Mathematical Models Reveal How Shrinking Breast Tumors Move During Therapy

Triple-negative breast cancer is one of the most aggressive and difficult-to-treat forms of the disease, lacking the three molecular receptors—estrogen receptor, progesterone receptor, and human epidermal growth factor receptor 2—that drive most targeted therapies. For these patients, neoadjuvant therapy, meaning chemotherapy delivered before surgery, is the standard of care because it can shrink tumors enough to make surgery more successful and can attack microscopic metastases before they take hold. Yet only about half of patients achieve a pathological complete response, defined as the complete absence of cancer in the surgically resected tissue and sampled lymph nodes. Adding immunotherapy can push that figure to roughly two-thirds, but at the cost of increased toxicity. Because patients who achieve a complete response are far less likely to relapse or die, researchers have invested heavily in mathematical models that can forecast, early in the treatment course, which patients are responding and which are not.

A team led by Casey E. Stowers and Thomas E. Yankeelov at the University of Texas at Austin and MD Anderson Cancer Center has now systematically tested how best to represent tumor cell movement in these predictive models. Their work, published in the Annals of Biomedical Engineering, draws on data from 141 patients with locally advanced triple-negative breast cancer enrolled in the ARTEMIS clinical trial at MD Anderson. Each patient received four cycles of Adriamycin and Cytoxan, with magnetic resonance imaging scans taken before treatment, after two cycles, and after four cycles. Patients whose tumors shrank by at least 70 percent on ultrasound went on to receive twelve cycles of Taxol, while others received experimental therapies according to trial guidelines. After the second course of therapy, all patients underwent surgery, at which point their pathological complete response status was determined.

The modeling framework rests on partial differential equations that track the number of tumor cells in every three-dimensional voxel of the breast over time. The number of cells in each voxel is estimated from diffusion-weighted MRI using an inverse relationship between the apparent diffusion coefficient of water in tissue and tumor cell density. The team compared four increasingly sophisticated models. The simplest, a reaction-only model, describes logistic proliferation of cells up to a carrying capacity and drug-induced cell death driven by a concentration map derived from dynamic contrast-enhanced MRI. The second adds a diffusion term, capturing cells migrating from regions of high density to low density. The third mechanically couples that diffusion term, damping the diffusion coefficient exponentially via the von Mises stress computed from a linear elastic mechanical model of the breast, reflecting the experimental observation that solid stress inhibits tumor cell invasion. The fourth and most complex model adds an advection term, in which cells are transported by a velocity field calculated as the time derivative of the mechanical displacement field—an approach originally developed for modeling glioma growth in the brain.

The rationale for exploring advection is subtle but important. In triple-negative breast cancer, tumors typically shrink during therapy rather than expand, and a diffusion term alone cannot physically represent the inward movement of cells that accompanies shrinkage. Advection, by contrast, can carry cells toward the tumor center as the mass effect diminishes with treatment-induced cell death. In the two example patients highlighted in the study—one who achieved a complete response and one who did not—the velocity maps revealed striking behavior. On days when the total tumor cell count was rising, the modeled velocities pointed outward, in the direction of growth. On days shortly after drug delivery, when cell counts were falling, the velocities reversed and pointed toward the tumor center, pulling cells inward and producing smaller, denser tumor maps that more closely matched the imaging data.

That mechanistic insight translated into measurable differences in model performance. For the patient who achieved a complete response, the advection model produced a concordance correlation coefficient of 0.44 at the calibration timepoint compared with 0.31 for the three simpler models, and a Dice score of 0.65 versus 0.58, indicating substantially better spatial overlap with the measured tumor. The advection model also predicted far fewer disconnected islands of tumor cells—two subvolumes instead of seven at the second imaging visit—producing tumor geometries that looked more biologically realistic. Similar patterns held for the non-responding patient, whose residual tumor was better captured by the advection model than by any of the alternatives.

Across the full cohort of 141 patients, however, the picture became more nuanced. At the calibration timepoint, the reaction–mechanically coupled diffusion–advection model was significantly more accurate than all three simpler models on every global and local metric examined, including the absolute difference in percent change of total tumor cell count and volume, voxel-wise mean square error, and concordance correlation coefficient. But at the prediction timepoint—the third MRI scan, which the models had never seen—no significant differences emerged among the four models. The median absolute difference between predicted and measured percent change in tumor volume across the cohort was 9.1 percent for both the reaction-only and reaction–diffusion models, and the advection model offered only a small, statistically insignificant improvement of roughly 4 to 5 percent in tumor volume error for the non-responding patients, who are precisely the group clinicians most need to forecast accurately.

The receiver operating characteristic analysis reinforced this equivalence. When the researchers used predicted total tumor cell count or tumor volume at the third scan to classify patients by their eventual pathological complete response status among the 102 patients who received Taxol as their second therapy, no significant differences appeared between the areas under the ROC curves for any of the four models—or, notably, between the model predictions and the measured imaging data themselves. In other words, even the simplest model predicted complete response status as well as the actual measurements did, and all four models achieved comparable accuracy in this clinically critical task.

Computational cost, however, separated the models dramatically. The reaction-only model calibrated in a median of about 0.31 minutes per patient, while the reaction–diffusion model took roughly 118 minutes, and the mechanically coupled diffusion and advection models required around 222 to 252 minutes each, because both demand a full mechanical equilibrium solve at every optimization iteration. Each calibration ran on a 24-core server, and the two calibrated parameters—global tumor cell proliferation rate and global drug efficacy—were identical across all four models, fitted using the Levenberg–Marquardt algorithm against the first two imaging visits. The authors suggest that reduced-order modeling techniques and higher-order numerical schemes could eventually narrow this efficiency gap for the most accurate model.

One of the study’s most consequential findings concerns the diffusion term itself. Adding diffusion, whether mechanically coupled or not, provided no measurable improvement in calibration or prediction accuracy compared with the reaction-only model. The researchers attribute this to the insensitivity of the diffusion coefficient in this disease setting, where tumors mostly shrink rather than expand; prior sensitivity analyses found the parameter’s influence fell below the threshold for importance even when varied over five orders of magnitude. Yet the authors caution against simply deleting the term: a reaction-only model cannot represent tumor invasion or any expansion in tumor volume, which would produce dangerously poor predictions for a patient whose disease progresses during therapy. Their recommendation is that, among models without advection, the reaction–diffusion model offers the best balance, capturing invasion while calibrating far faster than the mechanically coupled variant.

The implications extend well beyond breast cancer. Because the advection term was adapted from brain tumor modeling, and because the conclusions here apply to tumors that largely shrink under treatment, the framework could inform computational oncology across many solid tumor types. The authors outline several avenues for improvement: incorporating the Allee effect into the proliferation term to better capture the death of small tumors, subdividing tumor cells into sensitive and resistant populations, coupling hydrostatic stress to proliferation or drug resistance, and replacing the linear elastic tissue assumption with nonlinear hyperelastic or anisotropic material models. They also note that future validation should draw on multi-site datasets such as the I-SPY 2 trial, which spans multiple breast cancer subtypes. For now, the study delivers a clear practical message: the reaction term carries most of the predictive power for shrinking tumors, advection is the only term that can physically capture inward cell movement during treatment-induced shrinkage, and choosing among these formulations requires weighing calibration speed against the risk of missing a growing tumor.

Subject of Research: Partial differential equation modeling of tumor cell movement in triple-negative breast cancer receiving neoadjuvant therapy

Article Title: Characterizing Tumor Cell Movement in Partial Differential Equation Models of Triple-Negative Breast Cancer Receiving Neoadjuvant Therapy

Article References: Stowers, C. E., Wu, C., Lorenzo, G., Hormuth, D. A., II, Yam, C., Ma, J., Rauch, G. M., & Yankeelov, T. E. (2026). Characterizing Tumor Cell Movement in Partial Differential Equation Models of Triple-Negative Breast Cancer Receiving Neoadjuvant Therapy. Annals of Biomedical Engineering. https://doi.org/10.1007/s10439-026-04348-7

Image Credits: AI Generated

DOI: 10.1007/s10439-026-04348-7

Keywords: triple-negative breast cancer, neoadjuvant therapy, reaction-diffusion-advection, mathematical oncology, MRI, tumor forecasting, model calibration, advection, mechanical coupling, pathological complete response, partial differential equations, computational biology