Kareva Biology Direct (2016) 11:37 DOI 10.1186/s13062-016-0139-0 HYPOTHESIS Open Access Primary and metastatic tumor dormancy as a result of population heterogeneity Irina Kareva Abstract Existence of tumor dormancy, or cancer without disease, is supported both by autopsy studies that indicate presence of microscopic tumors in men and women who die of trauma (primary dormancy), and by long periods of latency between excision of primary tumors and diseaserecurrence(metastaticdormancy).Withindormant tumors, two general mechanisms underlying the dynamics are recognized, namely, the population existing at limited carrying capacity (tumor mass dormancy), and solitary cell dormancy, characterized by long periods of quiescence marked by cell cycle arrest. Here we focus on mechanisms that precede the avascular tumor reaching its carrying capacity, and propose that dynamics consistent with tumor dormancy and subsequent escape from it can be accounted for with simple models that take into account population heterogeneity. We evaluate parametrically heterogeneous Malthusian, logistic and Allee growth models and show that 1) time to escape from tumor dormancy is driven by the initial distribution of cell clones in the population and 2) escape from dormancy is accompanied by a large increase in variance, as well as the expected value of fitness-determining parameters. Based on our results, we propose that parametrically heterogeneous logistic model would be most likely to account for primary tumor dormancy, while distributed Allee model would be most appropriate for metastatic dormancy. We conclude with a discussion of dormancy as a stage within a larger context of cancer as a systemic disease. Reviewers: This article was reviewed by Heiko Enderling and Marek Kimmel. Keywords: Tumor dormancy, Quiescence, Population heterogeneity, Mathematical model Introduction A number of mechanisms have been proposed to ac- Long latency periods between treatment of primary count to explain the phenomenon of tumor dormancy. tumors and appearance of metastatic disease suggest These include cell cycle arrest [15–17], tumor manage- existence of tumor dormancy, which has been charac- mentbytheimmunesystem[18–20], age-dependent terized as “cancer without disease” [1]. It appears that decreased relative fitness with respect to somatic cells tumor dormancy can manifest itself not only in delayed [21], inability to secure blood supply [22–25], among metastatic recurrences [2–5] but also in primary tumors. others [15, 26, 27]. Unfortunately, there exist technical Evidence for primary tumor dormancy is commonly found difficulties with investigating many of the hypotheses. in autopsy studies, where men and women who died from Even if one is successful at isolating appropriate cell trauma have been found to have a surprisingly high num- lines [22, 23, 28, 29], the typical experimental models ber of microscopic clinically unapparent tumors [6–9]. involve mouse xenographs, which may or may not be Furthermore, evidence from donor transplants has shown applicable for these questions because of inter-species that microscopic tumors can be transmitted from donors compatibilities. Development of a strong theoretical who passed pre-transplantation screening as being free framework, including mathematical modeling, is thus from cancer for over 10–15 years [10–14], further impli- necessary to further our understanding of this phenomenon. cating the existence of cancer without disease. Two main mechanisms for tumor dormancy are gen- erally recognized: tumor mass dormancy and dormancy of solitary cancer cells, or quiescence [30–33]. Tumor Correspondence: [email protected] mass dormancy is characterized by a balance of prolif- Simon A. Levin Mathematical, Computational and Modeling Sciences Center eration and apoptosis and can be viewed as the tumor (SAL MCMSC), Arizona State University, Tempe, AZ, USA © 2016 The Author(s). Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated. Kareva Biology Direct (2016) 11:37 Page 2 of 18 having reached its pre-angiogenic carrying capacity can be applied to understanding one of the possible [15]. Changes in the carrying capacity, whether due to mechanisms underlying tumor dormancy. variations in nutrient availability [34], or acquisition of In our simulations, we will evaluate parameters necessary the ability to recruit blood vessels and avoid immune to reproduce dynamics consistent with escape from dor- surveillance [18], would allow for tumor escape. In con- mancy over approximately 10-year period (120 months). trast, solitary tumor dormancy, or quiescence, is char- Since we are focusing on modeling tumor dynamics in acterized by lack of proliferation due to long periods of the time preceding it reaching its carrying capacity, we cell cycle arrest. While both mechanisms can lead to are looking at the microscopic tumor reaching a size of the phenomenon of latent tumor growth, Wells et al. approximately 1mm3, which is a limit imposed on non- argue based on their computer simulation study [35] angiogenic tumor size by oxygen and nutrient con- that quiescence is a more likely mechanism to underlie straints [22]. We realize that there exist tissue and metastatic dormancy. tumor-specific variations, and would like to emphasize In our previous work, we have looked at several ex- that these estimates can be varied based on data. Our amples of tumor growth being impacted by changes in goal is to demonstrate that qualitative behavior of dynamic carrying capacity [36]. Here, we will focus on tumor dormancy and escape from it can be reproduced studying a possible mechanism of escape from dor- with these models. mancy that stems not from external limitations but from intrinsic growth laws. Specifically, we propose that Parameters and distributions behavior consistent with escape from dormancy can Choice of initial distribution come as a result of natural growth dynamics of hetero- According to the principle of maximum entropy (MaxEnt), geneous populations. Therefore, we will focus on look- if the mean value of the random variable is the only ing at the tumor dynamics before it reaches its carrying quantity that can be estimated from observations or capacity as allowed by its microenvironment. other data, then the most likely distribution of the vari- We will evaluate three different growth functions for able is exponential with the estimated mean [42]. Since heterogeneous populations. Specifically, we will look at no value of a biological characteristic, such as a growth parametrically heterogeneous populations that grow ac- rate, can be infinite, then we should choose the trun- cording to Malthusian, logistic and Allee growth laws. cated exponential distribution in this interval as the ini- Several of these examples have been investigated in a tial distribution. Any other distribution can of course different context by Karev [37]. We will demonstrate be used, if there exist data or theoretical considerations that a dynamics that is consistent with escape from underlying its choice. tumor dormancy after a long period of latency can be generated due solely to heterogeneous nature of the Parametrically heterogeneous Malthusian growth populations described by these three models. In addition Consider a population of cells xc(t), where each individ- to changes in total population size, we will look at ual cell is characterized by an intrinsic growth rate c.A changes in expected values of intrinsic parameters, as set of cells that are characterized by the same value of c well as variance and clone distributions, to demonstrate is referred to as c-clone.X The total size of the population what alterations occur in the time period preceding es- is given by NtðÞ¼ xcðÞt if c takes on discrete values cape from dormancy. We will conclude with a discussion Z A of how this stage of tumor growth fits into a larger con- and NtðÞ¼ xcðÞt dc if c is continuous. A denotes the text of cancer as a systemic disease. A range of possible values of parameter c. The rate of change of xc(t) is given by Materials and methods ðÞ0¼ ðÞ: ð Þ In order to investigate how dynamics consistent with xc t cxc t 1 tumor dormancy and subsequent escape from it can re- ct Solution to this equation is xc(t)=e xc(0). Distribution sult solely from population heterogeneity, we will require ðÞ of clones is by definition P ðÞ¼t xc t . From the defin- the use of the parameter distribution technique, also c NtðÞ known as the Reduction Theorem. It was well described ition of N(t) we can get that in [37–40], and main results have been summarized in Z Z ct [41]. We will apply it to describing the dynamics of para- NtðÞ¼ xcðÞt dc ¼ xcðÞ0 e dc A Z A metrically heterogeneous populations that grow accord- ¼ ðÞ ct ðÞ ¼ ðÞ ½; ð Þ ing to Malthusian, logistic and Allee growth laws. Our N 0 e Pc 0 dc N 0 M0 t 2 A goal here is to use methods developed by G. Karev in ct aforementioned publications to show how these results where M0[t]=∫Ae Pc(0)dc is by definition the moment Kareva Biology Direct (2016) 11:37 Page 3 of 18 generating function of the initial distribution of c-clones μþt in the population. t½¼ 1 − e : ð Þ Var c 2 μ 2 8 It can be shown [37] that the expected value of param- ðÞμ−t ðÞe −et eter c is given by As mentioned above, we are looking to reproduce the ½ dM0 t growth dynamics that commences with a period of very Et½¼c =M ½t ð3Þ dt 0 slow growth for a period of approximately 120–240 months (10–20 years), which then rapidly increases, and that variance of parameter c is given by reaching a size of ≈ 1mm3 as allowed by oxygen and nu- trient limitations [22].