Time Domain: Breathing Dynamics and Fractal Dimension - The Human Respiratory System - page 155

of how much information can be predicted about one time series point giving full
information about the other. The values of T d at which the mutual information has
a local minimum are equivalent to the values of T d at which the logarithm of the
correlation sum has a local minimum. Optimal time delay values based on the linear
Pearson correlation function is a straightforward and low computational method
adopted in our experiments.
Since the value of the correlation is between − 1 and 1, in order to have only

positive values, we will use its squared value.
From the correlation function, the first local minimum is detected and denoted
as the delay value T d . Consequently, the PPP two-dimensional plot results from
plotting the breathing pressure signal P(t) on the x -axis (in volts) and the shifted
breathing pressure signal P(t

T d ) on the y -axis (in volts).
The area inside the PV loop, and the slope of the axis of the minimal-to-maximal
points in the PV loop are used to evaluate the respiratory mechanics of the patient.
The interpretation of the PV loop is then made with respect to inspiratory and expi-
ratory parameters, such as airflow resistance and work of breathing.
In the phase plane representation, we have

+

T

Area = P(t) · P(t + T d )dt (8.20)

0

with P(t) the breathing pressure signal and T d the time delay estimated for each
patient. One may notice that ( 8.20 ) is nothing else but the definition of the correla-
tion function of two signals in time [ 136 ]. Since pressure and volume are related,
the position of the air in the lungs is determined by each of these signals. Assum-
ing that the pressure is a measure of the position of the air in the lungs, its delayed
component is also related to the position. In this framework, we conclude that the
PPP plot provides information on the position of air in the lungs between two time
instants.
As given in the introductory chapter, the fractal dimension F d is a quantity that
gives an indication of how completely a spatial representation appears to fill the
space. There are many specific methods to compute the fractal dimension. The most
popular and simple methods are the Hausdorff dimension and box-counting dimen-
sion [ 5 ]. The box-counting method is an iterative method. For each box size value
ε FD , follows a corresponding number of boxes N(ε FD ) which will be needed in or-
der to cover the area of the PPP loop. At the next iteration, another (bigger) size of
the box is assumed and again used to cover the area in the PPP loop. The sequence
of box-sizes and their corresponding total number used covering the area of the PPP
loop will yield a straight line on a log-log graph:

ln [ N(ε FD ) ]− ln (C)

ln ( 1 /ε FD )

F d = (8.21)

where C is a constant related to the total area, N(ε FD ) represents the minimal num-
ber of covering cells (e.g., boxes) of size ε FD required to cover the PPP graph. The
slope of the straight line in the log-log plane provides the estimate of the fractal
dimension F d :

N(ε FD ) = C( 1 /ε FD ) F d (8.22)