Differences in predictions of ODE models of tumor growth: a cautionary example

Murphy et al. BMC Cancer (2016) 16:163 DOI 10.1186/s12885-016-2164-x RESEARCH ARTICLE Open Access Differences in predictions of ODE models of tumor growth: a cautionary example Hope Murphy1 , Hana Jaafari2 and Hana M. Dobrovolny2* Abstract Background: While mathematical models are often used to predict progression of cancer and treatment outcomes, there is still uncertainty over how to best model tumor growth. Seven ordinary differential equation (ODE) models of tumor growth (exponential, Mendelsohn, logistic, linear, surface, Gompertz, and Bertalanffy) have been proposed, but there is no clear guidance on how to choose the most appropriate model for a particular cancer. Methods: We examined all seven of the previously proposed ODE models in the presence and absence of chemotherapy. We derived equations for the maximum tumor size, doubling time, and the minimum amount of chemotherapy needed to suppress the tumor and used a sample data set to compare how these quantities differ based on choice of growth model. Results: We find that there is a 12-fold difference in predicting doubling times and a 6-fold difference in the predicted amount of chemotherapy needed for suppression depending on which growth model was used. Conclusion: Our results highlight the need for careful consideration of model assumptions when developing mathematical models for use in cancer treatment planning. Keywords: Tumor growth, Mathematical model, Ordinary differential equation, Cancer, Chemotherapy Background Cancer is a leading cause of death and places a heavy burden on the health care system due to the chronic nature of the disease and the side effects caused by many of the treatments [1–3]. Much research effort is spent improving the efficacy of current treatments [4] and on developing new treatment modalitites [5–9]. As cancer treatment moves towards personalized treatment, mathematical models will be important component of this research, helping to predict the time course of the tumor and optimizing treatment regimens [10, 11]. Mathematical models are used in a number of ways to help understand and treat cancer. Models are used to understand how cancer develops [12] and grows [13–16]. They are used to optimize [17, 18] or even personalize [11, 19, 20] current treatment regimens; predict the efficacy of new treatments [21] or combinations of different therapies [22–24]; and give insight into the development of *Correspondence: 2 Department of Physics & Astronomy, Texas Christian University, 2800 S. University Drive, 76129, Fort Worth, TX, USA Full list of author information is available at the end of the article resistance to treatment [25, 26]. While models have great potential to improve development and implementation of cancer treatment, they will only realize this potential if they provide accurate predictions. The basis of any mathematical model used to study treatment of cancer is a model of tumor growth. This paper focuses on ordinary differential equation (ODE) models of tumor growth. A number of ODE models have been proposed to represent tumor growth [27, 28] and are regularly used to make predictions about the efficacy of cancer treatments [29]. Unfortunately, choice of a growth model is often driven by ease of mathematical analysis rather than whether it provides the best model for growth of a tumor [27]. Some researchers have attempted to find the “best” ODE growth model by fitting various models to a small number of experimental data sets of tumor growth [30–33]. Taken altogether, the results are rather inconclusive, with results suggesting that choice of growth model depends at least in part on the type of tumor [31, 32]. This leaves modelers with little guidance in choosing a tumor growth model. © 2016 Murphy et al. 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. Murphy et al. BMC Cancer (2016) 16:163 Page 2 of 10 Many researchers realize that improper choice of growth model is problematic [27] and can lead to differences in predictions of treatment outcomes [28, 29]. However, there has not yet been a study that compares and quantifies differences in predictions of the various models and how these differences affect predictions of treatment outcomes. This paper presents results of analysis of the various ODE growth models highlighting their predictions of tumor growth in the presence and absence of chemotherapy. We also fit the models to sample experimental tumor growth data sets and find a wide range of predicted outcomes based on the choice of growth model. Logistic: The logistic (or Pearl-Verhulst) equation was created by Pierre Francois Verhulst in 1838 [36]. This model describes the growth of a population that is limited by a carrying capacity of b. The logistic equation assumes that the growth rate decreases linearly with size until it equals zero at the carrying capacity. Linear: The linear model assumes initial exponential growth that changes to growth that is constant over time. In our formulation of the model, the initial exponential growth rate is given by a/b and the later constant growth is a. The model was used in early research to analyze growth of cancer cell colonies [16]. Methods Mathematical models Early studies of tumor growth were concerned with finding equations to describe the growth of cancer cells [13–16] and many of the models examined here were proposed at that time. The models predict the growth of a tumor by describing the change in tumor volume, V, over time. The model equations used in this analysis are presented in Table 1 and the models are described below. a, b, and c are parameters that can be adjusted to describe a particular data set. Exponential: In the early stages of tumor growth, cells divide regularly, creating two daughter cells each time. A natural description of the early stages of cancer growth is thus the exponential model [34], where growth is proportional to the population. The proportionality constant a is the growth rate of the tumor. This model was often used in early analysis of tumor growth curves [13–16] and appears to work quite well at predicting early growth. It is known to fail, however, at later stages when angiogenesis and nutrient depletion begin to play a role [27, 32]. Mendelsohn: A generalization of the exponential growth model was introduced by Mendelsohn [35]. In this model, growth is proportional to some power, b, of the population. Table 1 ODE models of tumor grow (...truncated)