Predicting survival in critical patients by use of body temperature regularity measurement based on approximate entropy

Abstract

Body temperature is a classical diagnostic tool for a number of diseases. However, it is usually employed as a plain binary classification function (febrile or not febrile), and therefore its diagnostic power has not been fully developed. In this paper, we describe how body temperature regularity can be used for diagnosis. Our proposed methodology is based on obtaining accurate long-term temperature recordings at high sampling frequencies and analyzing the temperature signal using a regularity metric (approximate entropy). In this study, we assessed our methodology using temperature registers acquired from patients with multiple organ failure admitted to an intensive care unit. Our results indicate there is a correlation between the patient’s condition and the regularity of the body temperature. This finding enabled us to design a classifier for two outcomes (survival or death) and test it on a dataset including 36 subjects. The classifier achieved an accuracy of 72%.

Appendix

1.1 Regularity estimation

Biomedical signal regularity measurement has proven to be an effective way to obtain new information from these signals that correlates well with clinical condition [28]. One of the most used mathematical tools to compute regularity in signals is ApEn [1]. This is a measure aimed at obtaining the regularity of a data series because it reflects the probability that patterns within the series are not followed by similar ones. Therefore, a data series containing many repetitive patters will have a low ApEn, whereas a less predictable one will have a higher ApEn [11].

The algorithm for computing ApEn is as follows. Given an input data series x[n] of length N, an epoch of valid temperature recordings, two input parameters must be chosen in order to compute its ApEn, the length of the pattern m, and the distance threshold r.

A data series pattern of length m is given by:

$$x_{m}(i)=\left \{x[i], x[i+1], \ldots, x[i+m-1]\right\}$$

that is, m refers to the number of consecutive temperature measures assumed to form a possible repetitive pattern within x[n], and starting at sample x[i].

The distance between two generic patterns x _m (i) and x _m (j) is given by:

$$d\left(x_{m}\left(i\right), x_{m}(j)\right)=\max\left(\left|x\left[i+k\right]- x\left[j+k\right]\right|\right), \quad 1 \leq k \leq m.$$

(A.1)

The distance threshold r determines if x _m (i) and x _m (j) can be considered similar when d(x _m (i), x _m (j)) ≤  r. Given the set of all possible patterns of length m, (x _m (1), x _m (2), ... , x _m (N−m + 1)), we define:

$$C_{r,m}(i)= \frac{k_{i,m}(r)} {N-m+1}$$

(A.2)

where k _r,m(i) is the number of patterns x _m (j) that are similar to x _m (i) according to the distance threshold r. Hence, C _r,m(i) is the fraction of patterns of length m starting at j, 1 ≤  j ≤  N−m + 1 whose distance to pattern starting at i, is below the threshold r, that is, they are considered to be similar to pattern x _m (i). This fraction is computed for each pattern, and then another quantity can be defined as:

$$\phi^{m}(r)=\frac{1}{N-m+1} \sum^{N-m+1}_{i=1}\log C_{r,m}(i).$$

Finally, the computation of the ApEn of a temperature epoch x[n], ApEn(m,r) is given by:

$$\hbox{ApEn}(m,r)=\left[\phi^{m}(r)-\phi^{m+1}(r)\right]$$

(A.3)

Namely, ApEn quantifies the relative prevalence of repetitive patterns of length m compared with patterns of length m + 1 [11]. ApEn is computed for all the epochs in the temperature register, and then the mean μ _y  = mean(ApEn(x _i [n])), ∀x _i [n] ∈Y _p , is obtained.

1.2 Hypothesis validation

ApEn was calculated for every temperature register in classes A and B as described in previous section, and the mean for both classes was obtained, $\mu_{A}=\hbox{mean}(\mu_{y_{i}}), \forall y_{i} \in A,$ and $\mu_{B}=\hbox{mean}(\mu_{y_{j}}), \forall y_{j} \in B.$ The objective of the hypothesis validation was aimed at assessing if μ _A and μ _B differences were statistically significant. There are several statistic tests for this validation but in order to consider all the possible scenarios, we chose two complementary tests [10]. The first one, the classical parametric Student’s t test [5], based on the assumptions of data normality and homoscedasticity, difficult to make when not many input instances are available, and the second one, the Mann–Whitney test [31], a non-parametric method that does not require the normality assumption.

For the Student’s t-test, the null hypothesis H ₀ is that the two ApEn means for classes A and B are considered to be equal, and then the objective is to decide whether to accept or reject such hypothesis. In order to be able to carry out this test, data normality, homoscedasticity, and independence must apply. Normality can be assured using the Shapiro–Wilks test [26, 33]. Homoscedasticity can be confirmed by means of the Bartlett test [15], and independency by a sample correlation study [10].

Taking a distribution as normal when not many observations are available may lead to incorrect conclusions. The Mann–Whitney U test (MW) [2], a non parametric test, can be carried out instead in order not to make such assumptions, and be able to assess if there are significative differences between the two populations with respect to their medians. Again, null hypothesis H ₀ states the two populations from which samples have been drawn have equal medians, and the alternative hypothesis H ₁ states medians are different.

To carry out the test, both groups are put together and observations are rank-ordered from lowest to highest. Then rankings are returned to the class, A or B, to which they belong. The test statistic U is given by Yue and Wang [31]:

$$U={\rm min} \{U_{1}, U_{2}\}$$

with:

$$U_{1}=n_{A} n_{B}+\frac{n_{A}(n_{A}+1)}{2}-W_{A}$$

$$U_{2}=n_{A}n_{B}+\frac{n_{B}(n_{B}+1)}{2}-W_{B}$$

and where U ₁ is the total number of class A observations preceding class B observations, and the other way round for U ₂. W _A and W _B are the rank sums for each class.

Finally, additional tests were carried out to accept or reject the assumptions of normality, homoscedasticity and independence for the data [26, 33].

1.3 ROC analysis

ROC analysis is a very useful tool to select a classifier and visualize its performance and behaviour [8]. It has been used in many medical diagnosis applications. If the previous statistical tests determine that both classes have different means, a classifier can be designed with this method. The input to the classifier is the regularity measure obtained with ApEn, and the output is a mapping to a predicted class.

Our classification problem consists of mapping an input instance (mean ApEn of a temperature epoch) to one of the classes in the discrete set {A,B }. If we call A the positive class, and B the negative class, we can define the following performance metrics for the classifier:

• True positive (TP): instance is A and it is classified as A.

• False positive (FP): instance is B but it is incorrectly classified as A.

• True negative (TN): instance is B and it is classified as B.

• False negative (FN): instance is A and it is incorrectly classified as B.

• Sensitivity: correctly classified instances of A divided by the total number of A instances.

• Sensitivity: correctly classified instances of A divided by the total number of A instances.

• Specificity: correctly classified instances of B divided by the total number of B instances.

• Accuracy: ratio of correctly classified instances: $\frac{({\rm TP}+{\rm TN})}{({P}+{N})},$ where P and N are the total number of positives and negatives, respectively.

A threshold is used to obtain a crisp classifier, that is, instances can only belong to a single class. If the score is greater than that threshold we map instance into class A, otherwise into class B. The objective is to find an optimal threshold that maximizes accuracy.

The ROC curve is plotted considering each possible threshold as a different classifier, obtaining a set of points in the ROC space that form the resulting curve, a step function. Only one of the possible classifiers is finally chosen, that considered optimal from the accuracy point of view. Finally, the area under the ROC curve (AUC) is computed in order to assess performance.