Modeling Spontaneous Metastasis following Surgery: An In Vivo-In Silico Approach

Abstract

Rapid improvements in the detection and tracking of early-stage tumor progression aim to guide decisions regarding cancer treatments as well as predict metastatic recurrence in patients following surgery. Mathematical models may have the potential to further assist in estimating metastatic risk, particularly when paired with in vivo tumor data that faithfully represent all stages of disease progression. Herein we describe mathematical analysis that uses data from mouse models of spontaneous metastasis developing after surgical removal of orthotopically implanted primary tumors. Both presurgical (primary tumor) and postsurgical (metastatic) growth was quantified using bioluminescence and was then used to generate a mathematical formalism based on general laws of the disease (i.e. dissemination and growth). The model was able to fit and predict pre-/post-surgical data at the level of the individual as well as the population. Our approach also enabled retrospective analysis of clinical data describing the probability of metastatic relapse as a function of primary tumor size. In these data-based models, inter-individual variability was quantified by a key parameter of intrinsic metastatic potential. Critically, our analysis identified a highly nonlinear relationship between primary tumor size and postsurgical survival, suggesting possible threshold limits for the utility of tumor size as a predictor of metastatic recurrence. These findings represent a novel use of clinically relevant models to assess the impact of surgery on metastatic potential and may guide optimal timing of treatments in neoadjuvant (presurgical) and adjuvant (postsurgical) settings to maximize patient benefit. Major findings A mathematical model was used to connect presurgical primary tumor volume and postsurgical metastatic burden and survival in two clinically relevant animal models of spontaneous metastasis and one clinical dataset of metastatic relapse probability in breast cancer patients. This model used one specific parameter to quantify differential metastatic aggressiveness, which could be of help for personalizing adjuvant therapy. Simulations revealed a highly nonlinear relationship between resected primary tumor size and metastatic recurrence. These results uncover a computable and patient-dependent threshold for evaluating the efficacy of surgery on overall survival.

On the growth and dissemination laws in a mathematical model of metastatic growth

“…We refer to [21] for a detailed description. Briefly, dynamics of the primary tumor volume V p (t) at time t is defined by the following Cauchy problem:…”

“…For calibration of the conversion ratio from number of cells to bioluminescence, we refer the reader to [21]. In (2), the first equation expresses conservation of the number of metastases when growing in size, the second equates the entering flux of new tumors with the rate of (successful) dissemination from the primary tumor and the last one is the initial condition.…”

“…Material and methods generating the data employed here are extensively described in [21]. For the growth of LM2-4 luc+ cells orthotopically xenografted in the mammary fat pad of severe combined immune-deficient mice, as employed here, it had been previously demonstrated that the growth kinetics can be accurately described using the Gompertz model [22], with expression…”

“…where λ is the in vitro proliferation rate, retrieved from preliminary experiments (see [21]). We then investigated what structural and parametrical shape of the secondary growth coefficient g(v) would better fit the data, while ensuring reasonable identifiability of its parameters.…”

“…Of note however, in 2000, Iwata, Kawasaki and Shigesada proposed a model particularly well suited for description of the time dynamics of a population of secondary tumors [20]. This model can be easily adapted to longitudinal data of total metastatic burden and has been recently shown able to fit such data in several animal models [18,21]. Based on one of this study, we will focus here on the identifiability aspects of particular coefficients of the model, respectively related to the growth and dissemination laws.…”