The diaphragm muscle is the most important inspiratory muscle involved in mammalian ventilation (1). Diaphragm dysfunction is a relatively frequent complication following cardiac surgery during which the phrenic nerve may be damaged due to cold injury or otherwise (2). The frequency of such events is about 10% in patients who undergo open-heart surgery. The injury is usually unilateral and asymptomatic in patients with normal pulmonary function, but this may not be the case in patients with borderline respiratory function. Unilateral phrenicotomy paralyzes the ipsilateral hemidiaphragm, and under resting conditions, the contralateral hemidiaphragm and other respiratory pump muscles increase their activity to maintain adequate ventilation (3). Diagnosis is based on clinical results, chest radiography presentation, pulmonary function tests, arterial blood gas composition, ultrasonography or fluoroscopy of the diaphragm, phrenic-nerve conduction measures, and the diaphragm EMG. Clinical signs of diaphragm dysfunction are difficult to diagnose in the early postoperative period. The aim of this study was to examine whether an innovative statistical approach, based on the pattern recognition theory, could aid an early recognition of diaphragm paralysis. We addressed this issue by analyzing the dependence between gradual unilateral paralysis of the diaphragm and features describing breathing pattern in acute animal experiments.
MATERIAL AND METHODS
Experiments were carried out in 10 anesthetized, tracheostomized, and spontaneously
breathing adult cats of either sex. Sequential unilateral sections of phrenic
nerve rootlets were performed yielding the experimental conditions of gradual
partial or complete diaphragm paralysis on the ipsilateral side. The following
six features were analyzed before (intact) and 1 hour after the phrenic nerve
sectioning: breathing frequency (f), time of inspiration and expiration (T
I,
T
E), tidal volume (V
T),
minute ventilation (V
E=f x V
T),
and the amplitude of integrated phrenic nerve activity (AMP
PHR).
The study obtained approval of an institutional ethical committee.
A data set of 199 measurements of the six features, representing breathing pattern,
was obtained. The set contained 90, 53 and 56 feature vectors from the intact
(Class I), partial (Class II), and complete (Class III) unilateral paralysis,
respectively. It was called the training set or reference set and its form is
shown in
Table 1. The range of values and median for each variable analyzed
are presented in
Table 2.
Table 1.
The form of the analyzed data set - reference set. |
|
Table 2.
Basic statistical description of variables in each class of the data matrix. |
|
|
Fig. 1.
The k-NN decision rule. |
Statistical methodology
We deal with the following question: is it possible to recognize the class of
a measurement vector knowing the values of the vector’s features? The probability
of a correct recognition or misclassification would tell us how strong the relation
between features and classes is. The pattern recognition theory estimates both
the classification and misclassification rates. The most powerful classification
rule is the standard
k nearest neighbor rule (
k-NN) (4). It assigns
the classified case to the class most heavily represented from among its
k
most similar, i.e. nearest, samples in the reference set. The value of
k
ought to be estimated for the all possible values of
k and
k-NN
rule in order to offer the lowest misclassification probability while classifying
new cases.
There are two basic methods of experimental error rate estimation: the testing
set method and the leave-one-out method. The former approach requires dividing
of the data set into two parts: the training and the testing set. If the classifier
derived from the training set classifies all
m samples from the testing
set and
r of them are misclassified then the error rate, understood as
ratio
r/m, estimates the probability of misclassification. The latter
approach allows using the whole data set as a training set. Each sample from
the training set is classified by the rule derived from the training set decreased
by the currently classified sample.
The leave-one-out method
The
1-NN rule, for the example presented in
Fig. 2, during the
course of the leave-one-out procedure, makes three misclassifications, since
the nearest neighbors of points 4, 5 and 6 come from the opposite class. But
the
3-NN rule misclassifies only points 5 and 6. This time the point
4 is correctly classified, since two out of its three nearest neighbors come
from the same class. The
3-NN rule offers a minimum misclassification
rate and this rule ought to be used to classified new samples.
|
Fig. 2.
One-dimensional example for the illustration of the leave-one-out method. |
The reference set reduction
Till now we have treated the terms “the training set” and “the reference set”
as the synonyms. Strictly speaking the training set is the set that is used
for the classifier construction while the reference set is the set, which must
be stored in computer memory during the classification phase. It can be noticed,
see
Fig. 3, that it is sufficient to store only the points lying close
to the class boundary. The reference set reduction algorithms have been developed
only for the
1-NN rule (5).
|
Fig. 3.
The Gowda-Krishna reference set reduction. |
The degree of dependence between classes and features
A minimum possible number of points in the reduced reference set is equal to
the number nc of classes and cannot exceed the number m of points in the original
reference set (the training set). The correlation approach, known from classical
statistics, concerns a pair of variables, and the correlation coefficient (of
any type) varies between 0 and 1. The reference set reduction algorithm can
be used to introduce a similar dependence measure between classes and features.
It can be defined as
dm=(
m-r)/(
m-nc), where m is the numerical
force of the training set, r is the size of the reduced reference set and nc
denotes the number of classes. If
r=m then
dm=0, if
r=nc
then
dm=1, since nc
r
m
then 0
dm
1.
RESULTS AND DISCUSSION
The leave-one-out method enables the calculation of the number
rij
of points from class
i assigned to class
j. This number allows
to compute the probability
pij that a
point from class
i will be classified to class
j and the probability
qij that a point assigned to class
i
is in fact from class
j:
pij=
rij/
jrij
and
qij=
rji/
jrji.
The optimum number of
k for the
k-NN rule is
k=1. It gives
7.04% of misclassifications. All three matrices
R=[
rij],
P=[
pij] and
Q=[
qij]
are presented in
Table 3. It can be noticed that Class I (innervated
muscle) is recognized with the highest probability of correct classification
and the most difficult for recognition is Class II (partial unilateral phrenicotomy).
The significance level can be calculated taking into account two qualitative
variables: the true and the assigned class. The assigned class represents “what
to say” features. Thus, the relation between the true and assigned classes corresponds
to the relation between features and classes.
Table 3.
Three types of matrices. |
|
From among the 6 features, considered separately, the best is AMP
PHR
and the worst is V
T (
Table 4A). Adding
sequentially next features in the best possible way, the best result, the error
rate equal to 7.04%, offers the set of all 6 features (
Table 4B). Subtraction
of any single feature increases the misclassification rate (
Table 4C).
The feature AMP
PHR is best, whether treated
separately or in the context of the remaining features; its subtraction causes
the error rate to increase from 7.04% to 13.07%. As a single feature, the worst
is V
T, but in the context of the 5 other features
the worst becomes V
E; its subtraction only slightly
increases the error rate as compared with the set of all 6 features.
Table 4.
Results of feature selection. |
|
The application of the Gowda-Krishna reference set reduction algorithm produces a reduced set containing only 47 points (out of 199) and then the dependence degree is equal to 0.78 (the counterpart of the correlation coefficient).
In conclusion, the statistical approach used can have a meaning for the development of diagnostic methods for diaphragm dysfunction, not only after the gradual unilateral paralysis of the muscle but also in other forms of impairment. To our opinion, the pattern recognition methods presented are powerful tools for statistical analysis of biomedical data and jointly with the classical statistics enable a better exploration of the knowledge contained in the data.
Acknowledgments:
This work was supported by the statutory budget of the Polish Academy of Sciences
Medical Research Center.
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