J. Biomedical Science and Engineering, 2013, 6, 1143-1154 JBiSE
http://dx.doi.org/10.4236/jbise.2013.612143 Published Online December 2013 (http://www.scirp.org/journal/jbise/)
Application of Van der Pol oscillator screening for
peripheral arterial disease in patients with diabetes
Jianxing Wu1*, Chianming Li1,2, Weiling Chen1, Chiahung Lin3, Tainsong Chen1*
1Department of Biomedical Engineering, National Cheng Kung University, Tainan City, Taiwan
2Division of Infectious Diseases, Department of Medicine of Chi Mei Medical Center, Tainan City, Taiwan
3Department of Electrical Engineering, Kao-Yuan University, Kaohsiung City, Taiwan
Email: *jian0218@gmail.com, 235813cmli@gmail.com, lynnchen.k@gmail.com, eechl53@gmail.com, *chents@mail.ncku.edu.tw
Received 7 November 2013; revised 4 December 2013; accepted 12 December 2013
Copyright © 2013 Jianxing Wu et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In accor-
dance of the Creative Commons Attribution License all Copyrights © 2013 are reserved for SCIRP and the owner of the intellectual
property Jianxing Wu et al. All Copyright © 2013 are guarded by law and by SCIRP as a guardian.
This paper proposes a Van der Pol (VDP) oscillator
screening for peripheral arterial disease (PAD) in
patients with diabetes mellitus. The long-term ele-
vated blood sugar levels produce a high risk of pe-
ripheral neuropathy and peripheral vascular disease,
especially in the foot of a diabetic. Early detection is
important, in order to prevent foot problems for dia-
betic patients with PAD. Photoplethysmography (PPG)
is a non-invasive method for the detection of blood
volume changes in peripheral arteries. Because of
changes in the resistance-compliance, the rise time
and transit time for the PPG signals increase and
change in their shape are highly correlated with PAD
severity. In this study, the Burg autoregressive (AR)
method is used to determine the characteristic fre-
quencies of the right- and left-side PPG signals. For
bilateral frequency spectra, the VDP oscillator uses
asynchronous signals. This produces damped sinu-
soidal responses and the oscillation overshoot (OS) is
an approximating function only of the damped factor.
This index is used to estimate the degree of PAD, in-
cluding normal the condition and diabetic patients
with PAD. The results show that the proposed me-
thod is efficient and more accurate in the estimation
of PAD.
Keywords: Van der Pol (VDP) Oscillator; Peripheral
Arterial Disease (PAD); Burg Autoregressive Method;
Oscillation Overshoot
Type 2 diabetes is a chronic, metabolic disease that is
prevalent in developed and developing countries, associ-
ated with mortality and morbidity due to increasing risk
factor for cardiovascular diseases. Patients with diabetes
mellitus incur substantantial medical expenditure to con-
trol their glycemic concentrations and complications.
Peripheral arterial disease (PAD) is highly prevalent in
patients with diabetes mellitus. However, it is usually
underestimated by patients as well as physicians, espe-
cially in the foot of diabetics, mostly contributing to the
slow progress of arterial atherosclerosis in lower limbs.
Manifestations of this peripheral arterial obstruction can
vary from a symptom to total occlusion or gangrene of
toe or foot [1]. Besides classic presentation of intermit-
tent claudication, foot ulceration or diabetic foot may
develop, signaling a significant degree of arterial occlu-
sion and high risk for future amputation. PAD is respon-
sible for 47% of amputations in diabetic patients [1]. In
recent years, many imaging techniques, such as Doppler
ultrasound, colour duplex ultrasound (CDU), X-ray an-
giography, computed tomographic angiography (CTA),
and magnetic resonance angiography (MRA), have been
used to diagnose PAD, verifying the presence of a steno-
sis inside the vasculature of diabetes [2-4]. Although
these methodologies are reliable and highly accurate, a
noninvasive and inexpensive technique for early diagno-
sis and monitoring of PAD remains unsettled in hospitals
and primary care settings. Modern ultrasound technique
is well-developed, compact, fast, and free of irradiation,
but they can only be correctly operated by a well-trained
technician clinical physician. Therefore, this study pro-
*Corresponding authors.
J. X. Wu et al. / J. Biomedical Science and Engineering 6 (2013) 1143-1154
poses a non-invasive means of measurement and an
automatic diagnostic algorithm to estimate the degree of
PAD, purporting for screening and monitoring of lower
limb vascular occlusion for high-risk groups like patients
of diabetes, dyslipidemia, and hypertension, and the eld-
Photoplethysmography (PPG) is an optical and nonin-
vasive technique, widely applied to measure oxygen sa-
turation, monitor heartbeat, and detect blood volume
changes in the vascular bed. The advantages of PPG like
compactness fast, and easy-to-operate [5-8] take the form
of a portable instrument for routine screening. PPG sig-
nals can be obtained from multiple-site measurements
including earlobe, forearm, fingertip, thumb, and toe [9-
17]. A typical waveform of PPG is triphasic. Biphasic or
monophasic form indicates a degree of arterial obstruc-
In signal processing, time-domain and frequency-do-
main analysis has been applied to assist diagnosis of
PAD. Previous research reports that frequency parame-
ters verify the differences between healthy and PAD
subjects, using high-frequency and low-frequency char-
acteristics [5-7,15]. The most common parameters of
PPG waveforms are transit time, amplitude, and shape
changes, which are helpful for evaluation of arterial com-
pliance single or bilateral measurements of lower limbs.
Previous research reports that these timing parameters
and shape characteristics reflect the degree of vascular
stenosis, contributing to time delay, damping, amplitude
reduction, and changes of resistance and compliance pro-
perties of occlusive peripheral arteries, with variations
depending on a subject’s age and measurement sites [11,
13,15]. Therefore, an alternative for applying PPG to
evaluate PAD is proposed, including bilateral or multi-
channel PPG techniques resulting from the findings of
significant bilateral asymmetry and increased difference
in timing and frequency parameters along with increased
PAD severity [13-15,18]. Multi-channel measurements
were used to study the age-related changes in the char-
acteristics of PPG shape at the ear, finger, and toe sites.
The largest changes with age were seen at the ear and
finger sites for the systolic rising edge region, and the
finger site for the dicrotic notch region. The smallest
changes with age are for the toe site [9-11]. Accordingly,
in this experiment, bilateral measurement at the right and
left great toes was designed for PAD estimation.
The objective of this study was to construct a fre-
quency-based Van der Pol (VDP) oscillator to evaluate
the degree of PAD in patients with diabetes mellitus, and
apply Burg autoregressive (AR) method as a parametric
method for time-domain and frequency-domain signal
processing [19-24], purporting to smooth the spectra and
find characteristic frequencies from the PPG signals re-
corded. Given a bilateral frequency spectrum, a second-
order VDP system has a step response that is character-
ized by damped oscillation [25-28]. The oscillation over-
shoot (OS) is an approximating function of the damped
factor, and determines the severity of PAD: normal con-
dition (Nor), mild-to-moderate disease (MD), and severe
disease (SD). With twenty-one subjects, the test results
demonstrate that the proposed method is efficient and
accurate in the estimation of PAD in the feet of diabetics.
In signal processing of plethysmographic waveforms,
higher frequency characteristics are preferred for fre-
quency analysis to identify normal pulse waves [5-7]. On
the other hand, low frequency is used to characterize
PAD in diabetic patients. Both frequency-based methods,
such as Fourier transforms (FTs), and frequency spec-
trum methods, are the non-parametric techniques that are
used to preprocess the PPG signals. They are disadvan-
tageous for their spectral leakage effect, because of the
size of sampling window. Wavelet transform is a para-
metric method and provides better time-frequency reso-
lution than non-parametric methods, and would be a
promising method to extract features from non-stationary
biosignals. However, significant frequencies are extracted
at specific wavelet coefficients with different wavelets,
and several frequency bands are analyzed using the cas-
caded low-pass or high-pass filters and down-sampling/
up-sampling operations through trial procedures.
In literatures [17-20], the Burg autoregressive (AR)
method is able to derive the frequency spectra by fitting
an AR model of a given specific order. Its advantages
over above-mentioned methods include smooth spectra,
high frequency resolution, stable AR model, and compu-
tationally efficient. The Burg AR method is also a para-
metric method for time-domain and frequency-domain
signal processes. It could smooth spectra in comparison
with frequency-based methods to find the characteristic
frequencies with distinguishing peak central/main spectra
by fitting an AR model of a given specific order.
The frequency spectra of PPG signals is expressed as a
linear combination of the previous samples and the re-
sidual values, resi. For a discrete set of n sampling points,
P coefficients are used to approximate the original data,
i, 1, 2, 3,,in
, where
is a discrete frequency spec-
trum of the PPG signal. The residual value is assumed to
be independent of the previous samples and is calculated
by [20,21]
ii pip
resi n
 
i is the frequency spectrum of the PPG signal, P
is the order of the AR model, and , and
p are the coefficients of the AR model.
1,2, 3,,pP
Copyright © 2013 SciRes. OPEN ACCESS
J. X. Wu et al. / J. Biomedical Science and Engineering 6 (2013) 1143-1154 1145
The optimal coefficients,
p, are used to minimize the
square error, , between the original data
and the approximated data. For the forward and back-
ward linear prediction, the object is to minimize Fp and
Bp, as shown in [22]
nn n
pitiiit p
ipip tip
 
 
01 0
np np
ipi ti
 
t, t[p, n], is a linear weighted combination of p
previous sampling data, and
t, t[0, ntp], is a linear
weighted combination of p subsequent sampling data.
The sum of the residual energy (SORE) in stage p is Ep =
Fp + Bp. The Levinson-Durbin recursion algorithm is
used to minimize the SOREs to estimate the model coef-
ficients. The coefficients of
p, p[1, P], are stored in a
vector and an inverted
order vector . Therefore,
1,,, ,, ,
 
the recursion formula is the following [22]
 (4)
 
11 0
pp inp
ip i
ip bi
 
  
The concept of Burg AR method adjusts the parameter
to minimize the SORE Ep using the final prediction
error criterion (PEC) [23,24]. Then update the Ap and
update fp(i) and bi(i) as
 
fifi bip
 (6)
 
bibi fip
  (7)
The optimal coefficients
p are chosen to minimize the
squared error, whereas the forward and backward linear
prediction equations attempt to minimize Fp and Bp. The
entries of
i represent samples of a discrete data, and P is
the integer order of an AR model, used in estimating the
power spectral density (PSD), log(
dB/kHz, . This property of the AR model
is used to smooth the frequency spectrum, which tends to
favor the peaky spectra. Qualitatively different PSDs are
observed by normalized process, which different values
within the specific ranges of normalized log-magnitude
and scaled frequency. Therefore, PSD can be used to
identify the characteristic frequencies and magnitudes in
each frequency spectra.
1,2, 3,,i
In the physiological measurements, PPG signals were
obtained using optical sensors (940 nm near infrared,
spectral bandwidth: 45 nm) with 1 KHz sampling rate, as
shown in Figure 1(a). The dashed line represents a nor-
mal subject and the solid line represents a diabetic pa-
tient [14,27,28]. It was found that the Fourier transform
based estimation methods has some drawbacks, such as
spectral leakage into sidelobes and characteristic spectral
broadening, as shown in Figures 1(b) and (c). In order to
find characteristic frequencies, the Burg AR method is
used to estimate the PSD by fitting an AR model with
given specific orders, P = 8, 12, and 16, to the PPG sig-
nals. The characteristic frequencies of the PPG signals
easily found in the frequency spectra between 0 Hz and
500 Hz. The PSDs are shown in Figures 1(d)-(f). The
peaky spectra for diabetic subjects and the smooth fre-
quency spectra for normal subjects fall into different
frequency bands, respectively. Depending on these bands,
frequency-based parameters provide information for the
estimation of PAD in diabetic patients.
3.1. Feature Extraction Using the Van der Pol
(VDP) Oscillator
PAD can develop in the arteries of most visceral organs
and extremities. Atherosclerosis in the brain and heart
draws much attention for clinical diagnosis and interven-
tion; by contrast, lower limb PAD is underestimated by
the patients, even in diabetics whose progress of vascular
atherosclerosis is accelerated by the metabolic disease.
PPG is a noninvasive technique capable to optically gen-
erate a plethysmograph at great toes, index fingers, and
ears, and monitor blood pressure, blood oxygen satura-
tion (SaO2), and blood volume changes in an artery or a
vascular bed of tissue. A PPG signal consists of AC (Al-
ternating Current) and DC (Direct Current) components.
The AC component reveals physiological information,
such as cardiac synchronous changes in the blood vol-
ume with each heartbeat and vasomotor activity [8-10,
15]. Time-domain and frequency-domain parameters
have been used to detect the degree of PAD severity for
diabetic patients with normal condition (Nor), mild-to-
moderate disease (MD), and severe disease (SD). For the
time-domain parameters, ECG R-peak as a timing refer-
ence is used to obtain pulse rise time (RT), pulse transit
time (PTT) from R-peak to pulse foot (PTTf), pulse tran-
it time from R-peak to pulse peak (PTTp), and PPG am- s
Copyright © 2013 SciRes. OPEN ACCESS
J. X. Wu et al. / J. Biomedical Science and Engineering 6 (2013) 1143-1154
Copyright © 2013 SciRes.
Figure 1. (a) Time-domain PPG signal analysis for normal subjects and diabetic subjects, (b) and (c) PSD estimation using the
frequency spectrum method, (d), (e), and (f) PSD estimation using the Burg method for an AR model with orders P = 8, 12, and 16
for normal subjects and diabetic subjects.
plitude (plus foot-to-plus peak amplitude), as shown in
Figure 2(a). The timing parameters, PTT and RT, are
prolonged as due to the resistance of peripheral vessels
increases, and the amplitude (AMP) and shape of pulse
waves are smoothed [11]. Parameters, PTT and AMP,
and shape differences vary with ages for individual sites,
but the smallest variations with age occur at the great toe
sites [9-15]. Thus, bilateral differences in the timing pa-
rameters, PTTf, PTTp, and RT, provide time delay
information for the estimation of PAD. Bilateral differ-
ences in frequency spectra are similar and are used to
distinguish between the normal subjects and diabetic
This study uses frequency-based parameters to esti-
mate the degree of PAD. The PSDs are estimated using
the Burg AR method. The Van der Pol (VDP) system is
used to quantify the features of the right- and left-side
PSDs. It is an oscillator with nonlinear damping that is
defined by a second-order differential equation. The
formulation has the form of an autonomous system with
two state variables. The state equations are evaluated as
below [23-26]:
 
  
 
 
) and
) are the functions with respect to
) is the PSD of the right-side PPG sig-
nal, except for
= 0, and
) is the PSD of the left-
side PPG signal, right and left are denoted by the sub-
scripts r and l. The parameter,
, is a control parameter,
> 0 reflects the degree of nonlinearity of the sys-
tem [21]. When the term, (
))2, becomes dominant,
the VDP system becomes a nonlinear equation with posi-
tive damping. The dynamics of the system are stable and
are restricted to a fixed point.
Equation (9) gives the frequency of self-oscillation, as
determined by a real parameter,
, and demonstrates
dissipation or damping. If the parameter,
= 0, then
Equation (9) reduces to that for a simple harmonic oscil-
lator. When the PSDs,
) and
), are different (not
) provides the damping in the VDP
oscillator that results in self-sustained oscillations. The
multiple peaks and amplitudes of these peaks demon-
strate the nonlinearity of the frequency spectrum. PSD
) is used to determine the response of the VDP
oscillator. The VDP system demonstrates a damped si-
nusoidal response for a general second-order system. The
transient response consists of a sinusoidal oscillating
waveform with an exponentially decaying amplitude.
The sinusoidal frequency is called the damped frequency
of oscillation.
In order to evaluate the discrete frequency spectrum,
the PSD is estimated using the Burg AR method. As-
suming a set of n points from the PSD,
i, ,
(n = 500 in this study), the continuous VDP system can
be modified as a discrete VDP system. Therefore, the
discrete VDP system that is proposed for the estimation
of PAD, as
1,2, 3,,in
J. X. Wu et al. / J. Biomedical Science and Engineering 6 (2013) 1143-1154 1147
 
lr rl
ii i
 
where the index, i, is the integer scale of the frequency,
. A bifurcation is a fundamental change in
the nature of a solution. For different initial conditions,
1,2, 3,,i
, the output,
li, given by
Equation (11), has a step response that is characterized
by a damped oscillation in the frequency domain. It ex-
hibits a rich variety of nonlinear dynamic behaviors and
generates the limit cycle for small values of
, and de-
velops into relaxation oscillations when
becomes large.
If the VDP oscillators use different PSDs, the oscillations
become unstable, as the amplitudes of spectra peaks in-
crease. The step response of the bilateral PSDs is shown
in Figure 2. A second-order step response is character-
ized by damped oscillations. From Figure 2(c), the per-
centage overshoot OS% is given by the following Equa-
tion [29]:
max min
OS c
The percentage overshoot OS% defines the amount by
which the oscillation overshoots the minimum value cmin
= b. The term cmax is determined by curve fitting the
function at the maximum value, as [29]
max 2
exp1 cossin
exp 1(13)
 
 
For the step response, substituting Equations (13) and
(14) into Equation (12) gives
% exp1100%OS b
 (15)
Equation (15) demonstrates that the percentage over-
shoot is a function only of the index,
, and allows the
determination of the OS%, given the index,
. The in-
verse of the equation allows a solution for
given the
OS%. The inverse is given by
ln %
ln %
  (16)
The relationship between the index,
, and the multi-
ple characteristic frequencies is clearly seen. The higher
the value of
, the more oscillatory is the response in the
frequency-domain from Equations (10), (11), (15), and
Figure 2. (a) ECG and PPG signals and pulse landmarks, (b)
The bilateral PSD functions for right-side and left-side PPG
signals, (c) The damped sinusoidal response.
(16). The increase in the resistance of peripheral vessels
produces multiple characteristic frequencies, so this
study uses the index,
, is used to estimate the degree of
PAD. The various ranges of the index
are obtained
from specific subjects, including those with normal con-
dition (Nor), mild-to-moderate disease (MD), and severe
disease (SD).
3.2. Physiological Measurement
PPG signals were collected from twenty-one subjects in
a hospital (Chi-Mei Medical Center, Department of In-
fectious Disease and Immunology, Tainan City, Taiwan).
The subjects, aged from 24 to 65 years, were divided into
three groups: Nor, MD and SD, in terms of diabetic dis-
ease. In a clinical examination, the ABPI (Ankle-Bra-
Copyright © 2013 SciRes. OPEN ACCESS
J. X. Wu et al. / J. Biomedical Science and Engineering 6 (2013) 1143-1154
chial Pressure Index) was used as an early detection
method to decide the degree of PAD. The degrees were
categorized by clinical manifestations, ABPI values, and
angiographic findings. For preliminary PAD estimation
using the APBI, the indices signify ABPI 0.9 for nor-
mal subjects and diabetic subjects with MD, ABPI < 0.9
(at least one leg) for diabetic subjects with SD, as shown
in Table 1 [28-30,32]. Despite the non-invasive nature of
the examination to assess death rate and cardiovascular
diseases in high-risk patients, the accuracy of the ABPI
measurement is reduced in patients with calcified blood
vessels, caused by conditions such as diabetes and chro-
nic renal failure. However, measurement of the ABPI
must be repeated several times (>10 minutes) and is of
limited use in routine vascular screening in a primary
care setting [13].
Bilateral timing parameters could offer a quick as-
sessment for the screening of PAD in primary care. Bi-
lateral measurements simultaneously acquire PPG sig-
nals from the right and left big toes. The absolute timing
differences are used to reference one side of the body
(right side) with the contra-lateral side (left side), in or-
der to calculate the parameters, PTTf, PTTp and RT,
so various ranges are obtained for specific groups. Each
parameter has a mean value and a specific range between
maximum (Max) and minimum (Min) values. A com-
parison of the bilateral differences demonstrates that
these parameters increase as the severity of the disease
increases. Therefore, clinical physicians would consider
timing differences to be a good reference for PAD as-
sessment. The three groups are divided into ten normal
subjects (No. 1 - No. 10), eight MD diabetic subjects (No.
11 - No. 18) and three SD diabetic subjects (No. 19 - No.
21), as shown in Table 1.
For these twenty-one subjects, Equations (12) and (16)
were used to compute the index,
. The indices,
, also
represent the specific ranges for diabetic patients, non-
diabetic patients with PAD and normal subjects as a con-
trol, as shown in Figure 3. Severe PAD causes more
asymmetrical PPG signals and the damped oscillation
increases as PAD increases, which is caused by the re-
sistance of the peripheral vessels. Therefore, the index,
also increases in diabetic patients with increased PAD.
The index,
, is less than 0.64 for normal subjects with
an ABPI 0.9, from 0.64 to 0.65 in diabetic patients
with an ABPI 0.9 and greater than 0.65 for diabetic
patients with an ABPI < 0.9. The standard indices,
, are
established, as shown in Table 1. The sensitivity is greater
than 85.7%, and positive predictivity is also greater than
80.0% to quantify the performance of the proposed mea-
surement method. These specific ranges also confirm the
degree of PAD noted by clinical physicians. The trend in
the degeneration of PAD is clearly demonstrated. De-
pending on the severity of the PAD assessment, these
specific ranges provide key information for the evalua-
tion of the degree of PAD.
4.1. Experimental Setup
Using the bilateral non-invasive measurement, two opti-
cal sensors (reflection mode), consisting of light sources,
photo-detectors, trans-impedance amplifiers, and high-
pass filters, were placed at the right and left great toes.
Near infrared (NIR) has large differences in the extinc-
tion coefficients of deoxyhaemoglobin and oxyhaemo-
globin. Thus, the light source 940 nm NIR was chosen in
this optoelectronic design. The reflection mode, includ-
ing light source and photo-detector, was positioned side
by side with 5 mm spacing, and the light is directed
down into the skin and is backscattered from the skin
adjacent to the photo-detector [10]. Two optoelectronic
probes (circle shape, diameter: 20.0 mm, height: 8.2 mm)
were synchronized using a data acquisition controller.
PPG signals are captured at a sampling rate of 1 kHz for
15 minutes [30]. A DAQ card (National Instruments
DAQ Card, 16 Channels, 1.25 MS/s) served as an ana-
log-to-digital (A/D) converter between the optical meas-
urement system and a computer. Locating each pulse
foot (PF)-pulse foot (PF) interval of PPG signal, 800
sampling data were acquired within a sampling window.
In this study, we used the Burg AR method to estimate
the PSDs of PPG signals. The suitable AR order can be
used to identify the peaky spectra. Figure 4 shows the
variation of residual energy versus AR model order. We
use model orders, from order 1 to order 30, to calculate
the SORE. To obtain the suitable order, Akaike’s final
prediction criterion is used to select the model order
[22,24]. For considering convergent condition, we con-
sidered SORE 101 to stop the Burg AR algorithm.
Then, the Burg method using model order P = 8 with
optimal coefficients was used to estimate the PSDs.
The VDP oscillator shows the step responses in the fre-
quency-domain, as shown in Figure 5. The oscillations of
the step responses are smaller than those for normal sub-
jects. The higher the value of
, the more oscillatory the
response are important information for the degree of
PAD assessment, which may be useful for determining the
trends of PAD in routine examination. The proposed algo-
rithms were developed on a PC AMD Celeron (R) CPU
2.40GHz with 2.39 GHz, 224 MB RAM GHz with 1.75
GB RAM and Matlab software. Portable optical meas-
urement was used to obtain the PPG signals in the labo-
ratory and medical center. To demonstrate the effective-
ness of the proposed method, the database of 21 subjects
were used to design the gold standard of PAD assess-
ment using the VDP oscillator from selected 6 subjects
Copyright © 2013 SciRes. OPEN ACCESS
J. X. Wu et al. / J. Biomedical Science and Engineering 6 (2013) 1143-1154
Copyright © 2013 SciRes.
Table 1. The ABPI, bilateral differences in timing parameters [30], and index
for PAD estimation.
ABPI 0.9
0.5 ABPI < 0.9
ABPI < 0.5
Normal subject Diabetic subject Diabetic subject
Parameter Normal (Nor) Mild-to-moderate disease (MD) Severe disease (SD)
PTTf (ms) Mean
0.3 - 7.4
5.1 - 23.7
23.6 - 34.8
PTTP (ms) Mean
0.4 - 22.3
14.3 - 56.5
46.2 - 57.8
RT (ms) Mean
1.3 - 15.6
3.4 - 32.3
11.5 - 35.3
0.616 0 .014
0.597 - 0.636
0.645 0.002
0.640 - 0.647
0.668 0.012
0.652 - 0.681
Note: The values of PTTf, PTTp, and RT are absolute values. (1) fRf
, (2)
Rp Lp
PTT PTTPTT  , (3) RL
RT RTRT , where
suffix words R and L are defined right and left legs, (4) ABPI: it is calculated using the highest of the right and left ankle systolic blood pressure divided by the
highest of the right and left arm brachial systolic blood pressure [13].
Figure 3. The specific ranges of the index
in normal subjects
and diabetic patients with PAD.
Figure 5. The step responses of the VDP Oscillator for severe
disease (SD), mild-to-moderate disease (MD), and normal con-
dition (Nor).
and diabetic patients with PAD were tested using physio-
logical measurement. There were two normal subjects
(No. 1 and No. 2), four diabetic patients, two of which
were MD subjects (No. 11 and No. 12) and two of which
were SD subjects (No. 19 and No. 20). The preliminary
diagnosis and ABPI classification were performed by
clinical physicians and the results are shown in Table 2.
However, the ABPI is significantly lower for normal
subjects and diabetic patients with MD (ABPI 0.9). For
bilateral PPG measurement, two optical sensors were
placed on the right and left great toes and PPG signals
were obtained from the three groups, as shown in Figure
6. The transit time is prolonged for an occluded leg and
gradually increases as the disease becomes more severe,
as seen in the interval between the dashed lines. For ex-
ample, the bilateral-timing differences are
Figure 4. Variation of residual energy versus AR model order.
(three groups: Nor, MD, and SD) were used to test, as shown
in Table 2. The result demonstrates the computational
efficiency and accurate diagnosis achieved by this study.
4.2. Experimental Results Normal Subject-No. 1: PTTf = 2.733, PTTp =
1.977, RT = 3.133,
In order to verify the proposed method, normal subjects
J. X. Wu et al. / J. Biomedical Science and Engineering 6 (2013) 1143-1154
Table 2. The experimental results of PAD estimation.
Mean values of bilateral differences
No. PTTf (ms) PTTp (ms) RT (ms)
Fuzzy logic
The proposed
1 2.7330% 7.05% 1.9970% 16.26% 3.1330% 0.19%1.06501.1138Nor Nor
2 1.1730% 2.78% 5.8700% 4.05% 4.6950% 3.97%1.14041.1333Nor Nor
3 2.3330% 12.07% 3.6660% 5.44% 13.330% 3.97%1.14401.1101Nor Nor
*Failure Nor
11 15.321% 16.92% 17.286% 10.50% 1.9640% 4.05%1.17881.2357MD MD MD
12 6.3850% 13.44% 16.923% 16.76% 17.556% 0.39%1.27351.3076MD MD MD
13 23.667% 13.62% 33.333% 7.58% 32.833% 16.63%1.0714 1.0446MD *SD MD
19 33.072% 10.06% 48.500% 14.18% 15.428% 8.57%1.08170.8941SD SD
20 23.606% 13.64% 57.700% 6.05% 35.000% 10.83%1.04420.8945SD SD
Note: ANFIS and Fuzzy Logic: 1 output variable with 3 triangular membership functions, the center values of the functions are 0.2, 0.4, and 0.6 for Nor, MD,
and SD.
Figure 6. Time-domain PPG signals of the three groups of patients.
MD Subject-No. 11: PTTf = 15.321, PTTp =
17.286, RT = 1.964,
SD Subject-No. 19: PTTf = 33.072, PTTp = 48.500,
RT = 15.428.
The shape of the PPG signals for a SD subject with a
unilateral occluded leg is substantially different. The
PPG signals at the right and left sites are asynchronous in
the time-domain. According to the standard established
in Table 1 [30], the degree of PAD from the preliminary
estimate is confirmed, as shown in Table 2. However,
the ABPI and bilateral timing differences are not the only
parameters that determine classification.
Using twenty bilateral PPG signals, the diagnostic re-
sults for six subjects are shown in Figure 7, where the
horizontal axis is the number of PPG signals and the ver-
tical axis is the index,
, for each PAD assessment. The
Burg AR method is used to estimate the frequency spec-
tra from the bilateral PPG signals. Using the bilateral
frequency spectra, the VDP oscillator shows the dynamic
characteristics to be step damped oscillation responses
for diabetic patients with PAD. In this study, the index,
is a parameter that represents the degree of PAD that is
Copyright © 2013 SciRes. OPEN ACCESS
J. X. Wu et al. / J. Biomedical Science and Engineering 6 (2013) 1143-1154 1151
Figure 7. Different indexes
for separating the diabetic PAD from the normal subjects.
assessed using the VDP oscillator, for the Nor, MD and
SD groups. A comparison of the results shown in Figure
7, demonstrates that the index,
, is a significant pa-
rameter for the separation of diabetic patients with PAD
from normal subjects. According to the standard values
in Table 1, the values are centered at 0.645 (±0.002) for
MD subjects and are greater than 0.65 for SD subjects
(0.668 ± 0.012). Within these specific ranges, the index,
, can also be used to estimate the degree of PAD.
Experimental tests show that the proposed VDP oscil-
lator can be used to assess the degree of severity and can
be used to monitor the trends in PAD degeneration at the
borders between Nor and MD (subjects No. 3, No. 6, and
No. 12), or between MD and SD (subject No. 21). In this
study, the index,
, is used to monitor the trends in the
degeneration of PAD to allow early detection or screen-
ing and monitoring. Diabetes mellitus is a chronic dis-
ease that may require therapy to prevent further problems.
In particular, for the feet of diabetics, adequate treatment
may improve the risk profile of chronic complications.
This research provides a potential method for early de-
tection, monitoring, and prevention of PAD by early in-
tervention to control its risk factors.
4.3. Discussions
Table 2 shows a comparison of the experimental results
using the proposed method, bilateral timing difference,
Fuzzy logic decision, and an adaptive network based
fuzzy inference system (ANFIS). Using the bilateral
timing difference, the timing parameters, PTTf, PTTp
and RT, are used to estimate the degrees of PAD sever-
ity. For example, subjects, No. 2 and No. 12, were con-
firmed as a healthy subject and a MD diabetic patient,
respectively. According to the Table 1, the timing pa-
rameters, PTTf = 6.3850 and PTTp = 16.923, signify
an overlap between Nor (standard: PTTf = 0.3 - 7.4 and
PTTp = 0.4 - 22.3) and MD (standard: PTTf = 5.1 -
23.7 and PTTp = 14.3 - 56.5). If there is overlapping of
the ranges of the timing parameters, the inferences are
affected by the timing reference (heart rate) and meas-
urement errors. This demonstrates that timing parameters
alone are insufficient for the estimation of the degree of
PAD. The clinical physicians decided the possible degree
using a combination of the variances in the three timing
parameters. However, this examination method required
off-line analysis to obtain a diagnosis.
According to the Table 1, each timing parameter has a
mean value and a specific range between the minimum
(Min) and maximum (Max) values. A triangular mem-
bership function is parameterized by its triplet values
(Min, Mean, Max) and is decomposed into three fuzzy
partitions, defined as Nor, MS and SD, as shown in Fig-
ure 8. Therefore, the Fuzzy logic decision has three input
variables with nine triangular membership functions and
1 output variable with three triangular membership func-
tions. The fourteen linguistic Fuzzy rules for the three
degrees are determined by professional physicians over
many examinations, as shown in Table 1. Inference uses
the centre of the mean of the maximal defuzzifier. For
adaptive and self-organizing applications, the literature
[30,31,33,34] also cites the gradient descent method, the
least-square algorithm and the modified least-square al-
gorithm for training a network structure and network
parameters. The ANFIS structure is immediately deter-
mined after the presentation of each input-output pair
and the inference rules. Updating of the parameters is
then performed after the network structure has been de-
cided. In this study, the least-square algorithm is used to
tune network parameters. For the same subjects, the
overall results are shown in Table 2. The sensitivity is
76.2% using Fuzzy logic deciion. The ANFIS has more s
Copyright © 2013 SciRes. OPEN ACCESS
J. X. Wu et al. / J. Biomedical Science and Engineering 6 (2013) 1143-1154
Figure 8. The value of certainty degree versus index,
, and bilateral timing
parameters for separating the diabetic PAD.
than 90% of sensitivity, but the adaptive mechanism is
difficult implement in hardware devices, because of
needs to assign and update network parameters [35].
The experimental results validate the effectiveness of
the proposed VDP oscillator. A comparison with the pro-
posed method and other examination methods, the results
of the 21 patients demonstrate the 85.7% of sensitivity
and allow early detection between Nor and MD or be-
tween MD and SD. However, the trends of the index,
increases as the PAD gradually becomes severity. The
proposed screening method provides a finding to evalu-
ate the orders of PAD for a routine examination. Its ad-
vantages are summarized as follows.
The Burg AR method overcomes spectral leakage and
smoothes the spectra to determine the characteristic
The proposed algorithm for the VDP oscillator is eas-
ily implemented in hardware devices, such as em-
bedded system (ES) and field-programmable gate ar-
ray (FPGA) chips.
The proposed VDP oscillator can also be imple-
mented using analog electronic circuits.
The proposed method potentially allows the use of a
portable monitor for the estimation of diabetic foot PAD
in daily homecare.
The Van der Pol (VDP) oscillator was able to detect pho-
Copyright © 2013 SciRes. OPEN ACCESS
J. X. Wu et al. / J. Biomedical Science and Engineering 6 (2013) 1143-1154 1153
toplethysgraphic signals to estimate severity of periph-
eral arterial disease (PAD) in patients with diabetes mel-
litus. In addition, bilateral non-invasive optical sensors
were helpful for acquiring the PPG signals at the right
and left great toes, combined with the Burg AR method
for determining the frequency spectra of the right- and
left-side PPG signals. Using the bilateral frequency spec-
tra, the VDP oscillator shows the step damped oscillation
responses for diabetic patients with mild and severe se-
verity, respectively, with the increasing amplitudes of
damped oscillation as PAD severity worsening. The os-
cillation amplitude is expressed as an index
to differen-
tiate between diabetic patients with PAD and normal
subjects. The numerical experiments reveal specific ranges
that allow confirmation of the degree, which allows
tracking the trends of PAD for screening and monitoring
of high-risk patients in primary settings as well as in hos-
pitals. For an improved diagnostic design, the proposed
method can be combined with a fuzzy inference system
to derive an index for the diagnosis of PAD. The pro-
posed method potentially allows for constructing a port-
able bio-monitor and the use of telemedicine in a variety
of clinical environments.
This work is supported in part by the National Science Council of Tai-
wan under contract number: NSC99-2221-E-244-007 (August 1 2010-
October 31 2011). The authors would like to thank Dr. Chian-Ming Li
for providing his valuable suggestion and help on experiments.
[1] I. S. Muller, M. L. Bartelink, W. J. C. Grauw, H. J. M.
Hooden, W. H. E. M. Gerwen and G. E. H. M. Rutten,
“Foot Ulceration and Low Limb Amputation in Type 2
Diabetic Patients in Dutch Primary Health Care,” Dia-
betes Care, Vol. 25, No. 3, 2002, pp. 570-574.
[2] J.-Y. David, S. A. Jones and D. P. Giddens, “Modern
Spectral Analysis Techniques for Blood Flow Velocity
and Spectral Measurements with Pulsed Doppler Ultra-
sound,” IEEE Transactions on Biomedical Engineering,
Vol. 38, No. 6, 1991, pp.589-596.
[3] C. Loewe, M. Schoder, T. Rand, U. Hoffmann, J. Sailer,
T. Kos, J. Lammer and S. Thurnher, “Peripheral Vascular
Occlusive Disease: Evaluation with Contrast-Enhanced
Moving-Bed MR Angiography versus Digital Subtraction
Angiography in 106 Patients,” American Journal of Ro-
entgenology, Vol. 179, No. 4, 2002, pp. 1013-1021.
[4] O. Pablo Vasquez, M. Marco Munguia and B. Manders-
son, “Arteriovenous Fistula Stenosis Detection Using
Wavelets and Support Vector Machines,” 31st Annual
International Conference of the IEEE EMBS, Mineapolis,
2-6 September 2009, pp. 1298-1301.
[5] M. Nitzan, A. Babchenko and B. Khonokh, “Very Low
Frequency Variability in Arterial Blood Pressure and
Blood Volume Pulse,” Medical & Biological Engineering
& Computing, Vol. 37, No. 1, 1999, pp. 54-58.
[6] E. D. Übeyli, D. Cvetkovic and I. Cosic, “AR Spectral
Analysis Technique for Human PPG, ECG and EEG
Signals,” Journal of Medical Systems, Vol. 32, No. 3,
2008, pp. 201-206.
[7] Y. M. Akay, M. Akay, W. Welkowitz, S. Lewkowicz and
J. L. Semmlow, “Non Invasive Acoustical Detection of
Coronary Artery Disease: A Comparative Study of Signal
Processing Methods,” IEEE Transactions on Biomedical
Engineering, Vol. 40, No. 6, 1993, pp. 571-578.
[8] J. Allen, “Photoplethysmography and Its Application in
Clinical Physiological Measurement,” Physiological Meas-
urement, Vol. 28, No. 3, 2007, pp. R1-R39.
[9] J. Allen and A. Murray, “Age-Related Changes in the
Characteristics of the Photoplethysmographic Pulse Shape
at Various Body Sites,” Physiological Measurement, Vol.
24, No. 2, 2003, pp. 297-307.
[10] R. Erts, J. Spigulis, I. Kukulis and M. Ozols, “Bilateral
Photoplethysmography Studies of the Leg Arterial Steno-
sis,” Physiological Measurement, Vol. 26, No. 5, 2005,
pp. 865-874.
[11] J. Allen and A. Murray, “Variability of Photoplethysmo-
graphy Peripheral Pulse Measurements at the Ears,
Thumbs, and Toes,” IEE Proceedings on Science, Meas-
urement and Technology, Vol. 147, No. 6, 2000, pp. 403-
407. http://dx.doi.org/10.1049/ip-smt:20000846
[12] P. A. Bonham, T. Kelechi, M. Mueller and J. Robison,
“Are Toe Pressures Measured by a Portable Photo-Phle-
thysmograph Equivalent to Standard Laboratory Tests,”
Journal of Wound, Ostomy and Continence Nurses Soci-
ety, Vol. 37, No. 5, 2010, pp. 475-486.
[13] J. Allen, K. Overbeck, A. F. Nath, A. Murray and G.
Stansby, “A Prospective Comparison of Bilateral Pho-
toplethysmography versus the Ankle-Brachial Pressure
Index for Detecting and Quantifying Lower Limb Pe-
ripheral Arterial Disease,” Journal of Vascular Surgery,
Vol. 47, No. 4, 2008, pp. 794-802.
[14] C.-H. Lin, “Assessment of Bilateral Photoplethysmogra-
phy for Lower Limb Peripheral Vascular Occlusive Dis-
ease Using Color Relation Analysis Classifier,” Com-
puter Method and Program in Biomedicine, Vol. 103, No.
3, 2011, pp. 121-131.
[15] J. Allen, C. P. oates, T. A. Lees and A. Murray, “Pho-
toplethysmography Detection of Lower Limb Peripheral
Arterial Occlusive Disease: A Comparison of Pulse Tim-
ing, Amplitude and Shape Characteristics,” Physiological
Measurement, Vol. 26, No. 5, 2005, pp. 811-821.
[16] M. Fallahpour, D. Megias and M. Ghanbari, “Reversible
Copyright © 2013 SciRes. OPEN ACCESS
J. X. Wu et al. / J. Biomedical Science and Engineering 6 (2013) 1143-1154
Copyright © 2013 SciRes.
and Highcapacity Data Hiding in Medical Images,” IET
Image Processing, Vol. 5, No. 2, 2011, pp. 190-197.
[17] D. M. Vázquez, J. J. Rubio and J. Pacheco, “A Charac-
terization Framework for Epileptic Signals,” IET Image
Processing, Vol. 6, No. 9, 2012, pp. 1227-1235.
[18] H. Heidrich, R. Wenk and P. Hesse, “Frequency of As-
ymptomatic Peripheral Arterial Disease in Patients En-
tering the Department of General and Internal Medicine
of a General Care Hospital,” Vasa, Vol. 33, No. 2, 2004,
pp. 63-67. http://dx.doi.org/10.1024/0301-1526.33.2.63
[19] I. Kauppinen, J. Kauppinen and P. Saarinen, “A Method
for Long Extrapolation of Audio Signals,” Journal of the
Audio Engineering Society, Vol. 49, No. 12, 2001, pp.
[20] A. Broadman, F. S. Schlindwein, A. P. Rocha and A.
Leite, “A Study on the Optimum Order of Autoregressive
Models for Heart Rate Variability,” Physiological Meas-
urement, Vol. 23, 2002, pp. 324-336.
[21] K. Roth, I. Kauppinen, P. A. A. Esquef and V. Valimaki,
“Frequency Warped Burg’s Method for AR-Modeling,”
IEEE Workshop on Applications of Signal Processing to
Audio and Acoustics, 19-22 October 2003, pp. 5-8.
[22] C. Collomb, “Linear Prediction and Levinson-Durbin Al-
gorithm,” 2009.
[23] N. Kannathal, U. Rajendra Acharya, P. Joseph and E. Y.
K. Ng, “Analysis of EEG Signals with and without Re-
flexology Using FFT and Auto Regressive Modeling
Techniques,” Chinese Journal of Medicine, Vol. 1, No. 1,
2006, pp. 12-20.
[24] M. Akay, J. L. Semmlow, W. Welkowitz, M. D. Bauer
and J. B. Kostis, “Detection of Coronary Occlusions Us-
ing Autoregressive Modeling of Diastolic Heart Sounds,”
IEEE Transactions on Biomedical Engineering, Vol. 37,
No. 4, 1990, pp. 366-373.
[25] S. B. Waluya and W. T. van Horssen, “On the Periodic
Solutions of a Generalized Nonlinear Van der Pol Os-
cillator,” Journal of Sound and Vibration, Vol. 268, No. 1,
2003, pp. 209-215.
[26] Q. S. Bi, “Dynamical Analysis of Two Coupled Paramet-
rically Excited Van der Pol Oscillators,” International
Journal of Nonlinear Mechanics, Vol. 39, No. 1, 2004, pp.
33-54. http://dx.doi.org/10.1016/S0020-7462(02)00126-9
[27] G. M. Mahmoud and A. A. M. Farghaly, “Chaos Control
of Chaotic Limit Cycles of Real and Complex Van der
Pol Oscillators,” Chaos, Solitons and Fractals, Vol. 21,
No. 4, 2004, pp. 915-924.
[28] Y. Abbas, J. Ann and A. Merna, “A Multistage Adomian
Decomposition Method for Solving the Autonomous Van
Der Pol System,” Australian Journal of Basic and Ap-
plied Sciences, Vol. 3, No. 4, 2009, pp. 4397-4407.
[29] N. S. Nise, “Control Systems Engineering,” 4th Edition,
John Wiley & Sons, INC., 2004.
[30] Y. C. Du and C. H. Lin, “Adaptive Network-Based Fuzzy
Inference System for Assessment of Lower Limb Peri-
pheral Vascular Occlusive Disease,” Journal of Medical
System, Vol. 36, No. 1, 2012, pp. 301-310.
[31] C. H. Lin, Y. F. Chen, Y. C. Du, J. X. Wu and T. S. Chen,
“Chaos Synchronization Detector Combining Radial Ba-
sis Network for Estimation of Lower Limb Peripheral
Vascular Occlusive Disease,” Lecture Notes in Computer
Science, Vol. 6165, 2010, pp. 126-136.
[32] P. A. Bonham, T. Kelechi, M. Mueller and J. Robison,
“Are Toe Pressures Measured by a Portable Photophle-
thysmograph Equivalent to Standard Laboratory Tests?”
Journal of Wound Ostomy & Continence Nursing, Vol.
37, No. 5, 2010, pp. 475-486.
[33] P. Angelov, E. Lughofer and X. Zou, “Evolving Fuzzy
Classifiers Using Different Model Architectures,” Fuzzy
Sets and Systems, Vol. 159, No. 23, 2008, pp. 3160-3182.
[34] J. J. Rubio, “SOFMLS: Online Self-Organizing Fuzzy
Modified Least Squares Network,” IEEE Transactions on
Fuzzy Systems, Vol. 17, No. 6, 2009, pp. 1296-1309.
[35] C.-H. Lin and G.-W. Lin, “FPGA Implementation of
Fractal Patterns Classifier for Multiple Cardiac Arrhyth-
mias Detection,” Journal of Biomedical Science and En-
gineering, Vol. 5, No. 3, 2012, pp. 120-132.