SCientifiC RepoRts | (2018) 8:3388 | DOI:10.1038/s41598-018-21734-x
A Simple 3-Parameter Model for
, Holger Fröhlich
, Dima Grigoriev
, Sergey Vakulenko
& Andreas Günter Weber
We propose a simple 3-parameter model that provides very good ts for incidence curves of 18 common
solid cancers even when variations due to dierent locations, races, or periods are taken into account.
From a data perspective, we use model selection (Akaike information criterion) to show that this
model, which is based on the Weibull distribution, outperforms other simple models like the Gamma
distribution. From a modeling perspective, the Weibull distribution can be justied as modeling the
accumulation of driver events, which establishes a link to stem cell division based cancer development
models and a connection to a recursion formula for intrinsic cancer risk published by Wu et al. For
the recursion formula a closed form solution is given, which will help to simplify future analyses.
Additionally, we perform a sensitivity analysis for the parameters, showing that two of the three
parameters can vary over several orders of magnitude. However, the shape parameter of the Weibull
distribution, which corresponds to the number of driver mutations required for cancer onset, can be
robustly estimated from epidemiological data.
Cancers arise aer accumulating epigenetic and genetic aberrations
. Earlier studies established a power law
model on the basis of multi-stage somatic mutation theory to explain age-dependent incidences
for several cancer
types. As noted by Hornsby et al.
in the context of classical epidemiological studies most cancers occur with the
same characteristic pattern of incidence, and the simplicity of this pattern is in contrast to the perceived complexity
of carcinogenesis. Orthogonal to these age stratication of dierent cancer types, Tomasetti and Vogelstein
) reported a signicant association between life time caner risk and stem cell divisions and concluded
the latter substantially contributes to the former. Challenging the conclusion of Tomasetti and Vogelstein
high-intrinsic cancer risk Wu et al.
subdivided cancer risk into extrinsic and intrinsic risk, arguing extrinsic factors
contribute more to cancer risks than intrinsic factors do. Based on a mechanistic model of accumulated mutations,
these authors provided a recursion formula for theoretical life time intrinsic risk (tLIR) parameterized by age a. is
recursion formula has the closed form solution
artLIR( )1(1 (1 (1 )))
, where S can be
interpreted as the numbers of stem cell, d as the stem cell division rate, k as number of driver events required for
cancer onset and r as the mutation rate per division. ey reported that tLIR goes outside of the plausible range of
empirical cancer risks by studying several pairs of values for two parameters (mutation rate and driver gene muta-
tions) concluding that there is a substantial contribution of extrinsic risk factors to cancer development. However,
this conclusion only holds in the studied parameter space and when parameters for all cancer types are treated
uniformly. By performing a systematic grid search in the space of biologically plausible parameter values we showed
that tLIR can be close to empirical risk for dierent cancer types (R
> 0.85). If the extrinsic risk factor is computed
by simply setting it to a complement of 1 for the intrinsic risk factor as performed by Wu et al. it will be concluded
that there is a possibility of high intrinsic risk, so that one of the presented arguments by Wu et al.
On a pure mathematical side, we show that a scaled Weibull function with 3 parameters approximates the
4-parameter mechanistic tLIR model. On an epidemiological data analytical side, this simple 3-parameter model
excellently agrees with age-dependent cancer incidence curves among 18 common solid cancers even when
variations due to dierent locations, races, or periods are taken into account. With this model, we study the
Bonn-Aachen International Center for Information Technology, Dahlmannstraße 2, Bonn, 53113, Germany.
Department of Medicine II, Klinikum Rechts der Isar, Technische Universität München, München, 81675, Germany.
German Cancer Consortium (DKTK), German Cancer Research Center (DKFZ), Heidelberg, 69120, Germany.
Biosciences GmbH, Alfred-Nobel-Straße 10, Monheim, 40789, Germany.
CNRS, Mathématiques, Université de Lille,
Villeneuve d’Ascq, 59655, France.
Institute for Mechanical Engineering Problems, Russian Academy of Sciences,
Saint Petersburg, Russia.
Saint Petersburg National Research University of Information Technologies, Mechanics
and Optics, Saint Petersburg, Russia.
Institut für Informatik II, Universität Bonn, Friedrich-Ebert-Allee 144, Bonn,
Germany. Correspondence and requests for materials should be addressed to A.G.W. (email: firstname.lastname@example.org)
Received: 30 June 2017
Accepted: 9 February 2018
Published: xx xx xxxx