Serum uPAR as Biomarker in Breast Cancer Recurrence: A Mathematical Model

Abstract

There are currently over 2.5 million breast cancer survivors in the United States and, according to the American Cancer Society, 10 to 20 percent of these women will develop recurrent breast cancer. Early detection of recurrence can avoid unnecessary radical treatment. However, self-examination or mammography screening may not discover a recurring cancer if the number of surviving cancer cells is small, while biopsy is too invasive and cannot be frequently repeated. It is therefore important to identify non-invasive biomarkers that can detect early recurrence. The present paper develops a mathematical model of cancer recurrence. The model, based on a system of partial differential equations, focuses on tissue biomarkers that include the plasminogen system. Among them, only uPAR is known to have significant correlation to its concentration in serum and could therefore be a good candidate for serum biomarker. The model includes uPAR and other associated cytokines and cells. It is assumed that the residual cancer cells that survived primary cancer therapy are concentrated in the same location within a region with a very small diameter. Model simulations establish a quantitative relation between the diameter of the growing cancer and the total uPAR mass in the cancer. This relation is used to identify uPAR as a potential serum biomarker for breast cancer recurrence.

Fig 1. Schematic network of breast cancer with uPA, PAI-1 and uPAR: Arrows means activation; block arrow means inhibition.

Fig 2. Average concentration of cytokines, average density of cells, and tumor radius R(t) for the first 600 days with R(0) = 10⁻² cm = 100 μm.

All the parameters are as in Tables 2 and 3.

Fig 3. The total mass of cells and cytokines for the first 600 days with R(0) = 10⁻² cm = 100 μm.

All the parameters are as in Tables 2 and 3.

Fig 4. Color map for R(t).

R₀ ranges from 0.01 to 0.05 cm and t ranges from t = 0 to t = 1000 days. Color represents the size of the radius of the cancer. All the parameters are as in Tables 2 and 3.

Fig 5. Color map for the total mass of uPAR.

R₀ ranges from 0.01 to 0.05 cm and t ranges from t = 0 to t = 1000 days. Color represents the total mass of uPAR. All the parameters are as in Tables 2 and 3.

Fig 6. Color map for the total mass of uPAR(t) v.s. R(t).

For any time t, 0 < t < 1000 days, measurement uPAR in gm/cm³ (on the horizontal axis) determines the size of the radius of the cancer in cm, using the column color. All the parameters are as in Tables 2 and 3.

Fig 7. The sensitivity analysis for the cytokine production rates.

The figure shows the partial rank correlation (PRCC) between the cytokine production rate and the radius of tumor. All the parameters are as in Tables 2 and 3.