# Mathematical model of perineural tumor spread: a pilot study

## Abstract

### Background

Perineural spread (PNS) of pelvic cancer along the lumbosacral plexus is an emerging explanation for neoplastic lumbosacral plexopathy (nLSP) and an underestimated source of patient morbidity and mortality. Despite the increased incidence of PNS, these patients are often times a clinical conundrum—to diagnose and to treat. Building on previous results in modeling glioblastoma multiforme (GBM), we present a mathematical model for predicting the course and extent of the PNS of recurrent tumors.

### Methods

We created three-dimensional models of perineurally spreading tumor along the lumbosacral plexus from consecutive magnetic resonance imaging scans of two patients (one each with prostate cancer and cervical cancer). We adapted and applied a previously reported mathematical model of GBM to progression of tumor growth along the nerves on an anatomical model obtained from a healthy subject.

### Results

We were able to successfully model and visualize perineurally spreading pelvic cancer in two patients; average growth rates were 60.7 mm/year for subject 1 and 129 mm/year for subject 2. The model correlated well with extent of PNS on MRI scans at given time points.

### Conclusions

This is the first attempt to model perineural tumor spread and we believe that it provides a glimpse into the future of disease progression monitoring. Every tumor and every patient are different, and the possibility to report treatment response using a unified scale—as “days gained”—will be a necessity in the era of individualized medicine. We hope our work will serve as a springboard for future connections between mathematics and medicine.

### Keywords

Mathematical model Perineural spread Lumbosacral plexopathy Treatment response Pelvic cancer## Notes

### Compliance with ethical standards

### Conflict of interest

All authors certify that they have no affiliations with or involvement in any organization or entity with any financial interest (such as honoraria; educational grants; participation in speakers’ bureaus; membership, employment, consultancies, stock ownership, or other equity interest; and expert testimony or patent-licensing arrangements), or non-financial interest (such as personal or professional relationships, affiliations, knowledge or beliefs) in the subject matter or materials discussed in this manuscript.

### Ethical approval

All procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional and/or national research committee (name of institute/committee) and with the 1964 Helsinki declaration and its later amendments or comparable ethical standards. For this type of study formal consent is not required.

## Supplementary material

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