By Dae Wook Kim
Humans have an endogenous timer, the circadian (~24-hour) clock that regulates behavioral and physiological processes like sleep and hormone secretion. This daily timekeeping system is organized in a hierarchical manner; the central pacemaker is located deep within the brain (in the hypothalamic suprachiasmatic nucleus) and regulates downstream cellular clocks in all of the peripheral tissues. The peripheral clocks (e.g., the heart circadian clock) control key cellular processes—such as cell proliferation, cell cycle progression, and apoptosis—that consequently display circadian rhythms. As a result, the efficacy and toxicity of a diverse array of drugs—including more than 50 anticancer drugs that manipulate cellular dynamics—largely depend on dosing time (see Figure 1) [1]. In fact, over 70 percent of clinical trials that examined the impact of dosing time found significant circadian variations in efficacy and toxicity [7]. A recent study notably revealed that timed immunotherapy for melanoma—the most serious type of skin cancer—can considerably improve survival outcomes, thus indicating the importance of treatment timing [6].
Figure 1. The circadian timing system controls key pharmacokinetic/dynamic factors, resulting in circadian variations in drug efficacy and toxicity. Figure courtesy of [5].
Despite these potential benefits, current standards of clinical practice largely ignore appropriate dose timing because accurately identifying the master clock’s state in real-world settings is challenging. The gold standard for calculating the internal circadian phase involves continuously measuring the concentration of melatonin—the sleep regulating hormone—under dim-light conditions and estimating its first touching time to a certain threshold concentration [5]. But while identification of the internal phase with dim light melatonin onset (DLMO) is well validated, this approach is not suitable for real-world circadian medicine. For example, one cannot endlessly measure DLMO over a long period of time because it requires labor-intensive continuous collection of blood or saliva in laboratory settings (not in free-living conditions) roughly every 30 minutes between evening and bedtime.
We can circumvent this issue by using wearable devices like the Apple Watch, which millions of people already own. These devices can continuously collect real-time information on physiological signals such as heart rate, movement, and temperature over a lengthy time period. Since all of the wearable devices measure the downstream circadian output from the master clock, we need subsequent mathematical analysis to infer the master clock phase from the indirect measurements.
A recent study explored the possibility of predicting the master clock phase with a Van der Pol limit cycle differential equation model of the human circadian clock that uses activity measurements from wearables as inputs [3]. Another new approach extracts circadian information from wearable heart rate measurements via a Bayesian inference framework with harmonic-regression-plus-autoregressive-noise models [2]. This technique filters out the measurement noise and the effects of external confounding factors like stress, activity, and hormones on heart rate, ultimately yielding an estimate of the peripheral heart clock’s phase. Yet despite this meaningful progress, limitations still exist. Specifically, researchers do not fully utilize the data from wearables when estimating the master clock phase. When inferring the master clock phase, the existing techniques that use mathematical models [3] do not exploit the phase of the peripheral heart clock that comes from wearable heart rate measurements, even if that phase is a key circadian output. Most importantly, there is currently no systematic methodology for the quantification of the master clock phase estimate’s uncertainty.
Figure 2. The level set Kalman filter framework accurately estimates the evolution of the internal circadian phase’s posterior distributions. 2a. Graphical illustration of the algorithm. 2b. A virtual scenario that captures a typical human lifestyle. 2c. Evolution of the internal circadian phase’s posterior distributions over 20 days. 2d. The posterior distribution on the last day. The triangle and dashed line denote the true internal phase. Figure courtesy of [4].
To address this dilemma, my colleagues and I first formulated a filtering problem with nonlinear continuous-discrete state-space models of the following form:
\[d\textbf{x}_t=\textbf{v}(\textrm{x}_t, t)dt+ \sqrt{\textbf{K}}dW_t\tag1\]
\[\textbf{z}_k=h(t_k)+ \epsilon_k,\tag2\]
where \(\textbf{x}_t\) denotes the \(n\)-dimensional state of the master clock at time \(t\); a nonlinear drift function \(\textbf{v}\) defines the dynamics of this clock [4]. Here, \(W_t\) represents the \(n\)-dimensional Brownian motion, \(\textbf{K}\) is the diffusion matrix that describes the process noise, \(\textbf{z}_k\) and \(h\) denote the measurement and the measurement function, and a Gaussian white noise process \(\epsilon_k\) describes the measurement noise. More details about the drift function \(\textbf{v}\) (which describes the dynamics of the master clock), the diffusion matrix \(\textbf{K}\), and the measurement function \(h\) (which describes the relationship between the master clock phase and peripheral phase) are available in the literature [4].
We efficiently solved this filtering problem by using a new extension of Kalman filtering called the level set Kalman filter, which computes a velocity of the Gaussian level set by reformulating the underlying Fokker-Planck equation as a system of ordinary differential equations [8]. In fact, this level set Kalman filter framework facilitates the accurate and precise estimation of the evolution of the master clock phase’s posterior distributions (see Figure 2) [4]. Importantly, utilizing this method allows us to systematically investigate the relationship between the phase estimate and the process noise in the circadian clock.
Figure 3. Analysis of wearable data via systems modeling and machine learning can enable personalized circadian medicine. Courtesy of [5].
The level set Kalman filter approach integrates information from multiple physiological measurements (collected by wearable devices) and efficiently processes the resulting data (see Figure 3). Doing so extracts more useful and reliable physiological proxies, even from the noisy wearable data in real-world settings. Combining this outcome with systems modeling and machine learning techniques can enable personalized chronotherapy, a therapeutic approach that considers patients’ individual internal circadian times. Our work clearly demonstrates the irreplaceable role of mathematical analysis to encourage the use of circadian medicine in free-living conditions.
Dae Wook Kim presented this research during a minisymposium session at the 2022 SIAM Conference on the Life Sciences, which took place concurrently with the 2022 SIAM Annual Meeting in Pittsburgh, Pa., this July.
Acknowledgments: This article describes the work of Dae Wook Kim, Minki P. Lee, and Daniel B. Forger. The corresponding paper is available online [4].
References
[1] Ballesta, A., Innominato, P.F., Dallmann, R., Rand, D.A., & Lévi, F.A. (2017). Systems chronotherapeutics. Pharmacol. Rev., 69(2), 161-199.
[2] Bowman, C., Huang, Y., Walch, O.J., Fang, Y., Frank, E., Tyler, J., … Forger, D.B. (2021). A method for characterizing daily physiology from widely used wearables. Cell Rep. Methods, 1(4), 100058.
[3] Huang, Y., Mayer, C., Cheng, P., Siddula, A., Burgess, H.J., Drake, C., … Forger, D.B. (2021). Predicting circadian phase across populations: A comparison of mathematical models and wearable devices. Sleep, 44(10), zsab126.
[4] Kim, D.W., Lee, M.P., & Forger, D.B. (2022). A level set Kalman filter approach to estimate the circadian phase and its uncertainty from wearable data. Preprint, arXiv:2207.09406.
[5] Kim, D.W., Zavala, E., & Kim, J.K. (2020). Wearable technology and systems modeling for personalized chronotherapy. Curr. Opin. Syst. Biol., 21, 9-15.
[6] Qian, D.C., Kleber, T., Brammer, B., Xu, K.M., Switchenko, J.M., Janopaul-Naylor, J.R., … Buchwald, Z.S. (2021). Effect of immunotherapy time-of-day infusion on overall survival among patients with advanced melanoma in the USA (MEMOIR): A propensity score-matched analysis of a single-centre, longitudinal study. Lancet Oncol., 22(12), 1777-1786.
[7] Ruben, M.D., Smith, D.F., FitzGerald, G.A., & Hogenesch, J.B. (2019). Dosing time matters. Science, 365(6453), 547-549.
[8] Wang, N., & Forger, D.B. (2021). The level set Kalman filter for state estimation of continuous-discrete systems. IEEE Trans. Signal Process., 70, 631-642.
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Dae Wook Kim is a James Van Loo Postdoctoral Fellow and an assistant professor in the Department of Mathematics at the University of Michigan. His research interests lie in the analysis of biological dynamics, development of mathematical frameworks to analyze biological data, and applications to real-world clinical data to ultimately enable personalized circadian medicine. |