A patient writes, “The Astros will win the World Series next year,” in bold print as Jaquelin Velasquez watches. The clinical nurse in Lymphoma and Myeloma isn’t interested in the accuracy of this prediction. She’s looking for signs of a serious side effect of immunotherapy.

“If a patient’s handwriting deteriorates, it’s a red flag that they’re developing neurotoxicity,” Velasquez says. She explains that the handwriting test is...