Gliomas are intra-axial tumors that manifest in the brain and spinal cord. They originate in glial cells — gluey supportive cells that surround nerve cells and aid in their function. When glial cells become cancerous, they produce tumors. While gliomas only appear in roughly five of every 100,000 patients, they make up 33 percent of all brain tumor cases. They are ranked on a grading system of I to IV, wherein the grade’s numerical value increases with the tumor’s severity. Grade I consists of only a soft, relatively benign tumor that a surgeon can typically remove in its entirety. Grades II to IV indicate the presence of both the soft tumor as well as isolated tumor cells in the surrounding tissue. In some cases, these cells have been found up to two centimeters away from the limit of the magnetic resonance imaging (MRI) abnormality. When a surgeon attempts to remove a tumor of grades II to IV, he/she nearly always leaves behind numerous tumor cells that then proliferate in the nearby tissue. This explains why systematic recurrence transpires even after treatment of a glioma. For this reason, gliomas are incurable. However, doctors routinely treat them with surgery, chemotherapy, and radiotherapy (RT).
Magnetic resonance imaging scans of various grades of gliomas.
Migration of the aforementioned cancerous cells plays a key role in glioma spread. During a scientific session at the 9th International Congress on Industrial and Applied Mathematics (ICIAM 2019), currently taking place in Valencia, Spain, Mathilde Badoual of Paris Diderot University presented a model of glioma evolution, including tumor response to RT in the instance of patient pregnancy. Badoual focused on diffuse, slow-growing, low-grade gliomas (of grade II). These gliomas are common in younger patients, including young adults. Though they are very invasive, patients can survive more than 10 years after initial diagnosis. While in the grade II stage, the mean tumor diameter and tumor volume grow at a fairly linear rate both without and then with symptoms. But as they transition from grade II to grade III, gliomas begin to grow more rapidly.
Badoual presented a graph demonstrating a decrease in tumor size with RT, and called attention to an interesting phenomenon. “There’s a large delay between the end of RT and regrowth of the tumor,” Badoual said. “We wanted to know why and be able to model this delay.” A strong tumor model requires one to virtually venture inside a tumor to observe its nuances. Badoual pointed out that the tissue towards the center of a tumor on an MRI becomes increasingly whiter. This coloring indicates the presence of liquid between the tissue fibers. This accumulation of fluid in the tissue is called edema. Badoual quantified edema for different patients via a model that includes an equation for cell density evolution and an equation for the edema fraction evolution.
Because she was specifically interested in studying the effect of RT, Badoual utilized a model that tracks cell density just before RT, cell density just after RT, cell density killed during RT, and overall cell proliferation. RT kills a large amount of tumor cells, which decreases the glioma’s radius. There are thus not enough tumor cells present to create edema, so that percentage decreases as well; this clearance of edema causes the delay in tumor regrowth at the end of RT.
Next, Badoual worked to incorporate new data into her model, specifically for patients who wish to become pregnant while diagnosed with a low-grade glioma. “More and more young women harboring diffuse low-grade gliomas now envision a pregnancy,” she said. A glioma diagnosis in a mother-to-be does not impact fetus development. Even so, Badoual was interested in the possible interactions between diffuse gliomas and pregnancy.
She began by examining the dynamics in patients who did not undergo RT before pregnancy. “During pregnancy, the mean tumor diameter increases faster than before,” Badoual said. “And the radial growth velocity comes back to normal at the end of the pregnancy.” However, the tumors in patients who experienced RT before becoming pregnant were shrinking prior to pregnancy. In this case, the slope of the decreasing tumor size can become less negative—or even positive—during pregnancy. After pregnancy, the diameter growth velocity can be either positive or negative.
These variables caused Badoual to further investigate the specifics of glioma evolution during pregnancy. She experimented with increasing the production coefficient of edema as well as the proliferation of tumor cells. She also realized a fatal flaw of her model: once the tumor cells are killed, the slope of the tumor’s growth cannot be negative again. This is not realistic, as the slope should decrease at some point after pregnancy. “We learned that this didn’t model RT in the right way,” she said, adding that the model was too simple to effectively capture the complex dynamics.
As a result, Badoual started again and established a new model with a two-cell population of damaged and non-damaged cells. This revised model was more effective in that the tumor experienced a delay in regrowth. It was also capable of effectively modeling RT treatment alone, the absence of RT prior to pregnancy, and RT treatment before pregnancy. Because Badoual has increasingly more patients, she’s currently working to mathematically fit her new patients into the model. She also hopes to further investigate damaged cells during pregnancy.
||Lina Sorg is the associate editor of SIAM News.