A mechanistic mathematical model of initiation and malignant transformation in sporadic vestibular schwannoma

Vestibular schwannoma (VS) is a benign Schwann-cell tumour of the eighth cranial nerve with lifetime risk ~1/1000. Most sporadic VS involve loss-of-function in NF2 plus at least one additional genetic alteration. Incidence patterns are consistent with a three-alteration (three-hit) process. Malignant transformation is rare (~0.2%), and a possible association with radiotherapy has been suggested in familial NF2, but quantitative risk estimates are lacking due to small numbers. The benign disease has richer genomic data (e.g., NF2 alterations, LOH on 22q), allowing a mechanistic mathematical model to link gene-level events to incidence and transformation risk. The study aims to: (1) build a mechanistic three-hit model for sporadic VS tied to specific genes and measurable processes, (2) estimate parameters such as mutation rate and LOH rate from genomic frequencies and incidence data, (3) extend the framework to model the risk of spontaneous and radiation-induced malignant transformation.

Prior multistage models captured VS incidence as a three-hit process but did not differentiate gene-specific mechanisms or mutation types. Mechanistic models in other cancers (e.g., colorectal) have tied parameters to cell populations, base substitution/indel rates, and selective effects, enabling interpretation without heavy fitting. VS genomics shows frequent NF2 inactivation, LOH on 22q, and occasional involvement of SMARCB1 and LZTR1. Reports suggest polyclonal VS foci, detailed growth measurements, and extremely rare malignant cases. Existing data on LOH (including copy-neutral) and variant frequencies enable constraining mechanistic parameters. Radiation-associated malignant transformation is controversial; empirical studies generally find no clear excess risk, but theoretical quantification is needed.

Incidence model: Construct a mechanistic three-hit model with two NF2 hits plus one additional alteration: either an oncogenic activation (hypothetical GFX) or inactivation of SMARCB1 (both on 22q considered; LZTR1 left for future work). All orders of occurrence are allowed, producing three neoplastic end-states with distinct alteration profiles (with/without LOH; with SMARCB1 alteration; double-mutant NF2). A mean-field system of linear ODEs tracks intermediate pre-neoplastic subpopulations, with transitions governed by gene-specific nonsense mutation rates (from base error rate u, cell division rate b, and number of sensitive sites n_gene) and an LOH rate on 22q (f_LOH). Initiating alterations are assumed neutral (no selective advantage). Probabilities to reach each neoplastic end-state are computed and combined allowing polyclonality (independent emergence events) to obtain overall tumour, LOH-positive tumour, and SMARCB1-deficient tumour probabilities. In the rare event limit, probabilities reduce to cubic power laws in age, enabling a compact At^3 form for cumulative incidence. Parameter estimation for incidence: Estimate precursor pool size N0 from anatomy (axons per vestibular nerve, nerve length, Schwann-cell spacing), yielding N0≈456,000. Set Schwann cell division rate b≈25.5/yr from proliferation after nerve injury. Compute sensitive sites for nonsense mutations: n_NF2 and n_SMARCB1 via counting substitution-sensitive codons (Jukes-Cantor assumption) plus indel-sensitive sites using an indel length distribution (geometric, q≈0.53) and positional counting; verify Bozic et al. approximation m≈0.74/L holds. Determine frequencies f_LOH (on 22q) and f_SMARCB1 (pathogenic variants) from literature and new Sanger sequencing (32 VS samples); apply additive smoothing for zero counts. Use model-predicted fractions f_LOH and f_SMARCB1 to solve for n_GFX and the LOH rate proxy, leaving u (per-base per-division) to be fit via incidence. Fit Pr_tumour(t)=A t^3 to mortality-corrected, NF2-associated incidence data (rescaled to 85% NF2 involvement) via non-linear least squares; relate A to parameters to solve for u. Use bootstrap resampling of the LOH/SMARCB1 datasets to obtain confidence intervals for n_GFX, u, and f_LOH. Model of sporadic malignant transformation: Within an expanding benign tumour, model a two-hit process on a hypothetical tumour suppressor TSX not on 22q: first hit via mutation (η_TSX sensitive sites times u), second via LOH with per-division probability P_LOH (derived from f_LOH and b). Assume haplosufficiency (neutral intermediate), edge-limited growth, and no explicit dependence on tumour age by expressing risks in terms of tumour cell number N (converted from MRI-measured volume using Schwann-cell fraction f_SC and cell volume V_SC). Derive risk of malignancy P ≈ 0.5 η_TSX u (η_TSX u + 2 P_LOH) N in the rare-event limit, implying a risk proportional to tumour volume. Constrain η_TSX by matching overall lifetime malignancy risk (~0.2%) at a typical surgical tumour size (~10^10 cells). LOH in malignant schwannoma: Predict excess LOH at the TSX locus among malignant tumours and derive an approximate size-independent relationship f_LOH ≈ 2 P_LOH / (2 P_LOH + η_TSX u N), enabling future estimation of η_TSX from LOH surveys. Radiation-induced excess risk: Model radiation-induced double-strand breaks (DSBs) causing TSX inactivation through misrepair-generated indels at indel-sensitive sites m_TSX ≈ 0.42 η_TSX. Use measured DSB induction probability per base pair P_DSB(D)=kD (k≈3.90×10^-7 Gy^-1 bp^-1), a conservative misrepair probability ε≈0.5 at high dose rates, and cell survival S(D)=exp(-αD-βD^2) with α=0.77 Gy^-1, β=0.31 Gy^-2. Derive excess risk E.R.(D) for single-fraction radiosurgery accounting for the probability of pre-existing malignant clones and for post-irradiation survival. Analyze fractionation: best-case (fractions within a tumour cell cycle so they approximate a single dose) vs. pessimistic worst-case (full recovery between fractions), yielding expressions where survival S(D/F) critically modulates risk as F increases. Uncertainty quantification: Bootstrap distributions for n_GFX, u, f_LOH from resampled LOH/SMARCB1 datasets; propagate to η_TSX via analytical formula and bootstrapping to provide 95% CIs.

• Mechanistic three-hit model fits sporadic VS cumulative incidence extremely well (R^2=0.989) when tying parameters to gene-level processes and allowing all orderings of hits.
• New parameter estimates for Schwann cells: • Base-pair mutation rate per division u = 4.48 × 10^-10 (95% CI: 3.36–8.74 × 10^-10). • LOH rate on 22q f_LOH = 2.03 × 10^-6 per year (95% CI: 1.29–4.93 × 10^-6 per year). • Sensitive sites for the third hit (oncogenic GFX): n_GFX ≈ 2002 (95% CI: 315–2442). • Sensitive sites for NF2 and SMARCB1 nonsense mutations: n_NF2 ≈ 135; n_SMARCB1 ≈ 85.
• LOH per division in Schwann cells is substantially lower than in colonic crypts even after controlling for cell division rates, indicating tissue-specific differences in LOH mechanisms.
• Spontaneous malignant transformation risk can be largely explained by a two-hit tumour-suppressor model within the benign tumour, with estimated η_TSX ≈ 1245 sensitive sites (95% CI: 604–1515). Risk scales linearly with tumour volume.
• Predicted excess LOH at the TSX locus among malignant tumours is size-insensitive and potentially detectable in small cohorts (e.g., ~22% expected; probability of no detections in 15 samples ≈ 2.4%).
• Radiotherapy-associated excess risk is negligible at therapeutic doses and dose rates when fractions are delivered within a tumour cell cycle (days). For single-fraction or well-scheduled fractionation, excess risk at doses >4 Gy is far below baseline lifetime risk; risk peaks at ~1–2 Gy and then declines due to cell killing.
• Worst-case, poorly spaced hyperfractionation could markedly increase risk by enhancing survival of induced mutants, underscoring the importance of appropriate fractionation intervals.

The mechanistic model links genomic alterations (NF2, SMARCB1, and an additional event) to epidemiologic incidence with gene-specific parameters, enabling direct inference of mutation and LOH rates from observed frequencies and incidence data. The derived u and f_LOH for Schwann cells align in magnitude with independent reports for brain tissue and suggest tissue-dependent LOH probabilities. The three-hit model’s strong fit supports the biological plausibility of two NF2 hits plus a third alteration (either GFX activation or SMARCB1 inactivation), while the large estimated n_GFX may reflect multiple oncogenes contributing to incidence. For malignant transformation, a neutral two-hit TSX model explains much of the observed lifetime risk and predicts risk proportional to tumour volume, together with an excess of LOH at the TSX locus that is largely independent of tumour size—offering a testable prediction. The relatively high η_TSX estimate could imply involvement of multiple tumour suppressors, though a single large gene cannot be excluded. Radiation modelling indicates that at clinically relevant doses and properly timed fractions, excess risk of malignant transformation is essentially negligible, consistent with observational studies. The analysis highlights the counteracting roles of mutagenesis and cell killing, with risk peaking at low doses and dropping rapidly at higher doses. A worst-case fractionation scenario illustrates the importance of avoiding long inter-fraction intervals that permit tumour cell recovery. Overall, the work demonstrates how mechanistic, gene-tied models can integrate genomic frequencies, incidence curves, and biophysical radiation data to produce clinically relevant risk estimates and testable predictions (e.g., LOH surveys in malignant schwannoma).

• A gene-specific, mechanistic three-hit model accurately reproduces sporadic VS incidence and yields new estimates for Schwann-cell mutation (u) and LOH rates, as well as the scale of the third-hit target (n_GFX).
• A simple two-hit, neutral model for malignant transformation involving an unidentified tumour suppressor (TSX) explains most of the observed lifetime malignancy risk and predicts risk proportional to tumour volume; the inferred η_TSX (~1245) suggests either a large single target or multiple tumour suppressors.
• Radiotherapy, when delivered as single-fraction radiosurgery or properly scheduled fractionation within a tumour cell cycle, is predicted to have a negligible excess malignancy risk for sporadic VS; care is required to avoid worst-case hyperfractionation scenarios. Future work should: incorporate LZTR1 and multiple third-hit genes into the initiation model; measure substitution spectra specific to schwannoma to refine u; perform LOH/copy-number surveys in malignant schwannoma to identify TSX and quantify excess LOH; and improve modelling of tumour cell survival and growth kinetics during therapy.

• Identity of the third-hit oncogene(s) (GFX) and the malignant-transforming tumour suppressor(s) (TSX) is unknown; η_TSX and n_GFX are inferred rather than directly measured, potentially reflecting multiple genes.
• Small sample sizes for LOH and SMARCB1 frequency estimates require additive smoothing and bootstrap methods, increasing uncertainty and potentially biasing parameter estimates.
• Assumptions of neutrality for initiating hits and TSX haplosufficiency may not hold universally; selection could alter allele frequencies and associations with age or tumour size.
• Mutation model simplifies substitution processes (Jukes-Cantor), ignores transition/transversion biases and positional effects (e.g., early truncations, frameshifts), due to lack of VS-specific data.
• LOH rate assumed uniform across chromosomes and constant per division; copy-neutral LOH mechanisms and detection limits could bias f_LOH.
• Mean-field approximation ignores stochastic clonal dynamics; tumour growth model assumes edge-limited growth and no cell death before malignancy.
• Radiation model uses simplified DSB induction, constant high misrepair probability, and standard linear-quadratic survival; dose homogeneity and recovery dynamics may vary, especially in multifocal disease.
• Familial NF2 cases excluded; findings may not generalize to hereditary contexts.