Capturing dynamic patterns in health care services

Capturing dynamic patterns in health care services

EFTYCHIOS PROTOPAPADAKIS, PANAGIOTIS MANOLITZAS, ANASTASIOS DOULAMIS,

EVANGELOS GRIGOROUDIS, NIKOLAOS MATSATSINIS

Department of Production Engineering and Management

Technical University of Crete

73100, Kounoupidiana, Chania, Creete

GREECE

[email protected], [email protected],[email protected], [email protected],

[email protected]

Abstract: - Hospitals across the globe face the challenge to respond to the public demand for more effective and

transparent health services. One valuable method, among many, to increase hospital’s effectiveness is

simulation. The scope of this paper is the development of a simulation model for the Emergency Department of

General Hospital of Chania. The simulation model, based on genetically optimized neural networks, and

inverse cumulative density functions, was used for the behavioral examination of the hospital in Chania city,

Crete, Greece. The simulating abilities of the model were compared against real life scenarios. Through this

model the hospital managers can further understand the system’s reactions and, therefore, further improve

various factors in order to minimize the total length of stay in the Emergency Department.

Key-Words: - emergency department, health services, neural networks, genetic algorithms, simulation models

1 Introduction In our days every health organization tries to

provide valuable and efficient health services to the

patients by taking into account some constraints like

budget, number of staff, waiting times etc. The most

common characteristic of the Emergency

Department is crowding. According to American

College of Emergency Physicians crowding occurs

when the identified need for emergency services

exceeds available resources for patient care in the

emergency department [1].

Many approaches, from the area of management

and information technology, can be adopted by a

health care organization in order for to optimize its

efficiency and effectiveness and to be competitive.

Many researchers use the Business Process

Reengineering (BPR) for optimizing the procedures

of health care organizations.

BPR is defined as the fundamental rethinking and

radical redesign of business processes to achieve

dramatic improvements in critical, contemporary

measures of performance such as cost, quality,

service and speed. It is obvious that the BPR is a

crucial methodology in order to examine the current

system via simulation [2]. Through the simulation

the management team will elucidate the weak points

of the department and will implement what-if

scenarios in order to examine the reaction of the

system.

Simulation analysis appears to be the right tool in

order to improve a business process and to identify

its bottlenecks. Many researchers have used various

simulation software tools in order to improve

various processes [3]–[6]. Other researchers use

mathematical techniques [7] in order to optimize the

department of emergency medicine.

The scope of this research is the development of a

simulator using neural networks and genetic

algorithms for the estimation of the time that a

patient will stay in the wards of the emergency

department taking into account the level of the

triage and the type of the incident.

The rest of the paper is organized as follows:

Section 2 provides a brief description about the

proposed methodology. Section 3 describes the

modeling assumptions. Section 4 refers to the

features extraction used by the simulator. Section 5

states all the steps for a creation of an adequate

estimator for the department exit time. Section 6

describes the simulation methodology phases.

Section 7 provides the experimental results.

2 The Proposed Methodology The main goal was the creation of an adequate

simulator that produces close to real life scenarios

for the Emergency Department of General Hospital

of Chania, Crete, Greece. The simulator is a

topologically optimized feed forward neural

network (FFNN) that takes as inputs specific

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arithmetic features and produces time estimations as

output. The combination of these features and time

estimations creates specific scenarios; each scenario

is described by 10 arithmetic features, which include

time differences, number of doctors, triage, etc.

The scenarios creation involves quantile

functions and neural networks, which require a large

training data set [8]. Therefore, an adequate amount

of observations had to be gathered in order to

support a smooth building process for the simulator.

The term observation refers to a variety of

parameters which fully describe an incident. These

parameters can be separated into two groups: time-

depended (time-differences among the different

treatment stages) and non-time-depended (triage,

incident category, number of available doctors).

Appropriate cumulative density functions were

created for the time differences and the rest of the

parameters. However, some of the parameters

appear singularities that made unfeasible the

calculation of the inverse of their cumulative

distribution function. Since these parameters were

too complex to be estimated using traditional

statistical methods, suitable nonlinear forecasters

were used (section 5). The methodology, presented

in this paper, consists of two phases.

In the first phase a nonlinear forecaster is trained

according to the gathered observations, in order to

provide the exit time of a patient given a specific

scenario’s parameters. The developed simulator

utilizes both the forecaster and the quantile

functions in order to generate random scenarios.

The second phase investigates if the simulation

procedure, of phase one, produces plausible

scenarios. A second forecaster is created and trained

using only the simulator’s scenarios. Once done, the

forecaster’s performance is evaluated on the real life

observations. The creation of such simulator can be

used by the hospital administration in order to

allocate the available personnel as good as possible.

3 Modeling the System To describe any system we need the concept of the

model, which is usually a mathematical description

of the features of interest. In our case multiple

random variables were used; most of them were

time-difference based, while others not.

A random variable x is defined by its set of possible

values Ω and its probability distribution

function . Assuming that time differences

among the hospital treatment stages follow a

specific distribution, the first step would be the

identification of the distribution or, if it is not

possible, an appropriate approximation.

Let be the probability density function for

the time differences between treatment stage i and

stage i-1. Given a sample of , … , real, non-

negative, observations, at stage i, the most

appropriate pdf, which describes the relative

likelihood for this random variable to take on a

given value, can be found. Once the appropriate pdf

is calculated, we can create the corresponding

cumulative distribution function (c.d.f.): Fx Px c,where c a known number, that describes

the sample population.

Inverse transform sampling (also known as the

inverse transformation method [9]) is a basic

method for pseudo-random number sampling, i.e.

for generating sample numbers at random from any

probability distribution given its c.d.f. The quantile

function is defined by:

F infx: Fx y , 0 y 1 (1)

Let F be the inverse distribution function, defined

by the observations at MED’s department i. Given a

continuous uniform variable 0,1!, the random

variable X FU has distribution Fx . The

inverse transform technique can be used to sample

from exponential, the uniform, the Weibull and the

triangle distributions. Of course, F cannot be

explicitly calculated all the times; many continuous

distributions don't have a closed-form inverse

function (e.g. normal distribution). Therefore, an

appropriate estimator for the X values should be

used. If we are willing to accept numeric solution,

inverse functions can be found.

The Inverse Transform Method for simulating

from continuous random variables have analog in

the discrete case. For instance, if we want to

simulate a random variable X having p.d.f. P$X x% P% then F$X%& ∑ P

%() . Discrete cases may

refer to the number of available doctors, current

department capacity, the triage, etc.

Although the aforementioned parameters have a

straightforward estimation procedure, the same does

not apply for the exit time. The time of exit from the

department depends on a plethora of parameters.

However, as we will see, using no more than 10

variables (continuous or discrete), in combination

with an appropriate neural network, is sufficient for

the creation of random time generator.

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4 Features Extraction Personnel of the hospital had to complete a specific

form during each patient arrival. These forms

(recorded observations) consists of the following

parameters: entry time at the hospital, registration

time, entrance time at the examination room,

diagnose time, exit time, and date of entrance end

departure. In addition, the number of treatment

facility (2 in our case), the number of the available

doctors at the treatment facility, the triage, and the

category of the event were recorded.

Feature vectors oi of size 10×1 were extracted,

were i denotes the number of the observation. These

feature vectors describe time differences as they

occur according to the observations and to the

hospital’s work flow plan. A brief description of the

vectors’ elements is shown at Error! Reference

source not found.

Table 1: Feature vectors’ elements description

Feature vector’s elements brief description

o1 Time difference between entrance time and 00:00 hours

o2 Time difference between arrival and registration

o3 Time difference between registration and examination

o4 Time difference between examination and diagnosis

o5 Category of the event (7 different cases)

o6 Triage

o7 Doctors availability 1st department

o8 Doctors availability 2nd department

o9 Treatment department

o10 Time difference between arrival and departure

The first element of vector o denotes the entry

time, while the elements 2 to 4 describe the time

differences among the registration time, the entrance

time at the examination room, the diagnose time and

the entry time at the hospital, respectively. The time

differences were calculated as follows: The entry

time is calculated as the difference with 00:00

hours. The rest, of the time differences, are

calculated using as a base the previous procedure

time. All the time differences are in hours. Elements

5 to 10 refer to the category of the event, the triage,

the number of the available doctors at the

emergency department no1 and no2, and the

department where the patient is treated, respectively.

5 Department Exit Time Generator In the following lines a brief description, for the

creation an appropriate neural network topology, is

given. FFNN are tested in various cases of complex

environments [10]. Thus, they are an appropriate

technique for the creation of the department’s exit

time generator. All the important characteristics of

the FFNN are created from techniques inspired by

nature (known as genetic algorithms).

The usefulness of the genetic algorithms (GAs) is

generally accepted [11]. The island GA uses a

population of alternative individuals in each of the

islands. Every individual is a FFNN. While eras

pass networks’ parameters are combined in various

ways in order to achieve a suitable topology.

A pair of FFNNs (parents) is combined in order

to create two new FFNNs (children). Children

inherit randomly their topology characteristics from

both their parents. Under specific circumstances,

every one of these characteristics may change

(mutation). The quartet, parents and children, are

then evaluated and the two best will remain,

updating that way the island’s population. An era

has passed when all the population members

participate in the above procedure. In order to bate

the genetic drift, population exchange among the

islands, every four eras. The algorithm terminates

when all eras have passed. Initially, the parameters’

range is described in Table 2 and the main steps of

the genetic algorithm are shown in Fig. 1. The

algorithm is used to parameterize the topology of

the non-linear classifier.

Individuals may mutate at any era. Mutation can

change any of the, previously stated, topology

parameters therefore individuals’ parameters outside

the initially defined range may occur. The fitness of

a network is evaluated using the mean square error

defined as:

*+, 1- .$/0 1 /&2

( (2)

Where 34 a vector of - predictions of the exit time

from the department and 3 the actual values

6 The Simulation Methodology Phases The simulation procedure is separated in two

phases. The first phase involves around the building

of an appropriate scenarios generator, while the

second phase examine the phase one generator

robustness. For each of the time differences the best

fitted cumulative density function is calculated, and

consequently, the inverse density function. Once the

input vectors are available the appropriate nonlinear

forecaster is created through the genetic island

algorithm.

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Random times are generated according to the

inverse density functions. Simultaneously to these

times generation various constrains are activated,

which allow the determination of the place of the

examination and of the treatment as well as the

number of the available doctors. All of the above

parameters are fed as an input to the forecaster and

the exit time is calculated. A significant amount of

plausible scenarios (over 400) is created.

For the quality check of the produced scenarios a

similar approach is followed. A new nonlinear

forecaster is trained using only the produced

scenarios. The performance of the new linear

forecaster is evaluated on the actual observations.

7 Experimental Validations A large number of observations concerning the

health services were collected for the month

February at the public hospital of Chania City,

Crete, Greece. The hospital has two emergency

departments ED1 and ED2. The second one is

running 24hrs/day. Generally, patients that arrive

between 08:00 and 23:00 have to pass through

registration. Depending on the triage (red case-

extremely important) patients can skip registration

and examination at ED1 and are sent directly at ED2

7.1. Experimental setup Initially, by using the 80% of the available

observations, the best possible network is produced

using the island genetic algorithm. The remaining

data were fed to the network, and the overall out of

sample performance is calculated. A vast amount of

possible scenarios was created (more than 400).

For all the time differences except entry time-

departure time the inverse transformation method

was used. As it is shown in Fig. 2, all of the time

differences used in the feature vectors 5, are

distributed exponentially. For these cases the

inverse transformation is described by the formula:

X 1ln 1 1 U λ⁄ , where 1/λ is the expected

value, U is a uniformly distributed number in [0, 1],

and X is a random time variable.

Regarding the department of the examination and

the number of the available doctors the following

constraints should be satisfied:

If the arrival time is before 08:00 hours or after

23:00 hours the patient goes straight to the 2nd

emergency department. The maximum number of

the available doctors is depending on the shift. The

available human resources are shown at Table 3.

Fig. 1: The island genetic algorithm flowchart.

Table 2: Island genetic algorithm parameters’ range.

Parameter Min value Max value

Training epochs 100 400

Number of layers 1 3

Number of neurons (per layer) 4 10

Number of islands 3 3

Number of eras 10 10

Population (per island) 16 16

Table 3: Availability of doctors for each sifts.

Operating hours Dep. 1 Dep.2

08:00-14:30 2 4

14:30-23:00 2 4

23:00-08:00 0 4

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Fig. 2: Time differences cumulative distributions for the

observations’ parameters.

The feature vector for each of these scenarios was

fed to the nonlinear forecaster and a corresponding

departure time was created. During the phase two

the newly created feature vectors and their

corresponding times were used to the formation of a

new forecaster that would be tasted on the original

data

7.2. Results An analytical presentation of the various parameters

impact at the simulation system’s performance is

provided at the following lines. The results are

based on a total of 237 independent simulations. It

appears that a similar pattern is suitable for both

nonlinear forecasters, at phase one and phase two,.

The various parameters refer to the number of

hidden, as well as, the training epochs. The

performance is calculated using well known

statistical errors (MSE, MAPE). Two hidden layers

(Fig. 5) and no more than 400 training epochs (Fig.

6) appear sufficient for the effective capturing of the

departments patterns.

8 Conclusions In this paper a simulation tool was developed for the

emergency department of general hospital of

Chania. The simulation model, based on genetically

optimized neural networks, and inverse cumulative

density functions, was used for the behavioral

examination of the hospital. Complex systems such

as health care facilities provide an excellent

opportunity for soft computing techniques to be

tested.

Fig. 3: ANN’s performance, for the in sample and out of

sample observations.

Fig. 4: ANN’s performance, trained with simulator’s

scenarios, on actual observations.

A combination of neural networks and inverse

transformation techniques is able to describe

sufficiently such dynamic patterns, as it was shown

above. It is possible that different descriptors (i.e.

alternative time differences, date of incident), or a

greater data sample cleared of time related patterns

can further improve the performance.

Through this model the management committee

of the hospital will have the advantage to use this

simulator in order to examine the total length of stay

at the emergency department and the estimated time

of exit from the emergency department. This model

in the future can be updated using more data like the

costs of the emergency department personnel.

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

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hrs

P(time spend ≤ hrs)

registration-examination

examination-diagnose

diagnose-departure

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Fig. 5: Impact of hidden layers num in exit time

estimations.

Fig. 6: Impact of training epochs range in exit time

estimations.

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