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Cancer Informatics 2009:7 183–197
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Cancer Informatics 2009:7 183
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Cancer Informatics
ORIGINAL RESEARCH
A Generalized Beta Model for the Age Distribution of Cancers:
Application to Pancreatic and Kidney Cancer
Tengiz Mdzinarishvili, Michael X. Gleason, Leo Kinarsky and Simon Sherman
Eppley Cancer Institute, University of Nebraska Medical Center, 986805 Nebraska Medical Center, Omaha, Nebraska,
68198-6805. Email: ssherm@unmc.edu
Abstract: The relationships between cancer incidence rates and the age of patients at cancer diagnosis are a quantitative basis for
modeling age distributions of cancer. The obtained model parameters are needed to build rigorous statistical and biological models
of cancer development. In this work, a new mathematical model, called the Generalized Beta (GB) model is proposed. Condence
intervals for parameters of this model are derived from a regression analysis. The GB model was used to approximate the incidence rates
of the rst primary, microscopically conrmed cases of pancreatic cancer (PC) and kidney cancer (KC) that served as a test bed for the
proposed approach. The use of the GB model allowed us to determine analytical functions that provide an excellent t for the observed
incidence rates for PC and KC in white males and females. We make the case that the cancer incidence rates can be characterized by a
unique set of model parameters (such as an overall cancer rate, and the degree of increase and decrease of cancer incidence rates). Our
results suggest that the proposed approach signicantly expands possibilities and improves the performance of existing mathematical
models and will be very useful for modeling carcinogenic processes characteristic of cancers. To better understand the biological
plausibility behind the aforementioned model parameters, detailed molecular, cellular, and tissue-specic mechanisms underlying the
development of each type of cancer require further investigation. The model parameters that can be assessed by the proposed approach
will complement and challenge future biomedical and epidemiological studies.
Keywords: cancer, incidence rates, aging, histopathology
Mdzinarishvili et al
184 Cancer Informatics 2009:7
Introduction
The number of newly diagnosed primary cancers
at particular organ sites occurring in a specied
population during a given time period (for instance,
one year or ve years), is called the cancer incidence
rate. The rate of cancer incidence for a specic age
group during this time period is called the age-specic
incidence rate. Usually, the age-specic incidence rates
are presented as the number of cancers per 100,000
persons in a specied age group. The sequence of age-
specic incidence rates for all specied age groups is
referred to as the age distribution of a given cancer.
A mathematical modeling of the age distribution of
cancer results in a simple analytical function, I(t), that
can approximate observed values of cancer incidence
rates and provides parameters of this function. The
obtained model parameters can be further used to
build rigorous statistical and/or biological models of
cancer development.
Development of mathematical models of age-
specic cancer incidence rates began more than
55 years ago. Analyzing cancer mortality rates in the
UK, Nordling,1 as well as Armitage and Doll,2 noticed
the existence of two age periods in which cancer
mortality manifests differently. In the initial age
period, a number of cancer mortalities per population
at a given age is equal or close to 0. For the majority
of adult onset cancers, this period is extended between
birth and an age when the cancer presentation begins
growing exponentially. In the second age period,
the cancer mortality per population at a given age is
exponentially growing with aging.
The rst mathematical model of cancer
presentation in aging was proposed by Armitage and
Doll (the AD model).2 This model can be presented
in the following way:
I(t) = ct
k - 1 (1)
where I(t) is the modeled cancer incidence rate at
age t; c is a parameter characterizing overall cancer
susceptibility in a population at cancer risk, and k is
the number of stages of cancer development. This
model describes the relationship between cancer
incidence rates and aging, when cancer development
is in the exponential growth phase.
Cook, Doll and Fellingham found a single
increasing linear trend for the logarithm of many cancer
incidences plotted as a function of the logarithm of
age at diagnosis,3 presumably reecting accumulated
lifetime carcinogenic risks and/or exposures. This led
to conclusions that: (i) the number of stages of cancer
presentation (parameter k) can vary between different
cancer tissues, but is constant for a given cancer
organ site, and (ii) the overall cancer susceptibility
(parameter c) may be dependent on the geographical
location (country of residency) of the population at
cancer risk.
Extrapolation of the AD model to ages older than
70 years (up to which observed data was considered
reliable at the time the AD model was proposed) can
lead to a statement that if a person lives long enough,
sooner or later he/she will get cancer. However, Cook,
Doll and Fellingham considered the possibility of
attening incidence rates of cancer in ages above 60.3
For this purpose, they assumed that only a very
limited and xed fraction of the whole population
is susceptible to a particular type of cancer. In this
case, the cancer-sensitive fraction of a population will
decrease with increasing age that, in turn, will cause a
attening of the cancer incidence rates at old ages.
In the second half of the 20th century, the quality of
the collected cancer incidence rate data has improved
markedly. Thanks to the implementation of the
Surveillance, Epidemiology, and End Results (SEER)
program, a lot of reliable data on the cancer incidence
rates for specic organ sites at different ages (including
the oldest ones) were collected.4 Using the SEER
database, Pompei and Wilson showed that patterns of
the age-specic incidence rate for the xed time period
(2000–2004) for most common adult cancers have a
turnover point (near the age of 80),5 after which these
patterns have a tendency to fall and may reach a value
of 0 as age increases toward the end of the human
life span. This observation encouraged Pompei and
Wilson to extrapolate the AD model beyond the age
of 70 to the life span by adding an additional term to
Equation 1, resulting in:
I(t) = (at)k - 1(1 - bt) (2)
where the a parameter is a constant for limiting
stage transitions; k reects the number of these rate-
limiting (slow) stages required to initiate cancer;
and b is a parameter, whose meaning can be easily
described by its reciprocal value, 1/b, that presents an
age at which I(t) becomes 0. Equation 2 is a special
form of the Beta function.
Modeling of cancer incidence rates in aging
Cancer Informatics 2009:7 185
Recently, Harding, Pompei, Lee and Wilson
modied Equation 2 to:
I(t) = ct
k - 1(1 - bt) (3)
where the c parameter characterizes a combined
rate constant for limiting stage transitions. Values
of the c, k and b parameters can be determined by
best-tting the age-specic incidence rates collected
in the SEER database.6 Below, we will refer to this
model as the Pompei and Wilson (PW) model.
By adjusting c, k and b of Equation 3, Harding and
coauthors performed curve tting for the age-specic
cancer incidence rates for 20 major organ sites listed
in SEER for males and 21 major organ sites listed for
females.6 A satisfactory data tting was shown for
many of the examined cancer sites. It was also shown
that the age-specic incidence rate distributions
demonstrated a common shape. In 36 of the
41 considered cancer sites (for males and females),
this common shape was characterized by the location
of the corresponding distribution peaks (the incidence
rate turnover near the age of 80), and relatively small
(10%) variability of 1/b (near age of 100). The
k values varied between 2.4 and 10.6. Very large
variations (more than ve orders of magnitude) were
found for the c values.
In all the aforementioned works, the modeling of
age distribution of cancer was performed without
considering time period and cohort effects. However,
ignorance of these effects could seriously distort
cancer presentation in aging.7–10 Recently, Meza,
Jihyoun, Moolgavkar, and Luebeck carried out an
adjustment of the observed age-specic incidence
rates of colorectal and pancreatic cancers for birth
cohort and time period effects and did not observe a
turnover point at old ages.11 These authors proposed to
present distribution of the adjusted cancer incidence
rates by a composition of two analytical functions:
(i) a power (or exponential) function that up to the
age of approximately 60 years remains the same
as in the AD model; and (ii) a linear function, after
the ages of 60+. Thus, Meza and coauthors not only
rejected the existence of the turnover points of the
age distribution of cancer at old ages (at least in the
cases of the colorectal and pancreatic cancers) but
also stated that for the ages beyond 60 years the linear
function approximates the adjusted observational
data better than other functions used before.
In addition to ignorance of the time period and
cohort effects, there are other potential problems in
the utilization of mathematical models for studies
of the age distribution of cancer. Most models use
raw incidence rate data without omitting cases
corresponding to second primary or secondary
tumors, and do not omit cases which have not been
microscopically conrmed. Also, these models do not
provide means to calculate condence intervals for
the determined model parameters.12
The present work is aimed at overcoming the
aforementioned shortcomings in the modeling of age
distribution in cancer.
Materials and Methods
Data preparation and ltration
To build mathematical models for age distribution in
PC and KC, we used data from the SEER 9 registries
that contain cancer data collected in the following
nine locations: Atlanta, Connecticut, Detroit, Hawaii,
Iowa, New Mexico, San Francisco-Oakland, Seattle-
Puget Sound, and Utah. In the SEER database, each
case record contains information on whether this is
the rst primary malignant case and whether the case
is histopathologically conrmed. Limiting inclusion
to cases where the patient was of known race and
whose case indicated a rst primary, microscopically
conrmed tumor is considered to be a ltered data,
and data where this ltering was not performed is
considered to be a raw data. For age distribution
modeling, we used the ltered data, which are expected
to be more reliable than raw data. We utilized the
incidence rate data expressed per 100,000 persons to
the nearest 0.0001 decimal place and age-adjusted by
the direct method to the 2000 United States standard
population.13
We used SEER 9 data collected during the 20-year
time period between 1985 and 2004. To smooth out
random uctuations, the data were combined in four
ve-year cross-sectional time periods: 1985–1989;
1990–1994; 1995–1999; and 2000–2004. For PC
and KC, the gender-specic incidence rates were
grouped into 18 ve-year age groups: 17 groups,
ranging from 0 to 84 years old, and the 18th group
that included all cases for ages 85 or over. For each of
these intervals, i, the corresponding I(ti) and standard
errors (SEi) were obtained by processing the SEER
data according to SEER’s rate algorithms.14 For each
Mdzinarishvili et al
186 Cancer Informatics 2009:7
age interval, the values of the coefcient of variance
were also determined as:
CV SE I t
i i i
=( ).
Birth cohort and time period adjustments
We assumed that each observed incidence rate, Iij(ti),
can be estimated as a product of the corresponding
coefcient of time period effect, vj, coefcient of
cohort effect, ul, and theoretical incidence rate
(or the hazard function depending only on age),
h(ti), i.e.
Iij(ti) = vjulh(ti) (4)
where i, j, and l are indexes of the age, time period
and birth cohorts, correspondingly; and t = ti is the
midpoint of the corresponding age group.11,15 Indices
i and j determine index l (see below). The birth cohort
and time period adjustments performed in this work
can be easily described by the use of Table 1 and Table 2.
Table 1 schematically presents the incidence
rate of data collected in 1985–1989, 1990–1994,
1995–1999, and 2000–2004. In this Table, the
incidence rates of the same cohorts are located along
diagonals. We used data for the age groups over age
30 (index i = 7, …, 18), because the incidence rates
for these age groups are statistically different from
0. We also limited our analysis by nine birth cohorts
(index l = 1, …, 9). The rst cohort includes patients
that were born in years of 1915–1919, while the ninth
cohort is formed from patients born in 1955–1959. In
Table 1, each of these nine birth cohorts are marked
by an arrow linking the diagonal cells, in which
the cancer incidence rates observed for this group
in each time period are presented.
Table 2 schematically presents the observed
incidence rates as a product of the hazard function,
h(ti), and the corresponding time period and birth
cohort coefcients, v and u. As can be seen on
the corresponding diagonals in Table 2, there are
four approximations of the observations related to the
rst cohort: v1u1h(t15), v2u1h(t16), v3u1h(t17), v4u1h(t18).
Analogously, there are four approximations for each
of the other eight cohorts. (Note that in this Table
we did not provide data corresponding to other
Table 1. Presentation of the observed age-specic incidence rates for nine birth cohort groups during four time periods.
Period of observation
Age group 1985–89 1990–94 1995–99 2000–04 Birth cohort
Index, iMid point, tij = 1 j = 2 j = 3 j = 4 Index, lYears
1 2.5
⋅⋅⋅ ⋅⋅⋅
7 32.5 I7,1(t7)I7,2(t7)I7,3(t7)I7,4(t7)
8 37.5 I8,1(t8)I8,2(t8)I8,3(t8)I8,4(t8)
9 42.5 I9,1(t9)I9,2(t9)I9,3(t9)I9,4(t9)
10 47.5 I10,1(t10)I10,2(t10)I10,3(t10)I10,4(t10) 9 1955–59
11 52.5 I11,1(t11)I11,2(t11)I11,3(t11)I11,4(t11) 8 1950–54
12 57.5 I12,1(t12)I12,2(t12)I12,3(t12)I12,4(t12) 7 1945–49
13 62.5 I13,1(t13)I13,2(t13)I13,3(t13)I13,4(t13) 6 1940–44
14 67.5 I14,1(t14)I14,2(t14)I14,3(t14)I14,4(t14) 5 1935–39
15 72.5 I15,1(t15)I15,2(t15)I15,3(t15)I15,4(t15) 4 1930–34
16 77.5 I16,1(t16)I16,2(t16)I16,3(t16)I16,4(t16) 3 1925–29
17 82.5 I17,1(t17)I17,2(t17)I17,3(t17)I17,4(t17) 2 1920–24
18 87.5+I18,1(t18)I18,2(t18)I18,3(t18)I18,4(t18) 1 1915–19
Modeling of cancer incidence rates in aging
Cancer Informatics 2009:7 187
cohorts: the cells that should be assigned for the
1900–04, 1905–09, 1910–14, 1960–64, 1965–69, and
1970–74 birth cohorts are empty. For these cohorts,
the numbers of observations are less than four and
their corresponding coefcients, u, should be treated
with a lower weight than ones for the considered
groups. Therefore, we used the most homogeneous
and reliable data.)
From Table 2, the relationship between indexes
i, j, and l can be presented as: l = j - i + 15. Now,
assuming the absence of the cohort effect (u = 1) and
using Equation 4 and Table 2, we can estimate the
ratios of the coefcients of the time period effect in
the following way:
v
v
I
I
v
v
I
I
v
v
I
I
i
i
i
i
i
i
i
i
1
2
1
2
8
15
2
3
2
3
3
4
9
16 3
1
8
1
8
1
8
= = =
= =
∑ ∑
,
,
,
,
,
,
; ;
44
10
17
i=
∑.
(5)
From this system of three Equations we can obtain
thee unknowns, v2, v3, and v4, by setting v1 = 1.
Analogously, assuming the absence of the time
effect (v = 1), we can obtain estimates of the ratios of
the coefcients of the cohort effect:
time period effect, v, using the system of equations
(Equation 5). Then, we xed the obtained time period
coefcients and corrected the observed incidence rates
by dividing them by the coefcients presented in
Equation 4. Continuing, we estimated the coefcients
of the cohort effect, u, from the system of equations
(Equation 6), in which the time effect-corrected
incidence rates were used. Assuming v = 1 in Equation 4
and using the estimated cohort effect coefcients u,
the incidence rates can be corrected one more time.
This adjusting procedure aims to correct possible
systematic errors in the observed age-specic incidence
rates, Iij. After such an adjustment, the incidence rates
mainly contain random errors that can be treated by
standard statistical approaches. In the calculations
presented below, we used age-specic incidence
rates adjusted for time period and cohort effects.
Generalized beta model
To t the ltered observational data on age-specic
incidence rates, we tested various models, such as:
a gamma function, a Weibull function, a special
variant of the Beta function proposed by Harding and
u
u
I
I
I
I
I
I
u
u
I
1
2
15 1
15 2
16 2
16 3
17 3
17 4
2
3
14
1
3
1
3
= + +
=
,
,
,
,
,
,
;,,
,
,
,
,
,
,
,
;
1
14 2
15 2
15 3
16 3
16 4
3
4
13 1
13 2
1
3
I
I
I
I
I
u
u
I
I
I
+ +
= + 114 2
14 3
15 3
15 4
4
5
12 1
12 2
13 2
13 3
1
3
,
,
,
,
,
,
,
,
;
I
I
I
u
u
I
I
I
I
+
= + ++
= + +
I
I
u
u
I
I
I
I
I
I
14 3
14 4
5
6
11 1
11 2
12 2
12 3
13 3
13
1
3
,
,
,
,
,
,
,
;
,,
,
,
,
,
,
,
;
4
6
7
10 1
10 2
11 2
11 3
12 3
12 4
1
3
= + +
u
u
I
I
I
I
I
I
= + +
=
;
;
,
,
,
,
,
,
u
u
I
I
I
I
I
I
u
u
I
7
8
9 1
9 2
10 2
10 3
11 3
11 4
8
9
8
1
3
1
3
,,
,
,
,
,
,
.
1
8 2
9 2
9 3
10 3
10 4
I
I
I
I
I
+ +
(6)
By setting u1 = 1, we can nd eight unknowns,
u2, u2, …, u9, from this system of eight Equations.
When the exact mathematical form of the hazard
function h(t) is unknown, we experience the well-
known “identiable problem.” In this case, simultaneous
evaluation of the time period and cohort effects can
only be performed using additional assumptions.8
To solve this problem, we used an iterative technique
proposed by Luebeck and Moolgavkar.15
Initially, we assumed that the cohort effect was
absent (u = 1) and evaluated coefcients of the
coauthors,6 and the Generalized Beta (GB) probability
distribution function dened as:
Ir(T ) = c(bT )k-1 (1 - bT )m-1 (7)
where T = (t - A), t is the age at cancer diagnosis; the
incidence rate Ir (T ) = I(t);
b B A= -1 ( );
A and B are the lower and upper age limits of cancer
development, respectively; c is a generalized rate
constant; k - 1 and m - 1 are the degrees of increase and
decrease in cancer incidence rates, correspondingly.
Mdzinarishvili et al
188 Cancer Informatics 2009:7
Our results suggest that the best t can be obtained
by using the GB function presented by Equation 7.
I(t) can be also presented as:
I t c t A
B A
B t
B A
k m
( )
1 1
=-
-
-
-
- -
(8)
For each age interval, j, the corresponding age-
adjusted incidence rates, I(tj), and their standard
errors (SEj), can be obtained from the SEER data; and
the coefcient of variance, CV, can be determined
as:
CV SE I t
j j j
=( ).
For each age interval, the SEER data presents
the incidence rates as well as the number of cases,
which can be considered as a Poisson distribution.10,14
For large case numbers, the incidence rates can be
viewed as variables that are approximately normally
distributed around expected I(tj) with standard error
SEj. To calculate incidence rates, we considered only
those age intervals that contain at least ve cases;
otherwise we assumed that in the corresponding age
interval the value of the incidence rate was 0.
Values of the A and B age limits of cancer
development can be considered as known a priori.
Traditionally, the lower age limit A has been
chosen as 0, assuming that the process of cancer
development starts from the birth.3 The upper age
limit, B, can be treated as an approximation of
the upper limit of the life span, or the age at which
the best curve-tting is obtained.5,6 It should be
noted that the model variables c, k and m are very
sensitive to variations of the A and B age limits.
In this case, the problem of the curve-tting becomes
a so-called “ill posed” problem. Therefore, the use
of a priori information is necessary to stabilize the
solution against variations of the input parameters,
A and B.16 In this work, for simplicity, we xed the
age interval of cancer development as: A = 0 year and
B = 100 years.
Thus, for each age interval, i, one can calculate
Ti = (ti - A) and obtain Ir (Ti
) = I(tj) and their standard
errors SE[Ir(Ti)] = SEi. Taking logarithms from both
sides of Equation 7 in each age interval, one can
obtain a system of linear Equations:
ln
Ir(Ti) = ln c + (k - 1) ln bTi
+ (m - 1) ln(1 - bTi), i = 1, 2, …, n. (9)
Where n is the number of the considered age
intervals.
Table 2. Presentation of the observed incidence rates as the product of the hazard function, h(t), and the corresponding
time period (v) and birth cohort (u) coefcients.
Period of observation
Age group 1985–89 1990–94 1995–99 2000–04 Birth cohort
Index, i Mid point, tij = 1 j = 2 j = 3 j = 4 Index, lYears
1 2.5
⋅⋅⋅ ⋅⋅⋅
7 32.5 v1u9 h(t7)
8 37.5 v1u8 h(t8)v2u9 h(t8)
9 42.5 v1u7 h(t9)v2u8 h(t9)v3u9 h(t9)
10 47.5 v1u6 h(t10)v2u7 h(t10)v3u8 h(t10)v4u9 h(t10) 9 1955–59
11 52.5 v1u5 h(t11)v2u6 h(t11)v3u7 h(t11)v4u8 h(t11) 8 1950–54
12 57.5 v1u4 h(t12)v2u5 h(t12)v3u6 h(t12)v4u7 h(t12) 7 1945–49
13 62.5 v1u3 h(t13)v2u4 h(t13)v3u5 h(t13)v4u6 h(t13) 6 1940–44
14 67.5 v1u2 h(t14)v2u3 h(t14)v3u4 h(t14)v4u5 h(t14) 5 1935–39
15 72.5 v1u1 h(t15)v2u2 h(t15)v3u3 h(t15)v4u4 h(t15) 4 1930–34
16 77.5 v2u1 h(t16)v3u2 h(t16)v4u3 h(t16) 3 1925–29
17 82.5 v3u1 h(t17)v4u2 h(t17) 2 1920–24
18 87.5+v4u1 h(t18) 1 1915–19
Modeling of cancer incidence rates in aging
Cancer Informatics 2009:7 189
According to Equation 8, the system (Equation 9)
can be rewritten as:
ln ( ) ln ( 1) ln
( 1) ln , 1,
I t c k t A
B A
mB t
B A i
i
i
i
= + - -
-
+ - -
-
=22, ,…n.
(10)
Three unknown parameters, ln c, (k - 1) and
(m - 1), can be determined from Equation 9 or 10
by minimizing the following function R* using a
weighted least square method:
min ( ) ,
1
2
R w O C
i
i
n
i i
*= -
=
∑
(11)
where wi is a weight of the i-th residual, (Oi - Ci),
which is the deviation between the observed value,
Oi, of the ln [Ir(Ti)] and its expected value calculated
by Equation 9 in the following way:
C c k bT m bT
c k t A
B A
i i i
i
= + -
( )
+ -
( )
-
( )
= + - -
-
+
ln ln ln
ln ( ) ln
1 1 1
1 (( ) lnmB t
B A
i
--
-
1
(12)
Based on the rules of the error propagation,17 one
can show that the variance of errors for ln [Ir(Ti)] is
approximately equal to the square of the coefcient
of a variation of the incidence rate:
CV SE I t
i i i
22
=
( )
( ) .
It also can be shown that, when
w CV
i i
=
( )
1 ,
2
the
weighted sum of residuals
R R w O C
i
i
n
i i
* * ( )
ν
= = -
=
∑
1
2
(13)
has a χν
2 distribution with ν = n – p degrees of
freedom, and p = 3 is the number of derived
parameters.18
By numerical experiments, we have shown that
for small variances of error in incidence rate data,
the distribution of errors of ln Ir
(Ti) is close to
normal. On the other hand, systems like systems
(Equations 9 and 10) with normally distributed errors
in the dependent variable, can be solved by multiple
linear regression analysis.19 Therefore, in this work
we used the multiple linear regression analysis to solve
the system (Equation 10). To estimate the goodness
of model tting, we used a standard χ2 test.19
Results and Discussion
Comparison of distributions of the raw
and ltered age-specic incidence rates
for PC and KC
As described in Materials and Methods, we extracted
the raw and ltered age-specic incidence rates for
PC and KC for white males and females collected in
the SEER 9 database during the years of 1985–2004
and combined these data in four time period subsets:
1985–1989, 1990–1994, 1995–1999, and 2000–2004.
The age distributions of the raw and ltered data,
gathered in each of these subsets, are shown in
Figure 1. As can be seen from this gure, the age
patterns corresponding to the raw and ltered incidence
rates for the same type of cancer have signicantly
different amplitudes and shapes. The large differences
in amplitudes are caused by the inclusion of cases
of non-rst primary and metastatic cancer, as well
as cases with microscopically unconrmed tumors
in the raw data. The number of cases excluded by
these criteria are detailed in Table 3. In addition, the
age patterns of the ltered incidence rates exhibit the
existence of a decline at old ages, while in the cases
of unltered data this fall is not evident. The obvious
deceleration/decline that the ltered incidence rates
exhibit at age 75 and over cannot be caused just by
a diagnostic bias at old ages, but rather it strongly
suggests an inuence of basic biological processes
on carcinogenesis and the rates of clinical cancer
manifestation at an old age.20 Because it is clear that
the ltered incidence rates represent more reliable
and more homogeneous statistical data than the raw
data, in the present work we exclusively used the
ltered data.
Figures 2 and 3 show how time period and
cohort adjustments affect the overall shape of age
distributions of PC and KC for white males and
females. Because all adjustments were made by
using data schematically shown in Table 2, these
gures presented the age distributions of nine cohorts
(1915–1919; 1920–1924; 1925–1929; 1930–1934;
1935–1939; 1940–1944; 1945–1949; 1950–1954; and
1955–1959) during four time periods (1985–1989;
1990–1994; 1995–1999; and 2000–2004). Therefore,
during the rst considered time period, 1985–1989,
these nine cohorts exhibit incidence rates in the
following nine ve-year age intervals: 30–34; 35–39;
Mdzinarishvili et al
190 Cancer Informatics 2009:7
40–44; 45–49; 50–54; 55–59; 60–65; 65–69; and
70–74. During the next considered time periods,
these nine cohorts exhibit incidence rates in the nine
ve-year age intervals shifted by ve years compared
to the previous time period.
Thus, for the 30–34 age interval, only one observation
made in the rst time period, 1980–1984, was used.
For the 35–39 age interval, two observations made
in the rst and second time periods, 1980–1984 and
1985–89, correspondingly, were used. Analogously,
three observations for the age interval 40–44 made in
the rst three time periods and four observations for
each of the 50–54, 55–59, 60–64, 65–69 and 70–74
age intervals, made during all considered time periods
were utilized. In the cases of the 75–79, 80–84, and
85+ age intervals, three, two and one observations
were used as shown in Table 2.
Figures 2 and 3 show inuence of adjustments
on PC and KC incidence rates, correspondingly.
Below, we demonstrate that the age distributions of
the adjusted incidence rates for PC and KC can be
very well approximated using the generalized Beta
function dened by Equation 7.
Mathematical models of age distribution
of PC and KC
To estimate the model parameters in Equation 8, we
minimized the weighted sum R* (Equation 11) by
30 40 50 60 70 80 90 100
0
40
80
120
Age (years)
Adjusted incidence rate per 100,000
30 40 50 60 70 80 90 100
0
25
50
Age (years)
Adjusted incidence rate per 100,000
30 40 50 60 70 80 90 100
0
60
120
Age (years)
Adjusted incidence rate per 100,000
30 40 50 60 70 80 90 100
0
50
100
Age (years)
Adjusted incidence rate per 100,000
1985–1989
1990–1994
1995–1999
2000–2004
BA
DC
Figure 1. Age-specic incidence rates of PC and KC obtained from the raw and ltered SEER data. (A) PC in white males; (B) PC in white females;
(C) KC in white males; (D) KC in white females.
Notes: Incidence rates obtained from the raw and ltered data are shown as dashed and solid lines, respectively. Time periods of data presented in
panel A are shown in the legend; time periods presented in panels B, C, and D are the same as in panel A.
Modeling of cancer incidence rates in aging
Cancer Informatics 2009:7 191
Table 3. Age distribution of number of cases (shown as numerators) excluded from the total number of cases (shown as denominators) of KC and PC
in white males and white females. “Not rst primary” denotes cases excluded due to secondary cancers or additional primary cancers. “Not conrmed”
denotes cases excluded because the diagnosis was not microscopically conrmed by a pathologist. “Other” denotes cases excluded due to diagnosis
by death certicate or autopsy record, in situ cancers, other non malignant tumors, and data records erroneously marked as rst primary. Percentage of
cases excluded is shown in parentheses.
Male KC Female KC
Age Not rst primary Not conrmed Other Not rst primary Not conrmed Other
30–34 13/213 (6.1%) 4/213 (1.9%) 18/213 (8.5%) 11/168 (6.5%) 1/168 (0.6%) 12/168 (7.1%)
35–39 27/521 (5.2%) 3/521 (0.6%) 18/521 (3.5%) 21/313 (6.7%) 2/313 (0.6%) 10/313 (3.2%)
40–44 71/1074 (6.6%) 23/1074 (2.1%) 53/1074 (4.9%) 51/605 (8.4%) 10/605 (1.7%) 25/605 (4.1%)
45–49 125/1801 (6.9%) 43/1801 (2.4%) 67/1801 (3.7%) 108/824 (13.1%) 12/824 (1.5%) 37/824 (4.5%)
50–54 213/2424 (8.8%) 66/2424 (2.7%) 110/2424 (4.5%) 163/1251 (13.0%) 34/1251 (2.7%) 72/1251 (5.8%)
55–59 398/3112 (12.8%) 114/3112 (3.7%) 195/3112 (6.3%) 238/1576 (15.1%) 42/1576 (2.7%) 79/1576 (5.0%)
60–64 667/3738 (17.8%) 168/3738 (4.5%) 219/3738 (5.9%) 394/1876 (21.0%) 94/1876 (5.0%) 125/1876 (6.7%)
65–69 938/4013 (23.4%) 203/4013 (5.1%) 281/4013 (7.0%) 482/2321 (20.8%) 146/2321 (6.3%) 164/2321 (7.1%)
70–74 1116/4061 (27.5%) 285/4061 (7.0%) 371/4061 (9.1%) 537/2415 (22.2%) 234/2415 (9.7%) 194/2415 (8.0%)
75–79 1029/3359 (30.6%) 438/3359 (13.0%) 321/3359 (9.6%) 602/2396 (25.1%) 326/2396 (13.6%) 224/2396 (9.3%)
80–84 690/2088 (33.0%) 466/2088 (22.3%) 234/2088 (11.2%) 398/1726 (23.1%) 413/1726 (23.9%) 202/1726 (11.7%)
Male PC Female PC
Age Not rst primary Not conrmed Other Not rst primary Not conrmed Other
30–34 2/64 (3.1%) 6/64 (9.4%) 3/64 (4.7%) 4/55 (7.3%) 1/55 (1.8%) 4/55 (7.3%)
35–39 11/186 (5.9%) 13/186 (7.0%) 6/186 (3.2%) 3/105 (2.9%) 2/105 (1.9%) 0/105 (0.0%)
40–44 12/410 (2.9%) 20/410 (4.9%) 12/410 (2.9%) 21/232 (9.1%) 9/232 (3.9%) 15/232 (6.5%)
45–49 40/776 (5.2%) 60/776 (7.7%) 22/776 (2.8%) 45/501 (9.0%) 21/501 (4.2%) 23/501 (4.6%)
50–54 68/1314 (5.2%) 117/1314 (8.9%) 30/1314 (2.3%) 85/843 (10.1%) 49/843 (5.8%) 28/843 (3.3%)
55–59 145/1826 (7.9%) 181/1826 (9.9%) 45/1826 (2.5%) 132/1292 (10.2%) 96/1292 (7.4%) 37/1292 (2.9%)
60–64 262/2497 (10.5%) 279/2497 (11.2%) 78/2497 (3.1%) 250/1792 (14.0%) 187/1792 (10.4%) 58/1792 (3.2%)
65–69 409/3139 (13.0%) 437/3139 (13.9%) 114/3139 (3.6%) 388/2742 (14.2%) 350/2742 (12.8%) 91/2742 (3.3%)
70–74 648/3556 (18.2%) 655/3556 (18.4%) 153/3556 (4.3%) 574/3491 (16.4%) 613/3491 (17.6%) 131/3491 (3.8%)
75–79 745/3243 (23.0%) 815/3243 (25.1%) 163/3243 (5.0%) 621/3871 (16.0%) 1044/3871 (27.0%) 223/3871 (5.8%)
80–84 563/2369 (23.8%) 886/2369 (37.4%) 153/2369 (6.5%) 588/3329 (17.7%) 1377/3329 (41.4%) 232/3329 (7.0%)
Mdzinarishvili et al
192 Cancer Informatics 2009:7
a least squares method. To do this, we utilized the
regress function of the MATLAB software package.21
For this function, we used n = 35 values of the adjusted
incidence rates Iij(ti), and their SEi as input data. The
rate for the 85 + age interval was not used due to an
uncertainty of its middle point position. This resulted
in three estimated model parameters, c, k, and m,
and their 95% condence intervals (CI), assuming
that A = 0 and B = 100. The obtained parameters
are supposed to determine the best curve tting that
system (Equation 10) can provide for the adjusted
incidence rates that were used as input data.
We examined the goodness of the curve t
by the χ
2 test for the values of the weighted sum
of residuals R*ν (Equation 13). The degrees of
freedom, ν = n – p, was dened by the number of
used data points (n = 35), less the number of derived
parameters ( p = 3). According to the standard
χ
2 test with the 0.05 signicance level, if the value
of R*
ν was outside the interval ( χ
2
0.025,ν; χ
2
0.975,ν),
we would reject the hypothesis that our model
ts the observed data. The two tail limit values of
the χ
2 test were χ
2
0.025,32 = 18.3 and χ
2
0.975,32 = 49.5,
correspondingly. Therefore, in the case when
the weighted sum of residuals, R*
ν
, is within the
interval 18.3 R*
ν 49.5, one can conclude that
the null hypothesis (that the distribution of the
modeled incidence rates obtained by Equation 10
30 40 50 60 70 80 90 100
0
20
40
60
Age (years)
Adjusted incidence rate per 100,000
30 40 50 60 70 80 90 100
0
20
40
60
Age (years)
Adjusted incidence rate per 100,000
30 40 50 60 70 80 90 100
0
20
40
60
Age (years)
Adjusted incidence rate per 100,000
30 40 50 60 70 80 90 100
0
20
40
60
Age (years)
Adjusted incidence rate per 100,000
1985–1989
1990–1994
1995–1999
2000–2004
B
A
D
C
Figure 2. Age-specic incidence rates for PC in white males and females obtained from the ltered SEER data. (A) Unadjusted for time period
and cohort effects incidence rates for white males; (B) adjusted for time period and cohort effects incidence rates for white males; (C) unadjusted for time
period and cohort effects incidence rates for white females; (D) adjusted for time period and cohort effects incidence rates for white females.
Notes: Time periods of data presented in panel A are shown in the legend. Time periods presented in panels B, C and D are the same as in panel A.
Modeling of cancer incidence rates in aging
Cancer Informatics 2009:7 193
ts the set of adjusted values) is not rejected at the
0.05 signicance level.
Table 4 presents the obtained values of model
parameters and descriptive statistics. Figure 4 shows
the results of modeling of the age-specic incidence
rates of PC and KC for the white male and female
populations. As was mentioned previously, the
incidence rates for the age interval 85+ were not used
as input data for the curve tting. These incidence
rates are shown on Figure 4 only for illustration
purposes. Figure 4 shows that the modeled incidence
rates well approximate the adjusted values of the
observed incidence rates (including the 85+ point).
This good visual t is strongly supported by the χ2
tests, which suggest that in all cases the values of the
weighted sum of residuals, R*
ν
, are within the given
interval, 18.3 R*ν 49.5 (see Table 4).
In Table 4 the parameters k and m assess the degrees
of the increase and decrease in cancer incidence rate,
correspondingly. The parameter c is related to an
overall risk of getting a cancer for a given population.
The point of inection indicates the point in which
the second derivative of I(t) (Equation 7) is equal
to 0, which corresponds to the age at which the
decrease begins to prevail over the increase in cancer
incidence rate. We did not consider the second point
of inection due to uncertainties of data at very old
ages. The maximum indicates the age at which the
30 40 50 60 70 80 90 100
0
10
20
30
Age (years)
Age-specific incidence rate/100,000
30 40 50 60 70 80 90 100
0
10
20
30
Age (years)
Adjusted incidence rate per 100,000
30 40 50 60 70 80 90 100
0
20
40
60
Age (years)
Adjusted incidence rate per 100,000
30 40 50 60 70 80 90 100
0
20
40
60
Age (years)
Adjusted incidence rate per 100,000
1985–1989
1990–1994
1995–1999
2000–2004
B
A
DC
Figure 3. Age-specic incidence rates for KC in white males and females obtained from the ltered SEER data. (A) Unadjusted for time period
and cohort effects incidence rates for white males; (B) adjusted for time period and cohort effects incidence rates for white males; (C) unadjusted for time
period and cohort effects incidence rates for white females; (D) adjusted for time period and cohort effects incidence rates for white females.
Notes: Time periods of data presented in panel A are shown in the legend. Time periods presented in panels B, C and D are the same as in panel A.
Mdzinarishvili et al
194 Cancer Informatics 2009:7
Table 4. Descriptive statistics and model parameters for pancreatic and kidney cancer.
Pancreatic cancer Kidney cancer
Male Female Male Female
Number of cancer cases (n) 13,919 13,306 20,054 12,106
Parameter c × 10,000†2.9 (2.1, 4.1) 1.5 (1.0, 2.4) 2.5 (1.9, 3.3) 0.3 (0.2, 0.5)
Parameter k†10.2 (9.8, 10.5) 10.2 (9.8, 10.7) 8.9 (8.6, 9.1) 7.8 (7.4, 8.2)
Parameter m†3.7 (3.6, 3.9) 3.4 (3.2, 3.6) 3.9 (3.8, 4.0) 3.2 (2.9, 3.4)
Weighted sum of residuals (R*ν
)25.0 39.0 24.5 46.0
Point (age) of inection 64.3 66.7 58.6 60.6
Age of maximum incidence 77.0 79.1 72.8 75.7
†Numbers in parentheses denote 95% condence intervals.
30 40 50 60 70 80 90 100
0
10
20
30
Age (years)
Adjusted incidence rate per 100,000
30 40 50 60 70 80 90 100
0
20
40
60
Age (years)
Adjusted incidence rate per 100,000
30 40 50 60 70 80 90 100
0
20
40
60
Age (years)
Adjusted incidence rate per 100,000
30 40 50 60 70 80 90 100
0
20
40
60
Age (years)
Adjusted incidence rate per 100,000
1985–1989
1990–1994
1995–1999
2000–2004
BA
DC
Figure 4. The GB approximation of the age-specic incidence rates adjusted for time period and cohort effects. (A) PC in white males; (B) PC in
white females; (C) KC in white males; (D) KC in white females.
Note: Error bars denote standard errors (SE).
Modeling of cancer incidence rates in aging
Cancer Informatics 2009:7 195
incidence rate reaches its maximum, after which the
cancer incidence rate declines.
Table 4 and Figures 4A and 4B suggest that the
parameter c for PC in white males is higher than for
white females. However, the condence intervals
of this parameter for males and females are slightly
overlapping. For PC, the values of the parameter k in
males and females are statistically indistinguishable,
while the values of parameter m for males are
statistically higher than for females. The higher
value of m in males is a result of the point (age) of
inection and maximum incidence rate, which occur
about two years earlier in males than in females.
This may suggest that biological mechanisms of PC
development differ in white males and females.
As can be seen from Table 4 and Figures 4C and 4D,
in the case of KC all three model parameters, c, k,
and m, for white males are statistically higher than
those for white females. The point of inection for
males appears two years earlier than that for females,
while the maximum of cancer incidence rate for
males appears about three years earlier than that for
females. Notable differences in all parameters that
characterize the age distributions of KC in males and
females may suggest distinct biological mechanisms
of KC development in white males and females.
A comparative analysis of the age patterns of the
PC and KC incidence rates (see Table 4) suggests that
for these types of cancer, the value of parameter m is
greater than 2. This is in contrast to the assumption
made by Harding and coauthors,6 where it was
postulated that for all types of cancers, m should
be equal to 2. As for parameter k, for PC in males
and females the values of k are statistically higher
than the ones for KC, which resulted in an earlier
presentation of inection points for KC incidence rates
than for the PC (see Table 4). Analogous comparisons
show that the peaks of the KC incidence rates appear
four years earlier than the corresponding ones of
the PC. These comparisons suggest the existence
of distinct organ-specic biological mechanisms of
the carcinogenesis in the pancreas and kidney.
Goodness of curve tting
for different model functions
Among existing models,2,3,5,6,11,15 only special types
of the Beta function as proposed by Pompei and
Wilson,5 and Harding and coauthors,6 have an ability
to describe the turnover of incidence rates at old age
(for other models, incidence rates are monotonically
increasing). There are other well-known statistical
models, such as the Gamma and Weibull functions,
which, in principle, can also be used to describe this
turnover. Therefore, using the weighted least squares
method, we compared the goodness-of-t of our
proposed GB function with the goodness-of-t of
the special type of the Beta function (PW model), as
well as the Gamma and Weibull functions. Figure 5A
20 30 40 50 60 70 80 90 100 110
0
25
50
Age (years)
incidence rate per 100,000
20 30 40 50 60 70 80 90 100
0
25
50
Age (years)
incidence rate per 100,000
BA
Figure 5. Comparison of goodness-of-t of the KC incidence rates in white males performed by the GB model (solid line) and other models. (A) The GB
model vs. the PW model (dotted line). (B) The GB model vs. models described by the Gamma (dotted line) and Weibull (dashed line) functions.
Note: Error bars denote standard errors (SE).
Mdzinarishvili et al
196 Cancer Informatics 2009:7
shows the comparison of tting of observational KC
data for white males using the GB model versus the
PW model. Figure 5B shows an analogous comparison
of the GB, Gamma, and Weibull functions.
As can be seen from Figure 5, the GB model ts
the pattern of the KC incidence rates in white males
much better than the other considered models, and
this visual appearance is well supported by the use
of χ
2 statistic. In fact, the data from Table 4 show that
the observed incidence rates of KC in white males
can be very well tted by the GB model. However,
for the PW model and the Gamma and Weibull
functions, the standard χ
2 test rejects the hypotheses
that the curves described by these functions t the
observed data with the 0.05 signicance level. In fact,
for the PW model, the value of this statistic is equal
to 565.0 with ν = 32 degrees of freedom (where
ν = n - p, n = 35 – the number of used data points,
and p = 3 - the number of parameters to be derived).
Analogously, for the Gamma function (ν = 33,
two estimated parameters) this statistic is equal to
1,720.6, and for the Weibull function this statistic is
equal to 208.0 (ν = 33, two estimated parameters). For
the special type of Beta function, the limit values of
the χ
2 test are χ
2
0.025,32 = 18.3 and χ
2
0.975,32 = 49.5, and
for the Gamma and Weibull functions, these values
are χ
2
0.025,33 = 19.0 and χ
2
0.975,33 = 50.7. For all of these
functions the values of the χ
2 statistic are outside of
the dened intervals of the χ
2 test. Analogous results
were obtained for KC incidence rates in white females,
as well as for PC incidence rates in both white males
and white females (data not shown). Therefore, these
results clearly show that the GB model has superior
performance compared to the other considered models.
Conclusion
In this work, we emphasized several general
shortcomings in the mathematical modeling of age
distribution of cancer, which include: the use of
“raw” cancer data (inclusion of cases which were not
microscopically conrmed or were not rst primary
cancers); the lack of consideration of time period and
cohort effects on the observed incidence rates; and
the omission of rigorous statistical evaluation of the
determined model parameters. To overcome these
shortcomings, we proposed a new approach, called
the Generalized Beta (GB) model. This model utilizes
observational data of age-specic incidence rates
and uses sound statistical criteria to assess model
parameters.
To test the performance of this model, we used
“ltered” data from the SEER 9 database during
the years of 1985–2004. We utilized these data to
estimate the incidence rates of the rst primary,
microscopically conrmed cases of pancreatic cancer
(PC) and kidney cancer (KC) in white males and
females. These incidence rates were adjusted for time
period and cohort effects. By the newly proposed GB
approach, we approximated the adjusted incidence
rates of the primary PC and KC in white males and
females. Condence intervals for model parameters
were estimated by regression analysis. We showed
that the age distributions of the KC and PC incidence
rates have turnover points within the age interval of
74–81, after which these distributions fall off and
reach the value of 0 (near the age of 100 years) at the
end of the human life span.
The results presented in this work suggest that our
approach signicantly expands the possibilities of
modeling of age distributions in PC and KC. We are
certain that this approach could be generalized for
many other organ-specic cancers and cancer
subtypes and provide distinct model parameters
that will be useful for the modeling of carcinogenic
processes characteristic to particular cancers. It should
be noted that in this work, we used the terms
degree of increase or degree of decrease in a purely
mathematical sense, because the precise mechanisms
causing the increase and decrease of cancer incidence
rates are not fully understood. To better understand
the biological plausibility of the model parameters
used in the proposed approach, detailed molecular,
cellular and tissue-specic mechanisms underlying
the development of each type of cancer will require
further investigation. The model parameters that can
be assessed by the proposed approach should challenge
future biomedical and epidemiological studies.
Disclosure
The authors report no conicts of interest.
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