Facility Location under Uncertainty and Spatial Data Analytics in Healthcare

by

Derya Demirta¸s

A thesis submitted in conformity with the requirements

for the degree of Doctor of Philosophy

Graduate Department of Mechanical and Industrial Engineering

University of Toronto

c

Abstract

Facility Location under Uncertainty and Spatial Data Analytics in Healthcare

Derya Demirta¸s

Doctor of Philosophy

Graduate Department of Mechanical and Industrial Engineering

University of Toronto

2016

Out-of-hospital cardiac arrest (OHCA) is a significant public health issue and treatment,

namely, cardiopulmonary resuscitation and defibrillation, is very time-sensitive. Public

access defibrillation programs, which deploy automated external defibrillators (AEDs) for

bystander use in an emergency, have been shown to reduce the time to defibrillation and

improve survival rates. The focus of this thesis is on data-driven decision making aimed

at improving survival from OHCA by analyzing cardiac arrest risk and optimizing AED

deployment. This work establishes a unique marriage of data analytics and facility

loca-tion optimizaloca-tion to address both the demand (cardiac arrest) and supply (AED) sides

of the AED deployment problem. In the demand side, we analyze the spatiotemporal

trends of OHCAs in Toronto and show that the OHCA risk is stable at the neighborhood

level over time. In other words, high risk areas tend to remain high risk, which supports

focusing public health resources for cardiac arrest intervention and prevention in those

areas to increase the efficiency of these scarce resources and improve the long-term

im-pact. In the supply side, we develop a comprehensive modeling framework to support

data-driven decision making in the deployment of public location AEDs, with the

ulti-mate goal of increasing the likelihood of AED usage in a cardiac arrest emergency. As a

part of this framework, we formulate three optimization models that consider

probabilis-tic coverage of cardiac arrests using AEDs and address specific, real-life scenarios about

AED retrieval and usage. Our models generalize existing location models and

and a contribution of this work lies in the development of mixed integer linear

formula-tion equivalents and tight and easily computable bounds. Next, we use kernel density

estimation to derive a spatial probability distribution of cardiac arrests that is used for

optimization and model evaluation. Using data from Toronto, Canada, we show that

optimizing AED deployment outperforms the existing approach by 40% in coverage and

substantial gains can be achieved through relocating existing AEDs. Our results suggest

that improvements in survival and cost-effectiveness are possible with optimization.

Dedication

To my parents and my sister

First and foremost, I would like to thank my co-supervisor Prof. Timothy Chan. Words

cannot describe my gratitude for your support. Thank you for your excellent guidance in

research and in professional development. Thank you for believing in me and encouraging

me to always aim for the top. Thank you for not only your research support but also the

emotional support and friendship that helped me get through some of the hardest times

of my life. It was a privilege to learn how to be a good advisor first-hand, and I hope to

become as good of a supervisor to my students as you have been for me.

I would also like to thank my co-supervisor Prof. Roy Kwon. I will be forever grateful

to you for accepting me as your PhD student. I truly appreciate the opportunity you

gave me to do research at the University of Toronto. Being in company of so many bright

and creative people was a privilege. Thank you for allowing me to pursue my academic

interests freely, and for your guidance and support throughout the years.

A sincere thank you to my committee members Profs. Oded Berman, Michael Carter

and Chi-Guhn Lee for the time and energy they spent providing helpful suggestions and

comments on this research. I am extremely thankful to my external examiner, Prof.

Armann Ingolfsson, for his invaluable feedback on this thesis. I am very grateful to Prof.

Michael Carter for giving me the opportunity to be his teaching assistant for four years

and making this experience very pleasant. I have gained invaluable knowledge in course

design and classroom teaching under your guidance. I would also like to thank you for

your support throughout my job search.

I owe a special thanks to Dr. Laurie Morrison for providing me with the medical

data, for her insightful comments and her full support for this research. Laurie, you are

an inspiring role model to me and many other women in science, thank you for being so

encouraging. I am also grateful to Dr. Steve Brooks for his meticulous and brilliant edits

on our medical papers, and our fruitful discussions.

I will be eternally grateful to Dr. Eugenia Tsao from UofT Academic Success Center

for her guidance and encouragement during the thesis writing process. Eugenia, thank

you for always being positive and having confidence in me, you made a huge difference

in my life. A big thanks to my thesis writing buddies, Jacob Hogan (History), Dianna

Roberts-Zauderer (Study of Religion), and Paula Karger (Comparative Literature).

I feel blessed to have been surrounded by wonderful friends during this journey. A

heart-felt thank you to my labmates: Velibor Mi˘si´c, Daria Terekhov, Auyon Siddiq,

Chris Sun, Justin Boutilier, Houra Mahmoudzadeh, Sarina Turner, Philip Mar, Taewoo

Lee, Heyse Li, Brendan Eagen, Ali Goli, Islay Wright, Aaron Babier, Neal Kaw, Iman

Dayarian, and Rafid Mahmood. I truly appreciate your time and effort in providing

me with feedback on papers and presentations, and brainstorming research ideas. Most

importantly, though, thank you for your friendship. Our gatherings at the cottage, lab

socials, numerous celebrations and board game nights were some of the most memorable

times I had in the last five years. Outside of the Applied Optimization Lab, I would like

to thank Jim Kuo and Kimia Ghobadi for their support and friendship. Thank you for

celebrating the happiest moments with me and cheering me up at the hardest times.

I also would like to thank my UTORG family: Jim Kuo, Kimia Ghobadi, Curtiss

Luong, Peter Zhang, Carly Henshaw, Shefali Kulkarni-Thaker, Auyon Siddiq, Sarina

Turner, and Taewoo Lee. Working together towards making UTORG better was one of

the most enriching experiences I had during my PhD. Thank you for being a great team.

To my parents and my sister, thank you for your boundless love and support. Thank

you mom for praying for me and believing in me. Feeling your support is what keeps me

going. Thank you dad, for being a huge supporter whichever career direction I take and

always being proud of me. To my dear sister, thank you for being there for me whenever

I needed, it gives me comfort to know that I can always count on you.

My deepest gratitude goes out to my fianc´e, Olivier Nguon. I am so glad you were by

my side at every step in this journey. Your love and support helped me through many

challenges, and I now know that there is no darkness that your smile could not brighten.

1 Introduction and Literature Review 1

1.1 Literature Review . . . 4

1.1.1 Relevant literature on stability of OHCAs . . . 4

1.1.2 Relevant literature on AED deployment . . . 5

1.1.3 Relevant literature on spatial distribution of OHCAs . . . 8

1.2 Contributions . . . 9

2 Spatiotemporal Stability of Cardiac Arrests 13 2.1 The Importance of Stability . . . 13

2.2 Methods . . . 14 2.2.1 Study design . . . 14 2.2.2 Study setting . . . 15 2.2.3 Study population . . . 15 2.2.4 Data sources . . . 16 2.2.5 Analyses . . . 17 2.3 Results . . . 19

2.3.1 Spatiotemporal stability of OHCAs . . . 19

2.3.2 Spatiotemporal stability of public OHCAs . . . 21

2.3.3 Spatiotemporal stability of OHCAs adjusted for population . . . . 23

2.4 Discussion . . . 23

2.4.1 Limitations . . . 30

2.5 Conclusion . . . 30

3 Facility Location Models for AED Deployment 31 3.1 Deterministic AED Location Model . . . 32

3.1.1 Motivating the need for a different coverage function . . . 34

3.1.2 Exponential coverage function . . . 34

3.2 Probabilistic Coverage Models for AED Location . . . 36

3.2.1 Model 1: Multiple-responder . . . 39

3.2.2 Model 2: Single-responder worst case . . . 44

3.2.3 Model 3: Single-responder best case . . . 47

3.2.4 Relationship between models 1, 2, 3 and MCLP . . . 50

4 Spatial Analysis of Cardiac Arrests 53 4.1 Cardiac Arrest Data . . . 53

4.2 Estimating a Spatial Distribution of Cardiac Arrest Risk . . . 53

4.3 Simulating from the Spatial Distribution of Cardiac Arrests . . . 56

5 Results from the Optimization Framework and Implications 58 5.1 AED Location Data . . . 58

5.2 Optimization Setup . . . 61

5.3 Results . . . 62

5.3.1 Comparing models and bounds . . . 62

5.3.2 The value of optimizing AED deployment . . . 65

5.3.3 The value of relocating existing AEDs . . . 70

5.3.4 Map-based geographical analysis of optimal AED locations . . . . 71

5.4 Implications for Public Access Defibrillator Deployment . . . 75

5.4.1 Lay responder behavior . . . 77

5.4.2 Coverage versus survival . . . 78

5.4.4 The need for a central decision maker . . . 82

5.5 Conclusion . . . 83

6 Conclusions 84

Bibliography 88

List of Tables

2.1 Demographic, clinical and EMS characteristics of the included OHCAs . 21

2.2 Ranking of the ten highest risk neighborhoods based on OHCA rate (2007–

2014) . . . 26

2.3 Ranking of the ten highest risk neighborhoods based on public OHCA rate

(2007–2014) . . . 29

4.1 Intra-class correlation comparison of split cardiac arrest sets . . . 55

5.1 Problem sizes for all optimization models using 11,701 AED locations

(10,032 candidate, 1,669 existing) and the 5,000 cardiac arrests from the

training set . . . 62

5.2 Bounds on Model 1 optimized and evaluated using the training set of 5,000

cardiac arrests . . . 63

5.3 Coverage results from Model 1, 2 and 3 on 100 testing sets, each with 300

cardiac arrests . . . 66

5.4 Sensitivity analysis on α in the decaying coverage function . . . . 67

5.5 Coverage values from actual, heuristic and optimized methods, calculated

over 100 testing sets . . . 69

5.6 Coverage values from single-stage versus multi-stage optimization,

calcu-lated over 100 testing sets . . . 70

lated over 100 testing sets . . . 71

5.8 Coverage versus predicted number of survival for Model 3 . . . 80

List of Figures

1.1 Automated external defibrillator (AED) . . . 2

2.1 Consort diagram for the study identifying patients included in the analysis 20

2.2 Toronto neighborhoods . . . 22

2.3 Number of out-of-hospital cardiac arrests per year . . . 23

2.4 Average number of OHCAs per year across neighborhoods . . . 24

2.5 Average annual number of OHCAs by neighborhood: Woburn (1), Moss

Park (2), and South Parkdale (3) . . . 25

2.6 The range (2007–2014) of out-of-hospital cardiac arrests by neighborhood 26

2.7 Number of public out-of-hospital cardiac arrests per year . . . 27

2.8 Average number of public OHCAs per year across neighborhoods . . . . 28

2.9 Average annual number of public OHCAs by neighborhood: The Bay

Street Corridor (1), West Humber-Clairville (2), and Waterfront

Communities-The Island (3) . . . 28

3.1 An exponentially decreasing coverage function . . . 35

3.2 Differences in AED deployment due to different coverage functions . . . . 36

4.1 Historical cardiac arrest locations (December 16, 2005 – July 15, 2010) . 54

4.2 Kernel density estimation of cardiac arrests in Toronto in 3D . . . 55

4.3 Kernel density estimation of cardiac arrests in Toronto in 2D . . . 56

4.4 Split kernel density estimation: simulated vs. historical cardiac arrests . 57

5.1 Existing AED locations, Toronto . . . 59

5.2 Candidate locations for new AEDs, Toronto . . . 60

5.3 Overview of our framework combining optimization models with kernel density estimated cardiac arrest risk . . . 62

5.4 Bounds on Model 1 using the training set of cardiac arrests . . . 64

5.5 Coverage results from Model 1, 2 and 3 on 100 testing sets, each with 300 cardiac arrests . . . 65

5.6 A comparison of coverage from the actual, heuristic, and optimized de-ployment methods . . . 69

5.7 Cumulative distance distribution of cardiac arrests to the closest AED . . 70

5.8 5000 cardiac arrest locations comprising the training set, Toronto . . . . 72

5.9 Optimal and existing AED locations, Toronto . . . 73

5.10 Optimal AED locations, downtown Toronto . . . 74

5.11 Differences between Model 2, Model 3 and MCLP solution . . . 75

5.12 Differences between Model 2 and Model 3 . . . 76

Abbreviations

AED Automated external defibrillator. AHA American Heart Association.

CI Confidence interval.

CPR Cardiopulmonary resuscitation.

EMS Emergency medical services. ERC European Resuscitation Council.

HSFC Heart and Stroke Foundation of Canada.

ICC Intraclass correlation coefficient.

KDE Kernel density estimation.

LB Lower bound.

MCLP Maximal covering location problem.

MEXCLP Maximum expected covering location problem.

OHCA Out-of-hospital cardiac arrest.

PAD Public access defibrillation.

ROC Resuscitation Outcomes Consortium.

SPARC Strategies for Post Arrest Care.

UB Upper bound. UN United Nations.

VaR Value at risk.

Chapter 1

Introduction and Literature Review

Out-of-hospital cardiac arrest (OHCA) is a significant public health issue, responsible for

approximately 400,000 deaths annually in North America (Mozaffarian et al. 2015, Heart

and Stroke Foundation of Canada 2015). Cardiac arrest occurs when the heart stops

pumping blood in a coordinated fashion due to abnormal heart rhythms. It is different

from (but may be caused by) a heart attack where the heart continues to beat but blood

flow to the heart is obstructed. The likelihood of survival from cardiac arrest decreases

by 7 to 10% for every minute of delay in treatment (Larsen et al. 1993, Valenzuela et al.

1997). In fact, less than 9% of cardiac arrest victims survive to hospital discharge (Nichol

et al. 2008, Brooks et al. 2010). It has been shown that the likelihood of survival can

be substantial (50–75%) with early cardiopulmonary resuscitation (CPR) and a

defibril-latory shock to the heart (Valenzuela et al. 2000, Page et al. 2000, Caffrey et al. 2002).

While emergency medical services (EMS) personnel are usually the first responders to a

cardiac arrest emergency, they often do not arrive in time to save the patient. Especially

in large metropolitan cities, heavy traffic, urban sprawl and road construction may result

in longer ambulance travel times (Trowbridge et al. 2009). This motivates the integration

of bystanders into the emergency response system by training and encouraging them to

provide timely CPR and defibrillation in a public cardiac arrest emergency. To facilitate

this integration, public access defibrillation (PAD) programs aim to place automated

external defibrillators (AEDs) in public locations for bystander use.

An AED (Figure 1.1) is a portable electronic device with vocal and visual prompts

that automatically diagnoses cardiac rhythms and delivers a shock to correct abnormal

activity in the heart if needed. AEDs are safe, easy to use, and can be used effectively by

lay responders with little or no training. In fact, it has been shown that untrained sixth

grade students can use them almost as well as trained paramedics (Gundry et al. 1999).

The study found that the average time to defibrillation in a simulated arrest situation

was only 90 seconds for the students, whereas it took 67 seconds on average for the

emergency medical technicians/ paramedics. Even though PAD programs are found to

be associated with markedly improved survival rates (Aufderheide et al. 2006, Hazinski

et al. 2005, Hallstrom et al. 2004), only a small percentage of OHCA victims have an

AED applied before EMS arrival (Culley et al. 2004). Success of PAD programs depends

on many factors, including public awareness and willingness of bystanders, but first and

foremost, AEDs need to be located well and they need to be accessible to have a chance

of being used in an emergency.

Figure 1.1: Automated external defibrillator (AED)

Chapter 1. Introduction and Literature Review 3

Foundation of Canada (HSFC), working with local PAD programs, assist communities in

obtaining AEDs. Current AHA guidelines suggest placing AEDs in locations with one or

more cardiac arrests in the last five years, which is cost prohibitive. In practice, priority

locations for AED deployment are typically public venues such as schools, convention

centers, community centers, and international airports. However, these placements are

often made without knowledge of the actual cardiac arrest risk. For instance, in Toronto,

the vast majority of registered AEDs are placed in elementary and secondary schools,

even though the relative risk of cardiac arrest in these locations is low (Brooks et al. 2013).

Furthermore, government funding for public AEDs is often targeted to specific venues

that are deemed appropriate by the government but are not necessarily the riskiest.

The focus of this thesis is on data-driven decision making aimed at improving

sur-vival from OHCA by analyzing cardiac arrest risk and optimizing AED deployment. This

work establishes a unique marriage of data analytics and facility location optimization to

address both the demand (cardiac arrest) and supply (AED) sides of the AED

deploy-ment problem. We begin by analyzing the spatiotemporal trends of OHCAs in Toronto

to better understand the demand side of this equation. We show that the OHCA risk

is stable at the neighborhood level over time, i.e., high risk areas tend to remain high

risk, which supports focusing public health resources for cardiac arrest intervention and

prevention in those areas. In the supply (AED) side, we develop a comprehensive

mod-eling framework to support data-driven decision making in the deployment of public

location AEDs, with the ultimate goal of increasing the likelihood of AED usage in a

car-diac arrest emergency. As a part of this framework, we build three optimization models

that consider probabilistic coverage of cardiac arrests using AEDs and address specific,

real-life scenarios about AED retrieval and usage. Model 1 considers the scenario where

multiple lay responders at the scene of a cardiac arrest fan out and search for nearby

AEDs independently. Models 2 and 3 examine the cases when there is only one lay

a general optimization model that encompasses all three models. Next, we estimate a

geographical distribution of cardiac arrest risk using historical cardiac arrest data and

kernel density estimation. Our framework uses the estimated distribution to generate

both a “training set” of cardiac arrests to be used as input to the optimization models,

as well as “testing sets” of cardiac arrests to be used to evaluate the optimization output.

Lastly, we demonstrate the value of applying our framework using data from Toronto,

Canada, and show that significant monetary savings and improvements in cardiac arrest

coverage may be possible through optimization.

1.1

Literature Review

1.1.1

Relevant literature on stability of OHCAs

Studies showed that OHCA incidence rates vary significantly from city to city (Becker

et al. 1993, Nichol et al. 2008, Berdowski et al. 2010), and even from neighborhood

to neighborhood within the same city (Lerner et al. 2005, Ong et al. 2008, Warden

et al. 2012). Several studies have analyzed the spatial distribution of cardiac arrests and

identified cardiac arrest hot spots (Soo et al. 2001, Lerner et al. 2005, Sasson et al. 2012).

Others have examined temporal trends in cardiac arrest across an entire city or county

(Rea et al. 2003, Brooks et al. 2010). However, there has been limited study of the

distribution of cardiac arrests in space and time simultaneously. One exception is Sasson

et al. (2010), which examined the stability of OHCA incidence rates within census tracts

in Fulton County, Georgia over a 3-year period. Their focus was to identify census tracts

with high rates of OHCA and low rates of bystander CPR. However, spatiotemporal

stability of OHCAs that occurred in public places (public OHCAs) has not been studied

before. Our study differs from Sasson et al. (2010) in several aspects: 1) We analyze eight

years of data, and therefore capture longer term trends. 2) We examine the stability of

Chapter 1. Introduction and Literature Review 5

4,150/km2_{, six times that of Fulton County, which is 675/ km}2_{. The 100 most populous}

cities in United States have population densities between 3,886/ km2 _{and 22,711/ km}2

(US Census Bureau 2010). Therefore, Toronto is more similar to other major cities and

urban settings than Fulton County. 4) Our study is the first to analyze the stability

of the subgroup of public OHCAs. Public cardiac arrests have a much higher chance

of receiving CPR and having AED applied by a bystander (Litwin et al. 1987, Jackson

and Swor 1997, Holmberg et al. 2000), therefore their stability is crucial for long-term

resource planning.

1.1.2

Relevant literature on AED deployment

Most of the research on AED location has been conducted in the medical community in

the last 15 years. One of the earliest efforts categorized buildings in Seattle, Washington

and determined the frequency of cardiac arrests that occur in each building type (Becker

et al. 1998). Similar studies were conducted in Kansas City, Missouri (Gratton et al.

1999), Windsor, Canada (Fedoruk et al. 2002), G¨oteborg, Sweden (Engdahl and Herlitz

2005), and Copenhagen, Denmark (Folke et al. 2009). Many of these studies identified

facilities having large daily flows of people, such as transportation hubs and shopping

malls, as high risk. Fedoruk et al. (2002) and Gratton et al. (1999) also found that

casinos are high risk for cardiac arrests. Interestingly, it was shown that placing AEDs

in casinos markedly increased the survival rate from OHCA (Valenzuela et al. 2000),

primarily due to faster response times made possible by the surveillance systems in place

for casino security. In general, these studies suggest that ranking locations based on

their risk of cardiac arrest is an appropriate method to guide the placement of public

AEDs. However, there are several limitations to this approach. First, identification of

high-risk locations is highly dependent on the demographics and infrastructure of the

studied cities and therefore not generalizable. For instance, while Becker et al. (1998)

(2005) listed golf courses among the low-incidence sites. Second, a substantial number

of cardiac arrests happen outdoors and cannot be categorized under any building type.

Lastly, a high-risk building type may have a large number of constituent facilities spread

across the city, making broad AED deployment in this building type cost prohibitive.

Mell and Sayre (2008) analyzed the “fire extinguisher model” of putting an AED next to

each fire extinguisher for the purpose of wide distribution and increased awareness, but

concluded that this approach is not cost-effective. These findings emphasize the need for

a generalizable model to guide cost-effective AED deployment.

Optimization has been used extensively to model location problems (Daskin and Dean

2005, Snyder 2006, Daskin 2008, Fallah et al. 2009, Ingolfsson 2013) and has been the

predominant method for solving emergency-related facility location problems for decades.

One of the earliest studies in this area used a set covering model to locate emergency

service facilities (Toregas et al. 1971). In a set covering model, the objective is to minimize

the number of facilities needed to cover all demand points. A demand point is covered

if and only if it is located within a certain distance (radius) of a facility. However, in

many real world problems resources are not sufficient to cover all demand points. In

a limited resource environment, the maximal covering location problem (MCLP) is a

natural alternative (Church and ReVelle 1974). The MCLP maximizes the number of

demand points covered within a specified coverage distance by a fixed number of facilities.

As opposed to the set covering problem, it is not required that all the demand points

are covered. The MCLP and its numerous extensions compose an important class of

problems in emergency facility location literature. Daskin (1983) extended the MCLP to

the maximum expected covering location problem (MEXCLP) by taking the availability

of the facilities into account. The expected value of the covered demand at each point was

modeled using Bernoulli trials with probability of success (availability) q. Daskin (1987)

and Berman et al. (2013) introduced MCLP variants with travel time uncertainty. Erkut

Chapter 1. Introduction and Literature Review 7

and MEXCLP-based models. Gendreau et al. (1997) extended the MCLP to a double

coverage problem where all the demand points have to be covered with respect to a large

coverage radius r2 and at least α% of them have to be covered with respect to a small

coverage radius r1 < r2. Berman et al. (2009) introduced a covering problem where

the coverage radius is variable and has a direct effect on the facility cost. Church and

Roberts (1983) developed a model that relates the quality of service of a facility to the

distance/service time to the customer using a piecewise linear step function. Berman

et al. (2003) and Karasakal and Karasakal (2004) formulated the MCLP in the presence

of partial coverage incorporating a decaying coverage function.

A decaying coverage function is relevant for the AED location problem because,

re-alistically, AED coverage decreases as the distance to the patient increases. However,

previous models do not adequately address other important aspects of the AED location

problem. First, previous models assume that the decision maker has perfect information

on locations of the facilities and that the demand is met by the closest non-busy facility.

While this may be reasonable for other location problems, lay responders in a cardiac

arrest emergency do not necessarily know where AEDs are located. They need to search

for them with the guidance of a 911 operator and available signage. This motivates us

to develop models for best-case and worst-case AED retrieval scenarios.

Second, in classical covering models it is assumed that each demand point is serviced

by only one facility. Even if there exist back-up facilities in case a preferred one is busy,

demand is still met by a single facility at the time of service. In the case of a cardiac arrest,

it is desirable to send out many lay responders to look for AEDs, which would increase the

likelihood of bringing an AED to the victim. Therefore, we develop a multiple-responder

model that maximizes coverage when more than one AED can contribute to the coverage

of a cardiac arrest. A similar model minimizing the sum of probabilities of non-coverage,

referred to as the “probabilistic partial set covering problem” was introduced by Sherali

bounds using a branch-and-bound procedure. Lee et al. (2006) removed the restriction on

the probabilities and solved cases up to 100 potential facility locations and 200 customers

using constraint generation. In Chapter 3, we introduce simple yet effective bounding

strategies based on reformulations and solve problems hundreds of times larger in size.

A limited amount of research has been conducted at the intersection of operations

research and AEDs. Mandell and Becker (1996) focused on the equitable distribution

of AEDs to ambulances using a multi-objective integer programming model. Rauner

and Bajmoczy (2003) developed a decision model to evaluate the cost-effectiveness of

placing AEDs in ambulances. Dao et al. (2011) optimized the locations of AEDs in an

indoor environment. Myers and Mohite (2008) used the MCLP to determine locations

for AEDs on a university campus. Chan et al. (2013) showed that an MCLP-driven

ap-proach to AED deployment outperforms an intuitive population-based method. Sun et al.

(2016) developed an MCLP-based model incorporating spatial and temporal availability

of AEDs.

1.1.3

Relevant literature on spatial distribution of OHCAs

Several researchers have studied the geographical distribution of cardiac arrest risk. Soo

et al. (2001) analyzed OHCA incidents in electoral districts of Nottinghamshire, United

Kingdom and found that districts with higher material deprivation scores (Townsend

1987) have significantly higher incidence rates. Sasson et al. (2012) analyzed census tracts

in Columbus, Ohio to identify those with both high OHCA risk and low bystander CPR

rates. Raun et al. (2013) used multivariable logistic regression to identify contiguous

geographic census tracts with high OHCA and low bystander CPR rates in Houston,

Texas. Lerner et al. (2005) used kernel analysis to identify OHCA clusters in Rochester,

New York. Similarly, Warden et al. (2012) used Poisson cluster analysis to identify OHCA

clusters in Columbus, Ohio. Moon et al. (2015) compared the locations of OHCAs and

Chapter 1. Introduction and Literature Review 9

correlation between OHCA events and deployed AEDs in Metropolitan Phoenix, Arizona.

In general, these studies conclude that spatial methods can identify areas for targeted

resource allocation, such as identifying the most appropriate areas for community CPR

training or AED placement. However, these studies focus on determining cardiac arrest

risk spatially and do not integrate their results within a prescriptive framework for AED

deployment.

1.2

Contributions

The specific contributions in this thesis are: (1) We develop the first comprehensive,

data-driven framework for public AED deployment, integrating cardiac arrest stability

and risk estimation with AED location optimization. (2) Our stability analysis,

pre-sented in Chapter 2 and in Demirtas et al. (2016), is the largest study to examine the

spatiotemporal stability of OHCAs, and the first to look at the spatiotemporal stability

of public OHCAs. (3) Our optimization models, presented in Chapter 3, generalize the

MCLP and represent the first application of a probabilistic coverage concept to AED

deployment. Moreover, these models are initially mixed integer nonlinear programs, and

a contribution of our work lies in the development of mixed integer linear formulation

equivalents and tight and easily computable bounds. (4) Also in Chapter 3, we derive

a theoretical result on the ordering of the optimal objective function values of the three

models that enables effective computation of bounds for the multiple-responder case

with-out the need to consider more complicated constraint generation techniques. (5) This is

also the first study to use kernel density estimation to estimate the locations of demand

points in facility location optimization (Chapter 4). (6) Lastly, we apply our framework

to real data to derive several new insights regarding AED deployment. In Toronto, we

show that gains in coverage, survival and cost-effectiveness are possible over the status

The rest of the document is organized as follows. In Chapter 2, we analyze the

spatiotemporal stability of OHCAs at the neighborhood level in Toronto, Canada. In

Chapter 3, we review the MCLP model and motivate the need for a more realistic

cov-erage model. We then introduce three probabilistic covcov-erage models for AED location,

develop tractable formulations and bounds, and establish relationships between these

three models and the MCLP model. In Chapter 4, we estimate a geographical

distribu-tion of cardiac arrests in Toronto using a nonparametric approach called kernel density

estimation, and generate data sets from this distribution to optimize and evaluate

differ-ent AED deploymdiffer-ent policies. In Chapter 5, we apply our models using real data from

Toronto, compare the performance of the different models and AED deployment policies,

and provide public policy insights and implications. Finally, Chapter 6 summarizes our

findings and provides concluding remarks.

The following are the contributions to the literature resulting from this research:

Journal Articles (Published/Accepted)

• Chan, T.C.Y., Demirtas, D., Kwon, R.H. 1 _{“Optimizing the deployment of public}

access defibrillators”, Management Science, 2016. Published online in articles in

advance.

– An earlier version of the paper received 2nd place at SPPSN best paper

com-petition at INFORMS 2012 Annual Conference.

Journal Articles (Submitted)

• Demirtas, D., Brooks, S.C., Morrison, L.J., Chan, T.C.Y. “Spatiotemporal stability of out-of-hospital cardiac arrests”. Submitted.

– The abstract was recognized by American Heart Association with a Young

Investigator Award at AHA Scientific Sessions 2015.

Chapter 1. Introduction and Literature Review 11

• Sun, C.L.F., Demirtas, D., Brooks, S.C., Morrison, L.J., Chan, T.C.Y. “Optimizing public defibrillator locations to overcome spatial and temporal accessibility

barri-ers”. Submitted.

– The poster was recognized by the National Association of EMS Physicians

(NAEMSP) with the Best Poster Abstract Presentation Award at NAEMSP

Annual Meeting 2016.

Conference presentations

• Demirtas, D., Kwon, R.H, Chan, T.C.Y. “Spatial analysis of cardiac arrests”, IN-FORMS Annual Conference, San Francisco, USA, 2014.

• Demirtas, D., Chan, T.C.Y., Kwon, R.H. “Where to place AEDs? : Using statistics and optimization to find optimal AED locations”, Resuscitation in Motion (RIM)

Conference, Toronto, Ontario, Canada, 2014.

• Demirtas, D., Chan, T.C.Y., Kwon, R.H. “A kernel density estimation approach to optimization of public access defibrillator locations”, INFORMS Annual

Confer-ence, Minneapolis, Minnesota, USA, 2013.

• Demirtas, D., Chan, T.C.Y., Kwon, R.H. “Optimizing the deployment of public access defibrillators”, Operational Research Applied to Health Services (ORAHS)

Annual Conference, Istanbul, Turkey, 2013.

• Demirtas, D., Chan, T.C.Y., Kwon, R.H. “Optimizing the deployment of public access defibrillators”, INFORMS Annual Conference, Phoenix, Arizona, USA, 2012.

• Demirtas, D., Chan, T.C.Y., Kwon, R.H. “Optimizing the locations of automated external defibrillators”, Canadian Operations Research Society (CORS) Annual

Poster presentations

• Demirtas, D., Brooks, S.C., Morrison, L.J., Chan, T.C.Y. “Spatiotemporal stability of public location cardiac arrests”, National Association of EMS Physicians Annual

Meeting, San Diego, USA, 2016.

• Demirtas, D., Brooks, S.C., Morrison, L.J., Chan, T.C.Y. “Spatiotemporal stability of out-of-hospital cardiac arrests”, American Heart Association Scientific Sessions,

Orlando, Florida, USA, 2015.

Other scholarly addresses

• Demirtas, D., Chan, T.C.Y., Kwon, R.H. “Optimizing the deployment of public access defibrillators”, Three Minute Thesis (3MT⃝R_{) Competition, University of}

Toronto, Ontario, Canada, 2014.

– Division IV (Life Sciences) finalist

• Demirtas, D., Chan, T.C.Y., Kwon, R.H. “Where should we locate defibrillators?: A challenging and exciting OR problem”, 4th Annual Toronto Operations Research

Challenge (TORCH), Toronto, Ontario, Canada, 2014.

• Demirtas, D., Chan, T.C.Y., Kwon, R.H. “Optimizing the deployment of public ac-cess defibrillators”, Healthcare Operations and Information Management (HOIM)

Summer School, Montreal, Quebec, Canada, 2013.

• Demirtas, D., Chan, T.C.Y., Kwon, R.H. “Automated external defibrillator loca-tion problem in public settings”, Annual Mechanical and Industrial Engineering

Research Symposium, University of Toronto, Ontario, Canada, 2012.

• Demirtas, D., Chan, T.C.Y., Kwon, R.H. “Optimizing the deployment of public access defibrillators”, University of Toronto Operations Research Group (UTORG),

Chapter 2

Spatiotemporal Stability of Cardiac

Arrests

In this chapter, we measure the spatiotemporal stability of OHCAs at the neighborhood

level in Toronto, Canada. We provide the stability analysis for the following groups:

all OHCAs (including both public and private locations), all OHCAs normalized by

population, daytime OHCAs, nighttime OHCAs, public OHCAs, public daytime OHCAs,

and public nighttime OHCAs. We interpret the results from our analyses, and discuss

insights and public health policy implications.

2.1

The Importance of Stability

Spatiotemporal stability of OHCAs is important because a sufficiently stable OHCA rate

can help justify neighborhood-based investment of public health resources for cardiac

arrest prevention and response. Cardiac arrest interventions are often planned with a

long-term horizon in mind. For instance, public access AEDs are deployed in public

locations to be used by lay responders during a cardiac arrest emergency. They are

often funded to be placed at a specific site and remain in their initial location for years.

As well, many CPR and educational outreach studies (Iwashyna et al. 1999, Mitchell

et al. 2009, Sasson et al. 2010) suggest targeting neighborhoods with a high risk of

cardiac arrest and provide them with resources over a prolonged period. Both practices

require that cardiac arrest risk (in other words, demand for AED and CPR “resources”)

remains stable over time in order for these interventions to be effective. In addition,

many studies (Chan et al. 2013, Siddiq et al. 2013, Chan et al. 2016, Becker et al.

1998, Frank et al. 2001, Engdahl and Herlitz 2005), as well as European Resuscitation

Council guidelines (Perkins et al. 2015), have suggested locating AEDs in high risk areas,

where the risk is measured by aggregating several years of cardiac arrest data. Similarly,

several studies (Lerner et al. 2005, Mitchell et al. 2009, Root et al. 2013, Chiang et al.

2014) have combined multiple years of cardiac arrest and CPR data to show the effect

of socioeconomic status or neighborhood characteristics on the provision of bystander

CPR, and advocated targeting CPR training efforts in certain areas/communities. By

aggregating several years of cardiac arrest data, these studies and practical applications

of AED and CPR interventions implicitly assume that there is no significant change in

spatiotemporal trends of cardiac arrest, i.e., high risk areas continue to be high risk and

low risk areas continue to be low risk.

The primary objective of this study was to measure the spatiotemporal stability of

OHCAs at the neighborhood level in Toronto, Canada. Accordingly, this study provides

comprehensive analysis to determine if the stability assumed by other studies is valid and

if long-term location-based cardiac arrest interventions can be justified.

2.2

Methods

2.2.1

Study design

This was a retrospective population-based cohort study using data from Rescu Epistry.

The Rescu Epistry database is a composite of two precursors: The Epistry–Cardiac

Chapter 2. Spatiotemporal Stability of Cardiac Arrests 15

for Post Arrest Care (SPARC) database (Morrison et al. 2008, Lin et al. 2011). Rescu

Epistry uses a web-based data management interface that links electronic data from EMS

and Fire Service call reports, device data from monitors, defibrillators, and clinical data

from hospital charts. The in-hospital data is entered into Rescu Epistry manually after

chart review with several built-in automated features to minimize errors (e.g. point of

entry logic and error checks). Epistry data collection was reviewed and approved by the

institutional review boards and/or research ethics boards at each participating site.

2.2.2

Study setting

Toronto is the fourth most populous city in North America, with a population of

approx-imately 2.8 million. It covers an area of 630.2 km2_{. The city is primarily served by a}

single emergency medical service, though neighboring EMSs may respond to emergencies

in Toronto if they are closer. The city uses a tiered response system: the fire department

and multiple EMS units are often dispatched to a single emergency call.

2.2.3

Study population

We considered all non-traumatic OHCA episodes occurring within the City of Toronto

from January 1, 2007 to December 31, 2014. Patients were identified as having an OHCA

if they were evaluated by EMS personnel and (a) attempts had been made at external

defibrillation by lay responders or emergency personnel, or at chest compressions by

organized EMS personnel; (b) or were pulseless and there were no attempts to defibrillate

or apply CPR by EMS personnel. “Non-traumatic” episodes refer to those not caused

by an obvious blunt or penetrating trauma or by burns. Eligible episodes were identified

using postal code, street address, and latitude/longitude. The episodes that occurred in

public transportation buildings, commercial, industrial and civic sites, hotels, schools,

public spaces or recreational areas are identified as “public cardiac arrests”. The ones

as “private cardiac arrests”. The cases where it was not possible to determine the type

of the location were marked as “unknown location type”.

2.2.4

Data sources

Cardiac arrests:

The ROC is a North American consortium of ten regional clinical centers across the

United States and Canada that provides the infrastructure for multiple collaborative

out-of-hospital clinical trials in the areas of cardiac arrest and severe injury. The ROC

Epistry–Cardiac Arrest database is an extensive registry of OHCAs attended by ROC

EMS providers (Morrison et al. 2008). This study used the cardiac arrest data from the

Toronto clinical center.

Neighborhood definitions:

Neighborhood definition files were obtained from the City of Toronto Open Data portal

(City of Toronto Open Data 2014). In the mid-1990s, the Social Development, Finance

and Administration Division of the City of Toronto defined 140 neighborhood planning

area boundaries with assistance from Toronto Public Health for service planning

pur-poses (City of Toronto Neighbourhood Profiles 2015). The neighborhoods were delineated

based on Statistics Canada Census Tracts respecting natural and man-made boundaries

such that no neighborhood comprised a single census tract, and the minimum

neighbor-hood population was at least 7,000. The neighborneighbor-hood boundaries have not changed

Chapter 2. Spatiotemporal Stability of Cardiac Arrests 17

2.2.5

Analyses

Geographic data analysis:

We first determined the coordinates of each historical OHCA using latitude, longitude,

and street addresses provided in the Rescu Epistry database. We also determined if

the cardiac arrest occurred during the daytime (8:00 am–7:59 pm) or nighttime (8:00

pm–7:59 am). We then plotted all the historical OHCA episodes and boundaries of

Toronto neighborhoods in ArcGIS (ESRI, Redlands, CA). We allocated each OHCA to a

neighborhood and counted the number of OHCAs in each neighborhood in each year. We

also calculated the counts in each neighborhood in each year for the following subgroups:

daytime OHCAs, nighttime OHCAs, public OHCAs, public daytime OHCAs, and public

nighttime OHCAs.

Spatiotemporal stability of cardiac arrests:

Once we determined the number of cardiac arrests by neighborhood by year, we

calcu-lated the intraclass correlation coefficient (ICC) (Fisher 1921, Shrout and Fleiss 1979)

to measure the relative variability of OHCA counts within and between neighborhoods

over time. ICC is a general measurement of agreement between units belonging to the

same group, and it ranges from 0 to 1. Shrout and Fleiss (1979) classified the intraclass

correlation into six forms and provided guidelines to select the correct form. The six

forms of ICC are (1,1), (2,1), (3,1), (1,k), (2,k), (3,k), where the first number in

paren-thesis represents the case (model) type, and the second number represents if the unit of

analysis is an individual entry or the mean of k entries. Case 1 is used when units are

interchangeable or randomly chosen. Case 2 is used when units considered is a subset of

a larger set of units. Case 3 is used when units considered are the only units of interest.

In our context, units are years, and the years we considered constitute a sample (i.e.,

value, not a mean of counts. Based on these guidelines, ICC (2,1) is the most appropriate

form for our study. The corresponding mathematical model is as follows: Let Yij denote

the cardiac arrest count of neighborhood (group) j in year (unit) i, and be expressed as:

Yij = µ + ai+ bj + ϵij. (2.1)

In this equation, µ is the overall population mean of cardiac arrest counts; ai is the

specific effect of year i; bj is the specific effect of neighborhood j; and ϵij is the residual

effect for year i within neighborhood j. In other words, µ + ai is the population mean

for the ith _{year and µ + b}

j is the population mean for the jth neighborhood. ai is

assumed to be normally distributed with a population mean of zero and population

variance σ2 (population within-group variance), bj is assumed to be normally distributed

with a population mean of zero and population variance τ2 _{(population between-group}

variance), and ϵij is assumed to be normally distributed with a population mean of zero

and population variance γ2_{. The total variance of Y}

ij is then equal to the sum of these

variances,

var(Yij) = σ2+ τ2+ γ2, (2.2)

and the intraclass correlation coefficient ρ is

ρ = population variance between groups total variance =

τ2

τ2_{+ σ}2_{+ γ}2. (2.3)

These variances can be estimated as:

τ2 = (BM S− EMS)/k (2.4a)

γ2 = EM S (2.4b)

Chapter 2. Spatiotemporal Stability of Cardiac Arrests 19

where n = number of neighborhoods, k = number of years, BMS = between-neighborhood

mean square, YMS = within-neighborhood mean square, and EMS = residual mean

square (values can be obtained from two way ANOVA without replication analysis).

Accordingly, ρ can be estimated as:

ICC(2, 1) = BM S− EMS

BM S + (k− 1)EMS + k(Y MS − EMS)/n. (2.5) The ICC in this context represents the degree of resemblance between OHCA counts

in different years in the same neighborhood. In other words, it measures the stability of

OHCA counts in each neighborhood over time. We calculated the ICC for all OHCAs

(including both public and private locations) and all OHCAs normalized by population

using the 2011 census data. We also determined the ICC values for daytime OHCAs

and nighttime OHCAs. In addition, we performed the stability analyses for the following

subgroups: public OHCAs, public daytime OHCAs, and public nighttime OHCAs.

2.3

Results

During the eight-year study period, there were a total of 24,605 non-traumatic OHCAs

recorded in the City of Toronto. The breakdown of these episodes is given in Figure 2.1.

Out of 24,605 OHCAs, 2,303 of them occurred in public locations and 22,240 occurred

in private locations. There were 62 cases where location type could not be identified.

Of the 2,303 public cardiac arrests, 1,677 (73%) occurred during daytime and 626 (27%)

occurred during nighttime. Demographic, clinical and EMS characteristics of the included

OHCAs are described in Table 2.1.

2.3.1

Spatiotemporal stability of OHCAs

Figure 2.2 shows the 140 neighborhoods of the City of Toronto. The average number of

Figure 2.1: Consort diagram for the study identifying patients included in the analysis

7%. The average number of OHCAs per neighborhood per year was approximately 21.97,

and varied substantially across neighborhoods, from 6.38 to 64.13, with an interquartile

range (IQR) of 14.16 to 28.38 (Figure 2.4). In total, 100 neighborhoods (71%) had 26

or fewer cardiac arrests per year on average. On the other hand, 21% of all OHCAs

occurred in the 14 highest risk neighborhoods (10%). Figure 2.5 shows the top three

neighborhoods identified as high risk: Woburn (1), Moss Park (2), and South Parkdale

(3). The highest risk neighborhood had an average number of 64.13 episodes per year

and remained the highest risk neighborhood during six out of the eight years of the study

period. Table 2.2 shows the ranking change of the ten highest risk neighborhoods over

the study period.

The intraclass correlation for OHCA variation between neighborhoods was 0.83 [95%

Confidence Interval (CI), 0.79 to 0.87]. Figure 2.6 shows the range of OHCA counts

Chapter 2. Spatiotemporal Stability of Cardiac Arrests 21

Table 2.1: Demographic, clinical and EMS characteristics of the included OHCAs

Characteristic* OHCAs (n = 24,605)

Average age (±SD), y 69.1±18.3

Male Sex, n (%) 15,169 (61.7)

Time of cardiac arrest, n (%)

Daytime (8:00 am–7:59 pm) 16,418 (66.7)

Nighttime (8:00 pm–7:59 am) 8,167 (33.2)

Witnessed by bystander, n (%) 5,786 (23.5)

Received bystander CPR, n (%) 5,601 (22.8)

Bystander applied AED, n (%) 347 (1.4)

Initial cardiac rhythm, n (%)

Shockable** 2,507 (10.2)

Not Shockable** 21,651 (88.0)

Survival to discharge, n (%) 958 (3.9)

Note. SD = standard deviation; y = year.*Number of missing entries: age (284), sex (70), witnessed by bystander (137), received bystander CPR (4), AED applied (260), initial cardiac rhythm (447), survival to discharge (16), time of cardiac arrest (20). **Shockable includes ventricular fibrillation, ventricular tachycardia and patients listed as shockable with an AED. Not shockable includes asystole, pulseless electrical activity, patients listed as not shockable by an AED, and patients whose initial rhythm was not obtained because resuscitation was stopped before rhythm analysis.

the same neighborhood demonstrates low temporal variation. For daytime OHCAs, the

ICC was 0.79 [95% CI, 0.74 to 0.83], which is 27% higher than the ICC for nighttime

OHCAs (0.62 [95% CI, 0.56 to 0.69]).

2.3.2

Spatiotemporal stability of public OHCAs

Similar results were obtained with the public OHCAs subgroup. The average number of

public OHCAs was 287.9 (±30.0) per year, or approximately two per neighborhood per year. The trend of public OHCAs over years was stable with a coefficient of variation of

10%, though slightly less stable than that of all OHCAs (Figure 2.7). The average number

of public OHCAs per year varied substantially across neighborhoods, from 0.13 to 12.63,

with an IQR of 0.88 to 2.31 (Figure 2.8). Compared to all OHCAs, public OHCA count

variance across neighborhoods was even higher. In total, 100 neighborhoods had two or

Figure 2.2: Toronto neighborhoods

cardiac arrests occurred in the 10 highest risk neighborhoods (7%). Figure 2.9 shows the

top three neighborhoods identified as high risk: The Bay Street Corridor (1), West

Humber-Clairville (2), and Waterfront Communities-The Island (3). The highest risk

neighborhood had an average number of 12.63 episodes per year and remained the highest

risk neighborhood during five out of the eight years. Table 2.3 shows the ranking change

of the ten highest risk neighborhoods over the study period. Table 2.2 and Table 2.3

have six neighborhoods in common. On the other hand, Table 2.2 has more residential

neighborhoods, whereas Table 2.3 contains neighborhoods with higher pedestrian traffic

such as Bay Street Corridor and Waterfront Communities. The ICC for public OHCAs

was found to be slightly lower than that for all OHCAs, 0.67 [95% CI, 0.62 to 0.73]. The

ICC for daytime OHCAs was 0.60 [95% CI, 0.54 to 0.67], 50% higher than the ICC for

Chapter 2. Spatiotemporal Stability of Cardiac Arrests 23

Figure 2.3: Number of out-of-hospital cardiac arrests per year

2.3.3

Spatiotemporal stability of OHCAs adjusted for

popula-tion

We also examined the OHCA rate after adjusting for population using the census data

from the last census conducted by Statistics Canada in 2011. The average annual number

of OHCAs in Toronto was 11.8 per 10,000 persons. The annual population-adjusted rate

varied substantially across neighborhoods, from 5.80 to 33.57 (IQR, 9.42 to 14.04), and

the ICC was 0.64 [95%, 0.58 to 0.70].

2.4

Discussion

In this chapter, we measured the relative variability of cardiac arrest rates within and

between neighborhoods over eight years. This is the largest study to examine the

spa-tiotemporal stability of OHCAs, considering 24,605 OHCA cases over eight years. It is

also the first to look at the spatiotemporal stability of public OHCAs.

Our study shows that the OHCA rate in Toronto is stable at the neighborhood level

Figure 2.4: Average number of OHCAs per year across neighborhoods

providing strong evidence of spatiotemporal stability. ICC is the ratio of

between-neighborhood variation to the total variation. Accordingly, our results indicate that most

of the variation was due to geographical differences (neighborhood-to-neighborhood

vari-ation) as opposed to temporal differences (year-to-year variation within the same

neigh-borhood). Figure 2.4 and Figure 2.5 demonstrate the high geographical variation across

neighborhoods, whereas Figure 2.6 demonstrates that the OHCA counts from the same

neighborhood tend to be similar, indicating high temporal stability within the

neighbor-hoods. Therefore, our results show that high risk neighborhoods tend to remain high

risk, and low risk neighborhoods tend to remain low risk over time.

Although our conclusion of temporal stability is similar, our results contrast with

those of Sasson et al. (2010) who reported intraclass correlation to examine the temporal

stability of cardiac arrests within 161 census tracts in Fulton County, Georgia over three

years. They found an ICC value of 0.29 when examining all (public and private location)

OHCAs, whereas we found an ICC value of 0.83, providing stronger evidence of

spa-tiotemporal stability. Moreover, we found an ICC value of 0.67 for population-adjusted

population-Chapter 2. Spatiotemporal Stability of Cardiac Arrests 25

Figure 2.5: Average annual number of OHCAs by neighborhood: Woburn (1), Moss Park (2), and South Parkdale (3)

adjusted OHCAs. The differences might be due to the differences in the lengths of periods

considered, sample sizes or urbanization.

Our results help to justify policies that target areas with a historical high risk for

cardiac arrest prevention and treatment since those areas are likely to continue to be

high risk in the future. Such policies could provide long-term benefits from allocating

public health resources in high risk areas. Several studies have suggested placing AEDs

at locations identified as high risk with historical data (Chan et al. 2013, Siddiq et al.

2013, ?, Becker et al. 1998, Frank et al. 2001, Engdahl and Herlitz 2005). Our results

provide support for this approach since the neighborhoods where AEDs are placed would

likely remain high risk for at least eight years and perhaps longer. Similarly, our findings

help to justify targeting CPR classes and public awareness campaigns in neighborhoods

Table 2.2: Ranking of the ten highest risk neighborhoods based on OHCA rate (2007– 2014) Overall 2007 2008 2009 2010 2011 2012 2013 2014 rank Woburn 1 3 1 1 1 1 1 1 3 Moss Park 2 1 3 3 3 2 2 5 1 South Parkdale 3 2 15 6 2 3 3 3 6 Islington-City Centre W. 4 5 2 9 12 5 4 4 2 South Riverdale 5 13 14 2 4 4 9 7 9 Mimico 6 10 9 7 10 8 5 8 10 Wexford/Maryvale 7 8 22 10 5 6 25 2 16 West Humber-Clairville 8 19 4 4 9 11 11 13 8 Annex 9 4 5 18 6 10 19 9 18 Bendale 10 12 10 5 18 7 26 6 17

Figure 2.6: The range (2007–2014) of out-of-hospital cardiac arrests by neighborhood

period.

Many studies in the literature combine multiple years of cardiac arrest data and

provide analyses based on the temporally aggregated data. However, such an approach

implicitly assumes no significant change in cardiac arrest rates over time. Our study

confirms that the temporal variability of the OHCA rate is low in Toronto, meaning that

the historical aggregated cardiac arrest distribution may be a good representative of the

future one. Although our study provides some evidence to justify such assumptions made

Chapter 2. Spatiotemporal Stability of Cardiac Arrests 27

Figure 2.7: Number of public out-of-hospital cardiac arrests per year

findings.

There are several phenomena that may be contributing to the spatial variability and

temporal stability observed in our analysis. Numerous studies have shown that there

is a correlation between socioeconomic factors, cardiac arrest and cardiovascular disease

rates (Kaplan and Keil 1993, Hendrix et al. 2010, Reinier et al. 2011, Ahn et al. 2011).

In general, socioeconomically disadvantaged neighborhoods are associated with poorer

health outcomes and higher incidence of cardiac arrest (Diez Roux and Mair 2010). Our

analysis on spatiotemporal risk by neighborhood can facilitate additional studies linking

socioeconomic factors with cardiac arrest. Spatiotemporal risk analysis at the right

geographical unit level can help to identify the role of socioeconomic factors on cardiac

arrest incidence and spatiotemporal cardiac arrest stability.

In this study, we use neighborhoods defined by the City of Toronto as opposed to

census tracts or postal codes, since neighborhood boundaries have been consistent over

time and are developed for service planning purposes (City of Toronto Neighbourhood

Profiles 2015, Toronto Strong Neighbourhoods Strategy 2014). The City of Toronto tracks

Figure 2.8: Average number of public OHCAs per year across neighborhoods

Figure 2.9: Average annual number of public OHCAs by neighborhood: The Bay Street Corridor (1), West Humber-Clairville (2), and Waterfront Communities-The Island (3)

(City of Toronto 2011). In addition, these neighborhood planning area boundaries have

been used by the City of Toronto and multiple other agencies to report on social wellbeing

Chapter 2. Spatiotemporal Stability of Cardiac Arrests 29

Table 2.3: Ranking of the ten highest risk neighborhoods based on public OHCA rate (2007–2014)

Overall 2007 2008 2009 2010 2011 2012 2013 2014

rank

Bay Street Corridor 1 1 1 4 1 1 1 6 7

West Humber-Clairville 2 3 3 1 3 2 3 1 1

Waterfront Comm.-The Island 3 7 7 5 2 4 2 4 3

South Riverdale 4 9 8 2 6 3 5 5 10 Church-Yonge Corridor 5 2 5 44 13 6 4 2 2 Islington-City Centre W. 6 5 2 6 4 8 20 3 6 Moss Park 7 4 4 3 24 10 7 16 9 Downsview-Roding-CFB 8 11 9 8 16 9 21 13 4 Wexford/Maryvale 9 13 37 14 7 5 6 7 11 Annex 10 20 6 7 15 18 23 18 18

us to provide meaningful results from a public health resource planning perspective. It

is important to measure cardiac arrest risk at a consistent geographical unit level that is

in parallel with city’s public health resource planning areas. Depending on the city, the

most meaningful geographical unit could be neighborhoods, census tracts or collections

of census tracts.

In this study, we report ICC values and risk based on both absolute number of OHCAs

and population-adjusted number of OHCAs. Both analyses reveal high ICC values and

spatiotemporal stability. However, we believe absolute numbers are more appropriate

because cardiac arrest intervention and prevention resources should be planned based on

absolute demand rather than population-adjusted demand to ensure the distribution of

resources (e.g. optimized EMS response time, cardiac arrest awareness, CPR training

and public access defibrillation) according to the absolute risk of the event in a given

area. Our objective in this analysis was to direct policy on the basis of geography, not

2.4.1

Limitations

First, our study is solely based on data from a large urban setting and may not be

representative of smaller cities. In addition, Toronto’s population structure is more stable

compared to fast-growing cities in Asia, the Middle East and Africa. According to United

Nations (UN) World Urbanization Prospects report (United Nations Population Division

2014), 10% of urban agglomerations1_{with at least 300,000 inhabitants (i.e., 169 cities and}

urban settings in the world) had 4.65% or more annual increase of population between

2010 and 2015, whereas Toronto only had 1.72%. Among these 169 locations, 159 of

them were in Asia, Middle East and Africa. Consequently, the cardiac arrest stability

results from Toronto may not be generalizable to cities with different urbanization trends,

and replications would be needed using region-specific data. Second, our study period

is limited to eight years. There might be longer term trends that we were not able

to observe due to the rather short time window. On the other hand, eight years is

an appropriate length of time to re-examine the public health investments. Third, our

population-adjusted OHCA analyses use 2011 census data, which is several years old.

Therefore, the changes in population in the last four years are not reflected in these

analyses.

2.5

Conclusion

The OHCA rate in Toronto is stable at the neighborhood level over time. High risk

neighborhoods tend to remain high risk, which supports focusing public health resources

in those areas to increase the efficiency of these scarce resources and improve the

long-term impact of health-related interventions in the community.

1_{UN defines “urban agglomeration” as the population contained within the contours of a contiguous}

territory inhabited at urban density levels without regard to administrative boundaries. It usually incorporates the population in a city or town plus that in the suburban areas lying outside of, but being adjacent to, the city boundaries.

Chapter 3

Facility Location Models for AED

Deployment

In this chapter, we first review the MCLP model that our probabilistic coverage models

are built upon. We then provide three novel and realistic facility location models tailored

for AED deployment in public settings. Model 1 considers the scenario where there are

many bystanders who witness a cardiac arrest and search independently for an AED

to bring back to the victim. Models 2 and 3 examine the worst and best cases when

there is only one lay responder available to retrieve an AED, who finds the furthest and

closest AED from victim, respectively. We develop a general optimization model that

encompasses all three models, which are presented as mixed integer nonlinear programs.

We employ three strategies to solve these models. For Model 1, we derive an

increas-ingly tighter sequence of linear upper and lower bounds. For Model 2, we develop a

mixed integer linear formulation that derives an equivalent optimal solution. For Model

3, we provide an exact mixed integer linear reformulation. We then characterize the

relationship between these three models and the MCLP model.

3.1

Deterministic AED Location Model

The traditional MCLP model seeks the maximum number of demand points which can

be served within a stated service distance given a limited number of facilities (Church

and ReVelle 1974). The deterministic AED location problem can be formulated as an

MCLP as follows: Let Ie denote the locations of existing AEDs, Icdenote the candidate

locations for new AEDs, I = Ie∪ Ic, and |I| = m. Assume Ie∩ Ic = ∅. Let J denote a

set of cardiac arrests that are used as the input demand points to be covered (|J| = n). Let Ij ={i ∈ I | dij ≤ 100}, where dij is the shortest distance from location i to cardiac

arrest j. We define binary decision variables xj to be 1 if cardiac arrest j is covered,

and 0 otherwise; and yi to be 1 if an AED is placed at location i, and 0 otherwise. The

standard MCLP model is:

ZM CLPstandard = maximize ∑ j∈J xj (3.1a) subject to ∑ i∈Ic yi ≤ N, (3.1b) yi = 1, ∀i ∈ Ie, (3.1c) xj ≤ ∑ i∈Ij yi, ∀j ∈ J, (3.1d) xj ∈ {0, 1}, ∀j ∈ J, (3.1e) yi ∈ {0, 1}, ∀i ∈ I. (3.1f)

The objective function (3.1a) maximizes the number of covered cardiac arrests from J .

Constraint (3.1b) limits the number of locations in which new AEDs are placed to N .

Constraint (3.1c) ensures that the existing AEDs are not moved (a real-world constraint).

Constraint (3.1d) allows xj to equal 1 only when one or more AEDs are placed at sites

in the set Ij (that is, one or more AEDs are located within 100 m of cardiac arrest j).

Chapter 3. Facility Location Models for AED Deployment 33

variable zij and a new parameter aij. Let zij to be 1 if an AED at location i is used to

cover cardiac arrest j, and 0 otherwise; and let aij be 1 if cardiac arrest j is within the

coverage radius of location i, and 0 otherwise. The equivalent MCLP is:

ZM CLP = maximize ∑ j∈J xj (3.2a) subject to ∑ i∈Ic yi ≤ N, (3.2b) yi = 1, ∀i ∈ Ie, (3.2c) zij ≤ aijyi, ∀i ∈ I, j ∈ J, (3.2d) xj = ∑ i∈I zij, ∀j ∈ J, (3.2e) xj, yi, zij ∈ {0, 1}, ∀i ∈ I, ∈ J. (3.2f)

Constraint (3.1d) in formulation (3.1) is substituted by constraints (3.2d) and (3.2e)

in formulation (3.2). Constraint (3.2d) ensures that an AED at location i can only

cover cardiac arrest j if j is within the coverage radius of i and an AED is placed at

i. Constraint (3.2e) sets xj to 1 when cardiac arrest j is covered by an AED at some

location i and to 0 otherwise. Model (3.2) is equivalent to Model (3.1), and introduced to

provide a stronger link with the probabilistic coverage models presented in the following

section.

Note that the objective function measures “in-sample” coverage—that is, coverage of

the demand points that were used as input. However, a decision maker does not know

where future cardiac arrests will occur when placing AEDs. Thus, the more relevant

measure of coverage that we examine in our computational results is “out-of-sample”

coverage—coverage of a separate set of demand points that were not used as input to the

optimization model (i.e., different from set J ).

Formulation (3.2) can be used to estimate the improvement potential of placing