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Mathematics to help diagnose cancer


Double Attack on Breast Cancer

Personalized medicine is a healthcare approach in which medical procedures are tailored to the individual patient's genetic, physiological, and biochemical type along with other characteristics. Cancer treatment is one type of care where a personalized approach is most needed and often applied.

A few classical mathematical models have been developed to describe the natural path of breast cancer, but they tend to address the primary tumor and secondary metastatic growth separately, according to the paper "Combined mathematical model of the growth of breast cancer" by Senior Research Fellow of HSE International Laboratory for Intelligent Systems and Structural Analysis Alexey Neznanov and Research Assistant of the Laboratory Ella Tyuryumina.

But a cancer's natural history does not always end with primary tumor removal; in many instances, secondary distant metastases can later develop and, if allowed to grow to a certain size, are likely to result in death. In addition to comprehensive treatment of the primary tumor, cancer patients today are assessed for secondary metastatic growth. Mathematical models used to facilitate such assessment should be capable of accurately describing the following stages:

  • latent period of primary tumor growth;
  • visible primary tumor, its diagnosis and removal;
  • latent period of distant metastatic growth; and
  • visible secondary metastases, their diagnosis and treatment, and the patient's outcome, including death.

The authors' goal was to design a comprehensive mathematical model capable of describing the entire natural history of breast cancer, including the development of primary tumor and secondary metastases according to histological stage, and predicting patient outcomes and survival at the time of the primary tumor treatment and removal.

Computing the Invisible

By developing a new mathematical model, Neznanov and Tyuryumina aimed to improve cancer growth prediction. In particular, they intended to:

  • review known mathematical models for describing the growth of primary tumor and secondary metastasis in breast cancer;
  • develop a comprehensive mathematical model of primary tumor and secondary metastasis development for this type of cancer (Combined Model);
  • identify critical periods in the course of primary tumor and secondary metastasis development, which can affect the prognosis of survival;
  • implement the Combined Model as a software solution; and
  • determine the scope of the Combined Model's applicability and areas for further research.

Having studied the available literature on breast cancer modeling published between 1930 and 2014, the HSE mathematicians proposed a new exponential growth model based on a set of deterministic linear and non-linear equations.

Their Combined Model describes both the primary tumor and secondary metastasis growth according to histological stages and helps predict survival prognosis. "By matching it to the official breast cancer classification, we found our model to be quite accurate in correlating the primary tumor size with patient survival prognosis," says Neznanov. "As was already known, the removal of primary tumor is often followed by the latent growth of secondary metastases; our model makes it clear how the five-year survival rate depends on the size of the primary tumor in breast cancer patients." In particular, the model can give an indication of when metastatic cells are likely to emerge, depending on the size of the primary tumor.

To facilitate the practical application of this Combined Model, a software tool was developed which is also expected to be used in further research — e.g. by enabling simple addition of new parameters for a more accurate prediction of patient survival prognosis. The software is integrated with a database of source data and prediction outcomes. Just two measurements of the primary tumor are required as the minimum source data for the model.

Both the model and the software implementing it can improve the accuracy of predicting breast cancer development and patient outcomes, which, in turn, can help with detecting secondary metastases.

Tested on a relatively small amount of clinical data, the new model showed better performance compared to existing tools. The study's early findings were presented at the "Information Technologies and Systems 2015" conference. However, according to Tyuryumina, this is only the first step. The researchers now plan to test the Combined Model on larger amounts of clinical data, consult with cancer specialists to further improve its applicability, and integrate the software with other tools for clinical data analysis used in oncology.


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Elena Sharova
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