Tracking glioblastoma progression after initial resection with minimal reaction-diffusion models.

We describe a preliminary effort to model the growth and progression of glioblastoma multiforme, an aggressive form of primary brain cancer, in patients undergoing treatment for recurrence of tumor following initial surgery and chemoradiation. Two reaction-diffusion models are used: the Fisher-Kolmogorov equation and a 2-population model, developed by the authors, that divides the tumor into actively proliferating and quiescent (or necrotic) cells. The models are simulated on 3-dimensional brain geometries derived from magnetic resonance imaging (MRI) scans provided by the Barrow Neurological Institute. The study consists of 17 clinical time intervals across 10 patients that have been followed in detail, each of whom shows significant progression of tumor over a period of 1 to 3 months on sequential follow up scans. A Taguchi sampling design is implemented to estimate the variability of the predicted tumors to using 144 different choices of model parameters. In 9 cases, model parameters can be identified such that the simulated tumor, using both models, contains at least 40 percent of the volume of the observed tumor. We discuss some potential improvements that can be made to the parameterizations of the models and their initialization.

Modeling the synergistic properties of drugs in hormonal treatment for prostate cancer.

Prostate cancer is one of the most prevalent cancers in men, with increasing incidence worldwide. This public health concern has inspired considerable effort to study various aspects of prostate cancer treatment using dynamical models, especially in clinical settings. The standard of care for metastatic prostate cancer is hormonal therapy, which reduces the production of androgen that fuels the growth of prostate tumor cells prior to treatment resistance. Existing population models often use patients' prostate-specific antigen levels as a biomarker for model validation and for finding optimal treatment schedules; however, the synergistic effects of drugs used in hormonal therapy have not been well-examined. This paper describes the first mathematical model that explicitly incorporates the synergistic effects of two drugs used to inhibit androgen production in hormonal therapy. The drugs are cyproterone acetate, representing the drug family of anti-androgens that affect luteinizing hormones, and leuprolide acetate, representing the drug family of gonadotropin-releasing hormone analogs. By fitting the model to clinical data, we show that the proposed model can capture the dynamics of serum androgen levels during intermittent hormonal therapy better than previously published models. Our results highlight the importance of considering the synergistic effects of drugs in cancer treatment, thus suggesting that the dynamics of the drugs should be taken into account in optimal treatment studies, particularly for adaptive therapy. Otherwise, an unrealistic treatment schedule may be prescribed and render the treatment less effective. Furthermore, the drug dynamics allow our model to explain the delay in the relapse of androgen the moment a patient is taken off treatment, which supports that this delay is due to the residual effects of the drugs.

Predictability and identifiability assessment of models for prostate cancer under androgen suppression therapy.

The past two decades have seen the development of numerous mathematical models to study various aspects of prostate cancer in clinical settings. These models often contain large sets of parameters and rely on limited data sets for validation. The quantitative analysis of the dynamics of prostate cancer under treatment may be hindered by the lack of identifiability of the parameters from the available data, which limits the predictive ability of the model. Using three ordinary differential equation models as case studies, we carry out a numerical investigation of the identifiability and uncer- tainty quantification of the model parameters. In most cases, the parameters are not identifiable from time series of prostate-specific antigen, which is used as a clinical proxy for tumor progression. It may not be possible to define a finite confidence bound on an unidentifiable parameter, and the relative uncertainties in even identifiable parameters may be large in some cases. The Fisher information ma- trix may be used to determine identifiable parameter subsets for a given model. The use of biological constraints and additional types of measurements, should they become available, may reduce these uncertainties. Ensemble Kalman filtering may provide clinically useful, short-term predictions of pa- tient outcomes from imperfect models, though care must be taken when estimating "patient-specific" parameters. Our results demonstrate the importance of parameter identifiability in the validation and predictive ability of mathematical models of prostate tumor treatment. Observing-system simulation experiments, widely used in meteorology, may prove useful in the development of biomathematical models intended for future clinical application.

Patient-specific parameter estimates of glioblastoma multiforme growth dynamics from a model with explicit birth and death rates.

Glioblastoma multiforme (GBM) is an aggressive primary brain cancer with a grim prog-nosis. Its morphology is heterogeneous, but prototypically consists of an inner, largely necrotic core surrounded by an outer, contrast-enhancing rim, and often extensive tumor-associated edema beyond. This structure is usually demonstrated by magnetic resonance imaging (MRI). To help relate the three highly idealized components of GBMs (i.e., necrotic core, enhancing rim, and maximum edema ex-tent) to the underlying growth "laws," a mathematical model of GBM growth with explicit motility, birth, and death processes is proposed. This model generates a traveling-wave solution that mimics tumor progression. We develop several novel methods to approximate key characteristics of the wave profile, which can be compared with MRI data. Several simplified forms of growth and death terms and their parameter identifiability are studied. We use several test cases of MRI data of GBM patients to yield personalized parameterizations of the model, and the biological and clinical implications are discussed.

Mathematical Analysis of Glioma Growth in a Murine Model.

Five immunocompetent C57BL/6-cBrd/cBrd/Cr (albino C57BL/6) mice were injected with GL261-luc2 cells, a cell line sharing characteristics of human glioblastoma multiforme (GBM). The mice were imaged using magnetic resonance (MR) at five separate time points to characterize growth and development of the tumor. After 25 days, the final tumor volumes of the mice varied from 12 mm3 to 62 mm3, even though mice were inoculated from the same tumor cell line under carefully controlled conditions. We generated hypotheses to explore large variances in final tumor size and tested them with our simple reaction-diffusion model in both a 3-dimensional (3D) finite difference method and a 2-dimensional (2D) level set method. The parameters obtained from a best-fit procedure, designed to yield simulated tumors as close as possible to the observed ones, vary by an order of magnitude between the three mice analyzed in detail. These differences may reflect morphological and biological variability in tumor growth, as well as errors in the mathematical model, perhaps from an oversimplification of the tumor dynamics or nonidentifiability of parameters. Our results generate parameters that match other experimental in vitro and in vivo measurements. Additionally, we calculate wave speed, which matches with other rat and human measurements.

Mathematically modeling the biological properties of gliomas: A review.

Although mathematical modeling is a mainstay for industrial and many scientific studies, such approaches have found little application in neurosurgery. However, the fusion of biological studies and applied mathematics is rapidly changing this environment, especially for cancer research. This review focuses on the exciting potential for mathematical models to provide new avenues for studying the growth of gliomas to practical use. In vitro studies are often used to simulate the effects of specific model parameters that would be difficult in a larger-scale model. With regard to glioma invasive properties, metabolic and vascular attributes can be modeled to gain insight into the infiltrative mechanisms that are attributable to the tumor's aggressive behavior. Morphologically, gliomas show different characteristics that may allow their growth stage and invasive properties to be predicted, and models continue to offer insight about how these attributes are manifested visually. Recent studies have attempted to predict the efficacy of certain treatment modalities and exactly how they should be administered relative to each other. Imaging is also a crucial component in simulating clinically relevant tumors and their influence on the surrounding anatomical structures in the brain.

Kalman filter data assimilation: targeting observations and parameter estimation.

This paper studies the effect of targeted observations on state and parameter estimates determined with Kalman filter data assimilation (DA) techniques. We first provide an analytical result demonstrating that targeting observations within the Kalman filter for a linear model can significantly reduce state estimation error as opposed to fixed or randomly located observations. We next conduct observing system simulation experiments for a chaotic model of meteorological interest, where we demonstrate that the local ensemble transform Kalman filter (LETKF) with targeted observations based on largest ensemble variance is skillful in providing more accurate state estimates than the LETKF with randomly located observations. Additionally, we find that a hybrid ensemble Kalman filter parameter estimation method accurately updates model parameters within the targeted observation context to further improve state estimation.

Accurate state estimation from uncertain data and models: an application of data assimilation to mathematical models of human brain tumors.

BACKGROUND: Data assimilation refers to methods for updating the state vector (initial condition) of a complex spatiotemporal model (such as a numerical weather model) by combining new observations with one or more prior forecasts. We consider the potential feasibility of this approach for making short-term (60-day) forecasts of the growth and spread of a malignant brain cancer (glioblastoma multiforme) in individual patient cases, where the observations are synthetic magnetic resonance images of a hypothetical tumor. RESULTS: We apply a modern state estimation algorithm (the Local Ensemble Transform Kalman Filter), previously developed for numerical weather prediction, to two different mathematical models of glioblastoma, taking into account likely errors in model parameters and measurement uncertainties in magnetic resonance imaging. The filter can accurately shadow the growth of a representative synthetic tumor for 360 days (six 60-day forecast/update cycles) in the presence of a moderate degree of systematic model error and measurement noise. CONCLUSIONS: The mathematical methodology described here may prove useful for other modeling efforts in biology and oncology. An accurate forecast system for glioblastoma may prove useful in clinical settings for treatment planning and patient counseling. REVIEWERS: This article was reviewed by Anthony Almudevar, Tomas Radivoyevitch, and Kristin Swanson (nominated by Georg Luebeck).

Universal and nonuniversal features in shadowing dynamics of nonhyperbolic chaotic systems with unstable-dimension variability.

An important quantity characterizing the shadowability of computer-generated trajectories in nonhyperbolic chaotic system is the shadowing time, which measures for how long a numerical trajectory remains valid. This time depends sensitively on an initial condition. Here, we show that for nonhyperbolic systems with unstable-dimension variability, the probability distribution of the shadowing time contains two distinct scaling behaviors: an algebraic scaling for short times and an exponential scaling for long times. The exponential behavior depends on the system details but the small-time algebraic behavior appears to be universal.

Measuring intense rotation and dissipation in turbulent flows.

Turbulent flows are highly intermittent--for example, they exhibit intense bursts of vorticity and strain. Kolmogorov theory describes such behaviour in the form of energy cascades from large to small spatial and temporal scales, where energy is dissipated as heat. But the causes of high intermittency in turbulence, which show non-gaussian statistics, are not well understood. Such intermittency can be important, for example, for enhancing the mixing of chemicals, by producing sharp drops in local pressure that can induce cavitation (damaging mechanical components and biological organisms), and by causing intense vortices in atmospheric flows. Here we present observations of the three components of velocity and all nine velocity gradients within a small volume, which allow us to determine simultaneously the dissipation (a measure of strain) and enstrophy (a measure of rotational energy) of a turbulent flow. Combining the statistics of all measurements and the evolution of individual bursts, we find that a typical sequence for intense events begins with rapid strain growth, followed by rising vorticity and a final sudden decline in stretching. We suggest two mechanisms which can produce these characteristics, depending whether they are due to the advection of coherent structures through our observed volume or caused locally.

Detectability of dynamical coupling from delay-coordinate embedding of scalar time series.

We address under what conditions dynamical coupling between chaotic systems can be detected reliably from scalar time series. In particular, we study weakly coupled chaotic systems and focus on the detectability of the correlation dimension of the chaotic invariant set by utilizing the Grassberger-Procaccia algorithm. An algebraic scaling law is obtained, which relates the necessary length of the time series to a key parameter of the system: the coupling strength. The scaling law indicates that an extraordinarily long time series is required for detecting the coupling dynamics.

kltool: A tool to analyze spatiotemporal complexity.

We announce the availability of a software package, called kltool, that can extract phase space information from complex spatiotemporal data via the Karhunen-Loeve analysis. Data generated by the periodic, quasiperiodic or chaotic evolution of a small number of spatially coherent structures can be processed. A key feature of kltool is that it allows the user to interact easily with the data processing and its graphical display. We illustrate the use of kltool on numerical data from the Kuramoto-Sivashinsky equation and laboratory data from a flame experiment.