More than 120 different tumor types can be found in the human brain. The glioma multiforme (GBM), on the other hand, is the most aggressive and incomprehensible brain tumor among them, and unfortunately, it possesses a life expectancy of approximately 15 months. Fortunately, the scientific community has been tirelessly continuing to work from all directions to uncover the mystery behind the elusive dynamics of GBM. In their research published in 2019, J. Jacobs et al. suggested an improved mathematical model prediction for GBM growth behavior. In this blog post, we will attempt to comprehend this study.

In a simplified mathematical model, tumors are classified by invasiveness and proliferation dynamics. These two parameters represent the “net” effects since they are estimated from volumetric measurements of clinical MRI scans, and should be thought of as a phenotypic compilation of biological qualities. However, this advancement does not provide adequate accuracy or updating of volumetric measurements for tumor growth over time throughout an anatomical area. From this point of view, a formula that adjusts derived parameters with respect to anatomical boundaries encountered has been presented by J. Jacobs et al.

Methods and results

J. Jacobs et al. have worked on the Fisher-Kolmogorov equation to model the growth of gliomas in this study. In their mathematical model, they have taken into account cancer cell density, invasion and proliferation dynamics, and tissue carrying capacity. It has also been addressed that the invasion coefficient, as the driving force in glioma growth behavior, moves differently with respect to tissue types, such as white matter, gray matter, and cerebrospinal fluid on an anatomical atlas. Because the invasion parameter varies significantly more than the proliferation parameter over time, the proliferation parameter has been held constant while the invasion coefficient has been varied. A volume definition that is within a detectable threshold has been presented for a simple boundary set by a cone, the point of which is at the center of a spherically symmetric tumor.

For the calibration algorithm, a patient’s total tumor volume has been segmented on both the T1-weighted gadolinium-enhanced (T1Gd) and the T2 weighted or Fluid Attenuation Inversion Recovery (FLAIR) MRIs, as in previous studies in the literature. Here, the thresholds for detection have been assumed to be a fraction of the carrying capacity of the tissue.

Both spherically symmetric and anatomical simulations have been executed along with an initial condition of a Gaussian distribution of cell density, and the peak of the Gaussian distribution has been thought of as being placed at the center of the tumor domain. Of course, the estimated center for the Gaussian distribution is a patient-specific value, as are the tumor invasion and proliferation parameters.

The solutions of the Fisher-Kolmogorov equation have been generated with a numerical diffusion-reaction solver employing a split-step method, with the reaction terms solved by a backwards Euler algorithms and the diffusion terms solved by a conjugate gradient method. To run these methods, Python programming language has been used in the research. For simplicity, the illustrations have been executed with isotropic diffusion. Besides, the McGill BrainWeb Atlas has been used for the anatomical simulations. It is known that the BrainWeb Atlas provides a spatial distribution of gray and white matter and cerebrospinal fluid for the entire brain at a resolution of 1 mm^{3} voxels. The fact remains that it may take several days to discover the sufficiently representative parameters in an anatomical atlas. However, examining the effect of anatomical boundaries has the potential to reduce the parameter space explored, the computational cost of discovering it, and the time required to build an adequately representative model.

The ansatz of the restrictive cone by considering six half-angles given with φ ∈ {0, π/12, π/6, π/4, π/3, 5π/12, π/2} that are centered in the initialized population has been tested with the constant invasion and proliferation parameters for the initial verification of the method that has been proposed. The question of how a brain tumor grows in harmony with a conical boundary can easily come to mind. Admittedly, it does not grow. Therefore, the numerical simulations have been performed around a pyramidal boundary and have used the conical boundary as an ansatz for the duration of measurements. Approximate results have been obtained for both. For each conical ansatz simulation, the volumetrically-derived radii, velocity, and observed invasion coefficient have been calculated. For the sake of clarity, the radius of the sphere with equivalent volume, as called a volumetrically-derived radius, is obtained from each segmented volume on clinical scans. In addition to these, the evolution of the volumetrically-derived radii has been looked into, and the apparent difference between an anatomical simulation and a spherically symmetric simulation has been illustrated using prescribed values of the parameters and a given initial condition in order to see how well the formula worked with patient data. Here, both ansatz and anatomical simulations have been run until the volumetrically-derived T1Gd radius reaches 40 mm. Thus, two sets of manufactured “data” have been generated by running the model with the prescribed parameter values. It has been concluded that the observed derived invasion approaches the predicted volumetrically-derived velocity definition in the conically restricted domain. It has also been noted that moderate perturbations in the shape of the boundary do not greatly affect the results. However, it has been remarked that if the ratio of surface area to volume becomes too great, the curvature of the advancing front along the boundary contributes to increased velocity and thus the volumetrically-derived invasion.

The underlying biology of invasion and proliferation phenomena in patients with GBM depends on the brain’s unique physiology, which is not yet well understood. To reach meaningful results, in this regard, the data from twenty patients has been used, and for each of them, three numerical simulations have been performed in this research in order to evaluate how well each parameter estimate was able to reproduce the observed measurements.

For the first step, taking three different time points into account, volumetrically-derived radial measurements have been sampled for both generated T1Gd radial profiles with the volumetrically-derived invasion coefficient and proliferation constant. Subsequently, the invasion parameter has been analyzed under both spherically-symmetric and anatomical simulations. Ultimately, an anatomical simulation has been performed with the estimated invasion coefficient updated according to the formula proposed in the research, and the unrestricted fraction of the evolving T1Gd threshold, which is called a, has been derived. This means that an updated invasion parameter has been derived. To make it clear, the active fraction, a, is the ratio of the T1Gd threshold surface area in white matter to the total surface area of the simulated tumor. It has been derived at a point when the anatomical simulation has an equivalent volume to the T1Gd tumor measurement in the first clinical scan of the patient.

An L2-norm, on the other hand, has been used to minimize the time obscurity between the simulation time and the measured time of clinical images. The effectiveness of each simulation to reproduce a given patient’s volumetric measurements has been evaluated with an L2-norm. The L2-errors for each numeric simulation, the relative change, conditions sufficient for improvement, and the mean of those quantities have also been presented. As a result, for eleven patients of the cohort, covering twenty patients, improvements ranging from 8% to 90% in reproducing the measurements have been observed with the applied L2-norm.

Since the updated volume formula in this work with respect to the anatomical boundaries is defined in linear form, it looks like it is not robust enough to make strong predictions; however, it is obvious that this research is a pathfinder for future work. In addition to what is covered in this blog post, there are other topics in the paper, namely, non-linearities in the simulation establishment phase and non-linearities due to a reduction in anatomical restriction, as well as many more valuable details. Some patient data and their tumor growth trends have also been pointed out in the paper.

For more detailed information about this inspiring research, you can access the paper by clicking on the black box under the title, and if any corrections are needed, you can reach me via the Contact page.