# Estimating and modelling the cure fraction in population based

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```					           Estimating and modelling the cure fraction in
population-based cancer survival analysis

Paul W Dickman1 , Paul C Lambert2 , Sandra Eloranta1 , Therese

1 Department   of Medical Epidemiology & Biostatistics, Karolinska Institutet, Stockholm,
Sweden
2 Centre   for Biostatistics and Genetic Epidemiology, University of Leicester, UK

11 Dec 2008

Dickman et al.   Cure models for cancer patient survival                           1/21

Speciﬁc aims of the research

1    Develop statistical methods for estimating and modelling the cure
fraction in population-based cancer survival analysis.
2    Apply the methods to data from Sweden, Finland, England, and the
USA with the joint aim of evaluating the new methodology as well as
studying temporal trends in cancer patient survival.

Dickman et al.   Cure models for cancer patient survival                           2/21

Background: Relative Survival

Observed Survival
Relative Survival =                                               R(t) = S(t)/S ∗ (t)
Expected Survival

Expected survival obtained from national population life tables
stratiﬁed by age, sex, year of diagnosis, other covariates.
Estimate of mortality associated with a disease without requiring
information on cause of death(1; 2; 3).
On hazard scale

λ(t) = h(t) − h∗ (t)

Excess          Observed         Expected
=                –
Mortality Rate   Mortality Rate   Mortality Rate

Dickman et al.   Cure models for cancer patient survival                           3/21
Deﬁnition of Cure (1)

For many cancers the excess mortality (hazard) rate returns to the
same level as that in the general population.
When this occurs the relative survival curve is seen to reach a plateau.
This is Population or Statistical Cure.
Information of cure at the individual level not available.
For the ‘uncured’ we can obtain a summary measure of survival.

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Deﬁnition of Cure (2)

1.0

0.8
Relative Survival

0.6

0.4

0.2

0.0
0            2               4             6          8          10
Years from Diagnosis

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Relative Survival for Cancer of the Colon in Finland

1.0
1953−1964
1965−1974
1975−1984
0.8                                                             1985−1994
1995−2003
Relative Survival

0.6

0.4

0.2

0.0
0            2               4             6          8          10
Years from Diagnosis

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Relative Survival for Cancer of the Colon in Finland

1.0

Interval Specific Relative Survival
0.8

0.6

0.4

1953−1964
0.2               1965−1974
1975−1984
1985−1994
1995−2003
0.0
0              2               4             6          8   10
Years from Diagnosis

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Relative Survival Models and Cure Models

Relative Survival Models
S(t) = S ∗ (t)R(t)
h(t) = h∗ (t) + λ(t)

When modelling cure we deﬁne an asymptote at the cure fraction, π,
for the relative survival function, R(t)(4; 5).

Mixture Cure Model
S(t) = S ∗ (t)(π + (1 − π)Su (t))

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Survival for the ‘Uncured’

As well as the cure fraction, summaries of the ‘uncured’ (those
‘bound to die’) are potentially useful.
For example, mean or median survival or some other percentile of the
survival distribution.
We need to choose parametric form for S(t).
For many scenarios the Weibull distribution provides a good ﬁt.

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Interpreting changes over time

(a) General Improvement
Survival of Uncured
(b) Selective
Improvement
(c) Improved palliative
care
(d) Inclusion of subjects
with no excess risk
Cure Fraction

Dickman et al.       Cure models for cancer patient survival                         10/21

Interpreting changes over time

(a) General Improvement
Survival of Uncured

(a)
(b) Selective
Improvement
(c) Improved palliative
care
(d) Inclusion of subjects
with no excess risk
Cure Fraction

Dickman et al.       Cure models for cancer patient survival                         10/21

Interpreting changes over time

(a) General Improvement
Survival of Uncured

(a)
(b) Selective
Improvement
(c) Improved palliative
care
(d) Inclusion of subjects
(b)                                      with no excess risk
Cure Fraction

Dickman et al.       Cure models for cancer patient survival                         10/21
Interpreting changes over time

(a) General Improvement
Survival of Uncured
(c)                          (a)
(b) Selective
Improvement
(c) Improved palliative
care
(d) Inclusion of subjects
(b)                                      with no excess risk
Cure Fraction

Dickman et al.       Cure models for cancer patient survival                                  10/21

Interpreting changes over time

(a) General Improvement
Survival of Uncured

(c)                          (a)
(b) Selective
Improvement
(c) Improved palliative
(d)                                   care
(d) Inclusion of subjects
(b)                                      with no excess risk
Cure Fraction

Dickman et al.       Cure models for cancer patient survival                                  10/21

The cure proportion is not aﬀected by lead time

Onset            Detectability                          Symptoms                      Death
DETECTABLE
PRECLINICAL
PHASE                                                          Time

TIME                      TIME
SURVIVAL
TIME
Early                           Clinical                   Postponed
diagnosis                        diagnosis                     death

Dickman et al.       Cure models for cancer patient survival                                  11/21
Time Trends for Cancer of the Colon

Cure Fraction
1.00
<50 years
50−59 years
60−69 years
0.80
70−80 years

0.60

Cure Fraction
0.40

0.20

0.00

1950            1960               1970               1980   1990   2000
Year of Diagnosis

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Time Trends for Cancer of the Colon

Median Survival of ’Uncured’
2.50
<50 years
50−59 years
60−69 years
2.00
70−80 years
Median Survival of Uncured

1.50

1.00

0.50

0.00

1950            1960               1970               1980   1990   2000
Year of Diagnosis

Dickman et al.        Cure models for cancer patient survival                  12/21

Quantifying Diﬀerences

Relative Odds of Cure (Age Group 70−79 / Age Group <50)
2.0

1.5

1.0
Odds Ratio

0.5

0.3

1950            1960              1970         1980          1990   2000
Years from Diagnosis

Dickman et al.        Cure models for cancer patient survival                  13/21
Quantifying Diﬀerences

Difference in Cure Fraction (Age Group <50 − Age Group 70−79)
0.2

0.1

Difference in Cure Fraction
0.0

−0.1

−0.2

−0.3
1950            1960              1970         1980          1990   2000
Years from Diagnosis

Dickman et al.        Cure models for cancer patient survival                  14/21

Time Trends for Cancer of the Rectum

Cure Fraction
1.00
<50 years
50−59 years
60−69 years
0.80
70−80 years

0.60
Cure Fraction

0.40

0.20

0.00

1950            1960               1970               1980   1990   2000
Year of Diagnosis

Dickman et al.        Cure models for cancer patient survival                  15/21

Time Trends for Cancer of the Rectum

Median Survival of ’Uncured’
2.50
<50 years
50−59 years

60−69 years
2.00
70−80 years
Median Survival of Uncured

1.50

1.00

0.50

0.00

1950            1960               1970               1980   1990   2000
Year of Diagnosis

Dickman et al.        Cure models for cancer patient survival                  15/21
Factors contributing to improvements

Surgical and anaestesiological techniques have become more
aggressive and sophisticated over time.
The age of the patient at which surgeons are prepared to operate has
increased over time, which may explain the reduced diﬀerences in the
proportion cured between age groups.
Better awareness among the public and doctors about the importance
of early diagnosis for cure.
Some of the trends over time are likely to be due to the ”learning
period” when gradually introducing new techniques within and
between hospitals.

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Greater improvements for rectal than colon cancer

Rectal cancer surgery is much more demanding than colon cancer
surgery, and the general improvements in anaesthesiology and
postoperative care seen in the late 1960s/early 1970s were relatively
much more important for rectal than colon cancer.
The steep increase during the 1990s seen in all age groups likely
reﬂects the marked decrease in the risk of a local failure after rectal
cancer surgery seen after the introduction of total mesorectal excision
(TME) and increased use of preoperative radiotherapy or
A local failure is a much less clinical problem in cancer of the colon
than in cancer of the rectum.
Metastatic disease has been the predominant cause of death among
colon cancer patients.
The natural course of local failure is longer than that of metastatic
disease (which often involves the liver).
Dickman et al.   Cure models for cancer patient survival       17/21

Time Trends for AML in Sweden

Dickman et al.   Cure models for cancer patient survival       18/21
Challenges and current/future research

How to assess goodness-of-ﬁt? Standard methods not appropriate.
How do the models behave when cure is not reached (e.g., female
breast cancer).
Preliminary evidence suggests the models perform best when
mortality is neither high nor low.

Dickman et al.   Cure models for cancer patient survival                          19/21

Cure models may ﬁt poorly when early mortality is high:
Cancer of the Colon: Weibull and Mixture of Weibulls

Age Group 80+
1.0
Ederer II
Weibull
Mixture of Weibulls
0.8
Relative Survival

0.6

0.4

0.2

0.0
0            2               4             6             8             10
Years from Diagnosis

Dickman et al.   Cure models for cancer patient survival                          20/21

Cure models may ﬁt poorly when early mortality is high:
Cancer of the Colon: Weibull and Mixture of Weibulls

Age Group 80+
1.0
Ederer II
Weibull
Mixture of Weibulls
0.8
Relative Survival

0.6

0.4

0.2

0.0
0            2               4             6             8             10
Years from Diagnosis

Dickman et al.   Cure models for cancer patient survival                          20/21
Cure models may ﬁt poorly when early mortality is high:
Cancer of the Colon: Weibull and Mixture of Weibulls

Age Group 80+
1.0
Ederer II
Weibull
Mixture of Weibulls
0.8

Relative Survival
0.6

0.4

0.2

0.0
0            2               4             6             8             10
Years from Diagnosis

Dickman et al.   Cure models for cancer patient survival                          20/21

[1] Dickman PW, Adami HO. Interpreting trends in cancer patient
survival. J Intern Med Aug 2006;260:103–117.
[2] Ederer F, Axtell LM, Cutler SJ. The relative survival rate: A statistical
methodology. National Cancer Institute Monograph 1961;6:101–121.
[3] Ederer F, Heise H. The eﬀect of eliminating deaths from cancer on
general population survival rates. Methodological note No. 11, End
Results Evaluation Section, National Cancer Institute, Bethesda MD,
1959.
[4] Lambert PC, Dickman PW, Osterlund P, Andersson T, Sankila R,
Glimelius B. Temporal trends in the proportion cured for cancer of the
colon and rectum: A population-based study using data from the
ﬁnnish cancer registry. Int J Cancer Nov 2007;121:2052–2059.
[5] Lambert PC, Thompson JR, Weston CL, Dickman PW. Estimating
and modeling the cure fraction in population-based cancer survival
analysis. Biostatistics Jul 2007;8:576–594.

Dickman et al.   Cure models for cancer patient survival                          21/21

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