Figure 1: Calipers can be used to measure tumor size in animal experiments, but they can only measure tumors above a certain size. Leaving out data for tumors that are too small to measure can skew the results of mathematical models. © LINDA BARTLETT, NATIONAL CANCER INSTITUTE/SCIENCE LIBRARY
Mathematical models for predicting how cancer tumors in mice grow over time can give distorted results if unmeasurable data is ignored, a team that includes two RIKEN researchers has shown1. This finding has important implications when applying mathematical models to medicine.
A self-confessed data geek, Catherine Beauchemin of the RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS) specializes in applying the power of math to everything from viruses to supernovae. "I love trying to figure out what data are telling us," she says.
Seeking new insights into cancer mechanisms, Beauchemin's team used mathematical models to analyze previously published data of tumor-size measurements over time for ten mice.
Initially, only a couple of mice had tumors large enough to be detected and measured with calipers. At later times, mice whose tumors had reached a certain size were euthanized to avoid discomfort, so only mice with the smallest tumors were still measurable.
The team applied five mechanistic models of increasing complexity to analyze these data. Their initial analysis included only the measured tumor sizes, as is normally done in the field.
The team repeated their analysis, this time including how many tumors could not be measured at each time point. They did this by considering the likelihood of having that many unmeasurable tumors at a particular time. This estimate is based on what the mathematical model predicts would be a reasonable range of tumor sizes at that time, along with the smallest and largest tumor sizes that can be measured experimentally.
When the team included only the measured data, all the mechanistic models overestimated the tumor size in the early stages of tumor development and underestimated it at later stages.
"If we only tell the mathematical model about the measurements we have, its results will be skewed upwards by the largest tumor sizes at early times, and downwards by the smallest tumor sizes at late times," explains Beauchemin. "This could lead to misleading conclusions when evaluating treatment effectiveness."
Her advice to researchers is to include all the information that they glean. "Don't leave any information on the cutting table," she says. "When a quantity is unmeasurable, that tells you it is too small or big to be measured. Ignoring this can change the result quite significantly."
The team provided an easy-to-use mathematical framework for including unmeasurable quantities to more accurately estimate parameters using mechanistic models.
Beauchemin is interested in applying these principles to other areas such as modeling infectious diseases. She is now collaborating with experimentalists to design new measurement protocols that can provide richer data to better inform mathematical models.
Catherine Beauchemin and co-workers have shown that results from mechanistic models for tumor growth vary significantly when data that cannot be measured is omitted. © 2025 RIKEN
