancer Research.

Clinical

Cancer

Research

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cured. Typically, the probability a patient is cured is mod-

eled with logistic regression. The second component of the

model is a survival model for patients who are not cured.

There are many options for this, but 2 common models are

the Weibull and the Cox models. In words, a mixture cure

model can be written as follows:

Probability alive at time t

¼ probability cured þ probability not cured?

probability alive at time t if not cured

ðAÞ

Standard survival models, such as the Cox model, do not

assume 2 different populations as the mixture cure model

does. Many variations of mixture cure models have been

proposed in the statistical literature (5–14, to name a few).

In our multiple myeloma analyses, we use the logistic

Weibull model (15). A nice feature of the logistic Weibull

model(andsomeothermixturemodels)isthatawiderange

of researchers understand how to interpret ORs and HRs.

The results of the model provide ORs for the probability of

beingcuredandHRsforthesurvivalforpatientswhoarenot

cured. A benefit of the mixture cure model is that it allows

covariates to have different influence on cured patients and

on patients who are not cured. For example, a therapy may

increase the proportion of patients who are cured (evi-

denced by a significant OR) but not affect survival for

patients who are not cured (evidenced by a nonsignificant

HR). A mixture cure model allows us to tease out that

relationship.

Nonmixture cure models take a different approach to

modeling survival. Many nonmixture cure models can be

thought of as Cox proportional hazards models that allow

for a cure fraction.

Nonmixture survival models can be written as follows:

Probability alive time t ¼ probability cured1?S?ðtÞ;

ðBÞ

where 1?S?(t) is an exponent of the probability of being

cured and S?(t) is a survival function. Equation B has a

very different form than the mixture cure model in equa-

tion A. Nonmixture cure models may fit some data better

than mixture cure models and vice versa.

For the nonmixture model, covariates can be incorpo-

rated both in the model for the probability of being cured

and in S?(t). The interpretation of covariates is different

with the nonmixture cure model than with the mixture

model. Covariates included in S?(t) characterize a "short-

term" effect, but the covariates do not describe the

survival for those who are not cured because the non-

mixture model does not directly model a mixture popu-

lation. We review the difference in interpretation between

mixture and nonmixture models in the data application

in the next section. A number of nonmixture cure models

have been proposed in the statistical literature (refs. 16–

22, among others). We use a Weibull nonmixture model

in the data application.

Using Cure Models to Investigate Survival in

Multiple Myeloma

We apply several survival models to multiple myeloma

survival data from the UAMS to elucidate information on

long-term survivors among patients with multiple myelo-

ma. The UAMS conducted 3 Total Therapy trials with the

intent of inducing some patients with multiple myeloma to

belong-termsurvivors.Thefirststudy,TT1,testedatandem

autotransplant approach (1, 2). The second study showed

that the addition of thalidomide, TT2þ, improved results

compared with the same regimen without thalidomide,

TT2?(23). TT3, the third study, incorporated both thalid-

omide and bortezomib in induction (24, 25).

Kaplan–MeierplotsofPFSfortheTotalTherapyregimens

(Fig.2)indicatethattheremaybesomelong-termsurvivors

on these regimens. Here PFS is defined from the time of

response to the first of death or progression, with patients

last known to be alive without progression censored at the

date of last contact. In each of the 4 curves in Fig. 2, there

appears to be evidence of an emerging plateau, indicating

thataproportionofpatientsfromeachoftheTotalTherapy

regimens may be long-term survivors. The more recent

regimens plateau at a higher level, indicating that the

proportion of long-term survivors may have increased over

the development of the regimens.

A standard survival analysis would use the Cox model to

test whether PFS has improved over the regimens. Results

foraCoxmodel[HRs,95%confidenceintervals(CI),andP

values] adjusting for the potential prognostic factors age

and presence of any cytogeneticabnormalities are provided

inTable1.Inthismodel,HRslessthan1indicateimproved

PFS. The results from this model suggest that PFS is signif-

icantly improved in TT2?, TT2þ, and TT3 compared with

TT1, which matches the interpretation of the PFS curves

© 2012 American Association for Cancer Research

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Figure1. PFScurveforTT1potentiallyindicatesaproportionoflong-term

survivors. Censoring is marked with a cross.

Othus et al.

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in Fig. 2. In addition, we can see that increasing age and

presence of cytogenetic abnormalities are associated with

decreased PFS.

The Cox model allows us to test whether PFS is the same

among the Total Therapy regimens while controlling for

other covariates and provides an overall summary of PFS.

Whensurvival curves plateau, as shown in Fig. 2,there is an

indication that a proportion of patients could be long-term

survivors of multiple myeloma. A cure model could esti-

mate the proportion of long-term survivors with each

therapy and could test whether the proportions have chan-

ged over the regimens. In addition, cure models can char-

acterize the survival of patients who are not long-term

survivors.

First, we summarize results from a Weibull mixture cure

model, with technical details of the parameterization pro-

vided in the Supplementary Material. In this model, ORs

greaterthan1indicateanincreaseintheproportionoflong-

termsurvivorsandHRslessthan1indicateanimprovement

insurvivalamongpatientswhoarenotlong-termsurvivors.

Results adjusted for age and cytogenetic abnormalities are

providedinTable1.Theresultsfromthismodelsuggestthat

the proportions of long-term survivors have increased over

the regimens and that PFS among those who are not long-

termsurvivorsisimprovedinTT2þcomparedwithTT1.PFS

among those who are not long-term survivors is not signif-

icantly improved in TT3 relative to TT1. This nonsignificant

result could be due to the more limited follow-up in TT3

compared with the other regimens or it could be that the

large improvement in PFS observed in Fig. 2 is due to TT3

having a larger proportion of patients who are long-term

survivors.Amongallregimens,olderageisassociatedwitha

decreased probability of being a long-term survivor but is

notsignificantlyassociatedwithshort-termPFS.Presenceof

cytogenetic abnormalities is associated with a decreased

probability of being a long-term survivor and worse PFS

for those who are not long-term survivors.

We can also use nonmixture cure models to study the

trends in the multiple myeloma data. We summarize a

Weibull nonmixture cure model, with technical details of

the parameterization provided in the Supplementary Mate-

rial. For the long-term part of the model, HRs less than 1

indicate an increase in the proportion of long-term survi-

vors, whereas for the short-term model HRs more than 1

indicate an improvement in short-term survival. We note

that, in contrast to the mixture model, the HRs for short-

termsurvivalinthismodelcannotbeinterpretedasHRsfor

patientswhoarenotcured.ResultsaresummarizedinTable

3. These results suggest that significantly more patients on

TT2þand TT3 are long-term survivors than on TT1 and

that there is no significant difference in the long-term

survivor proportions between TT2?and TT1. Higher age

and presence of cytogenetic abnormalities were both asso-

ciated with decreased probability of being a long-term

survivor. None of the covariates had a significant short-

term effect. Overall, the nonmixture model indicates that

Table 1. Cox model regression results

HR (95% CI)P

TT1 (ref.)

TT2?

TT2þ

TT3

Age

Cytogenetic abnormalities

0.65 (0.53–0.79)

0.45 (0.36–0.55)

0.29 (0.22–0.37)

1.02 (1.01–1.02)

1.72 (1.48–2.01)

<0.001

<0.001

<0.001

<0.001

<0.001

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TT2+

TT2–

TT1

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Figure 2. PFS plots for the Total Therapy regimens. Censoring is marked

with a cross.

Table 2. Weibull mixture cure model regression

results

Estimate (95% CI)P

Long-term survivor model (OR)

TT1 (ref.)

TT2?

TT2þ

TT3

Age

Cytogenetic

abnormalities

Short-term survival model (HR)

TT1 (ref.)

TT2?

TT2þ

TT3

Age

Cytogenetic

abnormalities

4.75 (2.22–10.18)

2.36 (1.06–5.26)

20.23 (9.11–44.90)

0.97 (0.95–0.97)

0.42 (0.25–0.70)

0.036

<0.001

<0.001

0.018

0.001

0.77 (0.59–1.00)

0.66 (0.48–0.93)

0.92 (0.52–1.64)

1.00 (0.999–1.02)

1.41 (1.15–1.73)

0.052

0.017

0.78

0.25

0.001

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the TT2þand TT3 patients had improved PFS compared

with TT1 patients and that the improvement is due to an

increase in the proportion of patients who are long-term

survivors.

AnalternativesummaryoftheresultsfromTables1and3,

on the coefficient scale rather than the HR and OR scale, is

provided in the Supplementary Material.

We use TT3 to emphasize the difference in interpretation

between the Cox model (Table 1) and the cure models

(Tables 1 and 3). From the Cox model results (Table 1), we

conclude that PFS is significantly improved in TT3 com-

pared with TT1 (HR ¼ 0.29, P < 0.0001). From the mixture

cure model results in Table 2, we conclude that a larger

proportion of patients are cured in TT3 than TT1 (OR ¼

20.23, P < 0.001), but there is no evidence that survival

among patients who are not cured is different in TT3

compared with TT1 (HR ¼ 0.92, P ¼ 0.78). From the

nonmixture curemodel (Table3),we concludethat alarger

proportion of patients are cured in TT3 compared with TT1

(HR ¼ 0.18, P < 0.001), but there is no evidence that short-

term survival is different in TT3 compared with TT1 (HR ¼

1.72, P ¼ 0.10). We note that the mixture cure model

compares cure proportions on the OR scale whereas the

nonmixture cure model uses the HR scale.

Results from mixture and nonmixture cure models can

provide estimates of the probability of being a long-term

survivor. An unadjusted logistic Weibull mixture cure

model (only including covariates for the Total Therapy

regimens in long- and short-term models) estimated the

proportion of cured patients in TT1, TT2?, TT2þ, and TT3

to be 10%, 16%, 29%, and 60%, respectively. An unad-

justed Weibull nonmixture cure model estimated the pro-

portion of cured patients in TT1, TT2?, TT2þ, and TT3 to be

9%, 11%, 27%, and 60%, respectively. The long-term

survivor estimates are similar in the 2 models, and both

models estimate an increasing trend in long-term survivor

proportions.

For each regimen, Fig. 3 has Kaplan–Meier survival plots

along plots based on the unadjusted Weibull mixture cure

model and Weibull nonmixture cure model. The model-

based survival curves fit the Kaplan–Meier curves so closely

that it is difficult to distinguish among the 3 curves on the

plot. Figure 3 indicates that both the mixture and nonmix-

turecuremodelsfitthemultiplemyelomadatawellandcan

be a useful tool to describe the trends across regimens.

Regimens TT1–TT3 were tested in sequential studies, and

thus the length of follow-up available and amount of

censoring differ between the studies, with TT1 having the

mostfollow-upandleastcensoringandTT3havingtheleast

follow-up and most censoring. As with other survival mod-

els, additional follow-up and less censoring will lead to

smaller standard errors relative to the sample size. The

increased variance associated with the shorter follow-up of

TT3 is reflected in the CIs in Tables 1–3. Cure models may

not be appropriate for data with too short of follow-up to

identify a plateau in the tail. In this application, TT3 has

the shortest follow-up, with under 6 years and has a sug-

gestion of a plateau only in the last year of follow-up. We

replicated the unadjusted analyses with just the TT3 subset

to evaluate the stability of the TT3 results. The estimates of

long-term survivors were nearly identical, and the model-

based survival curves were almost indistinguishable (data

not shown).

Discussion

Cure models are an underused statistical tool. Cure

models have been well developed in the statistical liter-

ature, but the models are not as common in the clinical

literature. For cancers in which some patients may be

long-term survivors, cure models can be an interesting

way to characterize and study patients’ survival. There are

2 general classes of cure models, mixture and nonmixture

models. Both classes can describe short-term and long-

term effects. Choosing between the 2 models is a matter

of preference and fit. In this application, both classes fit

the data well, so either class is useful for testing and

inference.

The cure model analyses for the multiple myeloma data

provide additional information beyond the standard

Cox model analysis. PFS has improved from TT1 to TT3,

and cure models indicate that the gains in survival were

primarily due to more patients being long-term survivors.

The short-term survival trends have not shown drastic

improvements.

While the application of this article focused on multiple

myeloma, the statistical tools reviewed above could be

useful for a wide range of cancers. Therapies for a number

ofcancer typesarebelieved toinduceacureamongasubset

of patients. Disease sites where this may be the case include

Burkitt lymphoma and Hodgkin disease (26), head and

neck cancer (27), colon cancer (28), melanoma (29), and

acute promyelocytic and myeloid leukemia (30–32).

Table 3. Weibull nonmixture cure model

regression results

Estimate (95% CI)P

Long-term survivor model (HR)

TT1 (ref.)

TT2?

TT2þ

TT3

Age

Cytogenetic abnormalities

0.85 (0.48–1.49)

0.50 (0.18–0.79)

0.18 (0.10–0.31)

1.13 (0.995–1.03)

1.17 (1.07–2.14)

0.57

0.003

<0.001

0.16

0.019

Short-term survival model (HR)

TT1 (ref.)

TT2?

TT2þ

TT3

Age

Cytogenetic abnormalities

0.51 (0.88–1.53)

0.72 (0.39–1.32)

1.72 (0.90–3.29)

1.00 (0.98–1.02)

1.18 (0.80–1.75)

0.66

0.29

0.10

0.80

0.40

Othus et al.

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Some statistical research has been done on testing

whether there is evidence of a cured proportion based on

mixture cure models (33, for example), but none of the

proposed statistical approaches has software available.

More research in this area is warranted.

Disclosure of Potential Conflicts of Interest

No potential conflicts of interest were disclosed.

Authors' Contributions

Conception and design: M. Othus, J.J. Crowley

Development of methodology: M. Othus

Analysis and interpretation of data (e.g., statistical analysis, biosta-

tistics, computational analysis): M. Othus, M.L. LeBlanc, J.J. Crowley

Writing, review, and/or revision of the manuscript: M. Othus, B. Barlo-

gie, M.L. LeBlanc, J.J. Crowley

Grant Support

This work was supported by the Hope Foundation (M. Othus) and the

National Cancer Institute (grant CA090998; J.J. Crowley, M.L. LeBlanc, and

M. Othus).

Received November 7, 2011;revised May4, 2012;accepted May 20,2012;

published OnlineFirst June 6, 2012.

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Figure 3. PFS plots for the Total

Therapy regimens. Kaplan–Meier

estimates are solid lines, mixture

model curves are dashed lines, and

nonmixture models are dash–dot

lines. Censoring is marked with a

cross.

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