Research Article
Mathematical Modeling of the Expert System Predicting
the Severity of Acute Pancreatitis

Maria A. Ivanchuk,1 Vitalij V. Maksimyuk,2 and Igor V. Malyk3

1 Department of Biological Physics and Medical Informatics, Bukovinian State Medical University, Kobyljanska Street 42,
Chernivtsi 58000, Ukraine

2 Department of Surgery, Bukovinian State Medical University, Golovna Street 137, Chernivtsi 58000, Ukraine
3 Department of the System Analysis and Insurance and Financial Mathematics, Chernivtsi National University of Yuriy Fedkovich,
Unversitetska Street 12, Chernivtsi 58012, Ukraine

Correspondence should be addressed to Maria A. Ivanchuk; mgracia@ukr.net

Received 26 December 2013; Accepted 22 May 2014; Published 9 June 2014

Academic Editor: Daniel Kendoff

Copyright © 2014 Maria A. Ivanchuk 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.

The method of building the hyperplane which separates the convex hulls in the Euclidean space 𝑅𝑛 is proposed. The algorithm of
prediction of the presence of severity in patients based on this method is developed and applied in practice to predict the presence
of severity in patients with acute pancreatitis.

1. Introduction

During the last decades, pronounced tendency to the
relentless increase in morbidity in acute pancreatitis is
observed. Thus, the depth of pathomorphological pancre-
atic parenchyma lesions can vary from the development of
edematous pancreatitis up to pancreatic necrosis. However,
accurate predicting of the probable nature of the lesion of the
pancreas in the early stages of acute pancreatitis is one of the
most difficult problems of modern pancreatology. Diagnostic
and the predictive probability of existing laboratory and
instrumental diagnostic markers and rating scales does not
exceed 70–80% [1–3]. Such situation is a major difficulty in
selecting the adequate treatment strategy in the initial stages
of acute pancreatitis. Thus the search for new methods of
accurate predicting of acute pancreatitis’ severity becomes an
urgent problem.

Development of mathematical approaches for prediction
in medicine was developed by Fisher, the father of the
linear discriminant analysis [4]. Currently, there are many
approaches to solving this problem: cluster analysis, the
construction of predictive tables, image recognition, and lin-
ear programming. Fundamentals of building the prognostic
tables and Wald serial analysis are described in [5]. Cluster

analysis is commonly used for solving the tasks of medical
prediction.

In the paper [6], the procedure of cluster analysis with
a study of the indices of the daily variability of cardiac
rhythm in patients with the ischemic disease of heart is
examined. In [7] using national data from the Scientific
Registry of Transplant Recipients authors compare transplant
and wait-list hospitalization rates. They suggest two marginal
methods to analyze such clustered recurrent event data; the
first model postulates a common baseline event rate, while
the second features cluster-specific baseline rates. Results
from the proposed models to those based on a frailty model
were compared with the various methods compared and
contrasted. Three major considerations in designing a cluster
analysis are described in [8]. The first relates to selection
of the individuals. The second consideration is selection of
variables for measurement and the third consideration is how
many variables to choose to enter into a cluster analysis. To
classify clinical phenotypes of anti-neutrophil cytoplasmic
antibody-associated vasculitis, cluster analysis was used in
[9]. Researches on the general theory of diagnosis, classifi-
cation, and application of optimization methods for pattern
recognition, solving applied problems in medicine and biol-
ogy, are conducted by Mangasarian et al. for many years [10].

Hindawi Publishing Corporation
Journal of Computational Medicine
Volume 2014, Article ID 532453, 4 pages
http://dx.doi.org/10.1155/2014/532453



2 Journal of Computational Medicine

But universal method for solving problems of recogni-
tion, identification, and diagnosis does not exist. Therefore,
development of methods for predicting in medicine still
remains relevant. One among the many challenges of recog-
nition is the task of constructing hyperplanes which separate
two convex sets. Many manuscripts [11–16] are devoted to the
solution of this problem.

We propose a methodology for constructing convex
hulls and their separation, which can be used for modeling
expert medical prognostic systems (e.g., to separate groups
of patients with different degrees of severity of the disease for
prediction of severity in patients).

2. Methods

2.1. Separation of the Convex Hulls. Let us have two sets of
points 𝐴 = {𝑎

𝑖
= (𝑎
1

𝑖
,𝑎
2

𝑖
, . . . ,𝑎

𝑛

𝑖
), 𝑖 = 1,𝑚

𝐴
} and 𝐵 =

{𝑏
𝑖
= (𝑏
1

𝑖
,𝑏
2

𝑖
, . . . ,𝑏

𝑛

𝑖
), 𝑖 = 1,𝑚

𝐵
} in Euclidean space 𝑅𝑛. Let

𝑚 be number of points in the set. We must find the separate
hyperplane:

𝐿
𝑝
= {𝑥 ∈𝑅

𝑛

: ⟨𝑝,𝑥⟩ = 𝛾}, 𝑝 ̸=0, (1)

where ⟨𝑝,𝑥⟩ is the scalar product of the vectors𝑝and𝑥 such
that sets 𝐴 and 𝐵 can be placed in the different half-spaces:

𝐿
+

𝑝
= {𝑥 ∈𝑅

𝑛

: ⟨𝑝,𝑥⟩ > 𝛾},

𝐿
−

𝑝
= {𝑥 ∈𝑅

𝑛

: ⟨𝑝,𝑥⟩ < 𝛾}.

(2)

To build the convex hull conv
𝐴
for the set 𝐴, for each of

𝐶
𝑛

𝑚
𝐴

points’ combinations from the set𝐴, if it is possible, build
the hyperplane

𝐻
𝑝
= {𝑥 ∈𝑅

𝑛

: ⟨𝑝,𝑥⟩ = 𝛽}, 𝑝 ̸=0. (3)

Coordinates of the vector 𝑝 = (𝑝1, . . . ,𝑝𝑛) are found as
minors(𝑛−1)order for elements of the first row of the matrix:

(

𝑥
1

−𝑎
1

1
𝑥
2

−𝑎
2

1
⋅ ⋅ ⋅ 𝑥
𝑛

−𝑎
𝑛

1

𝑎
1

2
−𝑎
1

1
𝑎
2

2
−𝑎
2

1
⋅ ⋅ ⋅ 𝑎
𝑛

2
−𝑎
𝑛

1

⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

𝑎
1

𝑛
−𝑎
1

1
𝑎
2

𝑛
−𝑎
2

1
⋅ ⋅ ⋅ 𝑎
𝑛

𝑛
−𝑎
𝑛

1

), (4)

where 𝑥 ∈ 𝑅𝑛, 𝑎
𝑖
∈ 𝐴, 𝑖 = 1,𝑛. Coefficient 𝛽 is determined

from the following equation:

𝛽=−(𝑎
1

1
𝑝
1

+𝑎
2

1
𝑝
2

+ ⋅ ⋅ ⋅ +𝑎
𝑛

1
𝑝
𝑛

). (5)

If all points of the set 𝐴 are in the one of half-spaces of
hyperplane 𝐻

𝑃
, then polygon 𝑎

1
𝑎
2
⋅ ⋅ ⋅𝑎
𝑛
is one of the convex

hull’s hyperfaces. The complex of all hyperfaces is the convex
hull conv

𝐴
.

Point 𝑏
𝑖
∈ 𝐵 is called outlier if point 𝑏

𝑖
is internal for

the conv
𝐴
. Point 𝑏

𝑖
is outlier if there is at least one hyperface

𝑎
1
𝑎
2
⋅ ⋅ ⋅𝑎
𝑛
∈ conv

𝐴
that
󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󳨀→
𝑐𝑓

󵄨󵄨󵄨󵄨󵄨󵄨󵄨
=

󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󳨀→
𝑐𝑏
𝑖

󵄨󵄨󵄨󵄨󵄨󵄨󵄨
+

󵄨󵄨󵄨󵄨󵄨󵄨󵄨

󳨀󳨀→
𝑏
𝑖
𝑓

󵄨󵄨󵄨󵄨󵄨󵄨󵄨
, (6)

where point 𝑐 ∈ int conv
𝐴
, point 𝑓 is the intersection point

of the hyperplane 𝐻
𝑝
(𝑎
1
𝑎
2
⋅ ⋅ ⋅𝑎
𝑛
∈𝐻
𝑝
), and line

𝑥
1

−𝑐
1

𝑏
1
−𝑐
1
=
𝑥
2

−𝑐
2

𝑏
2
−𝑐
2
= ⋅ ⋅ ⋅ =

𝑥
𝑛

−𝑐
𝑛

𝑏
𝑛
−𝑐
𝑛
. (7)

To find the point 𝑓 let us write (3) in parametric form:

𝑥
1

= 𝑐
1

+(𝑏
1

−𝑐
1

)𝑡

𝑥
2

= 𝑐
2

+(𝑏
2

−𝑐
2

)𝑡

⋅ ⋅ ⋅

𝑥
𝑛

= 𝑐
𝑛

+(𝑏
𝑛

−𝑐
𝑛

)𝑡.

(8)

Put (8) in the hyperplane equation (3) and find parameter 𝑡:

𝑡
∗

=
𝑝
1

𝑐
1

+𝑝
2

𝑐
2

+ ⋅ ⋅ ⋅ +𝑝
𝑛

𝑐
𝑛

+𝛽

𝑝
1
(𝑐
1
−𝑏
1
)+𝑝
2
(𝑐
2
−𝑏
2
)+ ⋅ ⋅ ⋅ +𝑝

𝑛
(𝑐
𝑛
−𝑏
𝑛
)
. (9)

To find coordinates of the point 𝑓 let us put (9) in (8):

𝑓
1

= 𝑐
1

+(𝑏
1

−𝑐
1

)𝑡
∗

𝑓
2

= 𝑐
2

+(𝑏
2

−𝑐
2

)𝑡
∗

...

𝑓
𝑛

= 𝑐
𝑛

+(𝑏
𝑛

−𝑐
𝑛

)𝑡
∗

.

(10)

After finding all outliers from the sets 𝐴 and 𝐵 eject
outliers from the set, with less number of outliers. Build the
new convex hulls and find the outliers. If there are outliers in
the new convex hulls, eject them. If there are not any outliers,
the convex hulls do not intersect. According to consequence
of Hahn-Banach theorem there is a nonzero linear functional
𝐿
𝑝
that separates conv

𝐴
and conv

𝐵
[17].

Find the separating functional 𝐿
𝑝
as hyperplane parallel

to one of convex hulls’ hyperfaces. Choose hyperface so that
convex hulls conv

𝐴
and conv

𝐵
are in different half-spaces

formed by hyperplane parallel to this hyperface. Find points
𝑎min ∈ conv𝐴 and 𝑏min ∈ conv𝐵 so that |

󳨀󳨀󳨀󳨀󳨀󳨀→
𝑎min𝑏min| =

min(|
󳨀󳨀→
𝑎
𝑖
𝑏
𝑗
| : 𝑎

𝑖
∈ conv

𝐴
, 𝑏
𝑗
∈ conv

𝐵
, 𝑖 = 1,𝑚

𝐴
, 𝑗 =

1,𝑚
𝐵
). Let 𝑑min ∈

󳨀󳨀󳨀󳨀󳨀󳨀→
𝑎min𝑏min. For each hyperface {𝐻𝐴min :

𝐻
𝐴min

⊂ conv
𝐴
;𝑎min ∈ 𝐻𝐴min} and {𝐻𝐵min : 𝐻𝐵min ⊂

conv
𝐵
;𝑏min ∈ 𝐻𝐵min} build the parallel hyperplane {𝐿𝑝 :

𝑑min ∈ 𝐿𝑝;𝐿𝑝‖𝐻𝐴minor 𝐿𝑝‖𝐻𝐵min}. If 𝑎𝑖 ∈ 𝐿
+

𝑝
, for all 𝑎

𝑖
∈ 𝐴,

𝑖 = 1,𝑚
𝐴
, and 𝑏

𝑗
∈ 𝐿
−

𝑝
, for all 𝑏

𝑗
∈ 𝐵, 𝑗 = 1,𝑚

𝐵
, then 𝐿

𝑝
is

separating hyperplane.

2.2. Modeling the Expert System of Predicting the Presence of
Severity in Patients. Let us have two groups of patients: 𝐴,
patients with severity, and 𝐵, patients without severity. There
are 𝑛
0
parameters (factors which affect the severity) known

for each patient.



Journal of Computational Medicine 3

During modelling we used the terms sensitivity (Se) and
specificity (Sp):

Se =
𝑎

𝑎+𝑐
, Sp =

𝑑

𝑏+𝑑
, (11)

where𝑎 is the true positives,𝑏 is the false positives (overdiag-
nosis errors), 𝑐 is the false negatives (underdiagnosis errors),
and 𝑑 is the true negatives. The sensitivity of a clinical test
refers to the ability of the test to correctly identify those
patients with the disease. The specificity of a clinical test refers
to the ability of the test to correctly identify those patients
without the disease [18].

We created an algorithm of modelling the expert system
in a way that uses the least amount of features for the
best result. Information of the parameters was found using
Kulback’s information measure [5]. We built convex hulls
for the most informative factor. If convex hulls intersect, we
found outliers—the points from the set 𝐴 that are internal to
conv
𝐵
and the points from the set𝐵 that are internal to conv

𝐴
.

The set𝐴outliers are underdiagnosis errors. The set𝐵outliers
are overdiagnosis errors. We built the prognostic system to
find the patients with severity, so we rejected the outliers from
the set𝐵. Let the set𝑂

𝐵
= {𝑜
𝑖
: 𝑜
𝑖
∈𝐵∩ int conv

𝐴
, 𝑖 = 1,𝑚

𝑂
𝐵
}

be the set of outliers from 𝐵. After rejecting, we get a new set
𝐵
󸀠

=𝐵/𝑂
𝐵
.

If you build the expert system for differential diagnosis,
you reject outliers out of the set where there are less of them.

If the percentage of rejected points is more than the
significance level

𝑚
𝑂
𝐵

𝑚
𝐵

>𝛼, (12)

the next (the most informative) factor was added. The space
dimension is increased by 1. In the new space convex hulls
were built and the outliers were rejected. The space dimension
was increased until preassigned significance level. If all
available diagnostic information was used, but preassigned
significance level was not reached, then decision of not suffi-
cient information was taken. When preassigned significance
level was reached, we found the separating hyperplanes. The
algorithm for modelling the prognostic system is represented
on the Figure 1. The results were checked in the control group
and the hyperplane with maximal sensitivity was chosen.

The complexity of this algorithm is 𝑂(𝑚𝑛+1) [19] if the
convex hulls are built by search of all combinations of points.
The complexity of this algorithm is 𝑂(𝑚2) if the convex hulls
are built by Jarvis march or “gift wrapping” algorithm [20].

3. Results

3.1. The Expert System of Predicting the Presence of Severity
in Patients with Acute Pancreatitis. The research involved 60
persons with severe and 28 patients with nonsevere acute
pancreatitis. Among them, there were 57 (64.8%) men and
31 (35.2%) women. The mean age was 48.54 years (±15.18)
in males and 56.21 (±17.91) in females. The most common
etiology was alcohol consumption (48.3%), followed by
gallstones (34.2%). In 17.5% no identifiable cause was found.

+

−

Find the outliers O 
+

−

Decision of not sufficient 

Begin

End

Reject the outliers B󳰀 = B/O

Find Lp

n := 1

mo
mB

> 𝛼 n := n + 1 n < n0

information
Choose Lmaxp : Se(Lmaxp )

Build convA, convB

Find convB󳰀

= max {Se(Lp)}

Figure 1: Algorithm for modelling the prognostic system.

The diagnostic criteria for acute pancreatitis were those
defined by the 2006 AP Guidelines, as the presence of at least
two of the following features: (1) characteristic abdominal
pain, (2) elevation over 3 times the upper normal limit of
serum amylase/lipase, and (3) characteristic features on com-
puter tomography (CT) scan [21]. Severe acute pancreatitis
was diagnosed according strictly to Atlanta criteria: Early
Prognostic Scores, APACHE II ≥ 8, Ranson ≥ 3; Organ
Failure, systolic pressure < 90 mmHg, creatinine > 2.0 mg/L
after rehydration, PaO

2
≤ 60 mmHg; Local Complications

(on CT scan), Necrosis, Abscess, and Pseudocyst [22].
Patients were divided into two samples—training (50

patients with severity and 20 without them) and control (10
patients with severity and 8 without). The level of significance
was 𝛼 = 0,01. The algorithm presented above was used for
patients with training set.

For 𝑛 = 1, the percentage of outliers was 29.5%.

For 𝑛 = 2, the percentage of outliers was 3%.

For 𝑛 = 3, the percentage of outliers was 1.4%.

For 𝑛 = 4, the percentage of outliers was 0%.

We got 8 hyperplanes which separate the convex hulls
of the training samples. Two of them had higher sensitivity
and specificity (we got only 1 (6%) of underdiagnosis errors



4 Journal of Computational Medicine

and there were no overdiagnosis errors for the control sample
with these hyperplanes):

−18937.5𝑥
1

−200.3𝑥
2

+5007.3𝑥
3

+42348𝑥
4

+310958.6 = 0,

−4802.5𝑥
1

−142.8𝑥
2

+5007.3𝑥
3

+2158.1𝑥
4

+177176.4 = 0,

(13)

where 𝑥1 is time before hospitalization, 𝑥2 is blood lipase, 𝑥3
is amylase urine, and 𝑥4 is BMI.

So, we built the expert system with sensitivity Se = 94%.
Statistical errors are seen only in 1 patient in the con-

trol group, who were diagnosed with interstitial edematous
pancreatitis development on the background severe diabetes
mellitus. According to Expert System the acute pancreatitis
without severity was predicted. This error, in our view, is
associated with late ambulation of the patient for medical care
as a result of atypical course of acute pancreatitis, increased
blood and urine amylase, and increased BMI, which is the
characteristic signs of diabetes mellitus. That is, in this case,
some of the most important prognostic parameters of acute
pancreatitis have been characterized by different diseases in
particular of diabetes mellitus, which caused the error.

4. Conclusions

The method of separation of convex hulls in Euclidean space
by constructing a separating hyperplane parallel to one of
the convex hulls hyperfaces is proposed. On the basis of
this method the algorithm for modelling the prognostic
system is stated. The proposed algorithm is applied in practice
to predict the presence of severity in patients with acute
pancreatitis and gives 94% correct results for the control
sample, while diagnostic and the predictive probability of
existing laboratory and instrumental diagnostic markers and
rating scales does not exceed 70–80%. Clinical application of
the developed mathematical model predicting the severity of
acute pancreatitis promotes proper choice of treatment tactics
and allows improving final results of these patients’ treatment.

Conflict of Interests

The authors declare that there is no conflict of interests
regarding the publication of this paper.

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Behavioural 
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Endocrinology
International Journal of

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Hindawi Publishing Corporation
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Disease Markers

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BioMed 
Research International

Oncology
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Oxidative Medicine and 
Cellular Longevity

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PPAR Research

The Scientific 
World Journal
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Immunology Research
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Journal of

Obesity
Journal of

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 Computational and  
Mathematical Methods 
in Medicine

Ophthalmology
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Diabetes Research
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Research and Treatment
AIDS

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Gastroenterology 
Research and Practice

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Parkinson’s 
Disease

Evidence-Based 
Complementary and 
Alternative Medicine

Volume 2014
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