Universal scaling laws rule explosive growth in human cancers

Mathematical modeling of brain metastases growth and response to therapies: A review

Brain metastases (BMs) are the most common intracranial tumor type and a significant health concern, affecting approximately 10% to 30% of all oncological patients. Although significant progress is being made, many aspects of the metastatic process to the brain and the growth of the resulting lesions are still not well understood. There is a need for an improved understanding of the growth dynamics and the response to treatment of these tumors. Mathematical models have been proven valuable for drawing inferences and making predictions in different fields of cancer research, but few mathematical works have considered BMs. This comprehensive review aims to establish a unified platform and contribute to fostering emerging efforts dedicated to enhancing our mathematical understanding of this intricate and challenging disease. We focus on the progress made in the initial stages of mathematical modeling research regarding BMs and the significant insights gained from such studies. We also explore the vital role of mathematical modeling in predicting treatment outcomes and enhancing the quality of clinical decision-making for patients facing BMs.

Spatial interactions modulate tumor growth and immune infiltration

Lenia, a cellular automata framework used in artificial life, provides a natural setting to implement mathematical models of cancer incorporating features such as morphogenesis, homeostasis, motility, reproduction, growth, stimuli response, evolvability, and adaptation. Historically, agent-based models of cancer progression have been constructed with rules that govern birth, death and migration, with attempts to map local rules to emergent global growth dynamics. In contrast, Lenia provides a flexible framework for considering a spectrum of local (cell-scale) to global (tumor-scale) dynamics by defining an interaction kernel governing density-dependent growth dynamics. Lenia can recapitulate a range of cancer model classifications including local or global, deterministic or stochastic, non-spatial or spatial, single or multi-population, and off or on-lattice. Lenia is subsequently used to develop data-informed models of 1) single-population growth dynamics, 2) multi-population cell-cell competition models, and 3) cell migration or chemotaxis. Mathematical modeling provides important mechanistic insights. First, short-range interaction kernels provide a mechanism for tumor cell survival under conditions with strong Allee effects. Next, we find that asymmetric interaction tumor-immune kernels lead to poor immune response. Finally, modeling recapitulates immune-ECM interactions where patterns of collagen formation provide immune protection, indicated by an emergent inverse relationship between disease stage and immune coverage.

Growth speed of large brain metastases between diagnostic and radiosurgical planning MRI and predictors of rapid tumor growth

PURPOSE

We aimed to assess volumetric changes of large brain metastases (≥ 2 cm) between their diagnosis and planning for treatment with fractionated stereotactic radiation surgery (fSRS). Predictors of rapid tumor growth were also analyzed.

MATERIALS AND METHODS

One hundred nine patients harboring 126 large brain metastases were retrospectively evaluated. Tumor characteristics were evaluated on diagnostic magnetic resonance imaging (dMRI) and MRI performed when planning fSRS (pMRI). Average tumor growth rate and percentage growth rate were calculated. Predictors of rapid growth (percentage growth rate > 5%) were determined using multivariate logistic regression.

RESULTS

Both tumor diameter and volume were significantly larger on pMRI than on dMRI (P < 0.001). Median tumor percentage growth rate was 2.6% (range, - 10.8-43.3%). Eighty-eight tumors (70%) were slow-growing (percentage growth rate < 5%) and 38 (30%) grew rapidly (percentage growth rate ≥ 5%). Major peritumoral edema and no steroids were predictors of rapid tumor growth.

CONCLUSION

Large brain metastases can grow considerably between the time of diagnosis and the time of fSRS treatment planning. We recommend the time between dMRI and fSRS treatment initiation be as short as possible.

Spatial interactions modulate tumor growth and immune infiltration

Direct observation of immune cell trafficking patterns and tumor-immune interactions is unlikely in human tumors with currently available technology, but computational simulations based on clinical data can provide insight to test hypotheses. It is hypothesized that patterns of collagen formation evolve as a mechanism of immune escape, but the exact nature of the interaction between immune cells and collagen is poorly understood. Spatial data quantifying the degree of collagen fiber alignment in squamous cell carcinomas indicates that late stage disease is associated with highly aligned fibers. Here, we introduce a computational modeling framework (called Lenia) to discriminate between two hypotheses: immune cell migration that moves 1) parallel or 2) perpendicular to collagen fiber orientation. The modeling recapitulates immune-ECM interactions where collagen patterns provide immune protection, leading to an emergent inverse relationship between disease stage and immune coverage. We also illustrate the capabilities of Lenia to model the evolution of tumor progression and immune predation. Lenia provides a flexible framework for considering a spectrum of local (cell-scale) to global (tumor-scale) dynamics by defining a kernel cell-cell interaction function that governs tumor growth dynamics under immune predation with immune cell migration. Mathematical modeling provides important mechanistic insights into cell interactions. Short-range interaction kernels provide a mechanism for tumor cell survival under conditions with strong Allee effects, while asymmetric tumor-immune interaction kernels lead to poor immune response. Thus, the length scale of tumor-immune interactions drives tumor growth and infiltration.

A Mathematical Model of Stroma-Supported Allometric Tumor Growth

Mounting empirical research suggests that the stroma, or interface between healthy and cancerous tissue, is a critical determinate of cancer invasion. At the same time, a cancer cell's location and potential to proliferate can influence its sensitivity to cancer treatments. In this paper, we use ordinary differential equations to develop spatially structured models for solid tumors wherein the growth of tumor components is coordinated. The model tumors feature two components, a proliferating peripheral growth region, which potentially includes a mix of cancerous and noncancerous stroma cells, and a solid tumor core. Mathematical and numerical analysis are used to investigate how coordinated expansion of the tumor growth region and core can influence overall growth dynamics in a variety of tumor types. Model assumptions, which are motivated by empirical and in silico solid tumor research, are evaluated through comparison to tumor volume data and existing models of tumor growth.

Challenges in radiological evaluation of brain metastases, beyond progression

MRI is the cornerstone in the evaluation of brain metastases. The clinical challenges lie in discriminating metastases from mimickers such as infections or primary tumors and in evaluating the response to treatment. The latter sometimes leads to growth, which must be framed as pseudo-progression or radionecrosis, both inflammatory phenomena attributable to treatment, or be considered as recurrence. To meet these needs, imaging techniques are the subject of constant research. However, an exponential growth after radiotherapy must be interpreted with caution, even in the presence of results suspicious of tumor progression by advanced techniques, because it may be due to inflammatory changes. The aim of this paper is to familiarize the reader with inflammatory phenomena of brain metastases treated with radiotherapy and to describe two related radiological signs: "the inflammatory cloud" and "incomplete ring enhancement", in order to adopt a conservative management with close follow-up.

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.

Radiation necrosis after radiation therapy treatment of brain metastases: A computational approach

Metastasis is the process through which cancer cells break away from a primary tumor, travel through the blood or lymph system, and form new tumors in distant tissues. One of the preferred sites for metastatic dissemination is the brain, affecting more than 20% of all cancer patients. This figure is increasing steadily due to improvements in treatments of primary tumors. Stereotactic radiosurgery (SRS) is one of the main treatment options for patients with a small or moderate number of brain metastases (BMs). A frequent adverse event of SRS is radiation necrosis (RN), an inflammatory condition caused by late normal tissue cell death. A major diagnostic problem is that RNs are difficult to distinguish from BM recurrences, due to their similarities on standard magnetic resonance images (MRIs). However, this distinction is key to choosing the best therapeutic approach since RNs resolve often without further interventions, while relapsing BMs may require open brain surgery. Recent research has shown that RNs have a faster growth dynamics than recurrent BMs, providing a way to differentiate the two entities, but no mechanistic explanation has been provided for those observations. In this study, computational frameworks were developed based on mathematical models of increasing complexity, providing mechanistic explanations for the differential growth dynamics of BMs relapse versus RN events and explaining the observed clinical phenomenology. Simulated tumor relapses were found to have growth exponents substantially smaller than the group in which there was inflammation due to damage induced by SRS to normal brain tissue adjacent to the BMs, thus leading to RN. ROC curves with the synthetic data had an optimal threshold that maximized the sensitivity and specificity values for a growth exponent β* = 1.05, very close to that observed in patient datasets.

Overcoming chemotherapy resistance in low-grade gliomas: A computational approach

Low-grade gliomas are primary brain tumors that arise from glial cells and are usually treated with temozolomide (TMZ) as a chemotherapeutic option. They are often incurable, but patients have a prolonged survival. One of the shortcomings of the treatment is that patients eventually develop drug resistance. Recent findings show that persisters, cells that enter a dormancy state to resist treatment, play an important role in the development of resistance to TMZ. In this study we constructed a mathematical model of low-grade glioma response to TMZ incorporating a persister population. The model was able to describe the volumetric longitudinal dynamics, observed in routine FLAIR 3D sequences, of low-grade glioma patients acquiring TMZ resistance. We used the model to explore different TMZ administration protocols, first on virtual clones of real patients and afterwards on virtual patients preserving the relationships between parameters of real patients. In silico clinical trials showed that resistance development was deferred by protocols in which individual doses are administered after rest periods, rather than the 28-days cycle standard protocol. This led to median survival gains in virtual patients of more than 15 months when using resting periods between two and three weeks and agreed with recent experimental observations in animal models. Additionally, we tested adaptive variations of these new protocols, what showed a potential reduction in toxicity, but no survival gain. Our computational results highlight the need of further clinical trials that could obtain better results from treatment with TMZ in low grade gliomas.

Evaluation of four tumour growth models to describe the natural history of meningiomas

BACKGROUND

The incidence of newly diagnosed meningiomas, particularly those diagnosed incidentally, is continually increasing. The indication for treatment is empirical because, despite numerous studies, the natural history of these tumours remains difficult to describe and predict.

METHODS

This retrospective single-centre study included 294 consecutive patients with 333 meningiomas who underwent three or more brain imaging scans. Linear, exponential, power, and Gompertz models were constructed to derive volume-time curves, by using a mixed-effect approach. The most accurate model was used to analyse tumour growth and predictors of rapid growth.

FINDINGS

The Gompertz model provided the best results. Hierarchical clustering at the time of diagnosis and at the end of follow-up revealed at least three distinct groups, which can be described as pseudoexponential, linear, and slowing growth with respect to their parameters. Younger patients and smaller tumours were more frequent in the pseudo-exponential clusters. We found that the more "aggressive" the cluster, the higher the proportion of patients with grade II meningiomas and who have had a cranial radiotherapy. Over a mean observation period of 56.5 months, 21% of the tumours moved to a cluster with a lower growth rate, consistent with the Gompertz's law.

INTERPRETATION

Meningiomas exhibit multiple growth phases, as described by the Gompertz model. The management of meningiomas should be discussed according to the growth phase, comorbidities, tumour location, size, and growth rate. Further research is needed to evaluate the associations between radiomics features and the growth phases of meningiomas.

FUNDING

No funding.

Knowledge-based mechanistic modeling accurately predicts disease progression with gefitinib in EGFR-mutant lung adenocarcinoma

Lung adenocarcinoma (LUAD) is associated with a low survival rate at advanced stages. Although the development of targeted therapies has improved outcomes in LUAD patients with identified and specific genetic alterations, such as activating mutations on the epidermal growth factor receptor gene (EGFR), the emergence of tumor resistance eventually occurs in all patients and this is driving the development of new therapies. In this paper, we present the In Silico EGFR-mutant LUAD (ISELA) model that links LUAD patients' individual characteristics, including tumor genetic heterogeneity, to tumor size evolution and tumor progression over time under first generation EGFR tyrosine kinase inhibitor gefitinib. This translational mechanistic model gathers extensive knowledge on LUAD and was calibrated on multiple scales, including in vitro, human tumor xenograft mouse and human, reproducing more than 90% of the experimental data identified. Moreover, with 98.5% coverage and 99.4% negative logrank tests, the model accurately reproduced the time to progression from the Lux-Lung 7 clinical trial, which was unused in calibration, thus supporting the model high predictive value. This knowledge-based mechanistic model could be a valuable tool in the development of new therapies targeting EGFR-mutant LUAD as a foundation for the generation of synthetic control arms.

Growth exponents reflect evolutionary processes and treatment response in brain metastases

Tumor growth is the result of the interplay of complex biological processes in huge numbers of individual cells living in changing environments. Effective simple mathematical laws have been shown to describe tumor growth in vitro, or simple animal models with bounded-growth dynamics accurately. However, results for the growth of human cancers in patients are scarce. Our study mined a large dataset of 1133 brain metastases (BMs) with longitudinal imaging follow-up to find growth laws for untreated BMs and recurrent treated BMs. Untreated BMs showed high growth exponents, most likely related to the underlying evolutionary dynamics, with experimental tumors in mice resembling accurately the disease. Recurrent BMs growth exponents were smaller, most probably due to a reduction in tumor heterogeneity after treatment, which may limit the tumor evolutionary capabilities. In silico simulations using a stochastic discrete mesoscopic model with basic evolutionary dynamics led to results in line with the observed data.

Metabolic activity grows in human cancers pushed by phenotypic variability

Different evolutionary processes push cancers to increasingly aggressive behaviors, energetically sustained by metabolic reprogramming. The collective signature emerging from this transition is macroscopically displayed by positron emission tomography (PET). In fact, the most readily PET measure, the maximum standardized uptake value (SUV_max), has been found to have prognostic value in different cancers. However, few works have linked the properties of this metabolic hotspot to cancer evolutionary dynamics. Here, by analyzing diagnostic PET images from 512 patients with cancer, we found that SUV_max scales superlinearly with the mean metabolic activity (SUV_mean), reflecting a dynamic preferential accumulation of activity on the hotspot. Additionally, SUV_max increased with metabolic tumor volume (MTV) following a power law. The behavior from the patients data was accurately captured by a mechanistic evolutionary dynamics model of tumor growth accounting for phenotypic transitions. This suggests that non-genetic changes may suffice to fuel the observed sustained increases in tumor metabolic activity.

Stochastic Fluctuations Drive Non-genetic Evolution of Proliferation in Clonal Cancer Cell Populations

Evolutionary dynamics allows us to understand many changes happening in a broad variety of biological systems, ranging from individuals to complete ecosystems. It is also behind a number of remarkable organizational changes that happen during the natural history of cancers. These reflect tumour heterogeneity, which is present at all cellular levels, including the genome, proteome and phenome, shaping its development and interrelation with its environment. An intriguing observation in different cohorts of oncological patients is that tumours exhibit an increased proliferation as the disease progresses, while the timescales involved are apparently too short for the fixation of sufficient driver mutations to promote explosive growth. Here, we discuss how phenotypic plasticity, emerging from a single genotype, may play a key role and provide a ground for a continuous acceleration of the proliferation rate of clonal populations with time. We address this question by combining the analysis of real-time growth of non-small-cell lung carcinoma cells (N-H460) together with stochastic and deterministic mathematical models that capture proliferation trait heterogeneity in clonal populations to elucidate the contribution of phenotypic transitions on tumour growth dynamics.

Growth dynamics of brain metastases differentiate radiation necrosis from recurrence

Background

Radiation necrosis (RN) is a frequent adverse event after fractionated stereotactic radiotherapy (FSRT) or single-session stereotactic radiosurgery (SRS) treatment of brain metastases (BMs). It is difficult to distinguish RN from progressive disease (PD) due to their similarities in the magnetic resonance images. Previous theoretical studies have hypothesized that RN could have faster, although transient, growth dynamics after FSRT/SRS, but no study has proven that hypothesis using patient data. Thus, we hypothesized that lesion size time dynamics obtained from growth laws fitted with data from sequential volumetric measurements on magnetic resonance images may help in discriminating recurrent BMs from RN events.

Methods

A total of 101 BMs from different institutions, growing after FSRT/SRS (60 PDs and 41 RNs) in 86 patients, displaying growth for at least 3 consecutive MRI follow-ups were selected for the study from a database of 1031 BMs. The 3 parameters of the Von Bertalanffy growth law were determined for each BM and used to discriminate statistically PDs from RNs.

Results

Growth exponents in patients with RNs were found to be substantially larger than those of PD, due to the faster, although transient, dynamics of inflammatory processes. Statistically significant differences (P < .001) were found between both groups. The receiver operating characteristic curve (AUC = 0.76) supported the ability of the growth law exponent to classify the events.

Conclusions

Growth law exponents obtained from sequential longitudinal magnetic resonance images after FSRT/SRS can be used as a complementary tool in the differential diagnosis between RN and PD.

Dynamic scaling of stochastic thermodynamic observables for chemical reactions at and away from equilibrium

Physical kinetic roughening processes are well-known to exhibit universal scaling of observables that fluctuate in space and time. Are there analogous dynamic scaling laws that are unique to the chemical reaction mechanisms available synthetically and occurring naturally? Here, we formulate an approach to the dynamic scaling of stochastic fluctuations in thermodynamic observables at and away from equilibrium. Both analytical expressions and numerical simulations confirm our dynamic scaling ansatz with associated scaling exponents, function, and law. A survey of common chemical mechanisms reveals classes that organize according to the molecularity of the reactions involved, the nature of the reaction vessel and external reservoirs, (non)equilibrium conditions, and the extent of autocatalysis in the reaction network. Varying experimental parameters, such as temperature, can cause coupled reactions capable of chemical feedback to transition between these classes. While path observables, such as the dynamical activity, have scaling exponents that are time-independent, the variance in the entropy production and flow can have time-dependent scaling exponents and self-averaging properties as a result of temporal correlations that emerge during thermodynamically irreversible processes. Altogether, these results establish dynamic universality classes in the nonequilibrium fluctuations of thermodynamic observables for well-mixed chemical reactions.

A Head-to-Head Comparison of 18F-Fluorocholine PET/CT and Conventional MRI as Predictors of Outcome in IDH Wild-Type High-Grade Gliomas

(1) Aim: To study the associations between imaging parameters derived from contrast-enhanced MRI (CE-MRI) and 18F-fluorocholine PET/CT and their performance as prognostic predictors in isocitrate dehydrogenase wild-type (IDH-wt) high-grade gliomas. (2) Methods: A prospective, multicenter study (FuMeGA: Functional and Metabolic Glioma Analysis) including patients with baseline CE-MRI and 18F-fluorocholine PET/CT and IDH wild-type high-grade gliomas. Clinical variables such as performance status, extent of surgery and adjuvant treatments (Stupp protocol vs others) were obtained and used to discriminate overall survival (OS) and progression-free survival (PFS) as end points. Multilesionality was assessed on the visual analysis of PET/CT and CE-MRI images. After tumor segmentation, standardized uptake value (SUV)-based variables for PET/CT and volume-based and geometrical variables for PET/CT and CE-MRI were calculated. The relationships among imaging techniques variables and their association with prognosis were evaluated using Pearson’s chi-square test and the t-test. Receiver operator characteristic, Kaplan−Meier and Cox regression were used for the survival analysis. (3) Results: 54 patients were assessed. The median PFS and OS were 5 and 11 months, respectively. Significant strong relationships between volume-dependent variables obtained from PET/CT and CE-MRI were found (r > 0.750, p < 0.05). For OS, significant associations were found with SUVmax, SUVpeak, SUVmean and sphericity (HR: 1.17, p = 0.035; HR: 1.24, p = 0.042; HR: 1.62, p = 0.040 and HR: 0.8, p = 0.022, respectively). Among clinical variables, only Stupp protocol and age showed significant associations with OS and PFS. No CE-MRI derived variables showed significant association with prognosis. In multivariate analysis, age (HR: 1.04, p = 0.002), Stupp protocol (HR: 2.81, p = 0.001), multilesionality (HR: 2.20, p = 0.013) and sphericity (HR: 0.79, p = 0.027) derived from PET/CT showed independent associations with OS. For PFS, only age (HR: 1.03, p = 0.021) and treatment protocol (HR: 2.20, p = 0.008) were significant predictors. (4) Conclusions: 18F-fluorocholine PET/CT metabolic and radiomic variables were robust prognostic predictors in patients with IDH-wt high-grade gliomas, outperforming CE-MRI derived variables.

Interdisciplinary approaches to metastasis

Interdisciplinary research is making a significant contribution to understanding metastasis - one of the grand challenges in cancer research. Examples drawn from apparently unconnected areas of physics, and described at a recent workshop on metastasis, illustrate the value of interdiscplinary thinking.

Simulations of tumor growth and response to immunotherapy by coupling a spatial agent-based model with a whole-patient quantitative systems pharmacology model

Quantitative systems pharmacology (QSP) models and spatial agent-based models (ABM) are powerful and efficient approaches for the analysis of biological systems and for clinical applications. Although QSP models are becoming essential in discovering predictive biomarkers and developing combination therapies through in silico virtual trials, they are inadequate to capture the spatial heterogeneity and randomness that characterize complex biological systems, and specifically the tumor microenvironment. Here, we extend our recently developed spatial QSP (spQSP) model to analyze tumor growth dynamics and its response to immunotherapy at different spatio-temporal scales. In the model, the tumor spatial dynamics is governed by the ABM, coupled to the QSP model, which includes the following compartments: central (blood system), tumor, tumor-draining lymph node, and peripheral (the rest of the organs and tissues). A dynamic recruitment of T cells and myeloid-derived suppressor cells (MDSC) from the QSP central compartment has been implemented as a function of the spatial distribution of cancer cells. The proposed QSP-ABM coupling methodology enables the spQSP model to perform as a coarse-grained model at the whole-tumor scale and as an agent-based model at the regions of interest (ROIs) scale. Thus, we exploit the spQSP model potential to characterize tumor growth, identify T cell hotspots, and perform qualitative and quantitative descriptions of cell density profiles at the invasive front of the tumor. Additionally, we analyze the effects of immunotherapy at both whole-tumor and ROI scales under different tumor growth and immune response conditions. A digital pathology computational analysis of triple-negative breast cancer specimens is used as a guide for modeling the immuno-architecture of the invasive front.

Integration of Heterogeneous Biological Data in Multiscale Mechanistic Model Calibration: Application to Lung Adenocarcinoma

Mechanistic models are built using knowledge as the primary information source, with well-established biological and physical laws determining the causal relationships within the model. Once the causal structure of the model is determined, parameters must be defined in order to accurately reproduce relevant data. Determining parameters and their values is particularly challenging in the case of models of pathophysiology, for which data for calibration is sparse. Multiple data sources might be required, and data may not be in a uniform or desirable format. We describe a calibration strategy to address the challenges of scarcity and heterogeneity of calibration data. Our strategy focuses on parameters whose initial values cannot be easily derived from the literature, and our goal is to determine the values of these parameters via calibration with constraints set by relevant data. When combined with a covariance matrix adaptation evolution strategy (CMA-ES), this step-by-step approach can be applied to a wide range of biological models. We describe a stepwise, integrative and iterative approach to multiscale mechanistic model calibration, and provide an example of calibrating a pathophysiological lung adenocarcinoma model. Using the approach described here we illustrate the successful calibration of a complex knowledge-based mechanistic model using only the limited heterogeneous datasets publicly available in the literature.

Allometric Scaling in Pharmacokinetic Studies in Anesthesiology

A clinical review is presented of basic allometric scaling theory and its application to pharmacokinetic models in anesthesia and other fields in the biologic sciences.

Cancer as a Model System for Testing Metabolic Scaling Theory

Biological allometries, such as the scaling of metabolism to mass, are hypothesized to result from natural selection to maximize how vascular networks fill space yet minimize internal transport distances and resistance to blood flow. Metabolic scaling theory argues two guiding principles—conservation of fluid flow and space-filling fractal distributions—describe a diversity of biological networks and predict how the geometry of these networks influences organismal metabolism. Yet, mostly absent from past efforts are studies that directly, and independently, measure metabolic rate from respiration and vascular architecture for the same organ, organism, or tissue. Lack of these measures may lead to inconsistent results and conclusions about metabolism, growth, and allometric scaling. We present simultaneous and consistent measurements of metabolic scaling exponents from clinical images of lung cancer, serving as a first-of-its-kind test of metabolic scaling theory, and identifying potential quantitative imaging biomarkers indicative of tumor growth. We analyze data for 535 clinical PET-CT scans of patients with non-small cell lung carcinoma to establish the presence of metabolic scaling between tumor metabolism and tumor volume. Furthermore, we use computer vision and mathematical modeling to examine predictions of metabolic scaling based on the branching geometry of the tumor-supplying blood vessel networks in a subset of 56 patients diagnosed with stage II-IV lung cancer. Examination of the scaling of maximum standard uptake value with metabolic tumor volume, and metabolic tumor volume with gross tumor volume, yield metabolic scaling exponents of 0.64 (0.20) and 0.70 (0.17), respectively. We compare these to the value of 0.85 (0.06) derived from the geometric scaling of the tumor-supplying vasculature. These results: (1) inform energetic models of growth and development for tumor forecasting; (2) identify imaging biomarkers in vascular geometry related to blood volume and flow; and (3) highlight unique opportunities to develop and test the metabolic scaling theory of ecology in tumors transitioning from avascular to vascular geometries.

Optimal adjuvant chemotherapy completion time for stage III colon cancer: a cohort study

Background

Adjuvant chemotherapy for 6 months following surgery is the standard treatment plan for stage III colon cancer. The aim of the present study was to determine whether the adjuvant chemotherapy completion time for stage III colon cancer had an effect on prognosis and cut-off time that affected the prognosis.

Methods

This was a retrospective study of stage III colon cancer patients who completed adjuvant chemotherapy at Guangzhou Red Cross Hospital from January 2010 to December 2017. Univariate and multivariate analyses were used to determine the association between adjuvant chemotherapy completion time and the 3-year disease-free survival (DFS). The restricted cubic spline model was used to analyze the cut-off time that affected the 3-year DFS.

Results

A total of 431 patients were included in the study. The 3-year DFS was associated with a combination of obstruction or perforation, preoperative serum carcino-embryonic antigen (CEA) concentration, T stage, N stage, pathological stage, and adjuvant chemotherapy completion time in the univariate analysis (P<0.05). A combination of obstruction or perforation, preoperative serum CEA concentration, N stage, and adjuvant chemotherapy completion time were independent prognostic factors in the multivariate analysis (P<0.05). The cut-off time was 28 weeks for adjuvant chemotherapy completion time in the restricted cubic spline model analysis. For those whose adjuvant chemotherapy completion time was >28 weeks, the risk of 3-year recurrence was 1.428 times higher compared with those whose adjuvant chemotherapy completion time was ≤28 weeks. [P=0.032, 95% confidence interval (CI): 1.034-2.055].

Conclusions

The 3-year DFS of stage III colon cancer was related to the adjuvant chemotherapy completion time. For those who completed adjuvant chemotherapy >28 weeks, the risk of 3-year recurrence increased.

A theoretical analysis of tumour containment

Recent studies have shown that a strategy aiming for containment, not elimination, can control tumour burden more effectively in vitro, in mouse models and in the clinic. These outcomes are consistent with the hypothesis that emergence of resistance to cancer therapy may be prevented or delayed by exploiting competitive ecological interactions between drug-sensitive and drug-resistant tumour cell subpopulations. However, although various mathematical and computational models have been proposed to explain the superiority of particular containment strategies, this evolutionary approach to cancer therapy lacks a rigorous theoretical foundation. Here we combine extensive mathematical analysis and numerical simulations to establish general conditions under which a containment strategy is expected to control tumour burden more effectively than applying the maximum tolerated dose. We show that containment may substantially outperform more aggressive treatment strategies even if resistance incurs no cellular fitness cost. We further provide formulas for predicting the clinical benefits attributable to containment strategies in a wide range of scenarios and compare the outcomes of theoretically optimal treatments with those of more practical protocols. Our results strengthen the rationale for clinical trials of evolutionarily informed cancer therapy, while also clarifying conditions under which containment might fail to outperform standard of care.

Lie point symmetries for generalised Fisher's equations describing tumour dynamics

A huge variety of phenomena are governed by ordinary differential equations (ODEs) and partial differential equations (PDEs). However, there is no general method to solve them. Obtaining solutions for differential equations is one of the greatest problem for both applied mathematics and physics. Multiple integration methods have been developed to the day to solve particular types of differential equations, specially those focused on physical or biological phenomena. In this work, we review several applications of the Lie method to obtain solutions of reaction-diffusion equations describing cell dynamics and tumour invasion.

Superlinear growth reveals the Allee effect in tumors

Integrating experimental data into ecological models plays a central role in understanding biological mechanisms that drive tumor progression where such knowledge can be used to develop new therapeutic strategies. While the current studies emphasize the role of competition among tumor cells, they fail to explain recently observed superlinear growth dynamics across human tumors. Here we study tumor growth dynamics by developing a model that incorporates evolutionary dynamics inside tumors with tumor-microenvironment interactions. Our results reveal that tumor cells' ability to manipulate the environment and induce angiogenesis drives superlinear growth-a process compatible with the Allee effect. In light of this understanding, our model suggests that, for high-risk tumors that have a higher growth rate, suppressing angiogenesis can be the appropriate therapeutic intervention.

A mesoscopic simulator to uncover heterogeneity and evolutionary dynamics in tumors

Increasingly complex in silico modeling approaches offer a way to simultaneously access cancerous processes at different spatio-temporal scales. High-level models, such as those based on partial differential equations, are computationally affordable and allow large tumor sizes and long temporal windows to be studied, but miss the discrete nature of many key underlying cellular processes. Individual-based approaches provide a much more detailed description of tumors, but have difficulties when trying to handle full-sized real cancers. Thus, there exists a trade-off between the integration of macroscopic and microscopic information, now widely available, and the ability to attain clinical tumor sizes. In this paper we put forward a stochastic mesoscopic simulation framework that incorporates key cellular processes during tumor progression while keeping computational costs to a minimum. Our framework captures a physical scale that allows both the incorporation of microscopic information, tracking the spatio-temporal emergence of tumor heterogeneity and the underlying evolutionary dynamics, and the reconstruction of clinically sized tumors from high-resolution medical imaging data, with the additional benefit of low computational cost. We illustrate the functionality of our modeling approach for the case of glioblastoma, a paradigm of tumor heterogeneity that remains extremely challenging in the clinical setting.

Dual-Target CAR-Ts with On- and Off-Tumour Activity May Override Immune Suppression in Solid Cancers: A Mathematical Proof of Concept

Simple Summary

(CAR)-T cell-based therapies have achieved substantial success against different haematological malignancies. However, results for solid tumours have been limited up to now, in part due to the fact that the immunosuppressive tumour microenvironment inactivates CAR-T cell clones. In this paper we study mathematically the competition of CAR-T and tumour cells, taking into account their immunosuppressive capacity. Using computer simulations, we show that the use of large numbers of CAR-T cells targetting the solid tumour antigens could overcome the immunosuppressive potential of cancer. To achieve such high levels of CAR-T cells we propose, and study in silico, the manufacture and injection of CAR-T cells targetting two antigens: CD19 and a tumour-associated antigen. This strategy lead in our simulations to the expansion of the CAR-T cells injected and the production of a massive army of CAR-T cells targetting the solid tumour, and potentially overcoming its immune suppression capabilities. Thus, our proposed strategy could provide a way to develop successful CAR-T cell therapies against solid tumours.

Abstract

Chimeric antigen receptor (CAR)-T cell-based therapies have achieved substantial success against B-cell malignancies, which has led to a growing scientific and clinical interest in extending their use to solid cancers. However, results for solid tumours have been limited up to now, in part due to the immunosuppressive tumour microenvironment, which is able to inactivate CAR-T cell clones. In this paper we put forward a mathematical model describing the competition of CAR-T and tumour cells, taking into account their immunosuppressive capacity. Using the mathematical model, we show that the use of large numbers of CAR-T cells targetting the solid tumour antigens could overcome the immunosuppressive potential of cancer. To achieve such high levels of CAR-T cells we propose, and study computationally, the manufacture and injection of CAR-T cells targetting two antigens: CD19 and a tumour-associated antigen. We study in silico the resulting dynamics of the disease after the injection of this product and find that the expansion of the CAR-T cell population in the blood and lymphopoietic organs could lead to the massive production of an army of CAR-T cells targetting the solid tumour, and potentially overcoming its immune suppression capabilities. This strategy could benefit from the combination with PD-1 inhibitors and low tumour loads. Our computational results provide theoretical support for the treatment of different types of solid tumours using T cells engineered with combination treatments of dual CARs with on- and off-tumour activity and anti-PD-1 drugs after completion of classical cytoreductive treatments.

Evolutionary dynamics at the tumor edge reveal metabolic imaging biomarkers

Human cancers are biologically and morphologically heterogeneous. A variety of clonal populations emerge within these neoplasms and their interaction leads to complex spatiotemporal dynamics during tumor growth. We studied the reshaping of metabolic activity in human cancers by means of continuous and discrete mathematical models and matched the results to positron emission tomography (PET) imaging data. Our models revealed that the location of increasingly active proliferative cellular spots progressively drifted from the center of the tumor to the periphery, as a result of the competition between gradually more aggressive phenotypes. This computational finding led to the development of a metric, normalized distance from 18F-fluorodeoxyglucose (18F-FDG) hotspot to centroid (NHOC), based on the separation from the location of the activity (proliferation) hotspot to the tumor centroid. The NHOC metric can be computed for patients using 18F-FDG PET-computed tomography (PET/CT) images where the voxel of maximum uptake (standardized uptake value [SUV]max) is taken as the activity hotspot. Two datasets of 18F-FDG PET/CT images were collected, one from 61 breast cancer patients and another from 161 non-small-cell lung cancer patients. In both cohorts, survival analyses were carried out for the NHOC and for other classical PET/CT-based biomarkers, finding that the former had a high prognostic value, outperforming the latter. In summary, our work offers additional insights into the evolutionary mechanisms behind tumor progression, provides a different PET/CT-based biomarker, and reveals that an activity hotspot closer to the tumor periphery is associated to a worst patient outcome.

Invasion front dynamics in disordered environments

Invasion occurs in environments that are normally spatially disordered, however, the effect of such a randomness on the dynamics of the invasion front has remained less understood. Here, we study Fisher's equation in disordered environments both analytically and numerically. Using the Effective Medium Approximation, we show that disorder slows down invasion velocity and for ensemble average of invasion velocity in disordered environment we have [Formula: see text] where [Formula: see text] is the amplitude of disorder and [Formula: see text] is the invasion velocity in the corresponding homogeneous environment given by [Formula: see text]. Additionally, disorder imposes fluctuations on the invasion front. Using a perturbative approach, we show that these fluctuations are Brownian with a diffusion constant of: [Formula: see text]. These findings were approved by numerical analysis. Alongside this continuum model, we use the Stepping Stone Model to check how our findings change when we move from the continuum approach to a discrete approach. Our analysis suggests that individual-based models exhibit inherent fluctuations and the effect of environmental disorder becomes apparent for large disorder intensity and/or high carrying capacities.