International Journal of Advances in Science and Technology,
Vol. 3, No.3, 2011
Stochastic Model for Blood Pressure
Analysis
M.Muthu Kumar
1
and K. Senthamarai Kannan
2
1
Department of Mathematics, Noorul Islam Centre for Higher Education , Tamil Nadu , India.
2
Department of Statistics, Manonmaniam Sundaranar University, Tirunelveli , Tamil Nadu , India
Abstract
The heart is an amazing organ. Over a 70 year life time, the heart beats over two billion
times in a human being. People exist in a fluctuating environment. During the course of the day, as
activity changes, the heart reacts to the changing demands. Blood pressure is essentially the
measurement of pressure exerted on the walls of the blood vessels by the blood flowing through them.
Typically depends on the level of blood pressure of the person. Blood pressure is measured by
sphygmomanometer with millimeters of mercury (mmHg) and is represented by systolic and diastolic
pressures. The systolic and diastolic pressures can be broadly divided in to LOW, NORMAL and
HIGH. In this paper deals with some experimental and clinical situations, deterministic differential
equations may give results that are qualitatively to consider statistically and stochastic analysis for
blood pressure.
Keywords: : Blood Pressure, Markov Chain, Transition Probability, Stationary, Chi-square
1. Introduction
The heart is an amazing organ. Over a 70 year life time, the heart beats over two billion times in a
human being. An interruption of this beating pattern often leads to serious neurological damage. This
interruption may occur for a time as brief as a few minutes. But it will be dangerous. Thus the heart
rhythm must be incredibly robust, able to sustain itself. There will be a change if a variety of changes
in the body that arise over a short term as a consequence of ones daily activities and over the long term
as a consequence of normal ageing and diseases [1,2]. Viewed from a theoretical perspective, one can
think of the heart rhythm as a stable limit-cycle oscillation, some of whose properties, such as the
period, may be modified to suit bodily demands that are conveyed to the heart by neural activities and
circulating hormones that regulate cardiac activities. In this chapter an argument is given to prove that
in some experimental and clinical situations, deterministic differential equations may give results that
are qualitatively incorrect and it is necessary to consider stochastic mathematical models in that
situations.
2. Blood Pressure
Arterial blood pressure is influenced by sleep- related breathing disorders. The cardiovascular
consequences can be diagnosed by an accurate recording and analysis of blood pressure, recording
methodologies and an approach for analysis are presented here [3]. The blood pressure chart has been
developed by the efforts of the International Society of Hypertension (ISH) and World Health
Organization (WHO). It is basically designed to guide you about high blood pressure and normal blood
pressure. High blood pressure may lead to fatal conditions such as stroke and heart attack, so it needs to
be monitored constantly.
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International Journal of Advances in Science and Technology,
Vol. 3, No.3, 2011
In the normal heart, electrical activity originates on each heartbeat in a specialized pacemaker
region called the sinus node. The activity then spreads through the upper chambers of the heart (the
atria), then through the atrioventricular node and the His-Purkinje system to the lower chambers of the
heart (the ventricles). At the cellular level, the heartbeat is associated with cyclic changes in the
electrical potential difference across the cell membrane, which separates the intracellular and
extracellular milieu.
The recognition of the presence of cardiac arrhythmias must have arisen in antiquity when people
felt abnormalities in the rhythm of the pulse. However, the analysis of arrhythmias has been enormously
aided by the electrocardiogram, which measures the potential difference arising between points on the
surface of the body as a consequence of the propagation of the action potential through the entire heart.
People exist in a fluctuating environment. During the course of the day, as activity changes, the
heart reacts to the changing demands. For example, everyone is familiar with the notion that physical
activity leads to a more rapid heartbeat. But the heart rate also typically increases somewhat during
inspiration and decreases during expiration. These changes are under the control of a large number of
feedback control systems and are mediated by the nervous system and circulating hormones. Activity of
a class of neurons called sympathetic neurons tends to increase the heart rate and the force of
contraction of the heart, whereas activity of another class of neurons, called parasympathetic neurons,
tends to decrease the heart rate. There are stochastic aspects of this influence. The firing (action
potential) of a nerve cell leads to the release of neurotransmitters in the vicinity of heart cells, which in
turn influence the heart. The neurotransmitters are released in discrete quintal packets called vesicles.
3. Blood Pressure Analysis
Before going further, it is required to know that blood pressure is essentially the measurement of
pressure exerted on the walls of the blood vessels by the blood flowing through them. Blood pressure
readings fall under normal to Hypertension (high blood pressure). Treatment typically depends on the
level of blood pressure of the person [4,5].
The clinical benefits that result from the treatment of uncontrolled hypertension have been
demonstrated in numerous studies [69]. However, questions remain as to what benefits may be
projected for different populations with different levels of blood pressure (BP) and cardiovascular
disease (CVD) risk and how they compare with other medical interventions. Controversies exist in
defining optimal BP treatment goals and the benefits obtained in treating below certain BP levels [10].
The relative roles of systolic and diastolic BP in determining CVD risk are also a subject of ongoing
research [11-14].
Blood pressure is determined by the force and amount of blood pumped. Blood pressure is
continuously changing depending on activity, temperature, diet, emotional, physical problems, diseases,
drugs and alcohol. Blood pressure is measured by sphygmomanometer with millimeters of mercury
(mmHg) and is represented by systolic and diastolic pressures. The systolic pressure is the maximum
pressure in an artery at the moment when the heart is beating. The diastolic pressure is the lowest
pressure in an artery in the moments between beats when the heart is beating. The systolic and diastolic
pressures can be broadly divided in to LOW, NORMAL and HIGH (Table 1).
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International Journal of Advances in Science and Technology,
Vol. 3, No.3, 2011
Table 1: Blood pressure classification
Classification
Diastolic pressure Systolic pressure
mmHg mmHg
LOW < 59 < 90
NORMAL 6089 90129
HIGH > 90 > 130
3.1 Dataset Description
The dataset related blood pressure was collected from Private hospitals in Kanayakumari District, Tamil
Nadu. This sample contains 250 patients record in the age group 30 to 60 years. The data collection was
carried out from October2009 to October 2010. All data records are divided into three categories LOW,
NORMAL and HIGH. In this study, day time blood pressure was only considered. Blood pressure taken in
day time is considered, because the variation in blood pressure can be viewed normally.
3.2 Statistical Analysis
The normal curve from the collection data for the blood pressure analysis are shown in Figure 1 There is a
5% level of significance in the probability distribution curve. When the probability distribution of a subject's
day time blood pressure (BP) is not known and the number of blood pressure measurements taken over the
course of the day is greater than or equal to 30 (ie, n 30), the probability distribution of the average of "n"
blood pressure measurements approximates a normal distribution. The area under the curve to the left of the
line (labeled avg BP, which is the average measured blood pressure in "n" readings) represents the
probability that a hypertensive subject will have an average measured blood pressure less than or equal to avg
BP.
Figure 1. Explanation of the sample distribution of blood pressure.
If this probability is small, then we have very little confidence that the null hypothesis is true; however, as
the probability increases, we have more confidence that the null hypothesis is true. Therefore, the degree of
confidence that we have in stating that a given subject is hypertensive may be quantified in terms of the area
under the probability curve to the left of avg BP (That is, the probability that the subject's "true" average
blood pressure is in the hypertensive range).
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International Journal of Advances in Science and Technology,
Vol. 3, No.3, 2011
The area under the sample distribution curve to the left of the measurement is performed on a test
subject, the probability that this average
blood pressure came from a hypertensive subject (That is, =140/90
mm
Hg). If this probability is small, then we have little confidence that the null hypothesis is true. As this
probability increases, we have greater confidence that the null hypothesis is true (That is, the subject is
"truly" hypertensive).
4. Stochastic Model
In this model a Markov Chain with three states LOW, NORMAL and HIGH. The architecture is shown
in Figure 2. Initial probability is equally distributed. Transition probability matrix is shown in Table 2. From
the below matrices, probability of Normal Blood Pressure is 0.2556; probability for Low Blood Pressure is
0.2222 and probability 0.4812 for High Blood Pressure patients
Table 2 Transition probability matrix
States LOW NORMAL HIGH
LOW 0.2222 0.4074 0.3704
NORMAL 0.0889 0.2556 0.6555
HIGH 0.0977 0.4211 0.4812
Figure 2: Markov Process for three states
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Vol. 3, No.3, 2011
Table : 3 Stationary Transition Probability values
Steps States LOW NORMAL HIGH
1
LOW 0.2222 0.4074 0.3704
NORMAL 0.0889 0.2556 0.6555
HIGH 0.0977 0.4211 0.4812
2
LOW 0.1218 0.3506 0.5276
NORMAL 0.1065 0.3776 0.5759
HIGH 0.1062 0.3501 0.5438
3
LOW 0.1098 0.3614 0.5288
NORMAL 0.1076 0.3572 0.5352
HIGH 0.1078 0.3617 0.5305
4
LOW 0.1082 0.3598 0.5320
NORMAL 0.1080 0.3605 0.5315
HIGH 0.1079 0.3598 0.5323
5
LOW 0.1080 0.3601 0.5319
NORMAL 0.1080 0.3600 0.5321
HIGH 0.1080 0.3601 0.5319
6
LOW 0.1080 0.3600 0.5320
NORMAL 0.1080 0.3600 0.5320
HIGH 0.1080 0.3600 0.5320
7
LOW 0.1080 0.3600 0.5320
NORMAL 0.1080 0.3600 0.5320
HIGH 0.1080 0.3600 0.5320
The steady state, the probability (A
7
) obtained from the Markov chain is shown in Table 3, obtained by
seven steps. The expected number of transitions needed to reach state j from state i for the first time is a
mean first passage time. The determination of probability
ij
f of at least one passage from state i to state j as
=
=
1 n
) n (
ij ij
f f where
) n (
ij
f is the probability of first passage from state i to state j in n transitions. Let
) n (
ij
f can be determined as
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International Journal of Advances in Science and Technology,
Vol. 3, No.3, 2011
=
= + =
1 n
1 k
k) - (n
ij
k
ij
) n (
ij
) n (
ij
1,2,3,.... n , P f f P
1. If
ij
f < 1 , it is not certain that the system will ever pass from state i to state j and =
ij
2. If
ij
f = 1 , the Markov chain is ergodic and the mean first passage time from state i to state j is
computed as
=
=
1 n
(n)
ij ij
f n
Mean of a first passage time for all the states in an m transition matrix P by matrix based formula
( ) i j 1 N - I
j ij
= =
Where I = (m-1) identity matrix
N
j
= transition Matrix P less its j the row and jth column of target j
1 = (m-1) column vector with all elements equal to 1
Let computation of the matrix
N
1
=
(
0.4812 0.4211
0.6555 2556 0.
(
=
(
6.7572 3.8225
5.9502 7094 4
0.5188 0.4211 -
0.6555 - 7444 0
1
1
1
. .
) N I (
|
|
.
|
\
|
=
|
|
.
|
\
|
(
=
|
|
.
|
\
|
10.5797
10.6596
1
1
6.7572 3.8225
5.9502 7094 4
31
21
.
From the above equation a clear claim can be conclude that, 11 times taken for a patient's blood pressure
from low state to normal state or to high state
N
2
=
(
0.4812 0.0977
0.3704 2222 0.
(
=
(
2.1174 0.2659
1.0083 4123 1
0.5188 0.0977 -
0.3704 - 7778 0
1
1
1
. .
) N I (
|
|
.
|
\
|
=
|
|
.
|
\
|
(
=
|
|
.
|
\
|
2.3834
2.4207
1
1
2.1174 0.2659
1.0083 4123 1
32
12
.
The equation gives a conclusion that, a sudden change (2 times) form normal state to low state or
high state, considering the patient's blood pressure.
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International Journal of Advances in Science and Technology,
Vol. 3, No.3, 2011
N
3
=
(
0.2556 0.0889
0.4074 2222 0.
(
=
(
1.4333 0.1638
0.7506 3715 1
0.7444 0.0889 -
0.4074 - 7778 0
1
1
1
. .
) N I (
|
|
.
|
\
|
=
|
|
.
|
\
|
(
=
|
|
.
|
\
|
1.5968
2.1221
1
1
1.4333 0.1638
0.7506 3715 1
23
13
.
The sudden changes (2 times) from high state to low state or normal state are seen after the anlaysis of
patient's blood pressure. This change may due to treatment by the doctors when patients admit in the hospital
with high blood pressure. He/she needs a sudden treatment or change in his blood pressure; otherwise he/she
may be in trouble
5. Test for Homogeneity
From Table 2 to test the hypothesis that transition of various level of Blood Pressure is homogenous. Under
the null hypothesis, the statistic is
-2 log =
i
ij
P )
. j
n (
ij
n
log
j
ij
n 2 = 5.2148
The _
2
value with 6 d.f at 5% level of significance is 12.969. {i.e., _
2
> 5.2148} the null hypotheses
is not rejected. The transitions of the various level of Blood pressure are homogenous.
Next we test the hypothesis that the transition of various levels of Blood Pressure and the different age
groups being homogenous. So we calculated Observed and Expected value form the levels of blood pressure
and age groups (Table 3.4).
Table 4 Observed and Expected value for the levels of blood pressure and age groups
.
Age 30 - 40 Age 41 - 50 Age 51 - 60
Observed Expected Observed Expected Observed Expected
Normal
84 75.6 94 92.1 72 72.3
Low
90 97.7 112 112.6 48 46.7
High
94 127.8 101 92.2 55 53
In this study three age groups are considered among a set of people with age 30 to 60. Under the
null hypothesis, the chi-square test.
( )
=
n
1 i
i
2
i i 2
E
E O
_ = 11.475
The table value of _
2
with 4 d.f, at 5% level of significance is 6.487 (P value < 0.05) It implies that
the null hypothesis is rejected, and then the transitions of the various levels of Blood pressure and
different age groups are not homogenous.
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Vol. 3, No.3, 2011
6. Conclusion
In general blood pressure changes during the day depending on the factors such as, temperature,
diet, emotional changes, diseases, alchoholic consumption and physical and psychological factors.
Analyzing the blood pressure of 250 patients, collected from various hospital records, came into a
conclusion that the transition probability of the blood pressure is homogenous, but among the different
age group of the patients the blood pressure is not always homogenous. The above analysis reveals that
there is 36% of probability for normal blood pressure,11% of probability for Low blood pressure and
53% probability for High blood pressure. Markov chain will give the same result when we statistically
analyze the results.
7. References
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