Optimal Control for Remote Patient Monitoring with Multidimensional Health States

TL;DR

This work extends RPM modeling to multidimensional health states and derives a DP-based framework to optimize monitoring intensity. The key finding is that the optimal control exhibits a threshold structure: the patient should switch to intensive monitoring when the multidimensional health indicators cross a switching curve or hypersurface, with the curve shaped by the geometry of the critical health set. The approach integrates cost considerations, transition dynamics, and asymptotic analyses to reveal how parameter choices and health-set geometry influence clinical decision-making and resource planning. The results offer a principled, interpretable guide for tailoring RPM policies to complex patient health profiles and varying medical conditions, and suggest directions for data-driven estimation and scalable approximations in practice.

Abstract

Selecting the right monitoring level in Remote Patient Monitoring (RPM) systems for e-healthcare is crucial for balancing patient outcomes, various resources, and patient's quality of life. A prior work has used one-dimensional health representations, but patient health is inherently multidimensional and typically consists of many measurable physiological factors. In this paper, we introduce a multidimensional health state model within the RPM framework and use dynamic programming to study optimal monitoring strategies. Our analysis reveals that the optimal control is characterized by switching curves (for two-dimensional health states) or switching hyper-surfaces (in general): patients switch to intensive monitoring when health measurements cross a specific multidimensional surface. We further study how the optimal switching curve varies for different medical conditions and model parameters. This finding of the optimal control structure provides actionable insights for clinicians and aids in resource planning. The tunable modeling framework enhances the applicability and effectiveness of RPM services across various medical conditions.

Figures (3)

• Figure 1: The evolution for a 2-dimensional model with $H=5$ and origin as the critical health state. The transitions are marked for states $(o,2,0), (o,5,2)$, and $(o,2,3)$ under action $o$. The transaction probabilities have been labeled on the arrows, and the cost $C_o$ is incurred in each case.
• Figure 2: Optimal controls for a two-dimensional model with $H=6$ for critical sets given by (a) $h^{(x)}=0$ or $h^{(y)}=0$, (b) $h^{(x)}+h^{(y)}\leq 2$, (c) $\max\{h^{(x)},h^{(y)}\}\leq 2$, and (d) $h^{(x)}=0$ or $h^{(y)}=0$ or $h^{(x)}+h^{(y)}\leq 2$. Larger black dots represent critical health states, and dotted black line represents the boundary of the critical set. Intensive monitoring is optimal for states marked as red, and ordinary monitoring is optimal for states marked as blue. The dotted red line represents the switching curve$f(\boldsymbol{h})=0$. For each of the plots: $\gamma=0.9, C_i=1, C_o=0, C_c=35, \lambda_{o,x}=\lambda_{o,y}=0.5-\mu_{o,x}=0.5-\mu_{o,y}=0.075$ and $\lambda_{i,x}=\lambda_{i,y}=0.5-\mu_{i,x}=0.5-\mu_{i,y}=0.2$.
• Figure 3: Optimal control for asymmetric two-dimensional models for critical sets given by (a) $h^{(x)}=0$ or $h^{(y)}=0$, and (b) $2h^{(x)}+3h^{(y)}\leq 6$. For both plots: $\gamma=0.9, C_i=1, C_o=0, C_c=35$. For plot (a): $\lambda_{o,x}=\lambda_{o,y}=0.5-\mu_{o,x}=0.5-\mu_{o,y}=0.1$ and $\lambda_{i,x}=0.5-\mu_{i,x}=0.3$ and $\lambda_{i,y}=0.5-\mu_{i,y}=0.25$. For plot (b): $\lambda_{o,x}=\lambda_{o,y}=0.5-\mu_{o,x}=0.5-\mu_{o,y}=0.1$ and $\lambda_{i,x}=\lambda_{i,y}=0.5-\mu_{i,x}=0.5-\mu_{i,y}=0.2$

Theorems & Definitions (3)

• Theorem 1
• proof
• proof