Glioblastoma brain tumours: estimating the time from brain tumour initiation and resolution of a patient survival anomaly after similar treatment protocols

Abstract

A practical mathematical model for glioblastomas (brain tumours), which incorporates the two key parameters of tumour growth, namely the cancer cell diffusion and the cell proliferation rate, has been shown to be clinically useful and predictive. Previous studies explain why multifocal recurrence is inevitable and show how various treatment scenarios have been incorporated in the model. In most tumours, it is not known when the cancer started. Based on patient in vivo parameters, obtained from two brain scans, it is shown how to estimate the time, after initial detection, when the tumour started. This is an input of potential importance in any future controlled clinical study of any connection between cell phone radiation and brain tumour incidence. It is also used to estimate more accurately survival times from detection. Finally, based on patient parameters, the solution of the model equation of the tumour growth helps to explain why certain patients live longer than others after similar treatment protocols specifically surgical resection (removal) and irradiation.

Numerical Study of Transport of Anticancer Drugs in Heterogeneous Vasculature of Human Brain Tumors Using Dynamic Contrast Enhanced-Magnetic Resonance Imaging

“…Both of them are increasingly being used for treatment of human brain tumors [25,26]. Value of cell proliferation rate has been taken for human brain tumors (glioblastoma) [27]. Pharmacokinetic parameters of LED and free drug and their AIF were taken from literature and are listed in Table 1.…”

A Review of Mathematical Models for Tumor Dynamics and Treatment Resistance Evolution of Solid Tumors

“…66 Setting f (c (x,t)) = ⋅ c (x,t), a similar model structure was also used to simulate the growth of glioblastoma based on previous reported parameters estimated from patients and estimated the survival times of patients under different parameter settings. 67 Likewise, the proliferation-invasion model with logistic growth function was also successfully applied in breast cancer patients to characterize and predict their tumor burden. 68 The model developed based on MRI data that were available from the early treatment phase was demonstrated to be able to predict patient response at the end of treatment.…”

“…35, where k d is the drug effect rate constant. 67 For radiotherapy, a linear-quadratic equation has been used to estimate the probability of tumor cell survival (Surv) after the administration of radiation with dose Dose (Eq. 36).…”

Applying Adomian Decomposition Method to Solve Burgess Equation with a Non-linear Source

“…With this solution we can given the density of cancer cells in every point of (r, τ) ∈ (0, 1]×[0, 1]. In figure 4, we can observe the approximate-linear tumour growth-profile after the patient is under chemotherapy treatment with Temozolomide in contrast with the fast exponential growth given by (15) and corresponding to free-growth tumour. The free-growth tumour profile was shown in [7], in which the value of η(r,t) is given for different values of the parameters D, p and k(t) = 0.…”

“…In order to see the effect of medical treatment, we can compare the radius of the tumour under medical treatment versus the radius of the untreated tumour. Using the solution (15) of the Burgess linear partial equation and solving for r (also see [15]) that accounts for free growth of an untreated tumour , we obtain…”