The biology and mathematical modelling of glioma invasion: a review.

rsif.royalsocietypublishing.org Review Cite this article: Alfonso JCL, Talkenberger K, Seifert M, Klink B, Hawkins-Daarud A, Swanson KR, Hatzikirou H, Deutsch A. 2017 The biology and mathematical modelling of glioma invasion: a review. J. R. Soc. Interface 14: 20170490. http://dx.doi.org/10.1098/rsif.2017.0490 Received: 6 July 2017 Accepted: 17 October 2017 Subject Category: Life Sciences–Mathematics interface Subject Areas: biomathematics, systems biology Keywords: glioma invasion, cell phenotypic plasticity, malignant progression, infiltrative tumour morphology, mathematical modelling Author for correspondence: A. Deutsch e-mail: [email protected] The biology and mathematical modelling of glioma invasion: a review J. C. L. Alfonso1,2, K. Talkenberger2, M. Seifert3,5, B. Klink4,5,6,7, A. Hawkins-Daarud8, K. R. Swanson8, H. Hatzikirou1,2 and A. Deutsch2 1 Department of Systems Immunology and Braunschweig Integrated Centre of Systems Biology, Helmholtz Centre for Infection Research, Braunschweig, Germany 2 Centre for Information Services and High Performance Computing, 3 Institute for Medical Informatics and Biometry, and 4 Institute for Clinical Genetics, Faculty of Medicine Carl Gustav Carus, Technische Universita ¨t Dresden, Germany 5 National Center for Tumor Diseases (NCT), Dresden, Germany 6 German Cancer Consortium (DKTK), partner site, Dresden, Germany 7 German Cancer Research Center (DKFZ), Heidelberg, Germany 8 Precision Neurotherapeutics Innovation Program, Mayo Clinic, Phoenix, AZ, USA JCLA, 0000-0003-2432-5953; AD, 0000-0002-9005-6897 Adult gliomas are aggressive brain tumours associated with low patient survival rates and limited life expectancy. The most important hallmark of this type of tumour is its invasive behaviour, characterized by a markedly phenotypic plasticity, infiltrative tumour morphologies and the ability of malignant progression from lowto high-grade tumour types. Indeed, the widespread infiltration of healthy brain tissue by glioma cells is largely responsible for poor prognosis and the difficulty of finding curative therapies. Meanwhile, mathematical models have been established to analyse potential mechanisms of glioma invasion. In this review, we start with a brief introduction to current biological knowledge about glioma invasion, and then critically review and highlight future challenges for mathematical models of glioma invasion. 1. Introduction Gliomas are the most common primary tumours of the central nervous system (CNS) in adults. They comprise a clinically, histologically and genetically very heterogeneous brain tumour category. Until recently, glioma classification was largely based on microscopic examination of histological sections of tumour specimens by expert pathologists, distinguishing tumours according to their microscopic similarities to different types of glial cells into astrocytomas, oligodendroglioma or ependymomas, the main subtypes of gliomas [1]. The current 2016 World Health Organization (WHO) Classification of Tumours of the CNS for the first time integrates molecular biomarkers together with classic histological features to define distinct glioma entities [1]. This paradigm shift in glioma diagnostics reflects the major progress in our understanding of the molecular biology of brain tumours, which has tremendously increased in the past two decades due to genome-wide molecular-profiling studies that have clarified the genetic basis of gliomas. For example, diffuse gliomas with histologically oligodendroglial features are genetically characterized by mutations in the IDH gene and 1p/19q codeletion, while the diagnosis astrocytoma is usually accompanied by mutations in IDH in combination with ATRX and/or TP53 mutations but intact 1p and 19q. By contrast, classical primary glioblastomas usually do not show mutations in the IDH genes and are therefore referred to as glioblastoma IDH wild-type. In addition, the WHO classification distinguishes four prognostic grades that reflect the degree of malignancy: WHO grade I is assigned to the more circumscribed, benign tumours with low proliferative potential that mainly occur during childhood and in young adults; WHO grade II–IV tumours are diffusely infiltrative with increased cellular abnormalities (diffuse gliomas); WHO grade III tumours also show dedifferentiation and mitotic cell activity; and WHO grade IV tumours exhibit, in addition to &2017 The Author(s) Published by the Royal Society. All rights reserved. the features present in the other grades, pathological proliferation of small vessels and/or necrosis. WHO grade II–IV tumours are characterized by extensive, diffuse infiltration of glioma cells into the host brain tissue, and are therefore referred to as diffuse gliomas. These aggressive brain tumours are typically associated with a poor prognosis, sharp deterioration in the patients’ quality of life and markedly low survival rates. In particular, IDH wild-type glioblastoma, the most common (approx. 45% of all gliomas) and malignant primary brain tumour (WHO grade IV), has a 5-year survival of about 5% from the time of diagnosis [2,3]. Even for patients with diffuse low-grade IDH-mutant gliomas (WHO grade II), although survival canbemorethan10years,theprognosis is unfavourable, as these tumours eventually progress to a high-grade malignant lesion (WHO grade III or IV) [4]. Despite significant advances in surgical and medical imaging techniques, as well as in adjuvant radio-, chemoand immunotherapy [5–10], the inherent tendency of glioma cells to widely disseminate within normal brain parenchyma severely limits treatment responses [11–13]. Therefore, a better understanding of the mechanisms thattriggerandgoverngliomainvasionisofhighclinicalimportance for the development of more effective and less toxic therapeutic strategies. Although there is a considerable amount of information about the clinical and biological behaviour of gliomas, the high complexity of the invasion mechanisms remains a major challenge in clinical neuro-oncology. Histologically, glioma cells closely resemble glial progenitor cells, which have the ability to proliferate and differentiate into different glial cell types. These cells have a high migratory behaviour in the developing CNS [11,12,14]. This suggests that mechanisms contributing to migration of neuroepithelial cells during embryogenesis are also relevant for glioma invasion [14]. In addition, accumulating evidence indicates that spatial and temporal variations in signalling pathways lead to functional and phenotypic changes in glioma cells, which then affect interactions with neighbouring malignant and non-malignant cells along with other components of the surrounding brain tissue. The exact consequences of the dynamic interplay between heterogeneous cellular entities and their response to alterations in the extracellular microenvironment have not yet been elucidated. Moreover, it remains unclear why metastases outside the CNS are extremelyrare in diffuse gliomas [15,16]. From a biological and medical perspective, it is difficult to investigate the connections between clinically observable glioma behaviour and the underlying molecular and cellular processes. The challenge is to integrate the theoretically and empirically acquired knowledge to better understand the mechanisms and factors that contribute to glioma invasion. In this context, mathematical models provide useful tools towards identifying dependencies and targets of cancer cell migration and invasion. Mathematical models and computational approaches have become increasingly abundant in cancer research to study tumour dynamics and responses to treatment modalities such as chemoand radiotherapy [17–20]. Mathematical modelling provides a useful theoretical framework to perform in silico experiments, as well as to evaluate assumptions and make predictions that can be experimentally tested [21–32]. In the last two decades, several mathematical models have been developed to investigate key mechanisms governing glioma growth and invasion [23,33–35]. Ten years ago, several of the current co-authors reviewed mathematical models of glioma development, growth and progression [34]. Since then, the field of glioma research has significantly grown. In this review we exclusively focus on mathematical models of glioma invasion. We first introduce current biological knowledge about glioma invasion. Then, we describe biological model systems, in particular, in vitro experiments and in vivo animal models for the analysis of glioma invasion, and medical imaging techniques. We then critically review mathematical models of glioma invasion, and highlight future challenges for mathematical and computational modellers in this research area. 2. Biology of glioma invasion Infiltration of the brain parenchyma is a prominent feature of diffuse gliomas, making complete surgical resection almost impossible [36]. Diffuse gliomas invade extensively as single cells anywhere within the host brain tissue, with some preference to infiltrate along white matter tracts and the periphery of blood vessel walls [16]. The infiltration of the surrounding brain tissue is determined by complex interactions between glioma cells and the extracellular microenvironment [37]. Here, we review cell intrinsic mechanisms and extrinsic factors that sustain and foster glioma invasion. 2.1. Intrinsic mechanisms: phenotypic plasticity and genetic variability 2.1.1. Epithelial–mesenchymal transition and migration Glioma cells have the ability to acquire a mesenchymal phenotype in response to microenvironmental cues and migrate through the extracellular matrix (ECM) exhibiting an elongated, often wedge-shaped phenotype [14,38,39]. Migration and invasion of glioma cells are related, multistep processes. Migration is defined as the movement of cells from one site to another, often in response to specific external signals such as chemical gradients or mechanical forces. Epithelial-to-mesenchymal transition (EMT) is an essential process in wound healing, embryonic development and tissue remodelling, consisting in the transdifferentiation of polarized epithelial cells into motile mesenchymal cells (originated from the mesodermal embryonic tissue which develops into connective and skeletal tissues). Accumulating evidence highlights the critical role of EMT during glioma progression and its association with increased glioma cell migration [40]. Individual glioma cells spread by active cell migration rather than by passive movement. Invasion encompasses glioma cell migration, but also involves degradation of the ECM [38]. It is a multifactorial process that consists of interactions between adjacent cancer cells with the ECM coupled with biochemical processes supportive of active cell migration. In general, glioma cell invasion involves four distinct steps [14,38,39]: (1) detachment of invading cells from the primary tumour mass, (2) adhesion to the ECM, (3) degradation of the ECM and (4) cell motility and contractility (active cell migration) (figure 1). At the subcellular level, secretion of proteases, cell adhesion molecules and related signals play an important role in glioma cell migration [37]. Detachment of glioma cells from the primary tumour mass involves several events, including destabilization and disorganization of cell–cell adhesion complexes (cadherin-mediated junctions), loss of expression of neural cell adhesion molecules and cleavage of CD44, a cell-surface protein which anchors the primary tumour mass to the ECM by the metalloproteinase ADAM [16,38]. Integrins rsif.royalsocietypublishing.org J. R. Soc. Interface 14: 20170490 2 are the most common molecules that allow glioma cells to adhere to the ECM, and matrix-metalloproteinases (MMPs) are the most common proteases that degrade the ECM creating migration routes. Several glioma-expressed molecular factors, such as focal adhesion kinase and urokinase-type plasminogen activator (an enzyme participating in ECM degradation), have been found to regulate their expression [38]. Glioma cells migrate similarly as non-transformed neural progenitor cells, with myosin II as the major source for cytoplasmic contractility [41]. Invading glioma cells alter their shape, extending a prominent leading cytoplasmic protrusion followed by a burst of forward movement of the cell body. The complex molecular mechanisms and changes in signalling pathways that occur during glioma invasion are still largely unknown. A major difficulty in understanding the oncogenomics of glioma cell invasion is to determine how and when genetic alterations and signalling cascades interact [42]. Only a few specific pathways have been consistently identified, and there exist multiple possible interactions along with additional unknown factors to be elucidated. 2.1.2. Migration–proliferation dichotomy At the time of diagnosis, gliomas are already widely disseminated, as they typically grow and invade extensively before the patient experiences any symptoms. This hidden dissemination is a major reason that makes gliomas difficult to treat successfully and a cure almost impossible. Current treatment strategies mainly focus on the highly proliferative tumour mass, but local invasion eventually leads to recurrence of the disease. Enormous efforts have been devoted to identifying the main signalling events that regulate glioma cell motility and invasion. However, therapeutically targeting invasion dynamics is complicated, because it has been observed that migratory and proliferative behaviours of glioma cells are mutually exclusive processes and inversely correlated [11,43,44]. In particular, highly migratory cells have a lower proliferation rate compared to actively proliferating cells that move slowly. This either–or behaviour of proliferative and invasive glioma cells is supported by both in vitro and in vivo experiments [11] and is referred to as the migration– proliferation dichotomy (or ‘Go-or-Grow’ mechanism) [43,44]. The ‘Go-or-Grow’ behaviour has been linked to metabolic stress by several experimental findings. Godlewski et al. [45] identified a glioma-expressed microRNA (small non-coding RNAs that regulate gene expression) that regulates the balance between glioma cell proliferation and migration in response to changes in the available energy. Their experimental data revealed that, in addition to inhibiting glioma cell migration, the microRNA expression also promotes cell proliferation. This suggests that migratory and proliferative events share common signalling pathways, defining a unique intracellular mechanism that regulates both phenomena. More recently, Ho ¨ring et al. [46] investigated the effects of carboxypeptidase E (CPE), a neuropeptide-processing enzyme, on glioma invasion by means of in vitro and in vivo studies. Their results indicate an oxygenand nutrient-dependent anti-migratory, but pro-proliferative role of CPE in glioma invasion. Additionally, experiments with glioblastoma-derived neurospheres provided further insight into the ‘Go-or-Grow’ mechanism. EphB2, a receptor of the tyrosine kinase family was found to have both pro-migratory and anti-proliferative effects in vivo [47]. These novel findings suggest that glioma invasion could be attacked by targeting specific cellular and molecular mechanisms associated with the migration–proliferation dichotomy. However, further investigations are required to unravel the underlying signalling pathways regulating glioma cell migration and proliferation. 2.1.3. Cell metabolic plasticity A common feature of cancer cells is their altered glucose metabolism [48,49]. Unlike non-neoplastic cells that rely on oxidative phosphorylation to generate the energy needed for cellular processes, cancer cells can shift their metabolism from respiration towards glycolysis producing lactic acid. Indeed, cancer cells tend to upregulate aerobic glycolysis even in the presence of sufficient oxygen, a phenomenon known as aerobic glycolysis or the Warburg effect, which is a characteristic metabolic hallmark of tumour development [50–52]. Although the Warburg effect has recently regained attention as a possible therapeutic target [53,54], its biological basis remains elusive. Compared to mitochondrial oxidative metabolism, aerobic glycolysis is an inefficient way to gain energy [55]. However, increased glycolysis creates a hostile acidic environment, in which cancer cells have an evolutionary advantage with respect to normal parenchyma [56]. There is accumulating evidence that acid-induced toxicity is an essential component required for tumour invasion, and therefore a hallmark of invasive cancers [56,57]. Glioma cells are specifically characterized by a high rate of glycolysis and lactate extrusion, with the ability to flourish in a relatively (1) (3)(2) (4) Figure 1. Glioma cell migration. Schematic of the process of glioma cell invasion into host brain tissue. Invasion of glioma cells involves four distinct steps: (1) detachment of invading cells from the primary tumour mass, a process triggered by downregulation of cell–cell adhesion molecules and microenvironmental changes, (2) integrin-mediated adhesion to the extracellular matrix (ECM), (3) secretion of proteases, which locally degrade ECM components creating routes along which glioma cells invade the brain and (4) migration by extending a prominent leading cytoplasmic protrusion, followed by a burst of forward movement of the cell body. Figure adapted from [39]. rsif.royalsocietypublishing.org J. R. Soc. Interface 14: 20170490 3 hypoxic environment [58]. In fact, many of the invasive features of gliomas may depend on distorted metabolic functions, which makes the study of metabolic alterations and their effects on invasion processes a growing field in cancer research with potential therapeutic benefits. 2.1.4. Intraand inter-tumoural heterogeneity Tumour heterogeneity contributes to disease progression and development of therapy resistance [59,60]. Extensive genetic and phenotypic variations exist among glioma cells within a single tumour (intra-tumoural heterogeneity) and between patients (inter-tumoural heterogeneity) due to extensive molecular diversity and microenvironmental heterogeneity [37,61–68]. It is commonly assumed that tumour heterogeneity arises either from a self-renewing cancer stem cell population or due to clonal competition for common resources driven by the acquisition and expansion of mutations in cancer cells [61,69–71]. Recent clinical and experimental findings have revealed extensive genetic variations in glioma cells due to intra-tumoural evolution [63,64,67,72]. Besides the large genetic heterogeneity, interactions between glioma cells and with the surrounding brain parenchyma lead to functional and phenotypic diversity. Evidence indicates that distinct clones within a tumour may harbor genetic and epigenetic alterations that promote cancer cell migration and invasion. Recently, experiments with mixtures of different cell types to mimic phenotypic heterogeneity have revealed that invasion is driven by the cooperation of multiple tumour-cell subpopulations [73,74]. Patterns of co-invasion were observed with inherently invasive cells acting as the leader and subpopulations of poorly invasive cells as followers. This ‘division of labour’ may facilitate not only tumour invasion, but also malignant progression. Although there is increasing appreciation that intra-tumoural heterogeneity is central to glioma behaviour, little is known about the temporal sequence of genetic and microenvironmental changes or how genomic instabilities and adaptation to microenvironmental conditions contribute to glioma invasiveness. 2.2. Cell-extrinsic factors 2.2.1. Guidance mechanisms The particular structures of the brain such as blood vessels, white matter tracts and brain parenchyma, and specific tumour cell–ECM adherence mechanisms are crucial factors in glioma invasion [75,76]. At any stage of invasion, glioma cells are confronted with non-neoplastic brain tissue composed of multiple cell types and various ECM components. The ECM in the CNS is different from the ECM in other tissues in that it has low fibrous protein content and high carbohydrate concentrations [77]. In particular, the brain ECM is mainly produced by astrocytes and oligodendrocytes, comprises an estimated 20% of the brain volume in adults and consists primarily of hyaluronic acid, except around blood vessels and at the pial surface (the boundary between grey matter and cerebrospinal fluid) [76]. The invasion of glioma cells into the adjacent brain tissue is guided by a combination of multiple molecular and physical mechanisms along pre-existing tracks of least resistance. The major invasion routes are basement membranes and intercellular tracks provided by myelinated axons and astrocyte processes [76,78]. Glioma cells migrate along blood vessels by using the outward vessel–parenchyma interface and the lumen of the perivascular space [76]. More precisely, blood vessels guide invading glioma cells via lamininand collagen-IV-mediated integrin engagement (ECM proteins mediated), whereas white matter tracks guide by cell–cell contacts and mechanisms regulating cell–ECM adhesion forces [76]. However, the specific guidance factors and related molecular mechanisms for most dissemination routes remain unclear. 2.2.2. Hypoxia-induced migration Uncontrolled glioma cell proliferation leads to the development of hypoxic regions. This microregional change produces a local milieu that favours certain glioma cell behaviours such as invasion [60]. A commonly held view is that constitutive upregulation of glycolysis is likely to be an adaptation to the lack of oxygen [57]. According to the WHO classification, detection of vascular proliferation with highly pathological blood vessels and tumour necrosis is essential for the diagnosis of grade IV gliomas [1]. These tumourinduced features are often spatially and temporally related, with sites of pathological neovascularization indicative of the formation of hypoxic and necrotic regions. Quantitative immunohistochemical analysis revealed that while highgrade gliomas may locally show a strong angiogenic activity, many regions of both lowand high-grade gliomas display vascular densities in the range of normal cerebral grey or white matter, indicating limited angiogenesis [79–81]. This supports the observation that in high-grade gliomas different invasive and pro-angiogenic tumour cell phenotypes coexist [39]. Overexpression of pro-angiogenic factors by tumour cells results in local vascular overgrowth with defective blood vessels, which have significantly larger diameters and thicker basement membranes than those in normal brain tissue [82]. It has been observed in vitro that under oxygen-limiting conditions due to vascular abnormalities, glioma cells actively migrate away from hypoxic regions [83]. Different pathological and experimental observations suggest that vaso-occlusion could readily explain the rapid peripheral expansion and diffusely infiltrative growth behaviour of high-grade gliomas [84,85]. Occlusion of vasculature mainly occurs due to increased mechanical pressure by either tumour cells or by intravascular pro-thrombotic mechanisms [83,84,86]. Occluded or collapsed blood vessels induce perivascular tumour hypoxia and necrosis in glioblastoma, which typically form lines with perifocally increased cell density, termed pseudopalisades [83–85,87]. Pseudopalisades around necrotic foci, a common feature of high-grade gliomas, are severely hypoxic and linked to waves of glioma cells actively migrating away from such oxygen-deficient regions [83–85,87]. Experimental studies further suggest that tumour hypoxia results in increased glioma cell migration and invasion, and strongly correlates with tumour malignancy [39,88,89]. The exact pathophysiological factors and mechanisms underlying hypoxia-induced cell migration in gliomas are still not known, and further investigation is required to understand the complex molecular pathways involved in hypoxic responses and metabolic control by glioma cells. 2.2.3. Blood–brain barrier An important structural component of the brain vasculature is the blood brain barrier (BBB), which is essential for supplying rsif.royalsocietypublishing.org J. R. Soc. Interface 14: 20170490 4 the brain tissue with oxygen and glucose, mediating efflux of waste products, and maintaining a precisely regulated microenvironment for reliable neuronal signalling [90]. The BBB is composed of tightly bound endothelial cells and perivascular astrocytes that restrict the exchange of molecules from the bloodstream much more than capillaries anywhere else in the body. High-grade gliomas have the ability to disrupt the integrity of the BBB, which is associated with increased tumour growth and diffuse invasion into the surrounding brain parenchyma [16,91–93]. Recent studies have shown that the BBB can be heterogeneously disrupted in high-grade gliomas, and the degree of BBB disruption is related to tumour malignancy [92]. However, the precise relationship between glioma-induced BBB dysregulation and cell invasion is still not clear [91]. On the other hand, experimental evidence shows that one of the major obstacles for standard anti-cancer drug delivery in the brain is the BBB, which limits the efficacy of chemotherapy [16,92]. In fact, the presence of an almost intact BBB is one of the main factors that makes the treatment of low-grade gliomas with chemotherapy challenging. Although the BBB may be disrupted at the core of high-grade gliomas, it can be relatively intact at the tumour periphery where invading glioma cells are located [92]. This permits infiltrative glioma cells to escape chemotherapy-induced death, which can result in tumour recurrence. 2.2.4. Immune system engagement Tumours have long been recognized as wounds that do not heal [94]. Both carcinogenesis and wound healing involve cell proliferation, migration, invasion, angiogenesis, inflammation and astrocyte activation in response to injuries [76,95]. The microenvironment of high-grade gliomas resembles in many ways a chronic wound [95]. In particular, the extensive cell migration and invasion observed in gliomas also accompanies reactive gliosis, a non-specific reaction where astrocytes are activated in response to injuries to the CNS as part of a healing process. The border of gliomas exhibits an increased number of reactive astrocytes, which together with glioma cells secrete various pro-migratory signalling molecules [76]. Recently, a protein (connective tissue growth factor) produced at high levels by reactive astrocytes has been identified to stimulate migration of glioma cells [95]. Indeed, the secreted factor modulates nearly all aspects of the signalling mechanisms that regulate cell invasion, such as modification of growth factor activities, ECM composition, integrin (transmembrane cell-matrix adhesion receptors) and E-cadherin expression (a transmembrane protein that mediates cell–cell adhesion). Thus, the identification of processes involved in glial reactivation may contribute to a better understanding of glioma cell invasion with potential therapeutic implications. Accumulated histopathological data have established that tissue-resident microglia and macrophages are the predominant infiltrating immune cells in gliomas, accounting for up to 30–50% of the total tumour mass [96]. Monocytes circulate in the bloodstream and are continuously recruited into tumours in response to several tumour-derived chemoattractants. Once inside the tumour, monocytes rapidly differentiate into tumour-associated macrophages (TAMs), and then accumulate in hypoxic/necrotic areas [96]. TAMs produce several factors that not only stimulate the survival and proliferation of tumour cells, but also suppress antitumour immunity [96,97]. There is growing evidence that TAMs are also involved in regulating glioma cell migration and invasion [96,98–100]. Moreover, TAMs contribute to malignant progression of gliomas through secretion of various chemical factors that affect angiogenesis and ECM remodelling, which promotes glioma invasion [96,97,101]. More precisely, TAMs can display two major phenotypes, the classically (M1) and alternatively (M2) activated, which can be viewed as two extreme phenotypes [102,103]. Unlike TAMs displaying an M1-like phenotype with proinflammatory and anti-tumour functions, M2 macrophages are immunosuppressive, produce pro-angiogenic factors, contribute to the ECM-remodelling, and thus create a favourable microenvironment for glioma growth and invasion. Although there is growing acceptance that the M1 and M2 states are not dichotomous but rather represent extremes of a continuum of phenotypes, a majority of macrophages in gliomas are macrophages with M2-like phenotype which exhibit pro-invasive properties, particularly in late stages of disease progression [104,105]. Recent evidence indicates that the colony stimulating factor-1 (CSF-1) secreted by glioma cells induces TAMs to promote invasion [104]. It has been observed that CSF-1 levels are elevated in high-grade gliomas, which is associated with the expression of M2 macrophage markers. In addition, the cytokine interleukin10 (IL-10) is also commonly associated with macrophages of the M2 phenotype and has been found to stimulate glioma cell invasion [104]. Together, chemokines, cytokines and growth factors secreted by TAMs activate different signalling pathways that can switch glioma cells towards more aggressive behaviour. 3. Biological model systems As it is currently impossible to monitor the entire process of glioma invasion in the human brain, different biological model systems have been introduced. In vitro cell cultures provide the opportunity to generate insights into molecular and cellular pathways related to glioma invasion under controlled conditions. Various in vivo models have been introduced which allow a more realistic representation and monitoring of the complex glioma dynamics. 3.1. In vitro experiments Experimental protocols in vitro are available to independently observe and control variables of interest at various scales, from single-cell movement to multicellular clonal growth and cancer cell population dynamics [106]. Invasion studies have been performed in both two-dimensional (2D) tumour monolayers and three-dimensional (3D) multicellular spheroids, combining glioma cell migration and proliferation assays, and considering different ECM compositions and substrate rigidities [107]. However, there are significant differences between tumour cell migration on a 2D surface and in a 3D matrix [108], and even more between in vitro and in vivo experiments [106]. Under controlled experimental conditions, glioma cells may exhibit a more fibroblastic shape with a broad lamellipodium and an undistorted nucleus, where the forward movement is continuous and unimpeded [41]. By contrast, glioma cells migrating through the complex heterogeneous brain parenchyma are highly polarized and elongated [38,39]. The mechanical constraints due to small intercellular spaces impede the forward movement of nucleus rsif.royalsocietypublishing.org J. R. Soc. Interface 14: 20170490 5 and cell body until necessary contraction forces are provided [41]. This ability of glioma cells to adapt their motility to the particular microenvironment reinforces the need for assays faithfully representing the environmental conditions of the brain to improve our understanding of invasion mechanisms. Using so-called normal brain cell aggregates derived from fetal rat brains or utilization of rodent or other mammalian brain slices might present a promising compromise to perform 3D in vitro studies that can be more standardized and avoid ethical issues attached to animal studies [109–112]. 3.2. In vivo models Animal models are essential for investigating the interactions between glioma cells and the complex brain microenvironment. The traditional use of animal models involves injecting established tumour cell lines either intravenously or at the target site and then waiting foratumour to develop before testing a therapy or a given hypothesis. While the complex microenvironments in the animal models mimic the human brain structure much better than in vitro studies, very few established cell lines are able to represent the histopathological characteristics of human gliomas, particularly their invasive nature. Recently, methods have been developed to harvest cells from patients for use in animal models. These are referred to as patient-derived tumour aggregates and are better able to recapitulate patterns of tumour cell invasion observed in patients [113]. Moreover, xenograft models have been extensively employed to assess the efficacy of therapies targeting glioma cells, such as the intra-tumoural administration of IL13-PE toxin (a fusion protein composed of IL-13 and a mutated form of Pseudomonas exotoxin), to monitor glioblastoma angiogenesis and to evaluate anti-angiogenic therapeutical approaches [114]. The main drawback of xenografts is that the histology and genetics of the original tumour are frequently not maintained. Moreover, highresolution imaging of single glioma cell invasion in xenografts remains laborious, costly and time-consuming. Tumours can also be induced in animals using retroviruses. Two notable cases use cells infected with retroviruses engineered to overexpress either constitutively activated epidermal growth factor receptor (EGFR) [115] or platelet-derived growth factor (PDGF) [116]. Both retroviruses are able to initiate tumour growth with human glioblastoma characteristics when injected into a rat or mouse brain. In addition to providing a good model system for drug therapy, they also shed light on the tumour initiating process. Another development in animal models is the creation of transgenic mice, where genetic engineering techniques are used to create mice with ubiquitous mutations that are predisposed to developing gliomas [117]. Many in vivo experimental techniques exist, the most common involves sacrificing the animal to allow for staining of the brain tissue, but this only provides a temporal snapshot of tumour composition. Another technique is referred to as ex vivo imaging where brain tissue is harvested and thin slices are placed on nutrient-filled media. This allows to microscopically observe cell movement for a time period up to 24 h [116]. A further option that allows longer observation is the use of bioluminescence (or biofluorescence) at the cell population scale, or multiphoton microscopy at the singlecell level. While any of the aforementioned animal models provides experimental conditions closer to the human brain, the ability of fully monitoring the cancer dynamics is still challenging. Additionally, the fruit fly Drosophila melanogaster has been considered as an alternative in vivo glioma model because many molecular pathways and cellular functions are fundamentally conserved [118,119]. Advantages include easy handling, a fully sequenced genome, a wide range of available genetic techniques and a well-known anatomical situation [118,120]. Model organisms such as D. melanogaster have been useful not only to visualize tumour cell migration and to investigate the effects of induced metastasis, but also to identify glioma signalling cascades via advanced genetic techniques. However, the D. melanogaster model also has some limitations. Invasion studies in D. melanogaster lack an accurate representation of the human brain parenchyma, including the absence of blood vessels and an adaptive immune system. 4. Medical imaging and histopathology Conventional computer tomography scanning reveals morphological information of gliomas, but tumours at early stages or small metastatic lesions are often not detected. This technique has been gradually replaced by magnetic resonance imaging (MRI), which is significantly more sensitive to the presence of tumours and has become the standard imaging modality in the evaluation of brain tumours [121,122]. MRI creates noninvasive images by exploiting the magnetic properties of water molecules in the body. By changing the intensity, timing and duration of radiofrequency pulses and directional gradients, myriad non-invasive images with varying contrasts and information can be created [123]. The more common T1and T2-weighted MRI sequences are mainly used for displaying general anatomic features of gliomas; however, the two types of sequences emphasize features differently (e.g. cerebral spinal fluid (CSF) is bright on T2-weighted images and dark on T1-weighted images). The T1-weighted image can be used to highlight the leaky blood vessels characteristic of glioblastoma by acquiring the image after administration of gadolinium, a contrast agent that appears bright on T1-weighted images. Gadolinium seeps out from the leaky vasculature haphazardly created by the tumour, highlighting what is believed to be the most active/aggressive tumour region on T1-weighted gadolinium-enhanced MRI (T1Gd). The T1Gd and T2 sequences are most commonly used for delineating tumour regions, but it is well known that neither of these sequences is able to provide a precise visualization of tumour abnormality due to the extensive invasion of the tumour cells [124]. In fact, in one study of high-grade gliomas, it was demonstrated that human gliomas grow invasively, with tumour cells demonstrable over 4 cm from the gross tumour [125]. While T1and T2-weighted MRI sequences remain the dominant images clinically used, there are many other advanced MRI techniques that are being explored such as vascular perfusion imaging, diffusion-weighted imaging (DWI) and proton magnetic resonance spectroscopy (MRS) [126]. Vascular perfusion imaging highlights regions of high vascularity and has been shown to be useful in predicting which patients are responding to anti-angiogenic therapies [127,128]. DWI can be used for various purposes, but for glioblastoma it is most commonly used to quantify the apparent diffusion coefficient, which is thought to be inversely rsif.royalsocietypublishing.org J. R. Soc. Interface 14: 20170490 6 correlated with cell density [129,130]. MRS uses the proton signals to determine relative concentrations of target metabolites rather than an anatomical image. In various studies, MRS has been shown capable of identifying regions of tissue enriched with stem-like cell-enriched foci [131], detecting tumours with mutations in the isocitrate dehydrogenase (IDH) genes [132,133], and assessing response to various therapies such as radiation and PI3K/mTOR inhibitors [134,135]. Further, the use of hyperpolarized (HP) contrast agents can significantly increase the sensitivity of MRS by enhancing the signal-to-noise ratio [136,137]. More generally, MRS has opened up the promising fields of metabolomics (the study of metabolomic signatures in tumours) [138] and radiomics (the use of imaging technology to extrapolate molecular tumour data) [139]. These studies are encouraging, but one must note that the resolution of a standard MRS is much lower than on a standard MRI, voxel sizes being approximately 10 10 10 mm 3 versus 1 11mm 3 . Thus, while there are many promising advanced magnetic resonance methods for determining various tumour characteristics, differentiating between normal and pathological tissue on the basis of MRI findings alone is complicated. Functional imaging techniques like positron emission tomography (PET) scans are useful in providing deeper insights into the biology of gliomas [140,141]. PET imaging is increasingly implemented in neuro-oncology, because it offers unique data about metabolic and physiologic processes such as glucose metabolism, protein/DNA synthesis, cell proliferation and apoptosis, as well as angiogenesis and hypoxia that can reflect the changes in a neoplasm. Assessment of the status of these processes has been shown helpful in delineation of tumour margins, and correlates with clinical metrics such as tumour grade, patient survival and therapy response [140,142,143]. Clinically, this type of information is viewed as complementary to the anatomical MRI, as PET scans generally lack anatomic context, and have a relatively low spatial resolution [144]. Diagnosis and classification of glioma is based on histopathology, referring to the microscopic examination of tissue sections by an experienced pathologist, e.g. neuropathologist. Additionally, histopathology analysis of tissue samples has some potential to provide further information at the singlecell level that is extremely important to quantify and classify intra-tumoural heterogeneity. Histological and immunohistochemical analyses are routinely performed by pathologists for confirmation of the presence or absence of disease, determining glioma grading and assessing disease progression [1,145,146]. However, tissue biopsies can be seen as ‘snapshot’-like frozen scenes of dynamic biological processes providing data severely limited in both space and time. Thus, despite the constant expansion of medical imaging technology, the identification of tumours at an early stage, assessment of intra-tumoural heterogeneity, reduction of radiation exposure and improvement of resolution are challenging for anatomic, functional and metabolic imaging alike. 5. Mathematical modelling of glioma cell migration and invasion A wide variety of mathematical models have been proposed to investigate the mechanisms of glioma invasion, which is characterized by invasive cell migration, phenotypic plasticity, infiltrative tumour morphologies and the ability of malignant progression. Model types include discrete and continuous approaches such as cellular automaton (CA), lattice-gas cellular automaton (LGCA), cellular Potts model (CPM), partial differential equations (PDE), agent-based models (ABM) and evolutionary game theory models (EGT) [17–20,35,147–153]. We subsequently review mathematical models for invasive cell migration, invasive effects of phenotypic plasticity, infiltrative tumour morphologies and malignant progression. Table 1 provides an overview of the reviewed models. 5.1. Invasive cell migration Based mainly on in vitro experiments, several mathematical models have been developed to investigate the effects of cell–cell adhesion strength under distinct microenvironmental conditions on the invasive behaviour of glioma cells. Khain et al. [27] investigated, both theoretically and experimentally, the effect of cell–cell adhesion on glioma front propagation and the structure of the invasive interface. Migration characteristics of U87-MG cells were measured using a scratch wound-healing assay, and invasion front patterns were simulated by means of both a 2D discrete lattice-based stochastic model and a continuum approach. Simulations of the continuum model show that a small effective cell–cell adhesion change does not influence the propagating front speed, and clusters of glioma cells were not formed (figure 2a). By contrast, the microscopic discrete model shows that a cell–cell adhesion strength exceeding a critical threshold leads to cluster formation in the invasive zone resulting in fingering-like front propagation patterns (figure 2b). The experimental time was characterized as a transient regime, which coincides with the period required for a relatively sharp initial cell density profile to develop into a propagating front. Although simulations successfully reproduced the maximal distance of migration of glioma cells on a plastic substrate, this model underestimated the migration of the main mass of tumour cells, suggesting the presence of chemotactic stimuli. Glioma cell migration on a substrate of collagen was investigated by Aubert et al. [162]. The proposed 2D CA model indicates that chemotaxis or cell–cell communication through gap junctions (specialized intercellular channels that permit direct cell–cell transfer of ions and molecules) is necessary to reproduce experimental density profiles of glioma cell distributions in tumour spheroids. In a followup study, migration patterns of glioma cells in the presence of astrocytes were studied by Aubert et al. [154]. An extended version of the model proposed in [162] was used to analyse the opposite effects of homotypic (between glioma cells) and heterotypic (between glioma cells and surrounding astrocytes) gap junction communication on the invasiveness of gliomas. Lowering glioma cell–cell interactions on a passive substrate of collagen was predicted to enhance the migratory potential, whereas the simultaneous inhibition of glioma cell–cell and glioma cell–normal astrocyte gap junction communication leads to reduced cell migration. This suggests that the interactions between glioma cells and astrocytes play an important role in glioma invasion, due to the effect of heterotypic gap junction inhibition which dominates that of homotypic inhibition. Model simulations are consistent with experimental data of glioma migration patterns in both rsif.royalsocietypublishing.org J. R. Soc. Interface 14: 20170490 7 Table 1. Overview of reviewed mathematical modelling approaches. Cellular automaton (CA), lattice-gas cellular automaton (LGCA), cellular Potts model (CPM), partial differential equations (PDE), agent-based model (ABM) and evolutionary game theory (EGT). mathematical modelling approach Khain et al. [27] Aubert et al. [154] Khain et al. [155] Szabo ´ et al. [156] Kim et al. [157] Kim [158] Tektonidis et al. [25] Hatzikirou et al. [31] Bo ¨ttger et al. [28] Pham et al. [29] Gerlee et al. [30] Alfonso et al. [32] Sander et al. [159] Martı ´nezGonza ´lez et al. [26] Frieboes et al. [21] Zhang et al. [160] Frieboes et al. [161] Jiao et al. [24] Basanta et al. [22] Swanson et al. [23] (2D) CA & (1D) PDE (2D) CA (2D) CA (2D) CPM (2D) PDE (2D) HYBRID (2D) LGCA (2D) LGCA (2D) LGCA (2D) PDE (2D) CA & (1D) PDE (1D) PDE (2D) HYBRID (1D) PDE (2D) PDE (3D) ABM (3D) HYBRID (2D) STATISTICAL EGT (1D) PDE main focus of mathematical models invasive cell migration 333333 phenotypic plasticity 33 3 333 3 3 3 tumour morphology 33 3 33 3 malignant progression 333 main biological assumptions cell–cell interactions 3333333 3 333333 cell–ECM interactions 333 33 hypoxia-induced migration 3333333 cell densitydependent invasiveness 3333 3 changes in metabolism 33 3 clonal heterogeneity 3 angiogenesis 33 3 3 rsif.royalsocietypublishing.org J. R. Soc. Interface 14: 20170490 8 homotypic and heterotypic situations by only introducing attractive contact between migrating tumour cells. However, the experiments considered in [154,162] only involved relatively small spheroids without central hypoxia or necrosis. Thus, to study migration patters of glioma cells in large spheroids, Aubert et al. [163] assumed a chemorepellent factor produced by cells submitted to stressful conditions in the hypoxic/necrotic microregions. A good agreement between model simulations and experiments allows to conclude the existence of repellent toxic cues that would promote glioma cell detachment and migration, although further work is needed to identify and characterize these chemorepulsive factors. The role of hypoxia in the regulation of glioma cell–cell adhesion and cell migration was investigated by Khain et al. [155]. A 2D discrete stochastic model was proposed to describe in vitro experiments of U87 glioma cell migration (i) away from tumour spheroids placed on a substrate and (ii) in typical scratch wound-healing assays. The distance migrated (i.e. invasive radius) by both normoxic and hypoxic glioma cells was measured. In the spheroid experiments, the overall migration rate of tumour cells under either normoxic or hypoxic conditions was similar. However, hypoxic glioma cells in the wound-healing assays migrated less than cells under normoxic conditions. This model suggests that lack of oxygen not only suppresses cell motility, but also substantially reduces the strength of cell– cell adhesion. Although oxygen deficiency resulted in reduced cell motility, the decreased cell–cell adhesion allows hypoxic cells to detach from the tumour mass, leading to enhanced glioma invasion. These model predictions are consistent with experimental data showing that hypoxia induces downregulation of E-cadherin, a transmembrane protein that positively regulates extension and strengthening of adhesive contacts, and promotes glioma cell invasion. To explore the influence of the ECM on glioma cell migration, Szabo ´et al. [156] considered not only cell–cell adhesion but also cell–ECM interactions. Cell aggregates were prepared from confluent cultures of two different glioblastoma cell lines (GBM1 and U87) placed within a 3D ECM of collagen I gel. The aim was to characterize the collective, large-scale invasion of glioma cells from tumour spheroids into the surrounding ECM. The interplay between haptotaxis, matrix degradation and active cell movement was investigated by means of a 2D CPM. Simulation results suggest that the complex interplay between space-constrained active cell motion, cell–ECM adhesion and degradation of the ECM determines the patterns of migration and increases persistence during cell invasion. In particular, haptotaxis and ECM degradation were observed to destabilize multicellular sprouts as each cell tries to invade the surrounding matrix. By contrast, when both haptotaxis and polarized motion are present, even a homogeneous glioma cell population may be organized into multicellular sprouts within an inhomogeneous ECM environment. Kim et al. [157] proposed a model that takes into account cell–cell adhesion, haptotaxis and chemotactic effects of a glucose gradient on glioma cell migration in vitro.Model simulations reveal that depending on the chemotactic and haptotactic sensitivities, and the strength of cell–cell adhesion, different migration patterns of glioma cells arise: dispersion, branching, island formation and a mixture of these patterns. In particular, this model reproduced the patterns observed in various invasion assays of in vitro spheroids generated from glioma U87 and mutant U87DEGFR cell lines. Moreover, changes of adhesion, haptotactic and chemotactic parameters result in a gradual shift from branching to dispersion as experimentally observed. The main finding was that the front of cell migration can be slowed down by both increasing cell–cell adhesion and blocking the ECM degradation effects of MMPs, a family of enzymes that are capable of breaking down all kinds of proteins, such as collagen, normally found in spaces between tissues (ECM proteins). In a follow-up study, Kim [158] developed a hybrid multiscale model in which glioma cell migration and proliferation are regulated by intracellular mechanisms in response to glucose availability and physical constraints in the microenvironment. In particular, a core control system of a single microRNA (miR-451) that regulates AMPK (the 50-adenosine monophosphate activated protein kinase) signalling [45,164] linked to extracellular glucose was simulated. Recent experimental evidence suggests that, in a glucose-rich environment, miR-451 is up-regulated by tumour cells, leading to AMPK pathway inhibition and in turn cell proliferation. Conversely, sustained AMPK activation under low glucose conditions results in suppression of miR-451, which induces phenotypic changes of glioma cells from a proliferative to a migratory phenotype [165]. Based on the assumption that glucose levels may induce phenotypic changes in glioma cells, fluctuations of the glucose concentration were predicted to trigger migration–proliferation cycles, which in turn increased gliomacell invasion and resulted in faster tumour growth. 5.2. Phenotypic plasticity The ability of glioma cells to switch their phenotype in response to local cell density and changes in the microenvironment allows adaptation and is believed to have important implications for glioma invasion. In particular, glioma cells can change from a proliferative to a migratory phenotype depending on microenvironmental conditions [43,44]. It is thus crucial to investigate the factors and conditions that drive the transition from the proliferative to the motile phenotype. Several mathematical models have been introduced to analyse implications of this ‘Go-or-Grow’ dichotomy on glioma invasion. Tektonidis et al. [25] proposed a latticeLGCA model to explain the spatio-temporal evolution of U87 tumour spheroids in vitro reported in [166]. It turns out that the ‘Go-or-Grow’ mechanism combined with selfrepulsion and a density-dependent phenotypic switch is required to quantitatively reproduce the experimental observations (figure 3). (a) (b) Figure 2. Invasive cell migration. Front interface for small (a) and high (b) effective cell–cell adhesion values. Shown are the simulations of a discrete stochastic lattice model; every black dot represents a cell, and every white dot corresponds to an empty site. The system size is 400 400 (in units of cell diameter). Figure reproduced with permission from [27]. rsif.royalsocietypublishing.org J. R. Soc. Interface 14: 20170490 9 9. 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