Descriptive and prognostic value of a computational model of metastasis in high-risk neuroblastoma

Abstract

High Risk Neuroblastoma (HRNB) is the second most frequent solid tumor in children. Prognosis remains poor despite multimodal therapies. Mathematical models have been developed to describe metastasis, but their prognosis value has yet to be determined and none exists in neuroblastoma. We established such a model for HRNB relying on two coefficients: α (growth) and μ (dissemination). The model was calibrated using diagnosis values of primary tumor size, lactate dehydrogenase circulating levels (LDH) and the meta-iodo-benzyl-guanidine (mIBG) SIOPEN score from nuclear imaging, using data from 49 metastatic patients treated according to the European HR_NBL1 protocol. The model was able to accurately describe the data for both total tumor mass (LDH, R^2>0.99) and number of visible metastasis (SIOPEN, R^2=0.96). Statistical analysis revealed significant association of LDH with overall survival (OS, p=0.0268). However, clinical variables alone were not able to generate a Cox-based model with sufficient prognosis ability (p=0.507). The parameter μ was found to be independent of the clinical variables and positively significantly associated with OS (p = 0.0175 in multivariate analysis). Critically, addition of this novel computational biomarker to the clinical data drastically improved the performances of predictive algorithms, with a concordance index in cross-validation going from 0.755 to 0.827. The resulting signature had significant prognosis ability of OS (p=0.0353). Mechanistic modeling was able to describe pathophysiological data of metastatic HRNB and outperformed the predictive value of clinical variables. The physiological substrate underlying these results has yet to be explored, and results should be confirmed in a larger cohort.

A Quasi Birth-and-Death Model For Tumor Recurrence

“…Starting with Eq. (12) and using the adjoint of the master equation operator (6), we can derive the backward Kolmogorov equation by means of a Kramers-Moyal expansion similar to that leading to the forward equation (14). It should be noted that the differential operators in the backward equation act on functions of the initial-state variables (x 0 , y 0 ):…”

“…The backward equation (17) is somewhat different from Eq. (14) in that the non-constant coefficients 185 appear outside the differential operators. It is subject to the final condition that some state x will be reached at time t, starting from anywhere ( x 0 ) in the state space.…”

“…15 In current clinical practice, chemotherapy is usually given at "maximum tolerated dose" for the "minimum possible duration", which is usually 3-6 months, in the belief that this will have the most benefit to the patient in the least possible time [13]. This is based on modeling tumor cells as continuously dividing at some fixed deterministic rate [14]. Norton and Simon [15] proposed that tumor growth follows a type of sigmoid curve known in the literature as the Gompertzian function, and also proposed the tumor-regression 20 hypothesis, that cell kill is proportional to the growth rate of the untreated tumor [12,16].…”

“…This proper definition of the boundary condition near the origin ensures that the Fokker-Planck equation (14) gives the correct probability conservation equation in its integral form. Indeed, using the boundarycondition scheme described above and integrating Eq.…”