

Basic principles of mathematical growth modeling applied to highgrade gliomas: A brief clinical review for clinicians
Correspondence Address: Source of Support: None, Conflict of Interest: None DOI: 10.4103/00283886.246238
Keywords: Brain tumors, glioblastoma multiforme, mathematical model, tumor growth
Mathematics and computation can help in solving several growing problems in medical research by proposing models that allow us to formalize the causeandeffect process and tie it to the biological observations. As cancer has become a leading cause of death and its different treatments produce several side effects,^{[1],[2]} therapy is moving towards personalized medicine.^{[3],[4]} The word “glioma” refers to a tumor originating from the glial cells (mainly from astrocytes and oligodendrocytes).^{[5]} Gliomas represent the most frequent type of brain tumors (30% of all central nervous system tumors and approximately 80% of all malignant brain tumors).^{[6]} The World Health Organization (WHO) classifies grade I astrocytomas (predominantly pilocytic astrocytomas) as lessinfiltrating, and usually curable tumors. Grade II gliomas are commonly referred to as lowgrade gliomas. Grades III and IV are known as highgrade gliomas.^{[7]} Glioblastoma multiforme (GBM), the most common malignant primary brain tumor, has an incidence of 3.19 cases per 100,000 personyears and a 26–33% survival rate at 2 years in clinical trials.^{[8]} Conventional imaging in glioblastoma multiforme The traditional imaging is considered as the “tip of the iceberg” in the current imaging protocol of a GBM, with a significant proportion of the GBM cells already invading the peripheral tissue.^{[9]} Advanced sequences of magnetic resonance imaging (MRI), especially the diffusion tensor imaging (DTI) sequence, can detect novel DTI biomarkers.^{[10],[11],[12]} The routine MRI examination for the diagnosis of gliomas around the world aims to identify the tumor size, the extension of edema and its boundaries, and evidence of local tumor infiltration, by using four primary MRI sequences: T1weighted (W) precontrast, T1W postcontrast, T2W precontrast, and fluidattenuated inversion recovery (FLAIR) precontrast images [Figure 1]a,[Figure 1]b,[Figure 1]c,[Figure 1]d.^{[13],[14]}
Tumor regions in glioblastoma multiforme It was reported in 2009 that some GBMs arise from the transformed astrocytelike neural stem cells in the subventricular zone (SVZ), whereas other GBMs initiate by neoplastic transformation of the nonSVZ progenitor cells or mature glial cells that have undergone dedifferentiation.^{[15]} A published MRIbased classification of GBMs that was based on the involvement of the SVZ and cortex promised a better understanding of the histogenetic and clinical heterogeneity of GBMs.^{[16]} Based on this classification of GBM, the Group I tumor is a contrastenhancing lesion that extends from the atrium SVZ to the pia. The Group II tumors refers to the a tumor making contact with the SVZ. It is a lesion that does not involve the cortex. In the Group III tumors, the contrastenhancing GBM is invading the cortex, reaching the pia, but does not touch the SVZ. In the Group IV GBMs, the tumor “respects” both the SVZ and the cortex. The application of the Cochran–Armitage trend test on GBM proved that an association exists between the incidence rate of GBMs and the group order and severity status it belongs to (thus, the incidence of Groups III–IV > Group II > Group I).^{[16]} We mention this information to make researcher aware that selected tumor regions in GBM might deserve specific tumor growth models. The response assessment utilizing the neurooncology response criteria versus the Macdonald criteria In 1990, Macdonald et al., introduced a response criteria based on a twodimensional measure of contrast enhancement on computed tomographic scans.^{[17]} The response assessment in neurooncology (RANO) response criteria replaced the Macdonald criteria to assess the changes determined on the postoperative MRI of glioblastomas and is currently considered the stateof theart criteria.^{[18]} The RANO response criteria defines progressive disease as an increase of the perpendicular diameter of the tumor more than 25% in postcontrast T1weighted sequences; 25% or more increase in enhancing lesions despite stable or increasing steroid dose; an increase (significant) in nonenhancing FLAIR/T2W lesions, not attributable to other nontumour causes; or the development of any new lesions.^{[18]} Assuming an isotropic tumor growth, this twodimensional increase of 25% in the tumor diameter would result in a 39.8% increase in the tumor volume. Even though the RANO criteria provided a quantitative and objective attempt to address radiographic changes in the course of glioblastoma, there are still several limitations due to the twodimensional evaluation, for example, the presence of irregularly shaped tumors, multifocal lesions, or cystic components.^{[18],[19]} Segmentation of tumor regions using magnetic resonance imaging of the brain Experts define image segmentation as the partitioning of an image into nonoverlapping constituent regions, which exhibit homogeneity related to some tumor characteristics, for example, related to the intensity or texture of the tumor. Several studies have shown that semiautomated segmentation of glioblastomas can be done reliably by different groups of raters using commercial software solutions based on a regiongrowing algorithm.^{[20]} Studies are in agreement on the fact that the collation of quantitative volumetric reports of contrastenhancing or fluid attenuated inversion recovery (FLAIR) hyperintense tumor compartments are essential requisites for an objective evaluation of the stable or progressive form of disease in patients harbouring a glioblastoma.^{[21]} The main problems in the segmentation of the cerebral images of MRI are that different tissues have or may have similar brightness; there are no absolute values in the intensity units; and, a characteristic noise related to each MRI sequence is associated with the technique. The conventional MRI techniques support their operation on the assumption that tissues of a type have a characteristic intensity signal. As this statement does not always hold true, it is not possible to segment the imaging of a tumor only using these techniques. Thus, the technique requires a postprocessing work to eliminate regions with an intensity similar to that of the object of interest but not belonging to it. There are different underlying techniques of segmentation,^{[22],[23],[24]} among which the most commonly used ones are:
Available software for segmentation of magnetic resonance imaging of the brain We found that BraTumIA is one of the most recent software tools for automatic segmentation of brain tumors.^{[25]} It can segment the neoplastic tissue including its subcompartments from the MRI of glioma patients. For carrying out this procedure, it requires four different MRI sequences (T1, T1 contrast, T2, and FLAIR) as an input; and, it outputs volumetric information about a tumor and its subcompartments (necrotic tissue, active enhancing tumor tissue, nonenhancing tumor tissue, and edema). In addition, the software can also segment healthy subcortical structures surrounding the tumor. Label maps of the segmented tissues are available as an overlay on the original images.^{[25]} The software calculates the tumorregion volumes in cm^{3} (necrotic tissue, active enhancing tumor tissue, and edema), according to the VASARI (Visually AcceSAble Rembrandt Images; https://wiki.nci.nih.gov/display/CIP/VASARI) guidelines of the National Cancer Institute of the American, National Institutes of Health (NIH).^{[26]} The images are processed using a pipeline approach, where skull stripping is performed first to generate a brain mask. Subsequently, all images are coregistered to ensure voxeltovoxel correspondence between the different MRI sequences. Based on these registered images, a segmentation of the patients’ images into healthy and tumor tissues is done based on a combined classification and regularization. This classification produces a label map and quantitative information about tissue volumes. Healthy subcortical structures are segmented using a deformable registration of an Atlas ^{More Details} to the patient image. In case of comparisons with followup MRIs, BraTumIA software uses the T1weighted contrast image of the first baseline scan of each patient as a reference template.^{[25]} Finally, the label maps can be transformed back into the original space of each image sequence so that they can be shown as an overlay on the original images [Figure 1]a,[Figure 1]b,[Figure 1]c,[Figure 1]d,[Figure 1]e, below the dashed line].^{[27]} Mathematical modeling applied to tumor growth The process of mathematical modeling may be utilized to describe an object that exists in any area of science, not only mathematics. Models can describe interactions between biological components, which allow researchers to deduce the consequences of the interactions in medical and natural processes.^{[28]} The basis of any mathematical model used to study the treatment of cancer is a model of tumor growth.^{[29]} The primary goals of a model of tumor growth are to predict the evolution of a tumor and optimize treatment regimens [Table 1]a.^{[3],[4]}
The simplest way to write a mathematical model is to assume a tumor to be spherically symmetrical and to occupy a region at each time “t” (this is our variable number 1). We need a second variable that explains the boundary of a tumor; it is given by the radius “r” which is an unknown function of t. A successful model is the one that depicts a high level of precision (its numerical structure) and its fidelity (the ability to concretize events and situations in the form of related variables). A mathematical model should be able to formalize the cause and effect process and create a linkage with associated observations. By analyzing mathematical models, researchers can describe and deduct the consequences of interactions among biological systems.^{[30],[31],[32]} An ideal model would consider all factors and conditions affecting the model. However, this goal is never reached because the researcher would be overwhelmed by the details, and an assembly of a model as complex as the system itself will be required [Figure 2].
Principles of modeling Basically, in a mathematical model, we notice three phases: (A) construction, a process in which the object is converted to mathematical language. (B) The analysis, the study of the constructed model. (C) The interpretation of this assessment, where the results of the study are applied to the object from which it was split.^{[31]} Understanding the dynamics of cancerous tumor growth may help to develop more effective treatment plans that lead to a better prognoses for patients [Table 1]b. Factors involved in tumor growth models Unfortunately, most mathematical models of tumor growth are based on the reaction–diffusion equations and mass conservation law, applied to the cellular level. However, it is possible to apply these principles to the macroscopic behavior of a growing tumor in humans.^{[32]} A consensus exists that the process of tumor growth has various stages, from the very early stage of a solid tumor without necrotic core inside,^{[33],[34],[35]} to the process of necrotic core formation.^{[36],[37]} As changes in the proliferation rate of tumor cells are related to apoptotic cell loss, and this apoptosis is mediated by growth factors expressed by the tumor cells,^{[37]} these facts led to the proposal of a timedelayed mathematical model for tumor growth. It has been observed that after an initial exponential growth phase that leads to tumor expansion, many tumors present a stage of growth saturation, which could be produced by internal (growth factors) or external conditions (e.g., administration of periodic chemotherapy affects the growth of avascular tumors).^{[32]} In an example like the above mentioned one, a mathematical expression should consider λ(t), and it would describe the rate of cell apoptosis caused by the periodic therapy. In the last two decades, several tumor growth models have been proposed [Table 2].
Mathematical modeling based on imaging data Mathematical models of tumor growth and treatment response use several quantities as parameters that are difficult to measure in intact organisms using a measurable spatial resolution. Required inputs for this kind of model are commonly obtained invasively or in isolated, idealized in vitro or ex vivo systems. However, as imaging data may be collected noninvasively and in a three dimensional form, quantitative imaging data from individual patients have been incorporated into mathematical models generating patientspecific results.^{[38]} In the last decade, several simulations and experiments have included imaging data into mathematical models of tumor growth and treatment response.^{[39]} In the evaluation of GBMs, the most aggressive primary brain tumor, most research studies focus on the reaction–diffusion model, which suggests that the diffusion coefficient and the proliferation rate can be related to derive clinically relevant information. As the estimation of the parameters of the reaction–diffusion model is difficult because of the lack of identifiability of the factors involved, the uncertainty in the tumor segmentation, and the model approximation, alternative models have been proposed that are based on the spherical asymptotic analysis of the reaction–diffusion model and on a derivativefree optimization algorithm.^{[40]} This approach is considered as Bayesian personalization that produces more informative results: Samples from the regions of interest highlight the presence of several models for some patients.^{[40]} Readers should be aware that tumor growth modeling in lower grade gliomas adopts a different approach. It makes the assumption that the first step in malignant transformation is the growth of the tumor density beyond a certain critical level that initiates a nonreversible damage to the microenvironment; beyond that point, hypoxia occurs, and angiogenesis is triggered.^{[5]} Most common equations used in highgrade gliomas growth modeling The first mathematical models that explained the dynamics of tumor growth were published more than 50 years ago.^{[41],[42],[43],[44]} The most recently accepted models to represent tumor growth use ordinary differential equations (ODEs).^{[45],[46]} ODEs describe systems that are deterministic, have few and many dimensions, and allow chaotic and complex behavior. ODEs have one independent variable and, if the systems are dynamic, the independent variable is time. Dependent variables can be the volume of a tumor, the fraction of a genetic alteration in a population, etc.^{[31]} They are also applied to make predictions about the efficacy of cancer treatments.^{[47]} All models predict the growth of a tumor by describing the change in tumor volume (V) over time [Table 3]. The assumption is that if there is a constant supply of drug C_{0} acting on the tumor, then the author subtracts the term C_{0}V from each equation.^{[47]}
Tumor growth modeling as a part of a brain's complex system Researchers now use multivariable models based on linear or even nonlinear relations to complement the shortcomings of simple linear relationships. However, these models still try to test the hypothesis that a group of risk factors causes a disease, considering or controlling other elements.^{[48]} We know that multiple factors are involved in the pathogenesis of GBMs. The involved parts are components of different systems such as chemical exposure, genes, epigenetic changes, and proteins. These methods receive various names such as exposome, genome, epigenome, or proteome.^{[49]} Latest trends in methodology show that the idea of simple relationships based on oneonone correspondence is obsolete, and therefore, the changes to a complex systems model based on a complex network/mathematical systems have been proposed for really understanding the disease processes.^{[48]} A mathematical system consists of a collection of assertions from which we derive consequences by logical arguments. The brain can be viewed as a system with various interacting regions that produce complex behaviors.^{[50]} A person cannot fully understand a complex system by only understanding its constituent parts; instead, an approach is needed to utilize knowledge about the complex interactions within a system to understand the overall behavior of the system.^{[50]} In this regard, network analysis allows the dissection of the complexities of the biological systems and provides information to understand the pathobiology of disorders, the pathways of disease processes, and the prioritization of candidate diseasecausing genes.^{[51]} We can design anatomical and functional networks using MRI data. Anatomical networks can be built from white matter tractography utilizing the DTI sequence, with voxels of gray matter treated as nodes and the fiber tracts between brain regions as links.^{[52]} Although some functional connectivity can be inferred from anatomical networks,^{[53]} the underlying structural connectivity alone does not determine functional connectivity, as two brain areas can be functionally connected without a direct anatomical connection between them [Figure 3].^{[54]}
Limitations in mathematical modeling Despite their high potential to improve the development and implementation of cancer treatment, mathematical models only realize their potential if they can provide accurate predictions.^{[29]} In many of the published articles on this topic, the choice of a growth model is often driven by the ease of mathematical analysis rather than whether or not it provides the best model for predicting the growth of a tumor.^{[45]} In addition, some researchers have attempted to find the “best” equations of growth model by fitting various models to a small number of experimental data sets of tumor growth.^{[55],[56],[57]} However, using that approach, their results are often inconclusive, suggesting that the choice of the growth model depends at least in part on type of the tumor.^{[55],[56]} Other significant limitations in the assembling of tumor growth models are the incorporation of several imaging modalities currently employed to probe different aspects of the tumor microenvironment, including the expression of molecules, cell motion, cellularity, vessel permeability, vascular perfusion, metabolic and physiological changes, apoptosis, and inflammation.^{[57],[58]} Among the imaging modalities reporting on this parameters, there are reports of utilization of the dynamic contrastenhanced MRI^{[59]} dataset and MRI angiography^{[60]} that are related to tumor vascularity. Challenges in mathematical growth modeling The challenges encountered might be classified in six areas: tumor initiation, progression, metastases, intratumoral heterogeneity, treatment responses, and tumor resistance.^{[31]} The classification of brain tumors published by the World Health Organisation has recently been changed;^{[7]} and now it integrates the use of phenotypic and genotypic parameters, which may improve the accuracy in diagnosis, in finding out the determinants of prognosis, and in the treatment response. In this direction, mathematical modeling will be an essential component of this new era of personalized treatment by helping to predict the time course of a tumor and in optimizing treatment regimens.^{[4]} Due to the complexity of the available models, modelers and clinicians must carefully consider their choice of growth model and how different growth assumptions might alter model predictions of the efficacy of treatment. Various decisions related to the construction of the growth model result in significant variations in model predictions; clinicians might provide either too much medication (causing more severe side effects) or too little drug (possibly resulting in continued growth of a tumor).^{[29]} A consensus is needed to define how the necrotic core, extra capsular extension, and angiogenesis are integrated into predictive models; therefore, the data used will more reliably reflect the dynamics at the time when the predictions are made.^{[61],[62],[63]} The increasing use of network science in neuroimaging utilizing graph metric tools will require the assessment of the anatomical network topology after the diagnosis of brain tumors and after their treatment.^{[50]} This usage invariably brings up the question of defining the best approaches for anatomical and functional network construction, and how this information should be integrated into a tumor growth model. Several examples of diseasefocused networks are now emerging, especially in the area of cancer research and neurological diseases.^{[64]} Development of specific diseasefocused models will ensure the validation of multiple biological markers and the understanding of their pathophysiology.^{[65],[66],[67],[68]} In conclusion, mathematical models do not have current relevance in a clinical setting to choose the best therapeutic options in GBMs; yet, biomedical imaging and mathematical modeling provide independent insights into cancer biology. It is the combination of both these fields that will allow the assembling of clinically relevant models of tumor growth and treatment response; the most appropriate model will depend on the premises and findings of each experiment. As we do not know all the factors implied in the GBM growth, the accuracy of a mathematical model may be weak. Only in the near future, after ongoing trials complete the validation or refuting of experimental models for specific tumors, will basic research provide us with new information on the genetic alterations or environmental factors that can impact the tumor growth. Researchers around the world could apply mathematical models for calculation of tumor growth and the distribution of tumor cells and reach a consensus on the prognostic information and the efficacy of personalized treatment. Acknowledgments Ana Karina CisnerosSanchez, B. Sc., was enrolled as a research fellow in the Directorate of Research, General Hospital of Mexico “Dr. Eduardo Liceaga”, Mexico City, Mexico; from January 2017 to December 2018. This article is derived from an institutional review board–approved study (protocol #DI/17/301/03/013).
[Figure 1], [Figure 2], [Figure 3]
[Table 1], [Table 2], [Table 3]


