Individualised recovery trajectories of patients with impeded mobility, using distance between probability distributions of learnt graphs

TL;DR

This paper provides a method for the learning of the recovery trajectory of an individual patient, as they undertake exercises as part of their physical therapy towards recovery of their loss of movement ability, following a critical illness.

Abstract

Patients who are undergoing physical rehabilitation, benefit from feedback that follows from reliable assessment of their cumulative performance attained at a given time. In this paper, we provide a method for the learning of the recovery trajectory of an individual patient, as they undertake exercises as part of their physical therapy towards recovery of their loss of movement ability, following a critical illness. The difference between the Movement Recovery Scores (MRSs) attained by a patient, when undertaking a given exercise routine on successive instances, is given by a statistical distance/divergence between the (posterior) probabilities of random graphs that are Bayesianly learnt using time series data on locations of 20 of the patient's joints, recorded on an e-platform as the patient exercises. This allows for the computation of the MRS on every occasion the patient undertakes this exercise, using which, the recovery trajectory is drawn. We learn each graph as a Random Geometric Graph drawn in a probabilistic metric space, and identify the closed-form marginal posterior of any edge of the graph, given the correlation structure of the multivariate time series data on joint locations. On the basis of our recovery learning, we offer recommendations on the optimal exercise routines for patients with given level of mobility impairment.

Figures (10)

• Figure 1: A cartoon of the basic framework of the method. The same patient plays a given exergame on successive instances - resulting in two successively recorded time series data on the locations of the 20 monitored joints in their body. The spatial coordinates of the three-dimensional location vector are periodically recorded on the e-platform MIRA. The Euclidean norm of this vector then gives the location of a joint, at a time point when a recording is done by MIRA. Thus, a time series (on joint location) data is produced as the patient plays this exergame. The inter-column correlation matrix of this data is estimated, and this estimated correlation used to learn a random graph variable - which is a Random Geometric Graph variable (or an RGG variable) in our work. Again, when this patient plays this exergame on the next instance, a new time series data is recorded; its inter-column correlation matrix estimated; and the RGG of this data is then learnt. A statistical distance/divergence measure between the posterior probabilities of the RGGs learnt given the (correlation of the) two datasets, is set proportional to the difference between the Mobility Recovery Scores attained (by this patient playing this exergame) at the next instance, and at the current instance.
• Figure 2: Sessions played by each subject for each of the 18 games.
• Figure 3: Figure displaying plots of performance of patient ID3071 playing the exergame Airplane, against the index of the instance of playing this exergame. On the left, the plot of MRS computed using Hellinger distance is plotted against the index of the instance of playing. In the middle panel, the MRS computed using Kullback–Leibler divergence is plotted, while on the right, points assigned by the MIRA e-platform for playing the game on 6 instances, are plotted against the instance index.
• Figure 4: Realisations of graphs learnt at a chosen $\tau$ of 0.2, given data recorded for patient ID3071 playing exergame Airplane on six successive instances, in the order of the instance of playing.
• Figure 5: Realisations of the graph variable, learnt at $\tau=0.3$, given data recorded for the patient ID3311 playing exergame Follow, on 35 successive instances - of which, results of playing on 6 instances are presented here, in the order of the instance of playing.

Theorems & Definitions (26)

• Definition 3.1
• Remark 3.1
• Definition 3.2
• Remark 3.2
• Definition 3.3
• Definition 3.4
• Definition 3.5
• Definition 3.6
• Remark 3.3
• Definition 3.7