Kristin R. Swanson

Continuous growth of mean tumor diameter in a subset of grade II gliomas

Serial magnetic resonance images of 27 patients with untreated World Health Organization grade II oligodendrogliomas or mixed gliomas were reviewed retrospectively to study the kinetics of tumor growth before anaplastic transformation. Analysis of the mean tumor diameters over time showed constant growth. Linear regression, using a mixed model, found an average slope of 4.1mm per year (95% confidence interval, 3.8–4.4mm/year). Untreated low‐grade oligodendrogliomas or mixed gliomas grow continuously during their premalignant phase, and their pattern of growth can be predicted within a relatively narrow range. These findings could be of interest to optimize patients management and follow‐up. Ann Neurol 2003;53:524–528

Virtual and real brain tumors: using mathematical modeling to quantify glioma growth and invasion

Over the last 10 years increasingly complex mathematical models of cancerous growths have been developed, especially on solid tumors, in which growth primarily comes from cellular proliferation. The invasiveness of gliomas, however, requires a change in the concept to include cellular motility in addition to proliferative growth. In this article we review some of the recent developments in mathematical modeling of gliomas. We begin with a model of untreated gliomas and continue with models of polyclonal gliomas following chemotherapy or surgical resection. From relatively simple assumptions involving homogeneous brain tissue bounded by a few gross anatomical landmarks (ventricles and skull) the models have recently been expanded to include heterogeneous brain tissue with different motilities of glioma cells in grey and white matter on a geometrically complex brain domain, including sulcal boundaries, with a resolution of 1 mm(3) voxels. We conclude that the velocity of expansion is linear with time and varies about 10-fold, from about 4 mm/year for low-grade gliomas to about 3 mm/month for high-grade ones.

A quantitative model for differential motility of gliomas in grey and white matter

We have extended a mathematical model of gliomas based on proliferation and diffusion rates to incorporate the effects of augmented cell motility in white matter as compared to grey matter. Using a detailed mapping of the white and grey matter in the brain developed for a MRI simulator, we have been able to simulate model tumours on an anatomically accurate brain domain. Our simulations show good agreement with clinically observed tumour geometries and suggest paths of submicroscopic tumour invasion not detectable on CT or MRI images. We expect this model to give insight into microscopic and submicroscopic invasion of the human brain by glioma cells. This method gives insight in microscopic and submicroscopic invasion of the human brain by glioma cells. Additionally, the model can be useful in defining expected pathways of invasion by glioma cells and thereby identify regions of the brain on which to focus treatments.

Virtual brain tumours (gliomas) enhance the reality of medical imaging and highlight inadequacies of current therapy

Gliomas are brain tumours that differ from most other cancers by their diffuse invasion of the surrounding normal tissue and their notorious recurrence following all forms of therapy. We have developed a mathematical model to quantify the spatio-temporal growth and invasion of gliomas in three dimensions throughout a virtual human brain. The model quantifies the extent of tumorous invasion of individual gliomas in three-dimensions to a degree beyond the limits of present medical imaging, including even microscopy, and makes clear why current therapies based on existing imaging techniques are inadequate and cannot be otherwise without other methods for detecting tumour cells in the brain. The models estimate of the extent of tumourous invasion beyond that defined by standard medical imaging can be useful in more accurately planning therapy regimes as well as predicting sites of potential recurrence without waiting for reemergence on follow-up imaging.

The Evolution of Mathematical Modeling of Glioma Proliferation and Invasion

Gliomas are well known for their potential for aggressive proliferation as well as their diffuse invasion of the normal-appearing parenchyma peripheral to the bulk lesion. This review presents a history of the use of mathematical modeling in the study of the proliferative-invasive growth of gliomas, illustrating the progress made in understanding the in vivo dynamics of invasion and proliferation of tumor cells. Mathematical modeling is based on a sequence of observation, speculation, development of hypotheses to be tested, and comparisons between theory and reality. These mathematical investigations, iteratively compared with experimental and clinical work, demonstrate the essential relationship between experimental and theoretical approaches. Together, these efforts have extended our knowledge and insight into in vivo brain tumor growth dynamics that should enhance current diagnoses and treatments.

Simulation of anisotropic growth of low-grade gliomas using diffusion tensor imaging

A recent computational model of brain tumor growth, developed to better describe how gliomas invade through the adjacent brain parenchyma, is based on two major elements: cell proliferation and isotropic cell diffusion. On the basis of this model, glioma growth has been simulated in a virtual brain, provided by a 3D segmented MRI atlas. However, it is commonly accepted that glial cells preferentially migrate along the direction of fiber tracts. Therefore, in this paper, the model has been improved by including anisotropic extension of gliomas. The method is based on a cell diffusion tensor derived from water diffusion tensor (as given by MRI diffusion tensor imaging). Results of simulations have been compared with two clinical examples demonstrating typical growth patterns of low‐grade gliomas centered around the insula. The shape and the kinetic evolution are better simulated with anisotropic rather than isotropic diffusion. The best fit is obtained when the anisotropy of the cell diffusion tensor is increased to greater anisotropy than the observed water diffusion tensor. The shape of the tumor is also influenced by the initial location of the tumor. Anisotropic brain tumor growth simulations provide a means to determine the initial location of a low‐grade glioma as well as its cell diffusion tensor, both of which might reflect the biological characteristics of invasion. Magn Reson Med, 2005.

Regional Hypoxia in Glioblastoma Multiforme Quantified with [18F]Fluoromisonidazole Positron Emission Tomography before Radiotherapy: Correlation with Time to Progression and Survival

Purpose: Hypoxia is associated with resistance to radiotherapy and chemotherapy and activates transcription factors that support cell survival and migration. We measured the volume of hypoxic tumor and the maximum level of hypoxia in glioblastoma multiforme before radiotherapy with [18F]fluoromisonidazole positron emission tomography to assess their impact on time to progression (TTP) or survival. Experimental Design: Twenty-two patients were studied before biopsy or between resection and starting radiotherapy. Each had a 20-minute emission scan 2 hours after i.v. injection of 7 mCi of [18F]fluoromisonidazole. Venous blood samples taken during imaging were used to create tissue to blood concentration (T/B) ratios. The volume of tumor with T/B values above 1.2 defined the hypoxic volume (HV). Maximum T/B values (T/Bmax) were determined from the pixel with the highest uptake. Results: Kaplan-Meier plots showed shorter TTP and survival in patients whose tumors contained HVs or tumor T/Bmax ratios greater than the median (P ≤ 0.001). In univariate analyses, greater HV or tumor T/Bmax were associated with shorter TTP or survival (P < 0.002). Multivariate analyses for survival and TTP against the covariates HV (or T/Bmax), magnetic resonance imaging (MRI) T1Gd volume, age, and Karnovsky performance score reached significance only for HV (or T/Bmax; P < 0.03). Conclusions: The volume and intensity of hypoxia in glioblastoma multiforme before radiotherapy are strongly associated with poorer TTP and survival. This type of imaging could be integrated into new treatment strategies to target hypoxia more aggressively in glioblastoma multiforme and could be applied to assess the treatment outcomes.

Predicting the efficacy of radiotherapy in individual glioblastoma patients in vivo: a mathematical modeling approach

Glioblastoma multiforme (GBM) is the most malignant form of primary brain tumors known as gliomas. They proliferate and invade extensively and yield short life expectancies despite aggressive treatment. Response to treatment is usually measured in terms of the survival of groups of patients treated similarly, but this statistical approach misses the subgroups that may have responded to or may have been injured by treatment. Such statistics offer scant reassurance to individual patients who have suffered through these treatments. Furthermore, current imaging-based treatment response metrics in individual patients ignore patient-specific differences in tumor growth kinetics, which have been shown to vary widely across patients even within the same histological diagnosis and, unfortunately, these metrics have shown only minimal success in predicting patient outcome. We consider nine newly diagnosed GBM patients receiving diagnostic biopsy followed by standard-of-care external beam radiation therapy (XRT). We present and apply a patient-specific, biologically based mathematical model for glioma growth that quantifies response to XRT in individual patients in vivo. The mathematical model uses net rates of proliferation and migration of malignant tumor cells to characterize the tumors growth and invasion along with the linear-quadratic model for the response to radiation therapy. Using only routinely available pre-treatment MRIs to inform the patient-specific bio-mathematical model simulations, we find that radiation response in these patients, quantified by both clinical and model-generated measures, could have been predicted prior to treatment with high accuracy. Specifically, we find that the net proliferation rate is correlated with the radiation response parameter (r = 0.89, p = 0.0007), resulting in a predictive relationship that is tested with a leave-one-out cross-validation technique. This relationship predicts the tumor size post-therapy to within inter-observer tumor volume uncertainty. The results of this study suggest that a mathematical model can create a virtual in silico tumor with the same growth kinetics as a particular patient and can not only predict treatment response in individual patients in vivo but also provide a basis for evaluation of response in each patient to any given therapy.

A mathematical modelling tool for predicting survival of individual patients following resection of glioblastoma: A proof of principle

The prediction of the outcome of individual patients with glioblastoma would be of great significance for monitoring responses to therapy. We hypothesise that, although a large number of genetic-metabolic abnormalities occur upstream, there are two ‘final common pathways’ dominating glioblastoma growth – net rates of proliferation (ρ) and dispersal (D). These rates can be estimated from features of pretreatment MR images and can be applied in a mathematical model to predict tumour growth, impact of extent of tumour resection and patient survival. Only the pre-operative gadolinium-enhanced T1-weighted (T1-Gd) and T2-weighted (T2) volume data from 70 patients with previously untreated glioblastoma were used to derive a ratio D/ρ for each patient. We developed a ‘virtual control’ for each patient with the same size tumour at the time of diagnosis, the same ratio of net invasion to proliferation (D/ρ) and the same extent of resection. The median durations of survival and the shapes of the survival curves of actual and ‘virtual’ patients subjected to biopsy or subtotal resection (STR) superimpose exactly. For those actually receiving gross total resection (GTR), as shown by post-operative CT, the actual survival curve lies between the ‘virtual’ results predicted for 100 and 125% resection of the T1-Gd volume. The concordance between predicted (virtual) and actual survivals suggests that the mathematical model is realistic enough to allow precise definition of the effectiveness of individualised treatments and their site(s) of action on proliferation (ρ) and/or dispersal (D) of the tumour cells without knowledge of any other clinical or pathological information.

Quantitative metrics of net proliferation and invasion link biological aggressiveness assessed by MRI with hypoxia assessed by FMISO-PET in newly diagnosed glioblastomas.

Glioblastoma multiforme (GBM) are aggressive and uniformly fatal primary brain tumors characterized by their diffuse invasion of the normal-appearing parenchyma peripheral to the clinical imaging abnormality. Hypoxia, a hallmark of aggressive tumor behavior often noted in GBMs, has been associated with resistance to therapy, poorer survival, and more malignant tumor phenotypes. Based on the existence of a set of novel imaging techniques and modeling tools, our objective was to assess a hypothesized quantitative link between tumor growth kinetics [assessed via mathematical models and routine magnetic resonance imaging (MRI)] and the hypoxic burden of the tumor [assessed via positron emission tomography (PET) imaging]. Our biomathematical model for glioma kinetics describes the spatial and temporal evolution of a glioma in terms of concentration of malignant tumor cells. This model has already been proven useful as a novel tool to dynamically quantify the net rates of proliferation (rho) and invasion (D) of the glioma cells in individual patients. Estimates of these kinetic rates can be calculated from routinely available pretreatment MRI in vivo. Eleven adults with GBM were imaged preoperatively with (18)F-fluoromisonidazole (FMISO)-PET and serial gadolinium-enhanced T1- and T2-weighted MRIs to allow the estimation of patient-specific net rates of proliferation (rho) and invasion (D). Hypoxic volumes were quantified from each FMISO-PET scan following standard techniques. To control for tumor size variability, two measures of hypoxic burden were considered: relative hypoxia (RH), defined as the ratio of the hypoxic volume to the T2-defined tumor volume, and the mean intensity on FMISO-PET scaled to the blood activity of the tracer (mean T/B). Pearson correlations between RH and the net rate of cell proliferation (rho) reached significance (P < 0.04). Moreover, highly significant positive correlations were found between biological aggressiveness ratio (rho/D) and both RH (P < 0.00003) and the mean T/B (P < 0.0007).