Stochastic growth pattern of untreated human glioblastomas predicts the survival time for patients

Ziwei Ma,Philip K Maini,Eric C Holland,Tuan Anh Phan,Jianjun Paul Tian,Anne Line Stensjøen,Tonghui Wang,Timothy Woodiwiss,Ben Niu,Chibawanye Ene

doi:10.1038/s41598-020-63394-w

Abstract

Glioblastomas are highly malignant brain tumors. Knowledge of growth rates and growth patterns is useful for understanding tumor biology and planning treatment logistics. Based on untreated human glioblastoma data collected in Trondheim, Norway, we first fit the average growth to a Gompertz curve, then find a best fitted white noise term for the growth rate variance. Combining these two fits, we obtain a new type of Gompertz diffusion dynamics, which is a stochastic differential equation (SDE). Newly collected untreated human glioblastoma data in Seattle, US, re-verify our model. Instead of growth curves predicted by deterministic models, our SDE model predicts a band with a center curve as the tumor size average and its width as the tumor size variance over time. Given the glioblastoma size in a patient, our model can predict the patient survival time with a prescribed probability. The survival time is approximately a normal random variable with simple formulas for its mean and variance in terms of tumor sizes. Our model can be applied to studies of tumor treatments. As a demonstration, we numerically investigate different protocols of surgical resection using our model and provide possible theoretical strategies.

Highlights

Can predict the tumor size at future time t after the first magnetic resonance imaging (MRI) scan, we can obtain more insight on the glioblastoma growth dynamics based on this model
If we make the simplifying assumption that tumor size is the only reason for patient death, the survival time for each patient can be expressed as the first time the tumor size reaches a specific value
Instead of growth curves predicted by deterministic models, we predict a band where the center curve is the mean of the solution, and its width is given by a prescribed probability that describes the variance of the tumor size deviating from the average over time

Summary

Introduction

Based on untreated human glioblastoma data collected in Trondheim, Norway, we first fit the average growth to a Gompertz curve, find a best fitted white noise term for the growth rate variance. Combining these two fits, we obtain a new type of Gompertz diffusion dynamics, which is a stochastic differential equation (SDE). In a recent clinical study, Stensjøen et al reported a data set of untreated human glioblastoma in vivo[3] They calculated the tumor growth rates for each patient, and found considerable variation among individual patients. We can predict the survival time for each patient based on a magnetic resonance imaging (MRI) scan with a prescribed probability before treatments

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Discussion

Conclusion