Computational Modelling of Metastasis Development in Renal Cell Carcinoma

Computational markers for personalized prediction of outcomes in non-small cell lung cancer patients with brain metastases

Intracranial progression after curative treatment of early-stage non-small cell lung cancer (NSCLC) occurs from 10 to 50% and is difficult to manage, given the heterogeneity of clinical presentations and the variability of treatments available. The objective of this study was to develop a mechanistic model of intracranial progression to predict survival following a first brain metastasis (BM) event occurring at a time [Formula: see text]. Data included early-stage NSCLC patients treated with a curative intent who had a BM as the first and single relapse site (N = 31). We propose a mechanistic mathematical model able to derive computational markers from primary tumor and BM data at [Formula: see text] and estimate the amount and sizes of (visible and invisible) BMs, as well as their future behavior. These two key computational markers are [Formula: see text], the proliferation rate of a single tumor cell; and [Formula: see text], the per day, per cell, probability to metastasize. The predictive value of these individual computational biomarkers was evaluated. The model was able to correctly describe the number and size of metastases at [Formula: see text] for 20 patients. Parameters [Formula: see text] and [Formula: see text] were significantly associated with overall survival (OS) (HR 1.65 (1.07-2.53) p = 0.0029 and HR 1.95 (1.31-2.91) p = 0.0109, respectively). Adding the computational markers to the clinical ones significantly improved the predictive value of OS (c-index increased from 0.585 (95% CI 0.569-0.602) to 0.713 (95% CI 0.700-0.726), p < 0.0001). We demonstrated that our model was applicable to brain oligoprogressive patients in NSCLC and that the resulting computational markers had predictive potential. This may help lung cancer physicians to guide and personalize the management of NSCLC patients with intracranial oligoprogression.

Mathematical and Machine Learning Models of Renal Cell Carcinoma: A Review

This review explores the multifaceted landscape of renal cell carcinoma (RCC) by delving into both mechanistic and machine learning models. While machine learning models leverage patients' gene expression and clinical data through a variety of techniques to predict patients' outcomes, mechanistic models focus on investigating cells' and molecules' interactions within RCC tumors. These interactions are notably centered around immune cells, cytokines, tumor cells, and the development of lung metastases. The insights gained from both machine learning and mechanistic models encompass critical aspects such as signature gene identification, sensitive interactions in the tumors' microenvironments, metastasis development in other organs, and the assessment of survival probabilities. By reviewing the models of RCC, this study aims to shed light on opportunities for the integration of machine learning and mechanistic modeling approaches for treatment optimization and the identification of specific targets, all of which are essential for enhancing patient outcomes.

Mechanistic modeling of metastatic relapse in early breast cancer to investigate the biological impact of prognostic biomarkers

BACKGROUND AND OBJECTIVE

Estimating the risk of metastatic relapse is a major challenge to decide adjuvant treatment options in early-stage breast cancer (eBC). To date, distant metastasis-free survival (DMFS) analysis mainly relies on classical, agnostic, statistical models (e.g., Cox regression). Instead, we propose here to derive mechanistic models of DMFS.

METHODS

The present series consisted of eBC patients who did not receive adjuvant systemic therapy from three datasets, composed respectively of 692 (Bergonié Institute), 591 (Paoli-Calmettes Institute, IPC), and 163 (Public Hospital Marseille, AP-HM) patients with routine clinical annotations. The last dataset also contained expression of three non-routine biomarkers. Our mechanistic model of DMFS relies on two mathematical parameters that represent growth (α) and dissemination (μ). We identified their population distributions using mixed-effects modeling. Critically, we propose a novel variable selection procedure allowing to: (i) identify the association of biological parameters with either α, μ or both, and (ii) generate an optimal candidate model for DMFS prediction.

RESULTS

We found that Ki67 and Thymidine Kinase-1 were associated with α, and nodal status and Plasminogen Activator Inhibitor-1 with μ. The predictive performances of the model were excellent in calibration but moderate in discrimination, with c-indices of 0.72 (95% CI [0.48, 0.95], AP-HM), 0.63 ([0.44, 0.83], Bergonié) and 0.60 (95% CI [0.54, 0.80], IPC).

CONCLUSIONS

Overall, we demonstrate that our novel method combining mechanistic and advanced statistical modeling is able to unravel the biological roles of clinicopathological parameters from DMFS data.

Practical identifiability analysis of a mechanistic model for the time to distant metastatic relapse and its application to renal cell carcinoma

Distant metastasis-free survival (DMFS) curves are widely used in oncology. They are classically analyzed using the Kaplan-Meier estimator or agnostic statistical models from survival analysis. Here we report on a method to extract more information from DMFS curves using a mathematical model of primary tumor growth and metastatic dissemination. The model depends on two parameters, α and μ, respectively quantifying tumor growth and dissemination. We assumed these to be lognormally distributed in a patient population. We propose a method for identification of the parameters of these distributions based on least-squares minimization between the data and the simulated survival curve. We studied the practical identifiability of these parameters and found that including the percentage of patients with metastasis at diagnosis was critical to ensure robust estimation. We also studied the impact and identifiability of covariates and their coefficients in α and μ, either categorical or continuous, including various functional forms for the latter (threshold, linear or a combination of both). We found that both the functional form and the coefficients could be determined from DMFS curves. We then applied our model to a clinical dataset of metastatic relapse from kidney cancer with individual data of 105 patients. We show that the model was able to describe the data and illustrate our method to disentangle the impact of three covariates on DMFS: a categorical one (Führman grade) and two continuous ones (gene expressions of the macrophage mannose receptor 1 (MMR) and the G Protein-Coupled Receptor Class C Group 5 Member A (GPRC5a) gene). We found that all had an influence in metastasis dissemination (μ), but not on growth (α).

Understanding biological mechanisms leading to metastasis development is a major challenge in order to prevent distant relapse of cancer. Classical methods to study associations of biomarkers with subsequent metastatic relapse rely on the analysis of metastasis free survival curves by means of statistical models such as proportional hazards Cox regression. These models act as black boxes and don’t provide detailed information about the specific mechanism involved. In our study, we propose to use a method based on mechanistic modeling of the metastatic development, that is, a mathematical model that simulates the biological process. The main challenge for these models is to implement the right level of complexity, because if too many parameters are included, these cannot be precisely identified from the data. We reduced the metastatic process to two main aspects: growth and dissemination. We then proposed a theoretical study of the identifiability of the two associated parameters from metastasis-free survival curves. Eventually, we applied our method to a clinical dataset in kidney cancer and illustrated how we could gain biological insights about the role of some diagnosis markers.

Experimental and computational modeling for signature and biomarker discovery of renal cell carcinoma progression

BACKGROUND

Renal Cell Carcinoma (RCC) is difficult to treat with 5-year survival rate of 10% in metastatic patients. Main reasons of therapy failure are lack of validated biomarkers and scarce knowledge of the biological processes occurring during RCC progression. Thus, the investigation of mechanisms regulating RCC progression is fundamental to improve RCC therapy.

METHODS

In order to identify molecular markers and gene processes involved in the steps of RCC progression, we generated several cell lines of higher aggressiveness by serially passaging mouse renal cancer RENCA cells in mice and, concomitantly, performed functional genomics analysis of the cells. Multiple cell lines depicting the major steps of tumor progression (including primary tumor growth, survival in the blood circulation and metastatic spread) were generated and analyzed by large-scale transcriptome, genome and methylome analyses. Furthermore, we performed clinical correlations of our datasets. Finally we conducted a computational analysis for predicting the time to relapse based on our molecular data.

RESULTS

Through in vivo passaging, RENCA cells showed increased aggressiveness by reducing mice survival, enhancing primary tumor growth and lung metastases formation. In addition, transcriptome and methylome analyses showed distinct clustering of the cell lines without genomic variation. Distinct signatures of tumor aggressiveness were revealed and validated in different patient cohorts. In particular, we identified SAA2 and CFB as soluble prognostic and predictive biomarkers of the therapeutic response. Machine learning and mathematical modeling confirmed the importance of CFB and SAA2 together, which had the highest impact on distant metastasis-free survival. From these data sets, a computational model predicting tumor progression and relapse was developed and validated. These results are of great translational significance.

CONCLUSION

A combination of experimental and mathematical modeling was able to generate meaningful data for the prediction of the clinical evolution of RCC.

A tipping point in cancer-immune dynamics leads to divergent immunotherapy responses and hampers biomarker discovery

BACKGROUND

Predicting treatment response or survival of cancer patients remains challenging in immuno-oncology. Efforts to overcome these challenges focus, among others, on the discovery of new biomarkers. Despite advances in cellular and molecular approaches, only a limited number of candidate biomarkers eventually enter clinical practice.

METHODS

A computational modeling approach based on ordinary differential equations was used to simulate the fundamental mechanisms that dictate tumor-immune dynamics and to investigate its implications on responses to immune checkpoint inhibition (ICI) and patient survival. Using in silico biomarker discovery trials, we revealed fundamental principles that explain the diverging success rates of biomarker discovery programs.

RESULTS

Our model shows that a tipping point-a sharp state transition between immune control and immune evasion-induces a strongly non-linear relationship between patient survival and both immunological and tumor-related parameters. In patients close to the tipping point, ICI therapy may lead to long-lasting survival benefits, whereas patients far from the tipping point may fail to benefit from these potent treatments.

CONCLUSION

These findings have two important implications for clinical oncology. First, the apparent conundrum that ICI induces substantial benefits in some patients yet completely fails in others could be, to a large extent, explained by the presence of a tipping point. Second, predictive biomarkers for immunotherapy should ideally combine both immunological and tumor-related markers, as a patient's distance from the tipping point can typically not be reliably determined from solely one of these. The notion of a tipping point in cancer-immune dynamics helps to devise more accurate strategies to select appropriate treatments for patients with cancer.

Spatial mechanistic modeling for prediction of the growth of asymptomatic meningiomas

BACKGROUND AND OBJECTIVE

Mathematical modeling of tumor growth draws interest from the medical community as they have the potential to improve patients' care and the use of public health resources. The main objectives of this work are to model the growth of meningiomas - slow-growing benign tumors requiring extended imaging follow-up - and to predict tumor volume and shape at a later desired time using only two times examinations.

METHODS

We develop two variants of a 3D partial differential system of equations (PDE) which yield after a spatial integration systems of ordinary differential equations (ODE) that relate tumor volume with time. Estimation of models parameters is a crucial step to obtain a personalized model for a patient that can be used for descriptive or predictive purposes. As PDE and ODE systems share the same parameters, they are both estimated by fitting the ODE systems to the tumor volumes obtained from MRI examinations acquired at different times. A population approach allows to compensate for sparse sampling times and measurement uncertainties by constraining the variability of the parameters in the population.

RESULTS

Description capabilities of the models are investigated in 39 patients with benign asymptomatic meningiomas who had had at least three surveillance MRI examinations. The two models can fit to the data accurately and more realistically than a naive linear regression. Prediction performances are validated for 33 patients using a population approach. Mean relative errors in volume predictions are less than 10% with ODE systems versus 12.5% with the naive linear model using only two times examinations. Concerning the shape, the mean Sørensen-Dice coefficients are 85% with the PDE systems in a subset of 10 representative patients.

CONCLUSIONS

Our strategy - based on personalization of mathematical model - provides a good insight on meningioma growth and may help decide whether to extend the follow-up or to treat the tumor.

Metastasis Initiation Precedes Detection of Primary Cancer-Analysis of Metastasis Growth in vivo in a Colorectal Cancer Test Case

Most cases of deaths from colorectal cancer (CRC) result from metastases, which are often still undetectable at disease detection time. Even so, in many cases, shedding is assumed to have taken place before that time. The dynamics of metastasis formation and growth are not well-established. This work aims to explore CRC lung metastasis growth rate and dynamics. We analyzed a test case of a metastatic CRC patient with four lung metastases, with data of four serial computed tomography (CT) scans measuring metastasis sizes while untreated. We fitted three mathematical growth models—exponential, logistic, and Gompertzian—to the CT measurements. For each metastasis, a best-fitted model was determined, tumor doubling time (TDT) was assessed, and metastasis inception time was extrapolated. Three of the metastases showed exponential growth, while the fourth showed logistic restraint of the growth. TDT was around 93 days. Predicted metastasis inception time was at least 4–5 years before the primary tumor diagnosis date, though they did not reach detectable sizes until at least 1 year after primary tumor resection. Our results support the exponential growth approximation for most of the metastases, at least for the clinically observed time period. Our analysis shows that metastases can be initiated before the primary tumor is detectable and implies that surgeries accelerate metastasis growth.

Mathematical modelling reveals cellular dynamics within tumour spheroids

Tumour spheroids are widely used as an in vitro assay for characterising the dynamics and response to treatment of different cancer cell lines. Their popularity is largely due to the reproducible manner in which spheroids grow: the diffusion of nutrients and oxygen from the surrounding culture medium, and their consumption by tumour cells, causes proliferation to be localised at the spheroid boundary. As the spheroid grows, cells at the spheroid centre may become hypoxic and die, forming a necrotic core. The pressure created by the localisation of tumour cell proliferation and death generates an cellular flow of tumour cells from the spheroid rim towards its core. Experiments by Dorie et al. showed that this flow causes inert microspheres to infiltrate into tumour spheroids via advection from the spheroid surface, by adding microbeads to the surface of tumour spheroids and observing the distribution over time. We use an off-lattice hybrid agent-based model to re-assess these experiments and establish the extent to which the spatio-temporal data generated by microspheres can be used to infer kinetic parameters associated with the tumour spheroids that they infiltrate. Variation in these parameters, such as the rate of tumour cell proliferation or sensitivity to hypoxia, can produce spheroids with similar bulk growth dynamics but differing internal compositions (the proportion of the tumour which is proliferating, hypoxic/quiescent and necrotic/nutrient-deficient). We use this model to show that the types of experiment conducted by Dorie et al. could be used to infer spheroid composition and parameters associated with tumour cell lines such as their sensitivity to hypoxia or average rate of proliferation, and note that these observations cannot be conducted within previous continuum models of microbead infiltration into tumour spheroids as they rely on resolving the trajectories of individual microbeads.

Tumour spheroids are an experimental assay used to characterise the dynamics and response to treatment of different cancer cell lines. Previous experiments have demonstrated that the localisation of tumour cell proliferation to the spheroid edge (due to the gradient formed by nutrient diffusing from the surrounding medium) causes cells to be pushed from the proliferative rim towards the nutrient-deficient necrotic core. This movement allows inert particles to infiltrate tumour spheroids. We use a hybrid agent-based model to reproduce this data. We show further how data from individual microbead trajectories can be used to infer the composition of simulated tumour spheroids, and to estimate model parameters pertaining to tumour cell proliferation rates and their responses to hypoxia. Since these measurements are possible using modern imaging techniques, this could motivate new experiments in which spheroid composition could be inferred by observing passive infiltration of inert particles.

Experimental and modeling study of the formation of cell aggregates with differential substrate adhesion

The study of cell aggregation in vitro has a tremendous importance these days. In cancer biology, aggregates and spheroids serve as model systems and are considered as pseudo-tumors that are more realistic than 2D cell cultures. Recently, in the context of brain tumors (gliomas), we developed a new poly(ethylene glycol) (PEG)-based hydrogel, with adhesive properties that can be controlled by the addition of poly(L-lysine) (PLL), and a stiffness close to the brain’s. This substrate allows the motion of individual cells and the formation of cell aggregates (within one day), and we showed that on a non-adhesive substrate (PEG without PLL is inert for cells), the aggregates are bigger and less numerous than on an adhesive substrate (with PLL). In this article, we present new experimental results on the follow-up of the formation of aggregates on our hydrogels, from the early stages (individual cells) to the late stages (aggregate compaction), in order to compare, for two cell lines (F98 and U87), the aggregation process on the adhesive and non-adhesive substrates. We first show that a spaceless model of perikinetic aggregation can reproduce the experimental evolution of the number of aggregates, but not of the mean area of the aggregates. We thus develop a minimal off-lattice agent-based model, with a few simple rules reproducing the main processes that are at stack during aggregation. Our spatial model can reproduce very well the experimental temporal evolution of both the number of aggregates and their mean area, on adhesive and non-adhesive soft gels and for the two different cell lines. From the fit of the experimental data, we were able to infer the quantitative values of the speed of motion of each cell line, its rate of proliferation in aggregates and its ability to organize in 3D. We also found qualitative differences between the two cell lines regarding the ability of aggregates to compact. These parameters could be inferred for any cell line, and correlated with clinical properties such as aggressiveness and invasiveness.

Disease progression model of 4T1 metastatic breast cancer

Cancer metastasis is the main cause of death in various types of cancer. However, in the field of pharmacometrics, cancer disease progression models focus on the growth of primary tumors with tumor volume or weight as target values, while the metastasis process is less mentioned. We propose a series of mathematical models to quantitatively describe and predict the disease progression of 4T1 breast cancer in the aspect of primary breast tumor, lung metastasis and white blood cell. The 4T1 cells were injected into breast fat pad of female BALB/c mice to establish an animal model of breast cancer metastasis. The number and volume of lung metastases at different times were measured. Based on the above data, a disease progression model of breast cancer lung metastasis was established and parameter values were estimated. The white blood cell growth and the primary tumor growth of 4T1 mouse are also modeled. The established models can describe the lung metastasis of 4T1 breast cancer in three aspects: (1) the increase in metastasis number; (2) the growth of metastasis volume; (3) metastasis number-size distribution at different time points. Compared with the prior metastasis models based on von Forester equation, our models distinguished the growth rate of primary tumor and metastasis and got parameter values for 4T1 mouse model. And the current models optimized the metastasis number-size distribution model by utilizing logistic function instead of the prior power function. This study provides a comprehensive description of lung metastasis progression for 4T1 breast cancer model, as well as an alternative disease progression model structure for further pharmacodynamics modeling.

A mathematical model for the immune-mediated theory of metastasis

Accumulating experimental and clinical evidence suggest that the immune response to cancer is not exclusively anti-tumor. Indeed, the pro-tumor roles of the immune system  -  as suppliers of growth and pro-angiogenic factors or defenses against cytotoxic immune attacks, for example  -  have been long appreciated, but relatively few theoretical works have considered their effects. Inspired by the recently proposed "immune-mediated" theory of metastasis, we develop a mathematical model for tumor-immune interactions at two anatomically distant sites, which includes both anti- and pro-tumor immune effects, and the experimentally observed tumor-induced phenotypic plasticity of immune cells (tumor "education" of the immune cells). Upon confrontation of our model to experimental data, we use it to evaluate the implications of the immune-mediated theory of metastasis. We find that tumor education of immune cells may explain the relatively poor performance of immunotherapies, and that many metastatic phenomena, including metastatic blow-up, dormancy, and metastasis to sites of injury, can be explained by the immune-mediated theory of metastasis. Our results suggest that further work is warranted to fully elucidate the pro-tumor effects of the immune system in metastatic cancer.

Quantitative mathematical modeling of clinical brain metastasis dynamics in non-small cell lung cancer

Brain metastases (BMs) are associated with poor prognosis in non-small cell lung cancer (NSCLC), but are only visible when large enough. Therapeutic decisions such as whole brain radiation therapy would benefit from patient-specific predictions of radiologically undetectable BMs. Here, we propose a mathematical modeling approach and use it to analyze clinical data of BM from NSCLC. Primary tumor growth was best described by a gompertzian model for the pre-diagnosis history, followed by a tumor growth inhibition model during treatment. Growth parameters were estimated only from the size at diagnosis and histology, but predicted plausible individual estimates of the tumor age (2.1-5.3 years). Multiple metastatic models were further assessed from fitting either literature data of BM probability (n = 183 patients) or longitudinal measurements of visible BMs in two patients. Among the tested models, the one featuring dormancy was best able to describe the data. It predicted latency phases of 4.4-5.7 months and onset of BMs 14-19 months before diagnosis. This quantitative model paves the way for a computational tool of potential help during therapeutic management.

Inferring rates of metastatic dissemination using stochastic network models

The formation of metastases is driven by the ability of cancer cells to disseminate from the site of the primary tumour to target organs. The process of dissemination is constrained by anatomical features such as the flow of blood and lymph in the circulatory system. We exploit this fact in a stochastic network model of metastasis formation, in which only anatomically feasible routes of dissemination are considered. By fitting this model to two different clinical datasets (tongue & ovarian cancer) we show that incidence data can be modelled using a small number of biologically meaningful parameters. The fitted models reveal site specific relative rates of dissemination and also allow for patient-specific predictions of metastatic involvement based on primary tumour location and stage. Applied to other data sets this type of model could yield insight about seed-soil effects, and could also be used in a clinical setting to provide personalised predictions about the extent of metastatic spread.

Heavy-tailed distributions in branching process models of secondary cancerous tumors

Recent progress in microdissection and in DNA sequencing has facilitated the subsampling of multi-focal cancers in organs such as the liver in several hundred spots, helping to determine the pattern of mutations in each of these spots. This has led to the construction of genealogies of the primary, secondary, tertiary, and so forth, foci of the tumor. These studies have led to diverse conclusions concerning the Darwinian (selective) or neutral evolution in cancer. Mathematical models of the development of multi-focal tumors have been devised to support these claims. We offer a model for the development of a multi-focal tumor: it is a mathematically rigorous refinement of a model of Ling et al. (2015). Guided by numerical studies and simulations, we show that the rigorous model, in the form of an infinite-type branching process, displays distributions of tumor size which have heavy tails and moments that become infinite in finite time. To demonstrate these points, we obtain bounds on the tails of the distributions of the process and an infinite series expansion for the first moments. In addition to its inherent mathematical interest, the model is corroborated by recent literature on apparent super-exponential growth in cancer metastases.

A Non-integrating Lentiviral Approach Overcomes Cas9-Induced Immune Rejection to Establish an Immunocompetent Metastatic Renal Cancer Model

The CRISPR-based technology has revolutionized genome editing in recent years. This technique allows for gene knockout and evaluation of function in cell lines in a manner that is far easier and more accessible than anything previously available. Unfortunately, the ability to extend these studies to in vivo syngeneic murine cell line implantation is limited by an immune response against cells transduced to stably express Cas9. In this study, we demonstrate that a non-integrating lentiviral vector approach can overcome this immune rejection and allow for the growth of transduced cells in an immunocompetent host. This technique enables the establishment of a von Hippel-Lindau (VHL) gene knockout RENCA cell line in BALB/c mice, generating an improved model of immunocompetent, metastatic renal cell carcinoma (RCC).

Mathematical Modeling of Tumor-Tumor Distant Interactions Supports a Systemic Control of Tumor Growth

Interactions between different tumors within the same organism have major clinical implications, especially in the context of surgery and metastatic disease. Three main explanatory theories (competition, angiogenesis inhibition, and proliferation inhibition) have been proposed, but precise determinants of the phenomenon remain poorly understood. Here, we formalized these theories into mathematical models and performed biological experiments to test them with empirical data. In syngeneic mice bearing two simultaneously implanted tumors, growth of only one of the tumors was significantly suppressed (61% size reduction at day 15, P < 0.05). The competition model had to be rejected, whereas the angiogenesis inhibition and proliferation inhibition models were able to describe the data. Additional models including a theory based on distant cytotoxic log-kill effects were unable to fit the data. The proliferation inhibition model was identifiable and minimal (four parameters), and its descriptive power was validated against the data, including consistency in predictions of single tumor growth when no secondary tumor was present. This theory may also shed new light on single cancer growth insofar as it offers a biologically translatable picture of how local and global action may combine to control local tumor growth and, in particular, the role of tumor-tumor inhibition. This model offers a depiction of concomitant resistance that provides an improved theoretical basis for tumor growth control and may also find utility in therapeutic planning to avoid postsurgery metastatic acceleration. Cancer Res; 77(18); 5183-93. ©2017 AACR.

An exactly solvable, spatial model of mutation accumulation in cancer

One of the hallmarks of cancer is the accumulation of driver mutations which increase the net reproductive rate of cancer cells and allow them to spread. This process has been studied in mathematical models of well mixed populations, and in computer simulations of three-dimensional spatial models. But the computational complexity of these more realistic, spatial models makes it difficult to simulate realistically large and clinically detectable solid tumours. Here we describe an exactly solvable mathematical model of a tumour featuring replication, mutation and local migration of cancer cells. The model predicts a quasi-exponential growth of large tumours, even if different fragments of the tumour grow sub-exponentially due to nutrient and space limitations. The model reproduces clinically observed tumour growth times using biologically plausible rates for cell birth, death, and migration rates. We also show that the expected number of accumulated driver mutations increases exponentially in time if the average fitness gain per driver is constant, and that it reaches a plateau if the gains decrease over time. We discuss the realism of the underlying assumptions and possible extensions of the model.