COMBINATORIAL DETECTION OF ARRHYTHMIA
Julien Allali
∗
, Pascal Ferraro
†
Laboratoire Bordelais de Recherche en Informatique, Bordeaux, France
∗
Pacific Institute for the Mathematical Sciences, University Simon Fraser, Vancouver, Canada
†
Pacific Institute For the Mathematical Sciences, University of Calgary, Canada
Costas S. Iliopoulos
‡
, Spiros Michalakopoulos
Dept. of Computer Science, King’s College London, Strand, London WC2R 2LS, U.K.
‡
Digital Ecosystems & Business Intelligence Institute, Curtin, Perth, Australia
Keywords:
Arrhythmia, Combinatorics, Electrocardiogram (ECG), Heart rate variability, Pattern matching.
Abstract:
Three problems that arise from electrocardiogram (ECG) interpretation and analysis are presented, followed
by algorithmic solutions based on a combinatorial model. First, the beat classification problem is discussed
and possible solutions are investigated. Secondly, given the R R -intervals, which can be determined using this
combinatorial model, or any Q R S detection algorithm, the heart rate is determined in a statistical manner from
which sinus bradycardia and sinus tachycardia are inferred. Finally, a new combinatorial method for measuring
heart rate variability (HRV) is presented and an algorithm for detecting atrial fibrillation is described. The
developed algorithms were implemented and tests were carried out on records from the MIT-BIH arrhythmia
database. The results of the tests are presented and discussed.
1 INTRODUCTION
1.1 The ECG and its Elements
A ECG is obtained by placing electrodes on the skin
and measuring the direction of electrical current dis-
charged by the heart. The current is plotted into wave-
forms and displayed as in Figure 1.
A lead provides a view of the heart’s electri-
cal activity between one positive and one negative
pole (Conover, 2002; Springhouse (Editor), 2007).
Most standard ECG recordings are obtained using a
12-lead device in clinical settings, and a 2-lead de-
vice in Holter (ambulatory) monitors. The different
leads provide alternate views of the heart and often a
combination of many leads are required to perform a
diagnoses; other times, one or two are enough.
The sample in Figure 1 is from a 2-lead reading, as
are all 48 records in the MIT-BIH (Massachusetts In-
stitute of Technology - Beth Israel Hospital) arrhyth-
mia database (Massachusets Institute of Technology,
1999) that was used for development and testing pur-
poses. The figure shows readings from modified limb
lead II and precordial lead V
1
.
Each beat of the heart corresponds to an EC G
complex, and each ECG complex is made up of ele-
ments, which are waves, intervals and segments. The
presence and configuration of these elements depends
on various factors, including the lead used to take the
reading, the physical device, the presence of noise,
the health of the heart and the age and physiology of
the subject (Thaler, 2006; Stein, 2000).
Figure 2 depicts a closeup view of the EC G com-
plex and its elements. Note that the figure is a free-
hand drawing of a ‘normal’ EC G complex, and the
intervals are not accurately depicted. The two ele-
ments of particular significance for this paper are the
Q R S complex and the P wave.
The Q R S complex comprises the Q , R and S
waves. Not all waves are always present. The P wave
appears before the Q R S complex and is smooth,
rounded and upright in contour in a healthy heart. The
interval between consecutive EC G complexes, usu-
ally measured between consecutive Q R S complexes,
determines the heart rate and rhythm of the heart.
180
Allali J., Ferraro P., S. Iliopoulos C. and Michalakopoulos S. (2010).
COMBINATORIAL DETECTION OF ARRHYTHMIA.
In Proceedings of the Third Inter national Conference on Bio-inspired Systems and Signal Processing, pages 180-187
DOI: 10.5220/0002696501800187
Copyright
c
 SciTePress

Figure 1: Section of ECG for record 101 from MIT-BIH
arrhythmia database.
1.2 Arrhythmias and HRV
Any cardiac irregularity is an arrhythmia, not every-
one of which indicates disease or requires treatment.
Nevertheless, some arrhythmias can indicate serious
cardiac danger and need immediate attention.
A normal heart rate is between 60 and 100 beats
per minute (bpm). Arrhythmia is caused by a dis-
turbance either of the heart rate, the regularity of the
beat, the site of origin or the conduction through the
heart, (Thaler, 2006).
Normally, the heart rate stays within the above
mentioned range, but is not entirely steady, due to
inspiration (expiration), which causes it to speed up
(slow down). The measuring of the “steadiness” of
the heart rate is heart rate variability (HRV).
The elements of the ECG, together with the HRV,
provide information as to what type of arrhythmia the
patient has. Some arrhythmias, though they may have
a different cause, manifest themselves in similar ways
on the ECG. The added study of the HRV can give
more indications and allow a more accurate diagnosis.
1.3 Paper Organization
The paper is organized as follows. Next, in Section 2,
the terms and parameters are defined. In Section 3 the
three problems are presented, suggested solutions are
discussed, and comments are made on the implemen-
tation and the experiments carried out. Each problem,
algorithmic solution and test results occupy their own
subsection, (3.1 to 3.3). Finally, Section 4 contains
further works and conclusions.
2 DEFINITIONS
A signal s is a k-tuple (t, p
MLII
, p
V 1
,...), where t is
the time in seconds, and p
E
is the electrical potential
at lead E in millivolts. When the signal read is from a
single lead, it is represented as a pair, a 2-tuple, (t, p).
P wave
Q wave
R wave
S wave
T wave
U wave
PR interval
QRS complex
QT interval
ST segment
Figure 2: A ‘normal’ EC G complex and its elements.
A sequence of signals s
1
,s
2
,. .., s
n
is a sequence of
pairs, (t
1
, p
1
), (t
2
, p
2
), .. ., (t
n
, p
n
).
A beat b is a pair (t,c), where t denotes the time in
seconds of occurrence of the beat, and c is a character
that describes the type of beat. The values that c can
take are described in Section 3.1.
A sequence of beats B = b
1
,. .., b
n
is a sequence
of pairs, (t
1
,c
1
), . .., (t
n
,c
n
). The R R -interval, rr, is
the time between two consecutive beats:
rr
i
= t
i
−t
i−1
(1)
The rate of change g is the difference between two
consecutive R R -intervals i.e., g
i
= rr
i
− rr
i−1
. The
cumulative rate of change for k beats is the sum of the
absolute rates of change of consecutive pairs i.e.,
|g
i
| + |g
i+1
| + ... + |g
i+k−1
| =
k−1
∑
j=0
|g
i+ j
|
Let
ˆ
Σ and Σ be the following alphabets:
ˆ
Σ = {−, 0,+}
Σ = {C
−
,C
0
,C
+
}
and the set of non-trivial pairs:
A = {(+,0), (−,0)}
(trivial pairs are {(0,0),(+,+),(−, −)}).
Let d be a symbol from the alphabet Σ, which de-
notes a decrease, no significant difference, or an in-
crease in the rate of change. Small differences, dis-
criminated by ν, are considered negligible, thus ν is a
threshold.
The value of d at position i, is determined by
Equation 2:
d
i
=



C
0
, if |rr
i
− rr
i−1
| ≤ ν
C
−
, if rr
i
− rr
i−1
< −ν
C
+
, if rr
i
− rr
i−1
> ν
(2)
The set of all strings over Σ is denoted by Σ
∗
.
Two consecutive rate changes have direction
equivalence, when there’s a smooth transition from
COMBINATORIAL DETECTION OF ARRHYTHMIA
181

one rate change to the other, without a significant
change of direction.
A consecutive sequence of direction equivalent
rate changes is defined as C [x,k] of C
x
, where x ∈
ˆ
Σ,
k ∈ N
+
and for some C
x
∈ Σ.
Formally, C
x
is equivalent to C
y
,
C
x
∼ C
y
(3)
if (x,y) ∈ A. Furthermore, the two sequences are
equivalent
C
x
1
C
x
2
...C
x
k
≈ C
y
1
C
y
2
...C
y
k
(4)
if C
x
i
∼ C
y
i
, ∀i.
For example, C
+
∼ C
0
, but C
−
 C
+
(not equiv-
alent to) and the sequence C
0
C
+
C
+
C
0
≈ C[+, 4],
but C
−
C
+
C
+
C
0
C
−
6≈ C [−,5]. Also note, C [+,k] ≈
C
x
1
C
x
2
...C
x
k
, where x
i
∈ {+,0}, for i ∈ [1..k], and
similarly for C [−,k].
A window is defined to be ω consecutive symbols
of Σ i.e., for i, j ∈ N and 0 ≤ i ≤ j, ω = j − i + 1
denotes the length of the string C
x
i
,C
x
i+1
,...,C
x
j
.
A change of direction occurs when, given
a sequence C
x
1
C
x
2
...C
x
k
all elements of which
are direction equivalent i.e., C [x, k] ≈ C
x
1
C
x
2
...C
x
k
,
the k + 1’th element in the sequence breaks this
equivalence i.e., C
x
1
C
x
2
...C
x
k
C
x
k+1
6≈ C [x,k].
The constant ξ is defined (φ is defined), to be the
maximum (minimum) allowable change of direction
within a window of length ω. For example, for a
window of size ω, and the sequence C [−,i
1
] C [+,i
2
]
.. .C [−,i
m
], the inequality m ≤ ξ (m ≥ φ) must hold.
Finally, ψ is the maximum cumulative rate of change
within a window of length ω i.e.,
∑
w
j=0
|g
i+ j
| ≤ ψ.
3 PROBLEMS & ALGORITHMS
The problems in this section are an attempt to for-
malize ECG interpretation and analysis in a combina-
torial way. In each subsection, the problem is given
with its practical setting, followed by a formal defini-
tion. Next, an algorithmic solution is suggested and
each subsection ends with discussion on the imple-
mentation and experimental results of the suggested
solution.
First, the beat classification problem identifies the
types of beats in an ECG, then the heart rate problem
identifies the rhythm of the heart and finally a new
method for HRV is developed and explained in the
heart rate variability problem.
All the implementations were done on a Win-
dows machine with an Intel Celeron M processor of
1.60GHz and 896MB RAM and the tests were run
through Cygwin, a Linux-like environment for Win-
dows. The code was written in C++ using the STL.
Table 1: Beat classes and types.
Type Description Class
· or N Normal N
A Atrial premature S
S Supraventricular premature S
V Premature ventricular contraction V
F Fusion of ventricular and normal F
E Ventricular escape V
/ Paced beat Q
Q Unclassifiable beat Q
3.1 Beat Classification Problem
The first problem is the classification of beats. The
(ANSI/AAMI, 1998b) standard recommends 5 gen-
eral heartbeat classes:
• Class N: normal and bundle branch block beats.
• Class S: supraventricular ectopic beats (SVEB).
• Class V : ventricular ectopic beats (VEB).
• Class F: fusion of a VEB and a normal beat.
• Class Q: paced and non-classifiable beats.
There are 15 different heartbeat types anno-
tated in the 48 records in the MIT-BIH arrhythmia
database (Goldberger et al., 2000; Massachusets In-
stitute of Technology, 1999). Table 1 depicts some of
the most commonly found types and their equivalent
classes.
Problem 1 (Beat Classification). Given an ECG as
a sequence of signals s
1
, ... , s
n
= (t
1
, p
1
), ... , (t
n
, p
n
)
and a sequence of times of occurrence of Q R S com-
plexes t
0
1
, . .., t
0
m
, where t
0
i
∈ t
1
.. .t
n
,∀ 1 ≤ i ≤ n, deter-
mine the class of the beats i.e., output the list of beats,
B = b
1
,b
2
.. .,b
m
= (t
0
1
,c
1
),(t
0
2
,c
2
). .., (t
0
m
,c
m
), where
c
j
∈ {N,S,V,F,Q}, ∀ 1 ≤ j ≤ m.
A number of methods to solve the beat classifica-
tion problem have been proposed. Most rely on sig-
nal processing, such as (Afonso et al., 1997) and use
varying computer science methodologies, for instance
neural networks (Hu et al., 1997). In (Chazal et al.,
2004), the authors achieved good accuracy results on
the MIT-BIH arrhythmia database. Recently, in (Il-
iopoulos and Michalakopoulos, 2009) a new model
for ECG interpretation was introduced. This model
can be used for beat classification; an algorithmic out-
line follows.
STEP 1
Identify the P waves and Q R S complexes. A
detailed description of identifying the Q R S com-
plexes is in (Iliopoulos and Michalakopoulos, 2009).
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182

V
N
0
2
1
N
V
3
4
5
6
7
V
8
N
V
N
N
V
9
V
V
10
11
12
Figure 3: Aho-Corasick automaton for keyword list
{NV NV,NNV NNV,VV,VVV }. Trivial failure links are not
shown.
The algorithm relies on searching for the pattern
C [++,2]C [−−, 2] in the ECG.
The P wave can be identified by searching for the
more complex pattern C [0,k
1
]C [+,k
2
]C [−,k
3
], where
k
1
,k
2
,k
3
∈ N
+
are determined during the algorithms
learning period.
STEP 2
If both the R R interval and the P wave are regular,
classify the beat as normal (N).
STEP 3
If the beat is not normal, then determine the beat type
from {S,V,F,Q} by examining the P wave, R R in-
terval and the Q R S complex and its duration.
Regardless of the method used, the output is
a string of characters, C = c
1
c
2
.. .c
m
, where c
i
∈
{N, S,V,F, Q}. Two types of ventricular arrhyth-
mia, namely ventricular bigeminy and ventricular
trigeminy can be identified by a simple pattern search.
For k ∈ N
+
and k ≥ 2, (NV )
k
indicates ventricular
bigeminy, while (NNV )
k
ventricular trigeminy.
A potentially more dangerous arrhythmia, ven-
tricular tachycardia manifests as a run of 3 or
more consecutive premature ventricular contractions
(PVC), (Thaler, 2006). Thus, ventricular tachycardia
is represented by the pattern V
`
,` ≥ 3.
Another interesting pattern are ventricular
couplets, VV . To identify these patterns on-
line, an Aho-Corasick automaton (Aho and
Corasick, 1975), is built for the dictionary
{NV NV, NNV NNV,VV,VVV } as shown in Fig-
ure 3.
Figure 4: Output of program for Problem 1, on record 106.
The above patterns can also be determined offline
in constant (O(1)) time, if the list of beats is indexed
using e.g. a suffix array or suffix tree (Crochemore
et al., 2007; Gusfield, 1997). For a prerecorded
ECG and a list of beats of size |m|, this indexing
can be done in O(m) time, given that the alphabet
{N, S,V,F, Q} is of constant size.
3.1.1 Experimental Results
The Aho-Corasick automaton of figure 3 was imple-
mented, and tests were carried out on the MIT-BIH
arrhythmia database. A short sample of the output for
record 106 can be seen in Figure 4.
Record 106 was chosen because it manifests many
occurrences of the above mentioned arrhythmias. The
program simply identifies the type of arrhythmia, and
outputs the information to the screen and to a text file,
for later processing.
3.2 Heart Rate Problem
The second problem identifies arrhythmias that man-
ifest as irregular heart rate. The cells of the heart
discharge electrical potential at different rates. The
dominant cells in a healthy heart are located in the
sinoatrial (SA) node. The term heart rate refers to
the sinus rhythm i.e., the rate that the SA node fires.
The ECG encapsulates information about the rates of
other parts of the heart, such as the atrial rate and the
ventricular rate.
The heart rate problem is concerned with calcu-
lating the sinus rhythm. A normal sinus rhythm is
between 60 and 100 bpm. Anything faster is termed
tachycardia, and anything slower bradycardia. Ac-
cording to (ANSI/AAMI, 1998a), sinus tachycardia
should be identified when the heart rate is faster than
110 bpm and sinus bradycardia for rates slower than
50 bpm, for 15 consecutive seconds.
It should be noted that these arrhythmias do not
always indicate irregularities. For example, a long-
distance runner will in rest have a heart rate below 60
bpm, and the same subject will have a heart rate of
well above 100 bpm during rigorous exercise. The al-
gorithm presented below does not thus claim to iden-
COMBINATORIAL DETECTION OF ARRHYTHMIA
183

tify dangerous conditions, but simply indicates when
the rhythm has fallen or risen to a level which should
be checked by a specialist.
Problem 2 (Heart Rate). Given a sequence of beats,
B = b
1
,b
2
.. .,b
n
= (t
1
,c
1
),(t
2
,c
2
). .., (t
n
,c
n
), calcu-
late the heart rate and detect sinus tachycardia and
sinus bradycardia.
A solution to this problem is depicted in Algo-
rithm 1. The series of beats are processed sequen-
tially. Each step can be visualized as a shift of a win-
dow of size ω by one beat to the right.
The heart rate is determined by first summing the
R R intervals within the window. This is only done
for valid windows. A valid window is one which con-
tains only normal (c
i
= N) beats.
The average R R interval of the window is cal-
culated on Line 16. This average is then converted
into beats per minute i.e., heart rate, on Line 17. If
the heart rate is below 50 bpm, the algorithm reports
sinus bradycardia, if it’s above 110 bpm, sinus tachy-
cardia is reported.
For example, if the average is found to be a beat
every 0.8 secs, the heart rate is 75 bpm, whereas if the
average is every 1.2 secs, the heart rate is 50 bpm and
sinus bradycardia is reported.
Algorithm 1. Process Heart Rate.
1: ave ← 0  average R R -interval in window of size ω
2: hr ← 0  heart rate in bpm
3: win count ← 0
4: win total ← 0
5: i ← 1
6: while not end of B do  for each beat in B
7: (t
i
,c
i
) ← B[i]
8: if c
i
6= ‘N’ then
9: win count ← 0, win total ← 0
10: else  beat is normal
11: win count ← win count + 1
12: win total ← win total + (t
i
−t
i−1
)
13: if win count ≥ ω then
14: if win count > ω then
15: win total ← win total −(t
i−ω
−t
i−ω−1
)
16: ave ← win total/ω
17: calculate hr from ave
18: if hr ≤ 50 bpm then
19: report SINUS BRADYCARDIA
20: else if hr ≥ 110 bpm then
21: report SINUS TACHYCARDIA
22: increment i  next beat
3.2.1 Experimental Results
Algorithm 1 was implemented, and tests were car-
ried out on the MIT-BIH arrhythmia database. The
PhysioNet utility ann2arr, see (Goldberger et al.,
2000), was used to first extract the beats from the
Figure 5: Output of program for Problem 2, on record 232.
ECG. This produces a text file with the beats in B =
(t
1
,c
1
),. .., (t
n
,c
n
) format, that serves as input to the
program.
The heart rate is calculated by the R R intervals as
in Algorithm 1. The test database averages the beats
over 3 consecutive R R intervals to calculate the heart
rate, and so the window length for the purpose of cal-
culating the normal sinus rhythm was set to ω = 3.
The heart rate was calculated for the beats marked
with N (normal), Q (unclassifiable), L (left bundle
branch block) and R (right bundle branch block).
The arrhythmias looked for are sinus bradycardia
and sinus tachycardia. The only records in the test
database with sinus bradycardia are records 202 and
232, whereas sinus tachycardia is not annotated in the
database. Figure 5 shows the output of the program
on record 232, where sinus bradycardia is correctly
identified and the heart rate is shown in the last line.
Software and ambulatory ECG devices should al-
low user-defined input parameters, as recommended
in (ANSI/AAMI, 1998a). In this implementation, the
identification of sinus bradycardia was set to 50 bpm,
and sinus tachycardia to 110 bpm, but it could eas-
ily be modified to allow these values to be input as
parameters.
Other input parameters seen in Figure 5 are w = 3,
which is the window length ω, n = 12, which is the ν
threshold and c = 8, which is the change of direction
parameter (see Section 3.3).
3.3 Heart Rate Variability Problem
Heart rate variability (HRV) measures the beat-to-
beat fluctuations in heart rate. It has been associated
with a number of cardiac and pathologic conditions.
A low HRV i.e., with few fluctuations, has been
used in clinical practice to predict risk after acute my-
ocardial infarction and is an early warning sign of di-
abetic neuropathy (Task Force of the European Soci-
ety of Cardiology the North American Society of Pac-
ing Electrophysiology, 1996). It has also been linked
to phobic anxiety, which in itself is a risk factor for fa-
tal coronary artery disease which can lead to sudden
cardiac death (Kawachi et al., 1995).
A high HRV on the other hand, together with other
ECG characteristics such as the absence of P waves,
may indicate atrial fibrillation (Thaler, 2006).
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184

Figure 6: Section of ECG for record 201. Each beat (R
wave peak) is annotated as a dot ‘•’, and the start of atrial
fibrillation as ‘(AFIB’.
Figure 6 shows a section of the ECG for record
201 in the MIT-BIH arrhythmia database. The onset
of atrial fibrillation is annotated as “(AFIB” in the fig-
ure. The characteristics of the ECG at this point can
be summed up as in (Thaler, 2006), as “irregularly ir-
regular tachycardia” and the absence of P waves.
Various methods of measuring HRV have
been proposed, including time-domain measures,
frequency-domain measures and geometric measures.
In this section, a combinatorial method is suggested.
An algorithm is proposed and presented, together
with experimental results of an implementation of the
model on the detection of atrial fibrillation.
Problem 3 (Heart Rate Variability). Given a se-
quence of beats, B = b
1
,. .., b
n
= (t
1
,c
1
),. .., (t
n
,c
n
),
detect irregularities in the heart rate variability and
report certain types of arrhythmia.
The definitions in Section 2 serve as the basis for
solving this problem. The series of beats B are trans-
formed into characters d
i
from the alphabet Σ. The
threshold value ν is important in determining d
i
, as
is illustrated in the last two columns in Table 2. The
data in the table is taken from record 101 in the test
database.
Note that the R R intervals, rr
i
, are measured in
number of readings of electrical potential per sec-
ond, and that the sample rate in the test database is
σ = 360. Thus, a R R interval of 360 is equivalent to
1 sec.
The same data is represented combinatorially in
Figure 7. A change of direction is intuitively a point
at which an up arrow follows a down arrow, or vice
versa.
Within a window of size ω, a range of change of
direction indicates a healthy heart; too few (< φ), or
too many (> ξ) indicate irregularities.
Next an outline of a generic algorithm for solving
the heart rate variability problem is presented. Pseu-
docode for identifying atrial fibrillation based on the
generic algorithm follows.
Table 2: Data from record 101.
i t
i
rr
i
rr
i
− rr
i−1
d
i
(ν = 5) d
i
(ν = 30)
427 6:21:964 370 18 C
+
C
0
428 6:23:019 380 10 C
+
C
0
429 6:23:978 345 -35 C
−
C
−
430 6:24:947 349 4 C
0
C
0
431 6:25:928 353 4 C
0
C
0
432 6:26:972 376 23 C
+
C
0
433 6:28:025 379 3 C
0
C
0
434 6:29:064 374 -5 C
−
C
0
435 6:30:092 370 -4 C
0
C
0
436 6:31:042 342 -28 C
−
C
0
437 6:32:017 351 9 C
+
C
0
438 6:33:039 368 17 C
+
C
0
STEP 1
Process series of beats B:
1. Convert into a string on Σ
∗
.
2. Calculate number of change of direction, for win-
dow of length ω.
3. Accumulate absolute total change of direction
within window.
STEP 2
If within the window the number of change of direc-
tion exceeds ξ, then report high HRV, and do further
checks on ECG.
Else if within the window the number of change of
direction is less than φ then report low HRV, and do
further checks on ECG.
STEP 3
If within the window the cumulative change of direc-
tion is greater than ψ, then indicate possible arrhyth-
mia and do further checks on ECG.
The type of arrhythmia sought, determines what
“do further checks” of Step 2 and 3 is. The pro-
gram could simply report an irregularity and advise
a cardiologist to take a closer look at the ECG con-
figuration, waves and intervals, at this point in time.
Alternatively, some automated task can be performed,
such as is presented in the pseudocode in Algorithm 2
for the identification of atrial fibrillation.
Algorithm 2 serves as an illustration of the po-
tentials of the model. The detection of atrial fibril-
lation was chosen because it is one of the most com-
mon types of arrhythmias, affecting 1% of the worlds
population, rising to 4% in the over 65’s (Garratt,
2001). The more important reason for this research
paper and model though, is that there is data in the
test database from patients presenting this type of ar-
rhythmia, which makes the program meaningful.
COMBINATORIAL DETECTION OF ARRHYTHMIA
185

428427 429 430 431
i
432 433 434 435 436 437 438
ν = 5
ν = 30
ω = 8
C
0
C
0
C
−
C
+
C
+
C
0
C
−
C
0
C
0
C
0
C
0
C
0
C
0
C
0
C
0
C
−
Figure 7: Combinatorial depiction of HRV for data from
Table 2.
As in Algorithm 1, only normal beats are taken
into account; the implementation details are omitted
from the pseudocode for reasons of simplicity.
The algorithm is an online algorithm and is O(n),
except for Line 13. This check however, can be
done in O(ω) time, using, for example, the algorithm
in (Portet, 2008), and given that ω is constant, the al-
gorithm remains O(n).
Algorithm 2. Identify Atrial Fibrillation.
1: cod count ← 0 . change of direction within window
2: i ← 1
3: while not end of B do . for each beat
4: (t
i
,c
i
) ← B[i]
5: if c
i
= ‘N’ then . if beat is normal
6: determine d
i
by Equations 1 and 2
7: if d
i−ω
 d
i−ω−1
then . equivalence test
8: cod count ← cod count − 1
9: if d
i
 d
i−1
then
10: cod count ← cod count + 1
11: if cod count > ξ then . high HRV
12: report high HRV
13: if P waves absent in window then
14: report ATRIAL FIBRILLATION
15: increment i . next beat
3.3.1 Experimental Results
Algorithm 2 was implemented and tests were carried
out on the records that present atrial fibrillation (201,
202, 203, 210, 217, 219, 221, 222), and randomly
chosen records (100, 104, 105, 119, 121, 205, 208,
230), that don’t present this type of arrhythmia.
Figure 8 shows the output on the program on
record 201 and for parameters ω = 16, ξ = 6 and
ν = 12. The program outputs the time that high HRV
is first observed. The time normal sinus rhythm re-
sumes is also displayed. Checking the ECG for the
absence of P waves at this point in time is a strong
indicator of atrial fibrillation. Figure 9 shows the pro-
gram run on records 105, 119 and 121 which do not
exhibit atrial fibrillation, for the same parameters as
the test above.
Figure 8: Output of program for Problem 3, on record 201.
Abnormal HRV is detected.
Figure 9: Output of program for Problem 3, on records 105,
119 and 121. These records do not demonstrate unusual
HRV and none is detected.
The ideal input parameters for the program de-
pend on each individual subject. Thus, the learning
period of the algorithm should be used to estimate
these optimal values. In the case of the test database,
this learning period is the first 5 mins of each record.
In the case of a real-world situation, a reasonable
learning period could be combined with user-defined
parameters and the length of the period itself may be
a user-defined parameter.
4 CONCLUSIONS & FURTHER
WORKS
The combinatorial formalization of some practical
ECG analysis and interpretation problems has been
attempted in this paper. The suggested algorithmic
solutions presented, are online and use pattern match-
ing and simple statistical techniques to determine a
subjects heart rate and heart rate variability (HRV).
The algorithms were implemented and certain
types of arrhythmia are reported on data from the
MIT-BIH arrhythmia database. The experimental re-
sults were promising, a fact that suggests that further
investigation and development of the model and pro-
grams are desirable.
Improvements can be made to the existing algo-
rithms and the program can be extended to identify
further types of arrhythmia. HRV measures more ac-
curately identify irregularities on longer recordings
than the 30 minutes in the records in the test database.
This suggests that the model would possibly better be
used in ambulatory (Holter) devices, and telemetry
communication could be employed for sending cru-
BIOSIGNALS 2010 - International Conference on Bio-inspired Systems and Signal Processing
186

cial data to the lab or clinic.
An additional improvement would be the appli-
cation of more sophisticated statistical analysis algo-
rithms, employed on the data in the learning period,
to determine ‘better’ parameters as input to the test
period of the program.
Another use of the model is for prerecorded ECG
data. Holter monitors are often used to gather ECG
data for an extended period of time, for later analy-
sis. Using the combinatorial model, this data could
be indexed using for example a suffix tree or suffix
array (Crochemore et al., 2007; Gusfield, 1997). For
an ECG of length n, this indexing can be done in O(n)
time.
Finally, further areas of investigation is to extend
the modeling of the ECG data to include more than
one lead, as well as to implement the beat classifica-
tion algorithm of Section 3.1.
REFERENCES
Afonso, V. X., Wieben, O., Tompkins, W. J., Nguyen, T. Q.,
and Luo, S. (1997). Filter bank-based ecg beat clas-
sification. In Proceedings of the 19th Annual Interna-
tional Conference of the IEEE, volume 1, pages 97–
100. Engineering in Medicine and Biology Society.
Aho, A. V. and Corasick, M. J. (1975). Efficient string
matching: an aid to bibliographic search. Commun.
ACM, 18(6):333–340.
ANSI/AAMI (1998a). EC38: Ambulatory Electrocardio-
graphs. Association for the Advancement of Medical
Instrumentation.
ANSI/AAMI (1998b). EC57: Testing and Reporting Per-
formance Results of Cardiac Rhythm and ST Segment.
Association for the Advancement of Medical Instru-
mentation.
Chazal, P. D., O’Dwyer, M., and Reilly, R. B. (2004). Auto-
matic classification of heartbeats using ecg morphol-
ogy and heartbeat interval features. IEEE Transac-
tions on Biomedical Engineering, 51:1196–1206.
Conover, M. B. (2002). Understanding Electrocardiogra-
phy. Mosby.
Crochemore, M., Hancart, C., and Lecroq, T. (2007). Algo-
rithms on Strings. Cambridge University Press.
Garratt, C. (2001). Mechanisms and Management of Car-
diac Arrhythmias. BMJ Books, London.
Goldberger, A. L., Amaral, L. A. N., Glass, L., Haus-
dorff, J. M., Ivanov, P. C., Mark, R. G., Mietus,
J. E., Peng, G. B. M. C. K., and Stanley, H. E.
(2000). PhysioBank, PhysioToolkit, and Phys-
ioNet: Components of a new research resource
for complex physiologic signals. Circulation,
101(23):e215–e220. Circulation Electronic Pages:
http://circ.ahajournals.org/cgi/content/full/101/23/
e215.
Gusfield, D. (1997). Algorithms on Strings, Trees and Se-
quences: Computer Science and Computational Biol-
ogy. Cambridge University Press.
Hu, Y. H., Palreddy, S., and Tompkins, W. J. (1997). A
patient-adaptable ecg beat classifier using a mixture
of experts approach. Biomedical Engineering, IEEE
Transactions on, 44(9):891–900.
Iliopoulos, C. S. and Michalakopoulos, S. (2009). A combi-
natorial model for ecg interpretation. In Proceedings
of ICMIBE 2009, International Conference on Med-
ical Informatics and Biomedical Engineering. World
Academy of Science.
Kawachi, I., Sparrow, D., Vokonas, P., and Weiss, S. (1995).
Decreased heart rate variability in men with phobic
anxiety (data from the normative aging study). The
American journal of cardiology, 75(14):882–885.
Massachusets Institute of Technology (1999). Mit-bih ecg
database. Available: http://ecg.mit.edu/.
Portet, F. (2008). P wave detector with pp rhythm tracking:
evaluation in different arrhythmia contexts. Physio-
logical Measurement, 29(1):141–155.
Springhouse (Editor) (2007). ECG Interpretation Made In-
credibly Easy! Lippincott Williams & Wilkins, 4
th
edition.
Stein, E. (2000). Rapid Analysis of Arrhythmias. Lippincott
Williams & Wilkins, 3
rd
edition.
Task Force of the European Society of Cardiology the
North American Society of Pacing Electrophysiology
(1996). Heart Rate Variability : Standards of Measure-
ment, Physiological Interpretation, and Clinical Use.
Circulation, 93(5):1043–1065.
Thaler, M. S. (2006). The Only EKG Book You’ll Ever Need.
Lippincott Williams & Wilkins, 5
th
edition.
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