SIAM News Blog

Mathematical Model Characterizes Crisis Patterns in Bipolar Disorder

By Lina Sorg

According to the World Health Organization, 46 million people suffer from bipolar disorder (BD): a mental illness that is characterized by alternating manic and depressive episodes. These episodes are separated by periods of normal moods called euthymia. During depressive episodes, patients experience feelings of sadness, irritability, and emptiness, as well as loss of interest and enjoyment in activities. Manic periods involve heightened feelings of euphoria, irritability, and distractibility; higher energy levels; racing thoughts; and increased tendencies towards impulsive reckless behavior. Individuals with BD are also at a higher risk of suicide. “It is estimated that mental disorders are attributable to 14.3 percent of deaths worldwide, or approximately eight million deaths each year,” Yury A. Jimenez Agudelo of CUNEF University in Madrid said.  “How can we contribute from our knowledge to improve the quality of mental health?”

During a minisymposium presentation at the 2023 SIAM Conference on Computational Science and Engineering, which took place this week, in Amsterdam, the Netherlands, Jimenez Agudelo utilized a mathematical model to characterize bipolar disorder (BD) and detect crises patterns in patients. “It’s difficult to know when someone is going through a crisis because the patterns and symptoms differ between people,” she said.

Figure 1. Patient data from smartwatches, biweekly medical reports, and daily self-reports collectively offered 170 unique variables for the model, which then had to be pared down. Figure courtesy of Yury A. Jimenez Agudelo.
First, Jimenez Agudelo gathered patient data from smartwatches, biweekly medical reports, and daily self-reports (see Figure 1). All three sources collectively contributed 170 unique variables, including things like wake-up time, heart rate, and daily step count (from smartwatches); irritability, motivation, impulsiveness, and concentration (from self-reports); and sleep quality, eating habits, health conditions, and personal thoughts (medical reports). After identifying all possible variables, Jimenez Agudelo used machine learning techniques to rank them. The most relevant variables have a high probability and low correlation, so she selected the highest-ranked variables for the study.

Next, Jimenez Agudelo fine-tuned the variables based on input from psychiatrists, who rely on two discrete indicators to evaluate a patient’s emotional state. Young’s indicator measures a manic emotional state, and the Hamilton indicator measures a depressive emotional state. When aggregating the control variables that characterize these states, Jimenez Agudelo modeled emotional state as a real function \(\hat{m}_P(t)\), where \(t\) is a real time variable and \(P\) denotes a specific patient. She hypothesized that the collected variables can change substantially throughout the seven-day week, meaning that people with BD react differently on different days. The resulting model of the weekly estimation of the discretized diagnosis provides an image of a patient’s emotional state that psychiatrists can then interpret.

Analysis of patient data and psychiatrist consensus allowed Jimenez Agudelo to identify when and why emotional changes occur. Using the results from the aforementioned \(\hat{m}_P(t)\) function (and in consensus with the psychiatrist), she defined the confidence bands of each emotional state:

  • \(\hat{m}_P(t) \in (\alpha_P, \beta_P)\) is an indicator of the euthymic state (see Figure 1)
  • \(\hat{m}_P(t) < \alpha_P\) is an indicator of depression
  • \(\hat{m}_P(t) > \beta_P\) is a mania crisis indicator.

Here, \(\alpha_P\) and \(\beta_P\) are specific indicators for each patient.

Figure 2. Confidence bands of the euthymic state for bipolar disorder. Figure courtesy of Yury A. Jimenez Agudelo.

Jimenez Agudelo concluded her talk by admitting that it is difficult to classify emotional states of BD because everyone is affected in unique ways. Her study is therefore not intended to replace medical care, but rather to serve as a support tool that helps psychiatrists better manage BD treatment. “The emotional state is determined by the characteristics of the individual and their interaction with the environment,” she said. “It is necessary to create a customized model that allows one to characterize emotional changes for each individual.” 

Lina Sorg is the managing editor of SIAM News.
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