Machine learning based framework to predict cardiac arrests in a paediatric intensive care unit

Abstract

A cardiac arrest is a life-threatening event, often fatal. Whilst clinicians classify some of the cardiac arrests as potentially predictable, the majority are difficult to identify even in a post-incident analysis. Changes in some patients’ physiology when analysed in detail can however be predictive of acute deterioration leading to cardiac or respiratory arrests. This paper seeks to exploit the causally-related changing patterns in signals such as heart rate, respiration rate, systolic blood pressure and peripheral cutaneous oxygen saturation to evaluate the predictability of cardiac arrests in critically ill paediatric patients in intensive care. In this paper we report the results of a framework constituting feature space embedding and time series forecasting methods to build an automated prediction system. The results were compared with clinical assessment of predictability. A sensitivity of 71% and specificity of 69% was obtained when the maximum value of Anomaly Index (12) in the 50 min (starting one hour and ending 10 min) before the arrest was considered for the case patients and a random 50 min of data was considered for the control set patients. A positive predictive value of 11% and negative predictive value of 98% was obtained with a prevalence of 5% by our method of prediction. While clinicians predicted 4 out of the 69 cardiac arrests (6%), the prediction system predicted 63 (91%) cardiac arrests. Prospective validation of the automated system remains.

Appendix

This appendix summarises the technical framework of signal space modelling using singular spectrum analysis and the stochastic assumptions regarding the residual data falling outside of the signal subspace.

Let $\varvec{y}(t)$ represent the observable time-series for a parameter and $\varvec{x}(t)$ the unobservable, latent variables generated by the biological dynamical system.

$$\begin{aligned} \mathcal {Y} = \{\varvec{y}^{1}(t), \ldots , \varvec{y}^{4}(t)\} \end{aligned}$$

If $\varGamma \in \mathbb {R}^{D}$ represents a low dimensional unobservable manifold which dictates the physiology of an individual, then the dynamical evolution of the state of health $\varvec{x}(t)\in \mathcal{A}$ of the individual patient is assumed to evolve on an attracting subset $\mathcal{A}$ of $\varGamma$.

Due to the physiological processes being autonomic and interacting with each other to maintain homeostasis, the state vector $\varvec{x}(t)$ represents the phase space evolution of the physiology of the patient system. The state space dynamics is described by an unknown mapping: $\varvec{y}(t) = f(\varvec{x}(t))$ plus dynamical noise. Let h represent an observation function which maps the latent space variable $\varvec{x}$ to the observation space: $\varvec{y}^{k}(t) = h^{k}(\varvec{x}^{k}(t)) + \eta ^{k}(t)$ including some observational noise $\eta ^{k}(t)$, for each separate parameter k.

The training process aims to understand the structure of the manifold $\mathcal {A}$ and transitions of $\varvec{x}(t)$ for a stable individual representative of the population in consideration independent of the characteristics that define each patient’s physiology given only knowledge of $\{ y^{k}(t) \}$. The test is to identify when there are significant departures of $\varvec{x}(t)$ away from the reconstructed manifold $\mathcal{A }$. The evolution of $\varvec{y}^{k}(t)$ can be visualised as a density model with multiple clusters occurring where the timescales of the dynamics slow down around different patient “states”.

The multiple processes are inter-dependent and affect each other’s function resulting in different sensors producing multiple time series $y^{k}(t)$. The information derived from the various $y^{k}(t)$ can be used to improve insight. This can be achieved by the data fusion of the four observed time series. Data fusion can be achieved at multiple levels, including:

• The fusion of the data from all the sensors at the sensor level and subsequent prediction of criticality based on a common model.

• Extraction of the underlying dynamics of the data from all the sensors separately. Fusion of the underlying dynamics and final prediction of criticality occurs in model space.

• Prediction of criticality from each time-series and fusion of the individual predictions to obtain a single value of criticality occurring at the decision level.

All the methods listed above require the detection of correlations within and between each time series. In the noise-free case, the trajectory of delay vectors from a given sensor formed from the measurements at time t

$$\begin{aligned} \varvec{y}_{t,n} = (y_{t},y_{t-1}, \dots , y_{t-n+1}) \in \mathbb {R}^{n} \end{aligned}$$

constitute a one-one mapping onto the topology of the latent dynamics $\{\varvec{x}_{t}\}$ provided n is large enough, according to Takens  [21]. The approach of Broomhead and King [2] using the SSA approach decomposes the delay vectors into projected signal and noise subspaces, thus generalising the noise-free case to real data.

The set of N delay vectors $\{ \varvec{y}^{k}_{t}\}_{t=1}^{N}$ form the delay embedding trajectory matrix $\varvec{Y}^{k}(t) \in \mathbb {R}^{N \times n}$ for parameter k. $\varvec{Y}^{k}(t)$ captures all the informative characteristics of the physiology represented by the parameter (including all the various noise processes), ie the delay embedding matrix of size $N\times n$ obtained from a time series of $N+n$ data points should include all relevant information on the sub-set of original manifold, and hence partial information of the patient’s health. The size of the trajectory matrix however is too large and includes redundant information that cannot be exploited. To obtain meaningful representations of the underlying data dynamics dimensionality reduction and feature extraction of this matrix needs to be conducted.

As we assume that the patient’s health is defined by the changes in the latent space the signal components of $\varvec{x}(t)$ describe the dynamics and transitions in the latent space. The factorisation of $\varvec{Y}(t)$ into separate orthogonal signal and noise subspaces using singular value decomposition produces the signal components of $\varvec{x}(t)$ [12]. The dominant d most significant right singular vectors of $\varvec{Y}$, say $\varvec{V}$ form a natural orthogonal topologically equal spanning basis set for the signal subspace and the remaining $n-d$ singular vectors span the noise subspace, $\varvec{Y} = \{\varvec{V} \cup \varvec{U}\}$. In terms of the right singular vectors, any embedding vector for sensor k$\tilde{\varvec{y}}^{k}(t)\in {\mathbb {R}}^{n}$ can then be expressed in terms of its projections onto the fixed row space basis vectors $\varvec{v}^{k}_{j}$, expanded as

$$\begin{aligned} \varvec{y}^{k}(t) = \sum _{j=1}^{d} \alpha ^{k}_{j}(t) \varvec{v}^{k}_{j} \end{aligned}$$

where we have selected the top d vectors defining the signal subspace. The remaining $n-d-1$ singular vectors define the noise subspace. While the projection coefficients remain modellable by the dominant d singular vectors—i.e. they conform to the given AR model, they are deemed to be in the signal subspace. However, as the AR model no longer describes the data, this means that additional singular vectors from the noise subspace need to be recruited to fit the data. This denotes leakage of the dynamic trajectories into the noise subspace [2].

The delay embedding vectors representing the observational space are now transcribed in terms of projections onto a spanning basis set defining an unobservable latent subspace, and so the dynamics have now been transferred into the projection coefficients $\varvec{\alpha }^{k} (t) = \{ \alpha ^{k}_{1}(t), \dots \alpha ^{k}_{d}(t) \}$ with a spread of $\sigma ^{k}_{d}$ since the spanning basis set is fixed. Visualisation of the noise sub-space leads to the identification of the patient state, stable or unstable.

An anomaly is detected when the total energy of the projection of the time series on to the noise subspace exceeds a pre-defined error threshold. We later determine this threshold via an ROC curve on representative data.

In order to predict the short term dynamical evolution of the patient state, we model the dynamics of the projection coefficients as separate linear AR processes, ie for each sensor k and each projection coefficient j we assume models of the form

$$\begin{aligned} \alpha ^{k}_{j} (t) = \sum _{\tau = 1}^{T} \beta _{\tau } \alpha ^{k}_{j} (t-\tau ) + \epsilon ^{k}_{j}(t) \end{aligned}$$

where $\beta _{\tau }$ represents the AR coefficients and the residual error $\epsilon ^{k}_{j} (t)$ is Gaussian-distributed:

$$\begin{aligned} p(\epsilon _{j}^{k}(t)) = \mathcal{N } \left[ \alpha ^{k}_{j} (t) - \sum _{\tau = 1}^{T} \beta _{\tau } \alpha ^{k}_{j} (t-\tau ),\tilde{\sigma }_{j}^{k}\right] \end{aligned}$$

$\tilde{\sigma }_{j}^{k}$ is the standard deviation of the spread of residual errors. Depending on the cause of the deterioration in the patient (cardiac/respiratory/other) the residual errors will differ vastly for each parameter, and each signal subspace projection direction for the patients who have experienced an ALTE. This difference was observed when we attempted to predict the occurrence of a cardiac arrest using only the deviations in the SpO2 values. A desaturation leading to a cardiac arrest presented with deteriorating trends in the SpO2 followed by rapid decline in the heart rate while in other cases the heart rate showed deteriorating trends followed by a sudden drop in the SpO2. A threshold defining the spread of the residual errors for each parameter is therefore estimated from the training data of ‘acceptable’ patients.

It was also observed that the physiology of the patients including those who experienced a cardiac arrest was near stationary for periods of 5 min most of the time. Figure 7 shows examples of the Anomaly Index of eight randomly chosen patients from the case set. The anomaly distance averaged over a minute is plotted for the 24 h leading up to the cardiac arrest. It can be seen that the values of the anomaly distance are constant over short intervals of time and change rapidly during periods of instability. Therefore once the local AR models have been estimated, assuming quasi-stationarity on the time-scale of the instantaneous patient physiology so the AR parameters $\beta _{\tau }$ remain constant, each model may then be extrapolated into the near-future to produce a prediction of each coefficient $\alpha _{j}^{k}(t+T)$T time steps into the future, which may then be recombined to estimate the future expected signal and noise components, if needed.

Fig. 7

Eight examples of Anomaly Index averaged over one minute interval from the set of cardiac arrest patients (case set)

We use the same approach for each one of the k sensors. This implicit independence assumption on the modelling of the projection coefficients gives great simplification for the data fusion process, since we can now assume that the joint distribution of the noise model over all sensors can be decomposed into the product of individual distributions. Although a more complex distribution may be considered if necessary, in this paper, each individual distribution is assumed to be a simple Gaussian (with different diagonal variance values) which therefore also decomposes into the product of individual one-dimensional Gaussian, ie the joint distribution is:

$$\begin{aligned} p(\varvec{\epsilon }^{1},\varvec{\epsilon }^{2},\varvec{\epsilon }^{3},\varvec{\epsilon }^{4}) = \prod _{k=1}^{4} p(\varvec{\epsilon }^{k}) \end{aligned}$$

and each single distribution is a simple product of 1-d Gaussians: $p(\varvec{\epsilon }^{k})=\prod _{j=1}^{d} \mathcal{N}[\tilde{\mu }^{k},\tilde{\sigma }^{k}_{j}]$.

This is an elegantly simple approach providing a preliminary baseline model to address the data fusion issue. However the above equation does suffer from the Veto Effect, i.e. if any one of the sensor models breaks (due to incorrectly measured data, sensors becoming detached, or major departures from expected norms of behaviour), then due to the exponential nature of the distributions, any (spurious) very small individual value would destroy the noise model across all sensors due to the multiplicative nature, and not just the faulty one. Therefore we operate with a distance measure in negative log probability space of the noise model, so that one rogue sensor will not obliterate the responses of the other sensors.

$$\begin{aligned} \mathcal {I}&= -log(p(\varvec{\epsilon }))\\&= -\sum _{k}\sum _{j} \log \mathcal{N } \left[ \alpha ^{k}_{j} (t) - \sum _{\tau = 1}^{T} \beta _{\tau } \alpha ^{k}_{j} (t-\tau ),\tilde{\sigma }_{j}^{k}\right] . \end{aligned}$$

Note that abnormality for a patient is tracked under the assumption of the above Gaussian independence model: if a new patient’s dynamics is significantly deviant from the signal space residual errors estimated from the training set of ‘not-abnormal’ patients, then the ‘distance’ between the new patient’s residual probability distribution and the distribution of not-abnormal patients’ residuals will be large.

In a metric space, because of the multiple gaussian assumption this approach becomes a simple weighted euclidean distance measure which can be checked against an a priori determined threshold derived from measuring typical index values $\mathcal {I}$ of training data. We use the ROC curve analysis based on training data of patients.

In effect $\mathcal {I}$ is measuring the distance between the pdfs of the model residual of the new patient and the pdfs of the model residuals of the training patients.The threshold of alarm is determined to balance sensitivity and specificity.

So our initial multivariate stochastic dynamical system is verified using a simple single scalar measure which takes into account all the sensors, and is patient-specific due to the projection of the patient‘s embedding vector onto the pre-determined singular vectors spanning the signal subspace.

Therefore the automated prediction system, or two-class classifier, has four input sensor time series and one output parameter (the Anomaly Index $\mathcal {I}$). This results in a data fusion system where multiple parameters are recorded from different sources and the fusion occurs at the probabilistic modelling level prior to decision-making.