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 ß* = 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.

The shape of cancer relapse: Topological data analysis predicts recurrence in paediatric acute lymphoblastic leukaemia.

Although children and adolescents with acute lymphoblastic leukaemia (ALL) have high survival rates, approximately 15-20% of patients relapse. Risk of relapse is routinely estimated at diagnosis by biological factors, including flow cytometry data. This high-dimensional data is typically manually assessed by projecting it onto a subset of biomarkers. Cell density and "empty spaces" in 2D projections of the data, i.e. regions devoid of cells, are then used for qualitative assessment. Here, we use topological data analysis (TDA), which quantifies shapes, including empty spaces, in data, to analyse pre-treatment ALL datasets with known patient outcomes. We combine these fully unsupervised analyses with Machine Learning (ML) to identify significant shape characteristics and demonstrate that they accurately predict risk of relapse, particularly for patients previously classified as 'low risk'. We independently confirm the predictive power of CD10, CD20, CD38, and CD45 as biomarkers for ALL diagnosis. Based on our analyses, we propose three increasingly detailed prognostic pipelines for analysing flow cytometry data from ALL patients depending on technical and technological availability: 1. Visual inspection of specific biological features in biparametric projections of the data; 2. Computation of quantitative topological descriptors of such projections; 3. A combined analysis, using TDA and ML, in the four-parameter space defined by CD10, CD20, CD38 and CD45. Our analyses readily extend to other haematological malignancies.

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.

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.

SUVmax to tumor perimeter distance: a robust radiomics prognostic biomarker in resectable non-small cell lung cancer patients.

OBJECTIVE: The purpose of this study was to evaluate the prognostic value of novel geometric variables obtained from pre-treatment [18F]FDG PET/CT with respect to classical ones in patients with non-small cell lung cancer (NSCLC). METHODS: Retrospective study including stage I-III NSCLC patients with baseline [18F]FDG PET/CT. Clinical, histopathologic, and metabolic parameters were obtained. After tumor segmentation, SUV and volume-based variables, global texture, sphericity, and two novel parameters, normalized SUVpeak to centroid distance (nSCD) and normalized SUVmax to perimeter distance (nSPD), were obtained. Early recurrence (ER) and short-term mortality (STM) were used as end points. Univariate logistic regression and multivariate logistic regression with respect to ER and STM were performed. RESULTS: A cohort of 173 patients was selected. ER was detected in 49/104 of patients with recurrent disease. Additionally, 100 patients died and 53 had STM. Age, pathologic lymphovascular invasion, lymph nodal infiltration, TNM stage, nSCD, and nSPD were associated with ER, although only age (aOR = 1.06, p = 0.002), pathologic lymphovascular invasion (aOR = 3.40, p = 0.022), and nSPD (aOR = 0.02, p = 0.018) were significant independent predictors of ER in multivariate analysis. Age, lymph nodal infiltration, TNM stage, nSCD, and nSPD were predictors of STM. Age (aOR = 1.05, p = 0.006), lymph nodal infiltration (aOR = 2.72, p = 0.005), and nSPD (aOR = 0.03, p = 0.022) were significantly associated with STM in multivariate analysis. Coefficient of variation (COV) and SUVmean/SUVmax ratio did not show significant predictive value with respect to ER or STM. CONCLUSION: The geometric variables, nSCD and nSPD, are robust biomarkers of the poorest outcome prediction of patients with NSCLC with respect to classical PET variables. KEY POINTS: • In NSCLC patients, it is crucial to find prognostic parameters since TNM system alone cannot explain the variation in lung cancer survival. • Age, lymphovascular invasion, lymph nodal infiltration, and metabolic geometrical parameters were useful as prognostic parameters. • The displacement grade of the highest point of metabolic activity towards the periphery assessed by geometric variables obtained from [18F]FDG PET/CT was a robust biomarker of the poorest outcome prediction of patients with NSCLC.

In vitro biomimetic models for glioblastoma-a promising tool for drug response studies.

The poor response of glioblastoma to current treatment protocols is a consequence of its intrinsic drug resistance. Resistance to chemotherapy is primarily associated with considerable cellular heterogeneity, and plasticity of glioblastoma cells, alterations in gene expression, presence of specific tumor microenvironment conditions and blood-brain barrier. In an attempt to successfully overcome chemoresistance and better understand the biological behavior of glioblastoma, numerous tri-dimensional (3D) biomimetic models were developed in the past decade. These novel advanced models are able to better recapitulate the spatial organization of glioblastoma in a real time, therefore providing more realistic and reliable evidence to the response of glioblastoma to therapy. Moreover, these models enable the fine-tuning of different tumor microenvironment conditions and facilitate studies on the effects of the tumor microenvironment on glioblastoma chemoresistance. This review outlines current knowledge on the essence of glioblastoma chemoresistance and describes the progress achieved by 3D biomimetic models. Moreover, comprehensive literature assessment regarding the influence of 3D culturing and microenvironment mimicking on glioblastoma gene expression and biological behavior is also provided. The contribution of the blood-brain barrier as well as the blood-tumor barrier to glioblastoma chemoresistance is also reviewed from the perspective of 3D biomimetic models. Finally, the role of mathematical models in predicting 3D glioblastoma behavior and drug response is elaborated. In the future, technological innovations along with mathematical simulations should create reliable 3D biomimetic systems for glioblastoma research that should facilitate the identification and possibly application in preclinical drug testing and precision medicine.

Dynamical properties of feedback signalling in B lymphopoiesis: A mathematical modelling approach.

Haematopoiesis is the process of generation of blood cells. Lymphopoiesis generates lymphocytes, the cells in charge of the adaptive immune response. Disruptions of this process are associated with diseases like leukaemia, which is especially incident in children. The characteristics of self-regulation of this process make them suitable for a mathematical study. In this paper we develop mathematical models of lymphopoiesis using currently available data. We do this by drawing inspiration from existing structured models of cell lineage development and integrating them with paediatric bone marrow data, with special focus on regulatory mechanisms. A formal analysis of the models is carried out, giving steady states and their stability conditions. We use this analysis to obtain biologically relevant regions of the parameter space and to understand the dynamical behaviour of B-cell renovation. Finally, we use numerical simulations to obtain further insight into the influence of proliferation and maturation rates on the reconstitution of the cells in the B line. We conclude that a model including feedback regulation of cell proliferation represents a biologically plausible depiction for B-cell reconstitution in bone marrow. Research into haematological disorders could benefit from a precise dynamical description of B lymphopoiesis.

Nonlinear science against the COVID-19 pandemic.

This special issue showcases recent uses of mathematical and nonlinear science methods in the study of different problems arising in the context of the COVID-19 pandemic. The sixteen original research papers included in this collection span a wide spectrum of studies including classical epidemiological models, new models accounting for COVID-19 specificities, non-pharmaceutical control measures, network models and other problems related to the pandemic.

A Mathematical Description of the Bone Marrow Dynamics during CAR T-Cell Therapy in B-Cell Childhood Acute Lymphoblastic Leukemia.

Chimeric Antigen Receptor (CAR) T-cell therapy has demonstrated high rates of response in recurrent B-cell Acute Lymphoblastic Leukemia in children and young adults. Despite this success, a fraction of patients' experience relapse after treatment. Relapse is often preceded by recovery of healthy B cells, which suggests loss or dysfunction of CAR T-cells in bone marrow. This site is harder to access, and thus is not monitored as frequently as peripheral blood. Understanding the interplay between B cells, leukemic cells, and CAR T-cells in bone marrow is paramount in ascertaining the causes of lack of response. In this paper, we put forward a mathematical model representing the interaction between constantly renewing B cells, CAR T-cells, and leukemic cells in the bone marrow. Our model accounts for the maturation dynamics of B cells and incorporates effector and memory CAR T-cells. The model provides a plausible description of the dynamics of the various cellular compartments in bone marrow after CAR T infusion. After exploration of the parameter space, we found that the dynamics of CAR T product and disease were independent of the dose injected, initial B-cell load, and leukemia burden. We also show theoretically the importance of CAR T product attributes in determining therapy outcome, and have studied a variety of possible response scenarios, including second dosage schemes. We conclude by setting out ideas for the refinement of the model.

Computational design of improved standardized chemotherapy protocols for grade II oligodendrogliomas.

Here we put forward a mathematical model describing the response of low-grade (WHO grade II) oligodendrogliomas (LGO) to temozolomide (TMZ). The model describes the longitudinal volumetric dynamics of tumor response to TMZ of a cohort of 11 LGO patients treated with TMZ. After finding patient-specific parameters, different therapeutic strategies were tried computationally on the 'in-silico twins' of those patients. Chemotherapy schedules with larger-than-standard rest periods between consecutive cycles had either the same or better long-term efficacy than the standard 28-day cycles. The results were confirmed in a large trial of 2000 virtual patients. These long-cycle schemes would also have reduced toxicity and defer the appearance of resistances. On the basis of those results, a combination scheme consisting of five induction TMZ cycles given monthly plus 12 maintenance cycles given every three months was found to provide substantial survival benefits for the in-silico twins of the 11 LGO patients (median 5.69 years, range: 0.67 to 68.45 years) and in a large virtual trial including 2000 patients. We used 220 sets of experiments in-silico to show that a clinical trial incorporating 100 patients per arm (standard intensive treatment versus 5 + 12 scheme) could demonstrate the superiority of the novel scheme after a follow-up period of 10 years. Thus, the proposed treatment plan could be the basis for a standardized TMZ treatment for LGO patients with survival benefits.

Morphological MRI-based features provide pretreatment survival prediction in glioblastoma.

OBJECTIVES: We wished to determine whether tumor morphology descriptors obtained from pretreatment magnetic resonance images and clinical variables could predict survival for glioblastoma patients. METHODS: A cohort of 404 glioblastoma patients (311 discoveries and 93 validations) was used in the study. Pretreatment volumetric postcontrast T1-weighted magnetic resonance images were segmented to obtain the relevant morphological measures. Kaplan-Meier, Cox proportional hazards, correlations, and Harrell's concordance indexes (c-indexes) were used for the statistical analysis. RESULTS: A linear prognostic model based on the outstanding variables (age, contrast-enhanced (CE) rim width, and surface regularity) identified a group of patients with significantly better survival (p < 0.001, HR = 2.57) with high accuracy (discovery c-index = 0.74; validation c-index = 0.77). A similar model applied to totally resected patients was also able to predict survival (p < 0.001, HR = 3.43) with high predictive value (discovery c-index = 0.81; validation c-index = 0.92). Biopsied patients with better survival were well identified (p < 0.001, HR = 7.25) by a model including age and CE volume (c-index = 0.87). CONCLUSIONS: Simple linear models based on small sets of meaningful MRI-based pretreatment morphological features and age predicted survival of glioblastoma patients to a high degree of accuracy. The partition of the population using the extent of resection improved the prognostic value of those measures. KEY POINTS: • A combination of two MRI-based morphological features (CE rim width and surface regularity) and patients' age outperformed previous prognosis scores for glioblastoma. • Prognosis models for homogeneous surgical procedure groups led to even more accurate survival prediction based on Kaplan-Meier analysis and concordance indexes.

Correction to: Morphological MRI-based features provide pretreatment survival prediction in glioblastoma.

The original version of this article, published on 15 October 2018, unfortunately contained a mistake. The following correction has therefore been made in the original: The name of Mariano Amo-Salas and the affiliation of Ismael Herruzo were presented incorrectly.

Tumor Surface Regularity at MR Imaging Predicts Survival and Response to Surgery in Patients with Glioblastoma.

Purpose To evaluate the prognostic and predictive value of surface-derived imaging biomarkers obtained from contrast material-enhanced volumetric T1-weighted pretreatment magnetic resonance (MR) imaging sequences in patients with glioblastoma multiforme. Materials and Methods A discovery cohort from five local institutions (165 patients; mean age, 62 years ± 12 [standard deviation]; 43% women and 57% men) and an independent validation cohort (51 patients; mean age, 60 years ± 12; 39% women and 61% men) from The Cancer Imaging Archive with volumetric T1-weighted pretreatment contrast-enhanced MR imaging sequences were included in the study. Clinical variables such as age, treatment, and survival were collected. After tumor segmentation and image processing, tumor surface regularity, measuring how much the tumor surface deviates from a sphere of the same volume, was obtained. Kaplan-Meier, Cox proportional hazards, correlations, and concordance indexes were used to compare variables and patient subgroups. Results Surface regularity was a powerful predictor of survival in the discovery (P = .005, hazard ratio [HR] = 1.61) and validation groups (P = .05, HR = 1.84). Multivariate analysis selected age and surface regularity as significant variables in a combined prognostic model (P < .001, HR = 3.05). The model achieved concordance indexes of 0.76 and 0.74 for the discovery and validation cohorts, respectively. Tumor surface regularity was a predictor of survival for patients who underwent complete resection (P = .01, HR = 1.90). Tumors with irregular surfaces did not benefit from total over subtotal resections (P = .57, HR = 1.17), but those with regular surfaces did (P = .004, HR = 2.07). Conclusion The surface regularity obtained from high-resolution contrast-enhanced pretreatment volumetric T1-weighted MR images is a predictor of survival in patients with glioblastoma. It may help in classifying patients for surgery.

Glioblastoma: does the pre-treatment geometry matter? A postcontrast T1 MRI-based study.

BACKGROUND: The potential of a tumour's volumetric measures obtained from pretreatment MRI sequences of glioblastoma (GBM) patients as predictors of clinical outcome has been controversial. Mathematical models of GBM growth have suggested a relation between a tumour's geometry and its aggressiveness. METHODS: A multicenter retrospective clinical study was designed to study volumetric and geometrical measures on pretreatment postcontrast T1 MRIs of 117 GBM patients. Clinical variables were collected, tumours segmented, and measures computed including: contrast enhancing (CE), necrotic, and total volumes; maximal tumour diameter; equivalent spherical CE width and several geometric measures of the CE "rim". The significance of the measures was studied using proportional hazards analysis and Kaplan-Meier curves. RESULTS: Kaplan-Meier and univariate Cox survival analysis showed that total volume [p = 0.034, Hazard ratio (HR) = 1.574], CE volume (p = 0.017, HR = 1.659), spherical rim width (p = 0.007, HR = 1.749), and geometric heterogeneity (p = 0.015, HR = 1.646) were significant parameters in terms of overall survival (OS). Multivariable Cox analysis for OS provided the later two parameters as age-adjusted predictors of OS (p = 0.043, HR = 1.536 and p = 0.032, HR = 1.570, respectively). CONCLUSION: Patients with tumours having small geometric heterogeneity and/or spherical rim widths had significantly better prognosis. These novel imaging biomarkers have a strong individual and combined prognostic value for GBM patients. KEY POINTS: • Three-dimensional segmentation on magnetic resonance images allows the study of geometric measures. • Patients with small width of contrast enhancing areas have better prognosis. • The irregularity of contrast enhancing areas predicts survival in glioblastoma patients.

Transfer of Drug Resistance Characteristics Between Cancer Cell Subpopulations: A Study Using Simple Mathematical Models.

Resistance to chemotherapy is a major cause of cancer treatment failure. The processes of resistance induction and selection of resistant cells (due to the over-expression of the membrane transporter P-glycoprotein, P-gp) are well documented in the literature, and a number of mathematical models have been developed. However, another process of transfer of resistant characteristics is less well known and has received little attention in the mathematical literature. In this paper, we discuss the potential of simple mathematical models to describe the process of resistance transfer, specifically P-gp transfer, in mixtures of resistant and sensitive tumor cell populations. Two different biological hypotheses for P-gp transfer are explored: (1) exchange through physical cell-cell connections and (2) through microvessicles released to the culture medium. Two models are developed which fit very well the observed population growth dynamics. The potential and limitations of these simple "global" models to describe the aforementioned biological processes involved are discussed on the basis of the results obtained.

Hypoxia in Gliomas: Opening Therapeutical Opportunities Using a Mathematical-Based Approach.

This chapter explores the use of mathematical models as promising and powerful tools to understand the complexity of tumors and their, frequently, hypoxic environment. We focus on gliomas, which are primary brain tumors derived from glial cells, mainly astrocytes and/or oligodendrocytes. A variety of mathematical models, based on ordinary and/or partial differential equations, have been developed both at the micro and macroscopic levels. The aim here is to describe in a quantitative way key physiopathological mechanisms relevant in these types of malignancies and to suggest optimal therapeutical strategies. More specifically, we consider novel therapies targeting thromboembolic phenomena to decrease cell invasion in high grade glioma or to delay the malignant transformation in low grade gliomas. This study has been the basis of a multidisciplinary collaboration involving, among others, neuro-oncologists, radiation oncologists, pathologists, cancer biologists, surgeons and mathematicians.

Resistance to DNA Damaging Agents Produced Invasive Phenotype of Rat Glioma Cells-Characterization of a New in Vivo Model.

Chemoresistance and invasion properties are severe limitations to efficient glioma therapy. Therefore, development of glioma in vivo models that more accurately resemble the situation observed in patients emerges. Previously, we established RC6 rat glioma cell line resistant to DNA damaging agents including antiglioma approved therapies such as 3-bis(2-chloroethyl)-1-nitrosourea (BCNU) and temozolomide (TMZ). Herein, we evaluated the invasiveness of RC6 cells in vitro and in a new orthotopic animal model. For comparison, we used C6 cells from which RC6 cells originated. Differences in cell growth properties were assessed by real-time cell analyzer. Cells' invasive potential in vitro was studied in fluorescently labeled gelatin and by formation of multicellular spheroids in hydrogel. For animal studies, fluorescently labeled cells were inoculated into adult male Wistar rat brains. Consecutive coronal and sagittal brain sections were analyzed 10 and 25 days post-inoculation, while rats' behavior was recorded during three days in the open field test starting from 25th day post-inoculation. We demonstrated that development of chemoresistance induced invasive phenotype of RC6 cells with significant behavioral impediments implying usefulness of orthotopic RC6 glioma allograft in preclinical studies for the examination of new approaches to counteract both chemoresistance and invasion of glioma cells.

Glucose-lactate metabolic cooperation in cancer: insights from a spatial mathematical model and implications for targeted therapy.

A recent study has hypothesised a glucose-lactate metabolic symbiosis between adjacent hypoxic and oxygenated regions of a developing tumour, and proposed a treatment strategy to target this symbiosis. However, in vivo experimental support remains inconclusive. Here we develop a minimal spatial mathematical model of glucose-lactate metabolism to examine, in principle, whether metabolic symbiosis is plausible in human tumours, and to assess the potential impact of inhibiting it. We find that symbiosis is a robust feature of our model system-although on the length scale at which oxygen supply is diffusion-limited, its occurrence requires very high cellular metabolic activity-and that necrosis in the tumour core is reduced in the presence of symbiosis. Upon simulating therapeutic inhibition of lactate uptake, we predict that targeted treatment increases the extent of tissue oxygenation without increasing core necrosis. The oxygenation effect is correlated strongly with the extent of wild-type hypoxia and only weakly with wild-type symbiotic behaviour, and therefore may be promising for radiosensitisation of hypoxic, lactate-consuming tumours even if they do not exhibit a spatially well-defined symbiosis. Finally, we conduct in vitro experiments on the U87 glioblastoma cell line to facilitate preliminary speculation as to where highly malignant tumours might fall in our parameter space, and find that these experiments suggest a weakly symbiotic regime for U87 cells, thus raising the new question of what relationship might exist between symbiosis and tumour malignancy.

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.