2.2.1. Determination of the Duration of Each Phase of the Cardiac Cycle
First, the duration of a cardiac cycle, denoted as
, is determined. The resting heart rate of a normal subject is in the range of 60 to 100 beats per minute (bpm) [
29]. To account for more extreme cases, a range between 56 and 102 bpm is established, resulting in
ranging from 0.58 s to 1.07 s. A Gaussian distribution with a mean of 0.82 and a standard deviation of 0.08 is used to generate a sequence of heartbeat cycle duration data. In general, systole tends to be shorter than diastole. The systolic and diastolic durations are represented by
and
. The ratio of systole to diastole, denoted as
r, is set in the range [0.7, 0.9] using a uniform distribution. Once
is determined, the systolic and diastolic durations can be calculated. The ratio of the IVC to the systole, denoted as
, is established between 0.09 and 0.13. It is generated via a uniform distribution. The duration of IVC is represented as
. The occurrence of IVC is during the RS intervals within the QRS complex. The duration of the QRS complex, denoted as
, is established within the range of
. To ensure that the maximum duration of the QRS complex does not exceed 120 ms, constraints are implemented on
.
In the absence of a clear demarcation between the rapid ejection periods (RPEs) and reduced ejection periods, the modeling approach, which is based on the measured data, no longer divides the ejection period into these two phases. An analysis of the signals from both the camera and radar indicates that the changes in velocity during the ejection period are rapid and complex. As a result, a more granular temporal division during the ejection period has been justified. Upon referring to
Figure 3 and
Figure 4, it becomes evident that identifying the locations of the S-wave, J-wave, the onset of the T-wave, and the peak of the T-wave are crucial. The J-point, which is the junction where the QRS complex meets the ST segment, signifies the approximate end of depolarization and the beginning of repolarization. However, the specific time interval between the S-wave and the J-point is not usually measured nor reported in standard ECG interpretation. This time duration, represented as
, was analyzed from the available camera and radar data and was found to be within the range
. By employing a uniform distribution,
is randomly generated within this range.
The positions of the T-wave start and T-wave peak are denoted by and , respectively. The ST segment, which connects the QRS complex and the T-wave, begins at the J-point and ends at the initiation of the T-wave. The typical duration of the ST segment is around 0.08 s (80 ms). To account for individual differences, the duration of the ST segment, represented as , is randomly generated between 70 ms and 90 ms using a uniform distribution.
The duration of the isovolumetric relaxation (IVR) phase is proportionally generated, with its proportion to the diastolic period falling within the range . This generation is executed through a uniform distribution. In the absence of a clear demarcation between the RPF periods and reduced-filling periods (RDFs), the duration for the RPF phase is set between , denoted as , and is generated using a uniform distribution. The proportion of the atrial systole phase to the diastolic phase is established within the range of 16% to 22%, denoted as , and is generated using a uniform distribution. Simultaneously, it is ensured that the duration of the atrial systole significantly exceeds that of the QRS complex. The total duration of the diastole, which comprises the IVR phase, RPF phase, reduced-filling phase, and atrial systole phase, is calculated. With the durations of the other three phases known, the duration of the reduced-filling phase, denoted as , can be calculated using the formula . Finally, a physiologically realistic distribution of the duration for each phase of the cardiac cycle is generated using either a Gaussian distribution or a uniform distribution.
2.2.2. Fitting the Velocity Using a Piecewise Function
Atrial contraction commences near the peak of the P-wave. In line with the pattern previously analyzed, the velocity following the peak of the P-wave displays an initial increase, followed by a decrease. Within the PQ interval, the process of the velocity change is characterized by a point of significant magnitude and a corresponding point of considerable minima. The point of great minima, typically located near the Q-wave, carries a negative value, denoted as
b. In the modeling process, the position of the Q-wave is directly set to the location of the
c-point. However, the position of the Q-wave is not fixed and depends on the lengths of the atrial systole (AS) phase
, the QRS complex
, and the IVC phase
. The duration of the PQ interval, denoted as
, is calculated using the equation
. A sinusoidal function is utilized to model the velocity change over this period, expressed as Equation (
1).
where the initial phase,
, is generated by a uniform distribution with a range of values in
. The frequency,
, is determined by Equation (
2).
where the coefficient,
, is generated by a uniform distribution with a range of values in
. The ventricular volume expands due to AS, with this expansion velocity being less than the maximum expansion velocity during the RPF, denoted as
. The maximum velocity during the RPF period is generated by a uniform distribution within the range [0.02, 0.08]. Given
, the peak atrial systole velocity
falls within the range
, with this ratio being randomly generated by a uniform distribution from [0.2, 0.5]. The velocity
at the minima point Q is randomly generated by a uniform distribution, taking values in the range
. Finally, the velocity of this phase is adjusted within the interval of the minimum value
and the maximum value
. During IVC, the ventricular volume is maintained constant, while the myocardium continues exhibiting motion. Based on actual radar measurements, the characteristics of myocardial motion are found to vary among individuals, but commonalities are observed. Typically, myocardial velocity is observed to gradually increase less than 0, reaching a local maximum point in the RS interval. This maximal point is conveniently denoted as
c. Given that the location and size of the maximal point
c are not fixed,
c is randomly generated within the RS interval. The duration of the
interval is denoted by
. The maximal point has a velocity within the range of
, generated by a uniform distribution. To describe this process, a sigmoid function is employed, expressed as
where
determines the rate of change. To ensure both the
b and
c points are within the saturation region of the sigmoid function, a condition is set such that
, allowing for the calculation of the value of
. The maximum velocity at point
c is denoted by
, taking values within the range of
, randomly generated from a uniform distribution. The velocity is obtained by substituting the sampling time into Equation (
3). Given the known minimum velocity
and the maximum velocity
, the values fit to the function are adjusted for magnitude. Upon the end of IVC, the ventricles are entered into a phase of rapid ejection initiated by the opening of the aortic and pulmonary valves. As the ventricular volume decreases rapidly, the rate of chest wall surface motion experiences a significant increase. This phenomenon is observable in the velocity of the CWM, as illustrated in
Figure 3 and
Figure 4. Examining the curve, it can be noticeable that the velocity decreases from point
c, with a potential local minimum at point
J. The duration of the
interval, denoted as
, can be calculated from the position of
c and
. Within the
interval, an exponential decay function is employed to describe the process, expressed mathematically as:
where
determines the decay rate. A condition is set such that the velocity within the
interval decreases to 0.01, allowing for the determination of the value of
. From the measured data, it is observed that the velocity at point
J varies widely within the range of
, where
denotes the maximum motion velocity of the thoracic surface during the ejection period. From the analysis of the radar and camera data, this value was found to be within the range of 0.04 to 0.14. A random value is generated by uniformly distributing between 0.04 and 0.14 as
. The range of values for
is set to be
, and values within this range are randomly generated using a uniform distribution. The sampling time is determined based on the duration of this interval, and the velocity is obtained by substituting them into Equation (
4). Finally, adjustments are made within this interval based on the maximum value
and the minimum value
.
Following point
J, the velocity undergoes a rapid increase, followed by a deceleration, culminating in a localized peak at point
e. Subsequently, the velocity decreases to a minimum value at
. After the peak at point
e, the velocity experiences another rapid decline, featuring a minimum at
. The duration of the
interval, which ranges between 0.02 s and 0.03 s, is randomly generated through a uniform distribution. To simulate the velocity change in the
interval, the rising part of the Riley decay is employed, expressed as follows:
where the standard deviation
equals
. The maximum velocity at
e is denoted as
, with the requirement that
. Thus, the range of values for
is set to
, and
is randomly generated from a uniform distribution. The sampling time is determined, and upon substituting it into Equation (
5), the velocity within the interval is obtained. Finally, adjustments are made to the amplitude based on the minimum
and maximum
values.
In the segment from point
J to
, two local minima are observed, occurring near the onset and the peak of the T-wave. For the ease of description, the first extreme point is labeled
f, and the second extreme point is labeled
t. Typically, the velocity near the initiation of the T-wave surpasses that at the peak of the T-wave. Given the duration of the ST segment,
, the SJ segment,
, and the
segment,
, the duration of the
segment can be computed. An exponential decay model is employed to describe the
segment, with the mathematical expression as Equation (
5). The attenuation coefficient
is derived to ensure that the value at the end of the interval is below
,
.
is in the range of [0.001, 0.1]. From the measured data, it is evident that there is no consistent relationship between the velocity at
f and the velocity at
J. In some cases, the velocity at
J exceeds that at
f, and vice versa. To account for this variability, the range of values for the velocity at
f is set to be
. Utilizing a uniform distribution, a random velocity value within this range is generated, denoted as
. Upon confirming the sampling time, the velocity is determined by substituted it in Equation (
5). Ultimately, the amplitude is adjusted based on the minimum
and maximum
values within the interval.
During the interval from the onset to the peak of the T-wave, the ejection velocity exhibits an increasing and then decreasing trend. Notably, there are an extreme value point
g before the T-wave peak and an extreme value point
t near the T-wave peak. To characterize this process, a sigmoid function is used for fitting. Initially, the location of the extreme point
g needs to be determined. Since point
g is in the first half of the T-wave and its position is not fixed, a uniform distribution in the interval
is used to randomly generate the position of point
g, where the length of the T-wave,
, is calculated over
,
,
. Next, the duration
of the interval from point
f to point
g is determined, which can be calculated from the position of point
g and
. According to the properties of the sigmoid function, if
f and
g are in the saturation region of the function, i.e.,
, the steepness of the velocity
can be calculated. According to the results of the analysis of the measured data, the velocity
at point
g is expected to be greater than
. Therefore, the value of
is randomly generated using a uniform distribution that takes values in the interval
. Upon substituting the sampling time and
into Equation (
3), the value of the velocity in the current interval is obtained. Finally, the amplitude is adjusted according to the minimum value
and the maximum value
.
Near the peak of the T-wave, a minimum point
t is observed. Its location is randomly determined by a uniform distribution with values in the interval
. The length from point
g to point
t is denoted as
. Based on the analysis of the actual collected radar data, it was found that the velocity
at point
t is smaller than the velocity at point
f. Therefore, the value of
is generated from a uniform distribution with values in the range
. Given the relatively slow rate of decay of the waveform, this process is modeled using a cosine signal, as in Equation (
6). Here, the frequency
is determined by
and a factor
in Equation (
7). The range of values for
is restricted to
. Finally, the value of
is adjusted by taking the last value of the
segment as the maximum value and
as the minimum value.
During the final phase of systole, i.e., the interval from point
t to
, the velocity is observed to gradually decrease to near zero. The end point is assumed to be located at
h, considering that the location of
h varies from person to person and is mainly located in the second half of the T-wave. The location of
h is randomly generated using a uniform distribution taking values in the range
. At the same time, the velocity
of
h is randomly generated by a uniform distribution taking values in the range
. Once the position of the point
t with respect to
h is determined, the duration
can be calculated. To simulate the change in velocity over this interval, a sinusoidal signal is used. Let the period of the sinusoidal signal be
, where the coefficients
are randomly generated from a uniform distribution taking values in the range
. To ensure that the initial value of the sine is a minimum, the signal is phase-shifted by
. The mathematical expression is given in Equation (
8). Finally, the value of
is adjusted according to the interval of the maximum and minimum. This completes the modeling of the systolic phase of the cardiac cycle.
Following the onset of ventricular diastole, the velocity during IVR exhibits a specific pattern of increasing and then decreasing, with distinct peaks and valleys. The magnitudes of these two values vary among individuals. Therefore, the peak velocity
is randomly generated from a uniform distribution taking values in the range
, and the trough velocity
is randomly generated from a uniform distribution taking values in the range
to produce the valley velocity value. Let the peak point be
i and the trough point be
k. Considering the skewed distribution property of the waveform in the
region, a skewed distribution function is used to simulate the velocity change during this period. The mathematical expression is as follows:
where
is the peak position. It is within the IVR period, and its value is randomly generated from a uniform distribution with a range of
. The standard deviation,
, takes values in the range [0.01, 0.2] and is also generated from a uniform distribution. The parameter
determines the direction of the skewness, less than 0 to the left and greater than 0 to the right. Considering that the measured data are mainly skewed to the left, its value is set in the range
, randomly generated by a uniform distribution. Due to the wide range of variation in the location of the valley point
k, it may occur even in the filling zone. Therefore, the
segment consists of three parts: the first part is from point
h to
; the second part is
; the third part is from the second half of the IVR period to half of the filling period. The position of
k falls in the third part and is randomly generated by a uniform distribution. Finally, the sampling time, mean
, standard deviation
, and
are substituted into Equation (
9) for the calculation. Adjustments are made for the minimum value
and the maximum value
in the interval. This completes the modeling of the IVC phase of the cardiac cycle.
During the filling phase of the heart, the ventricles gradually increase in volume as they fill with blood. This results in a tendency for the velocity at the chest wall surface to initially increase and, subsequently, decrease. To model this process, the Rayleigh distribution (RD) is used to describe the change in velocity at the chest wall surface, as shown in Equation (
5). The variance, denoted as
, determines the shape of the distribution and the location of the peak. It takes values in the range
and is generated by a uniform distribution. However, it is also constrained to lie between [0.03, 0.06]. The maximum chest wall surface velocity during the RPF phase, denoted as
, is smaller than the maximum chest wall surface velocity during the fast ejection phase. The peak velocity
is set to take a value in the range [0.02, 0.08] and is randomly generated from this range using a uniform distribution. The sampling time is determined based on the position of point
k,
, and
, and the velocity is obtained by substituting these into Equation (
5). Finally, the values are adjusted according to the maximum value
and the minimum value of 0 within the interval. The chest wall motion (CWM) velocity for the entire cardiac cycle is simulated by the above steps, allowing us to obtain the CWM velocity curve for one cardiac cycle. The CWM parameters associated with the model are listed in
Table 2. This completes the modeling of the diastolic phase of the cardiac cycle.