Neurology India
menu-bar5 Open access journal indexed with Index Medicus
  Users online: 3756  
 Home | Login 
About Editorial board Articlesmenu-bullet NSI Publicationsmenu-bullet Search Instructions Online Submission Subscribe Videos Etcetera Contact
  Navigate Here 
 Resource Links
  »  Similar in PUBMED
 »  Search Pubmed for
 »  Search in Google Scholar for
 »Related articles
  »  Article in PDF (1,739 KB)
  »  Citation Manager
  »  Access Statistics
  »  Reader Comments
  »  Email Alert *
  »  Add to My List *
* Registration required (free)  

  In this Article
 »  Abstract
 »  References
 »  Article Figures
 »  Article Tables

 Article Access Statistics
    PDF Downloaded60    
    Comments [Add]    

Recommend this journal


Table of Contents    
Year : 2018  |  Volume : 66  |  Issue : 6  |  Page : 1575-1583

Basic principles of mathematical growth modeling applied to high-grade gliomas: A brief clinical review for clinicians

1 Directorate of Research, General Hospital of Mexico “Dr. Eduardo Liceaga”, Mexico City, Mexico
2 Department of Neurosurgery, General Hospital of Mexico “Dr. Eduardo Liceaga”, Mexico City, Mexico
3 Directorate of Research, General Hospital of Mexico “Dr. Eduardo Liceaga”, Mexico City, Mexico; I.M. Sechenov First Moscow State Medical University (Sechenov University), Department of Radiology, Moscow, Russia

Date of Web Publication28-Nov-2018

Correspondence Address:
Dr. Ernesto Roldan-Valadez
Directorate of Research, General Hospital of Mexico “Dr. Eduardo Liceaga”, Mexico City, Mexico; I.M. Sechenov First Moscow State Medical University (Sechenov University), Department of Radiology, Moscow, Russia

Login to access the Email id

Source of Support: None, Conflict of Interest: None

DOI: 10.4103/0028-3886.246238

Rights and Permissions

 » Abstract 

The battle against cancer has intensified in the last decade. New experimental techniques and theoretical models have been been proposed to understand the behavior, growth, and evolution of different types of brain tumors. Unfortunately, for glioblastoma multiforme (GBM), except for methylation of the O6-methylguanine-DNA methyltransferase (MGMT) promoter that has some benefit in the local control of tumors using alkylating agents such as temozolomide, to date personalized treatments do not exist. In this article, we present a comprehensive review of different aspects intertwined in the mathematical growth modeling applied to high-grade gliomas. We briefly cover the following fundamental aspects related to the conventional imaging in GBM: defining the tumor regions in GBM, segmentation of the tumor regions using magnetic resonance imaging (MRI) of the brain, response assessment using the neuro-oncology response criteria versus the Macdonald criteria, availability of software for the segmentation of MRI of the brain, mathematical modeling applied to tumor growth, principles of mathematical modeling, factors involved in tumor growth models, mathematical modeling based on imaging data, most common equations used in high-grade glioma growth modeling, integration of mathematical growth models in computer simulators, tumor growth modeling as a part of brain's complex system, and challenges in mathematical growth modeling. We conclude by saying that it is the combination of biomedical imaging and mathematical modeling that allows the assembling of clinically relevant models of tumor growth and treatment response; the most appropriate model will depend on the premise and findings of each experiment.

Keywords: Brain tumors, glioblastoma multiforme, mathematical model, tumor growth
Key Message: A combination of biomedical imaging and mathematical modeling permits clinically relevant models of tumor growth and treatment response to develop that are intimately related to the management of glioblastomas.

How to cite this article:
Cisneros-Sanchez AK, Flores-Alvarez E, Melendez-Mier G, Roldan-Valadez E. Basic principles of mathematical growth modeling applied to high-grade gliomas: A brief clinical review for clinicians. Neurol India 2018;66:1575-83

How to cite this URL:
Cisneros-Sanchez AK, Flores-Alvarez E, Melendez-Mier G, Roldan-Valadez E. Basic principles of mathematical growth modeling applied to high-grade gliomas: A brief clinical review for clinicians. Neurol India [serial online] 2018 [cited 2022 May 17];66:1575-83. Available from: https://www.neurologyindia.com/text.asp?2018/66/6/1575/246238

Mathematics and computation can help in solving several growing problems in medical research by proposing models that allow us to formalize the cause-and-effect 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 less-infiltrating, and usually curable tumors. Grade II gliomas are commonly referred to as low-grade gliomas. Grades III and IV are known as high-grade gliomas.[7] Glioblastoma multiforme (GBM), the most common malignant primary brain tumor, has an incidence of 3.19 cases per 100,000 person-years 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: T1-weighted (W) precontrast, T1-W postcontrast, T2-W precontrast, and fluid-attenuated inversion recovery (FLAIR) precontrast images [Figure 1]a,[Figure 1]b,[Figure 1]c,[Figure 1]d.[13],[14]
Figure 1: Conventional MRI of the brain sequences used in the visualization of brain tumors. Above the dashed line: (a), T1-W precontrast. (b), T1-W postcontrast. (c), FLAIR precontrast. (d), T2-W precontrast. Below the dashed line: Images a–d represent the corresponding segment of the tumor regions after the application of segmentation software BraTumIA. (e) The output from the software showing the calculation of seven volumes; notice that each color corresponds to a specific tumor region explained in the article

Click here to view

Tumor regions in glioblastoma multiforme

It was reported in 2009 that some GBMs arise from the transformed astrocyte-like neural stem cells in the subventricular zone (SVZ), whereas other GBMs initiate by neoplastic transformation of the non-SVZ progenitor cells or mature glial cells that have undergone dedifferentiation.[15] A published MRI-based 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 contrast-enhancing 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 contrast-enhancing 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 neuro-oncology response criteria versus the Macdonald criteria

In 1990, Macdonald et al., introduced a response criteria based on a two-dimensional measure of contrast enhancement on computed tomographic scans.[17] The response assessment in neuro-oncology (RANO) response criteria replaced the Macdonald criteria to assess the changes determined on the postoperative MRI of glioblastomas and is currently considered the state-of- the-art criteria.[18] The RANO response criteria defines progressive disease as an increase of the perpendicular diameter of the tumor more than 25% in postcontrast T1-weighted sequences; 25% or more increase in enhancing lesions despite stable or increasing steroid dose; an increase (significant) in non-enhancing FLAIR/T2W lesions, not attributable to other non-tumour causes; or the development of any new lesions.[18] Assuming an isotropic tumor growth, this two-dimensional 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 two-dimensional 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 non-overlapping 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 semi-automated segmentation of glioblastomas can be done reliably by different groups of raters using commercial software solutions based on a region-growing algorithm.[20] Studies are in agreement on the fact that the collation of quantitative volumetric reports of contrast-enhancing 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 post-processing 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:

  • Manual segmentation: It is based on the selection of the pixels that belong to the object of interest interactively, either by selecting pixel-by-pixel that object, or by using semiautomatic tools such as thresholding and the growth of regions
  • Thresholding: It is one of the most straightforward and most used techniques in the segmentation of images in general. It consists of the determination of thresholds that delimit the range of intensities that characterize and discriminate the different objects of the image
  • Contour detection: Separation boundaries between different tissues are used to perform segmentation. The objective is to select those pixels of the image that delimit the border of the object of interest
  • The growth of regions: This technique consists in the selection of a seed spot in the picture around which new pixels are added that fulfill a particular predicate, such as being in geometric contact with other pixels classified as being of interest and having an intensity within a given range.

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 tumor-region volumes in cm3 (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 co-registered to ensure voxel-to-voxel 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 follow-up MRIs, BraTumIA software uses the T1-weighted 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]
Table 1: (a) Most common applications for mathematical models related to cancer. (b) Desirable criteria for an ideal mathematical model in a real-world situation

Click here to view

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].
Figure 2: Existing dissonance between mathematical models and reality (biological data). (a-c) MRIs are depicting a real tumor growth in a patient with GBM. (d), A schematic representation of the most straightforward mathematical model in GBM assuming the the tumor is spherically symmetric with a constant growth rate

Click here to view

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 time-delayed 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].
Table 2: Some proposals for tumor growth modeling at the micro- and macrocellular level

Click here to view

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 patient-specific 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 derivative-free 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 high-grade 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 C0 acting on the tumor, then the author subtracts the term C0V from each equation.[47]
Table 3: Most recent ordinary differential equations (ODE) used to represent tumor growth. Last two columns in the table show the differences in volume predictions if chemotherapy is absent or if it is added to the model

Click here to view

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 one-on-one 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 disease-causing 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]
Figure 3: Simplified diagram of the authors’ view of tumor growth modeling as part of a brain's complex system in a patient with a GBM

Click here to view

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 contrast-enhanced 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 disease-focused networks are now emerging, especially in the area of cancer research and neurological diseases.[64] Development of specific disease-focused 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.


Ana Karina Cisneros-Sanchez, 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).

 » References Top

Hanly P, Pearce A, Sharp L. The cost of premature cancer-related mortality: A review and assessment of the evidence. Expert Rev Pharmacoecon Outcomes Res 2014;14:355-77.  Back to cited text no. 1
Carlotto A, Hogsett VL, Maiorini EM, Razulis JG, Sonis ST. The economic burden of toxicities associated with cancer treatment: Review of the literature and analysis of nausea and vomiting, diarrhoea, oral mucositis and fatigue. Pharmacoeconomics 2013;31:753-66.  Back to cited text no. 2
Agur Z, Elishmereni M, Kheifetz Y. Personalizing oncology treatments by predicting drug efficacy, side-effects, and improved therapy: Mathematics, statistics, and their integration. Wiley Interdiscip Rev Syst Biol Med 2014;6:239-53.  Back to cited text no. 3
Agur Z, Vuk-Pavlovic S. Mathematical modeling in immunotherapy of cancer: Personalizing clinical trials. Mol Ther 2012;20:1-2.  Back to cited text no. 4
Bogdanska MU, Bodnar M, Piotrowska MJ, Murek M, Schucht P, Beck J, et al. A mathematical model describes the malignant transformation of low grade gliomas: Prognostic implications. PLoS One 2017;12:e0179999.  Back to cited text no. 5
Ostrom QT, Gittleman H, Liao P, Rouse C, Chen Y, Dowling J, et al. CBTRUS statistical report: Primary brain and central nervous system tumors diagnosed in the United States in 2007-2011. Neurooncol 2014;16(Suppl 4):iv1-63.  Back to cited text no. 6
Louis DN, Perry A, Reifenberger G, von Deimling A, Figarella-Branger D, Cavenee WK, et al. The 2016 World Health Organization Classification of Tumors of the Central Nervous System: A summary. Acta Neuropathol 2016;131:803-20.  Back to cited text no. 7
Batash R, Asna N, Schaffer P, Francis N, Schaffer M. Glioblastoma multiforme, diagnosis and treatment; recent literature review. Curr Med Chem 2017;24:3002-9.  Back to cited text no. 8
Wang CH, Rockhill JK, Mrugala M, Peacock DL, Lai A, Jusenius K, et al. Prognostic significance of growth kinetics in newly diagnosed glioblastomas revealed by combining serial imaging with a novel biomathematical model. Cancer Res 2009;69:9133-40.  Back to cited text no. 9
Cortez-Conradis D, Favila R, Isaac-Olive K, Martinez-Lopez M, Rios C, Roldan-Valadez E. Diagnostic performance of regional DTI-derived tensor metrics in glioblastoma multiforme: Simultaneous evaluation of p, q, L, Cl, Cp, Cs, RA, RD, AD, mean diffusivity and fractional anisotropy. Eur Radiol 2013;23:1112-21.  Back to cited text no. 10
Roldan-Valadez E, Rios C, Cortez-Conradis D, Favila R, Moreno-Jimenez S. Global diffusion tensor imaging derived metrics differentiate glioblastoma multiforme vs. normal brains by using discriminant analysis: Introduction of a novel whole-brain approach. Radiol Oncol 2014;48:127-36.  Back to cited text no. 11
Cortez-Conradis D, Rios C, Moreno-Jimenez S, Roldan-Valadez E. Partial correlation analyses of global diffusion tensor imaging-derived metrics in glioblastoma multiforme: Pilot study. World J Radiol 2015;7:405-14.  Back to cited text no. 12
Seidel C, Dorner N, Osswald M, Wick A, Platten M, Bendszus M, et al. Does age matter? - A MRI study on peritumoral edema in newly diagnosed primary glioblastoma. BMC Cancer 2011;11:127.  Back to cited text no. 13
Roldan-Valadez E, Rios C, Motola-Kuba D, Matus-Santos J, Villa AR, Moreno-Jimenez S. Choline-to-N-acetyl aspartate and lipids-lactate-to-creatine ratios together with age assemble a significant Cox's proportional-hazards regression model for prediction of survival in high-grade gliomas. Br J Radiol 2016;89:20150502.  Back to cited text no. 14
Holland EC. Gliomagenesis: Genetic alterations and mouse models. Nat Rev Genet 2001;2:120-9.  Back to cited text no. 15
Lim DA, Cha S, Mayo MC, Chen MH, Keles E, VandenBerg S, et al. Relationship of glioblastoma multiforme to neural stem cell regions predicts invasive and multifocal tumor phenotype. Neurooncol 2007;9:424-9.  Back to cited text no. 16
Macdonald DR, Cascino TL, Schold SC Jr., Cairncross JG. Response criteria for phase II studies of supratentorial malignant glioma. J Clin Oncol 1990;8:1277-80.  Back to cited text no. 17
Wen PY, Macdonald DR, Reardon DA, Cloughesy TF, Sorensen AG, Galanis E, et al. Updated response assessment criteria for high-grade gliomas: Response assessment in neuro-oncology working group. J Clin Oncol 2010;28:1963-72.  Back to cited text no. 18
Sorensen AG, Batchelor TT, Wen PY, Zhang WT, Jain RK. Response criteria for glioma. Nat Clin Pract Oncol 2008;5:634-44.  Back to cited text no. 19
Huber T, Alber G, Bette S, Boeckh-Behrens T, Gempt J, Ringel F, et al. Reliability of semi-automated segmentations in glioblastoma. Clin Neuroradiol 2017;27:153-61.  Back to cited text no. 20
Chow DS, Qi J, Guo X, Miloushev VZ, Iwamoto FM, Bruce JN, et al. Semiautomated volumetric measurement on postcontrast MR imaging for analysis of recurrent and residual disease in glioblastoma multiforme. AJNR Am J Neuroradiol 2014;35:498-503.  Back to cited text no. 21
Menze BH, Jakab A, Bauer S, Kalpathy-Cramer J, Farahani K, Kirby J, et al. The Multimodal Brain Tumor Image Segmentation Benchmark (BRATS). IEEE Trans Med Imaging 2015;34:1993-2024.  Back to cited text no. 22
Rana M, Modrow D, Keuchel J, Chui C, Rana M, Wagner M, et al. Development and evaluation of an automatic tumor segmentation tool: A comparison between automatic, semi-automatic and manual segmentation of mandibular odontogenic cysts and tumors. J Craniomaxillofac Surg 2015;43:355-9.  Back to cited text no. 23
Gordillo N, Montseny E, Sobrevilla P. State of the art survey on MRI brain tumor segmentation. Magn Reson Imaging 2013;31:1426-38.  Back to cited text no. 24
Porz N, Bauer S, Pica A, Schucht P, Beck J, Verma RK, et al. Multi-modal glioblastoma segmentation: Man versus machine. PLoS One 2014;9:e96873.  Back to cited text no. 25
Gutman DA, Cooper LA, Hwang SN, Holder CA, Gao J, Aurora TD, et al. MR imaging predictors of molecular profile and survival: Multi-institutional study of the TCGA glioblastoma data set. Radiology 2013;267:560-9.  Back to cited text no. 26
Bauer S, Fejes T, Meier R, Reyes M, Slotboom J, Porz N, et al. BraTumIA-A software tool for automatic brain tumor image analysis. 2013. Availbale from: https://pdfs.semanticscholar.org/3806/1483bf02e073b6ff9e475932e2b40c006f4d.pdf. [Last accessed on 2018 Nov 16].  Back to cited text no. 27
de Vries G, Hillen T, Lewis M, Muller J, Schonfisch B. The Modeling Process. In: de Vries G, Hillen T, Lewis M, Muller J, Schonfisch B. editors. A Course in Mathematical Biology. Quantitative Modeling with Mathematical and Computational Methods. Philadelphia, PA, USA: SIAM; 2006. p. 3-4.  Back to cited text no. 28
Murphy H, Jaafari H, Dobrovolny HM. Differences in predictions of ODE models of tumor growth: A cautionary example. BMC Cancer 2016;16:163.  Back to cited text no. 29
Olinik M. Mathematical modeling in the social and life science. Wiley; 2014.  Back to cited text no. 30
Altrock PM, Liu LL, Michor F. The mathematics of cancer: Integrating quantitative models. Nat Rev Cancer 2015;15:730-45.  Back to cited text no. 31
Xu S, Wei X, Zhang F. A time-delayed mathematical model for tumor growth with the effect of a periodic therapy. Comput Math Methods Med 2016;2016:3643019.  Back to cited text no. 32
Byrne HM, Chaplain MA. Growth of nonnecrotic tumors in the presence and absence of inhibitors. Math Biosci 1995;130:151-81.  Back to cited text no. 33
Cui S, Friedman A. Analysis of a mathematical model of the effect of inhibitors on the growth of tumors. Math Biosci 2000;164:103-37.  Back to cited text no. 34
Friedman A, Reitich F. Analysis of a mathematical model for the growth of tumors. J Math Biol 1999;38:262-84.  Back to cited text no. 35
Byrne HM, Chaplin MA. Growth of necrotic tumors in the presence and absence of inhibitors. Math Biosci 1996;135:187-216.  Back to cited text no. 36
Bodnar M, Forys U. Time delay in necrotic core formation. Math Biosci Eng 2005;2:461-72.  Back to cited text no. 37
Atuegwu NC, Colvin DC, Loveless ME, Xu L, Gore JC, Yankeelov TE. Incorporation of diffusion-weighted magnetic resonance imaging data into a simple mathematical model of tumor growth. Phys Med Biol 2012;57:225-40.  Back to cited text no. 38
Atuegwu NC, Gore JC, Yankeelov TE. The integration of quantitative multi-modality imaging data into mathematical models of tumors. Phys Med Biol 2010;55:2429-49.  Back to cited text no. 39
Le M, Delingette H, Kalpathy-Cramer J, Gerstner ER, Batchelor T, Unkelbach J, et al. MRI based Bayesian personalization of a tumor growth model. IEEE Trans Med Imaging 2016;35:2329-39.  Back to cited text no. 40
Laird AK. Dynamics of tumour growth: Comparison of growth rates and extrapolation of growth curve to one cell. Br J Cancer 1965;19:278-91.  Back to cited text no. 41
Laird AK. Dynamics of tumor growth. Br J Cancer 1964;13:490-502.  Back to cited text no. 42
Summers WC. Dynamics of tumor growth: A mathematical model. Growth 1966;30:333-8.  Back to cited text no. 43
Dethlefsen LA, Prewitt JM, Mendelsohn ML. Analysis of tumor growth curves. J Natl Cancer Inst 1968;40:389-405.  Back to cited text no. 44
Gerlee P. The model muddle: In search of tumor growth laws. Cancer Res 2013;73:2407-11.  Back to cited text no. 45
Wodarz D, Komarova N. Towards predictive computational models of oncolytic virus therapy: Basis for experimental validation and model selection. PLoS One 2009;4:e4271.  Back to cited text no. 46
Usher JR, Henderson D. Some drug-resistant models for cancer chemotherapy. Part 1: Cycle-nonspecific drugs. IMA J Math Appl Med Biol 1996;13:99-126.  Back to cited text no. 47
Hong YC. Time to change from a simple linear model to a complex systems model. Environ Health Toxicol 2016;31:e2016008.  Back to cited text no. 48
Auffray C, Chen Z, Hood L. Systems medicine: The future of medical genomics and healthcare. Genome Med 2009;1:2.  Back to cited text no. 49
Telesford QK, Simpson SL, Burdette JH, Hayasaka S, Laurienti PJ. The brain as a complex system: Using network science as a tool for understanding the brain. Brain Connect 2011;1:295-308.  Back to cited text no. 50
Stevens A, De Leonibus C, Hanson D, Dowsey AW, Whatmore A, Meyer S, et al. Network analysis: A new approach to study endocrine disorders. J Mol Endocrinol 2014;52:R79-93.  Back to cited text no. 51
Hagmann P, Cammoun L, Gigandet X, Meuli R, Honey CJ, Wedeen VJ, et al. Mapping the structural core of human cerebral cortex. PLoS Biol 2008;6:e159.  Back to cited text no. 52
Honey CJ, Kotter R, Breakspear M, Sporns O. Network structure of cerebral cortex shapes functional connectivity on multiple time scales. Proc Natl Acad Sci U S A 2007;104:10240-5.  Back to cited text no. 53
Honey CJ, Sporns O, Cammoun L, Gigandet X, Thiran JP, Meuli R, et al. Predicting human resting-state functional connectivity from structural connectivity. Proc Natl Acad Sci U S A 2009;106:2035-40.  Back to cited text no. 54
Sarapata EA, de Pillis LG. A comparison and catalog of intrinsic tumor growth models. Bull Math Biol 2014;76:2010-24.  Back to cited text no. 55
Benzekry S, Lamont C, Beheshti A, Tracz A, Ebos JM, Hlatky L, et al. Classical mathematical models for description and prediction of experimental tumor growth. PLoS Comput Biol 2014;10:e1003800.  Back to cited text no. 56
Hartung N, Mollard S, Barbolosi D, Benabdallah A, Chapuisat G, Henry G, et al. Mathematical modeling of tumor growth and metastatic spreading: Validation in tumor-bearing mice. Cancer Res 2014;74:6397-407.  Back to cited text no. 57
Serres S, O'Brien ER, Sibson NR. Imaging angiogenesis, inflammation, and metastasis in the tumor microenvironment with magnetic resonance imaging. Adv Exp Med Biol 2014;772:263-83.  Back to cited text no. 58
Yankeelov TE, Lepage M, Chakravarthy A, Broome EE, Niermann KJ, Kelley MC, et al. Integration of quantitative DCE-MRI and ADC mapping to monitor treatment response in human breast cancer: Initial results. Magn Reson Imaging 2007;25:1-13.  Back to cited text no. 59
van Vliet M, van Dijke CF, Wielopolski PA, ten Hagen TL, Veenland JF, Preda A, et al. MR angiography of tumor-related vasculature: From the clinic to the micro-environment. Radiographics 2005;25(Suppl 1):S85-98.  Back to cited text no. 60
Kolobov AV, Kuznetsov MB. Investigation of the influence of angiogenesis on tumor growth with the use of a mathematical model. Biofizika 2015;60:555-63.  Back to cited text no. 61
Cooper MD, Tanaka ML, Puri IK. Coupled mathematical model of tumorigenesis and angiogenesis in vascular tumours. Cell Prolif 2010;43:542-52.  Back to cited text no. 62
Emerick KS, Leavitt ER, Michaelson JS, Diephuis B, Clark JR, Deschler DG. Initial clinical findings of a mathematical model to predict survival of head and neck cancer. Otolaryngol Head Neck Surg 2013;149:572-8.  Back to cited text no. 63
Lones MA, Smith SL, Tyrrell AM, Alty JE, Jamieson DR. Characterising neurological time series data using biologically motivated networks of coupled discrete maps. Biosystems 2013;112:94-101.  Back to cited text no. 64
Vlahou A. Network views for personalized medicine. Proteomics Clin Appl 2013;7:384-7.  Back to cited text no. 65
Collins VP. Observation on growth rates of human tumors. Am J Roentgenol 1956;76:988-1000.  Back to cited text no. 66
Patt HM, Blackford ME. Quantitative studies of the growth response of the Krebs ascites tumor. Cancer Res 1954;14:391-6.  Back to cited text no. 67
Vaidya VG, Alexandro FJ Jr. Evaluation of some mathematical models for tumor growth. Int J Biomed Comput 1982;13:19-36.  Back to cited text no. 68


  [Figure 1], [Figure 2], [Figure 3]

  [Table 1], [Table 2], [Table 3]


Print this article  Email this article
Online since 20th March '04
Published by Wolters Kluwer - Medknow