A Spoonful of Math Helps the Medicine Go Down: An Illustration of How Healthcare can Benefit from Mathematical Modeling and Analysis

<p>Abstract</p> <p>Objectives</p> <p>A recent joint report from the Institute of Medicine and the National Academy of Engineering, highlights the benefits of--indeed, the need for--mathematical analysis of healthcare delivery. Tools for such analysis have been developed over decades by researchers in Operations Research (OR). An OR perspective typically frames a complex problem in terms of its essential mathematical structure. This article illustrates the use and value of the tools of operations research in healthcare. It reviews one OR tool, queueing theory, and provides an illustration involving a hypothetical drug treatment facility.</p> <p>Method</p> <p>Queueing Theory (QT) is the study of waiting lines. The theory is useful in that it provides solutions to problems of waiting and its relationship to key characteristics of healthcare systems. More generally, it illustrates the strengths of modeling in healthcare and service delivery.</p> <p>Queueing theory offers insights that initially may be hidden. For example, a queueing model allows one to incorporate randomness, which is inherent in the actual system, into the mathematical analysis. As a result of this randomness, these systems often perform much worse than one might have guessed based on deterministic conditions. Poor performance is reflected in longer lines, longer waits, and lower levels of server utilization.</p> <p>As an illustration, we specify a queueing model of a representative drug treatment facility. The analysis of this model provides mathematical expressions for some of the key performance measures, such as average waiting time for admission.</p> <p>Results</p> <p>We calculate average occupancy in the facility and its relationship to system characteristics. For example, when the facility has 28 beds, the average wait for admission is 4 days. We also explore the relationship between arrival rate at the facility, the capacity of the facility, and waiting times.</p> <p>Conclusions</p> <p>One key aspect of the healthcare system is its complexity, and policy makers want to design and reform the system in a way that affects competing goals. OR methodologies, particularly queueing theory, can be very useful in gaining deeper understanding of this complexity and exploring the potential effects of proposed changes on the system without making any actual changes.</p

Optimizing Equitable Resource Allocation in Parallel Any-Scale Queues with Service Abandonment and its Application to Liver Transplant

We study the problem of equitably and efficiently allocating an arriving resource to multiple queues with customer abandonment. The problem is motivated by the cadaveric liver allocation system of the United States, which includes a large number of small-scale (in terms of yearly arrival intensities) patient waitlists with the possibility of patients abandoning (due to death) until the required service is completed (matched donor liver arrives). We model each waitlist as a GI/MI/1+GI queue, in which a virtual server receives a donor liver for the patient at the top of the waitlist, and patients may abandon while waiting or during service. To evaluate the performance of each queue, we develop a finite approximation technique as an alternative to fluid or diffusion approximations, which are inaccurate unless the queue's arrival intensity is large. This finite approximation for hundreds of queues is used within an optimization model to optimally allocate donor livers to each waitlist. A piecewise linear approximation of the optimization model is shown to provide the desired accuracy. Computational results show that solutions obtained in this way provide greater flexibility, and improve system performance when compared to solutions from the fluid models. Importantly, we find that appropriately increasing the proportion of livers allocated to waitlists with small scales or high mortality risks improves the allocation equity. This suggests a proportionately greater allocation of organs to smaller transplant centers and/or those with more vulnerable populations in an allocation policy. While our motivation is from liver allocation, the solution approach developed in this paper is applicable in other operational contexts with similar modeling frameworks.Comment: 48 Page

MARKOV DECISION PROCESS MODELS FOR IMPROVING EQUITY IN LIVER ALLOCATION

In the United States, end-stage liver disease (ESLD) patients are prioritized primarily by their Model for End-stage Liver Disease score (MELD) to receive organ offers. Therefore, patients are required to update their MELD score at predefined frequencies that depend on the patient's last reported score. One aim of this dissertation is to mitigate inequities that stem from patients' flexibility regarding MELD score updates. We develop a Markov decision process (MDP) model to examine the degree to which an individual patient can benefit from the updating flexibility, and provide a menu of updating requirements that balance inequity and data processing more efficiently than the current updating requirements. We also derive sufficient conditions under which a structured optimal updating policy exists. As the coordinator of the harvesting Organ Procurement Organization (OPO) extends offers according to MELD score prioritization, the organ becomes less desirable. To avoid not placing the organ, the OPO coordinator can initiate an expedited placement, i.e., offer the organ to a transplant center, which can then allocate it to any of its patients. A second aim of this dissertation is to mitigate inequities induced by the OPO coordinator's premature departure from the prioritized list of patients via an expedited placement. As a preliminary step to studying the inequity induced by expedited liver placement, we conduct an extensive analysis of the current expedited liver placement practice based on recent data. We investigate different aspects of extending offers, e.g., the number of offers extended concurrently, and patients' response characteristics. Several of the results from this analysis serve as inputs for a second MDP model that examines how many concurrent offers the OPO coordinator should extend and when the coordinator should initiate an expedited placement. Numerical experimentation reveals a structured optimal policy, and we test the sensitivity of the model outcomes with respect to changes in model inputs. Lastly, we examine how our model outputs compare to the analogous measures observed in current practice and how they can be used to improve current practice

IMPROVING HEALTHCARE DELIVERY: LIVER HEALTH UPDATING AND SURGICAL PATIENT ROUTING

Growing healthcare expenditures in the United States require improved healthcare delivery practices. Organ allocation has been one of the most controversial subjects in healthcare due to the scarcity of donated human organs and various ethical concerns. The design of efficient surgical suites management systems and rural healthcare delivery are long-standing efforts to improve the quality of care. In this dissertation, we consider practical models in both domains with the goal of improving the quality of their services. In the United States, the liver allocation system prioritizes among patients on the waiting list based on the patients' geographical locations and their medical urgency. The prioritization policy within a given geographic area is based on the most recently reported health status of the patients, although blood type compatibility and waiting time on the list are used to break ties. Accordingly, the system imposes a health-status updating scheme, which requires patients to update their health status within a timeframe that depends on their last reported health. However, the patients' ability to update their health status at any time point within this timeframe induces information asymmetry in the system. We study the problem of mitigating this information asymmetry in the liver allocation system. Specifically, we consider a joint patient and societal perspective to determine a set of Pareto-optimal updating schemes that minimize information asymmetry and data-processing burden. This approach combines three methodologies: multi-objective optimization, stochastic programming and Markov decision processes (MDPs). Using the structural properties of our proposed modeling approach, an efficient decomposition algorithm is presented to identify the exact efficient frontier of the Pareto-optimal updating schemes within any given degree of accuracy. Many medical centers offer transportation to eligible patients. However, patients' transportation considerations are often ignored in the scheduling of medical appointments. In this dissertation, we propose an integrated approach that simultaneously considers routing and scheduling decisions of a set of elective outpatient surgery requests in the available operating rooms (ORs) of a hospital. The objective is to minimize the total service cost that incorporates transportation and hospital waiting times for all patients. Focusing on the need of specialty or low-volume hospitals, we propose a computationally tractable model formulated as a set partitioning based problem. We present a branch-and-price algorithm to solve this problem, and discuss several algorithmic strategies to enhance the efficiency of the solution method. An extensive computational test using clinical data demonstrates the efficiency of our proposed solution method. This also shows the value of integrating routing and scheduling decisions, indicating that the healthcare providers can substantially improve the quality of their services under this unified framework

Dynamic Decision Models for Managing the Major Complications of Diabetes

Diabetes is the sixth-leading cause of death and a major cause of cardiovascular and renal diseases in the U.S. In this dissertation, we consider the major complications of diabetes and develop dynamic decision models for two important timing problems: Transplantation in prearranged paired kidney exchanges (PKEs) and statin initiation. Transplantation is the most viable renal replacement therapy for end-stage renal disease (ESRD) patients, but there is a severe disparity between the demand and supply of kidneys for transplantation. PKE, a cross-exchange of kidneys between incompatible patient-donor pairs, overcomes many difficulties in matching patients with incompatible donors. In a typical PKE, transplantation surgeries take place simultaneously so that no donor may renege after her intended recipient receives the kidney. We consider two autonomous patients with probabilistically evolving health statuses in a PKE, and model their transplant timing decisions as a discrete-time non-zero-sum stochastic game. We explore necessary and sufficient conditions for patients' decisions to form a stationary-perfect equilibrium, and formulate a mixed-integer linear programming (MIP) representation of equilibrium constraints to characterize a socially optimal stationary-perfect equilibrium. We calibrate our model using large scale clinical data. We quantify the social welfare loss due to patient autonomy and demonstrate that the objective of maximizing the number of transplants may be undesirable. Patients with Type 2 diabetes have higher risk of heart attack and stroke, and if not treated these risks are confounded by lipid abnormalities. Statins can be used to treat such abnormalities, but their use may lead to adverse outcomes. We consider the question of when to initiate statin therapy for patients with Type 2 diabetes. We formulate a Markov decision process (MDP) to maximize the patient's quality-adjusted life years (QALYs) prior to the first heart attack or stroke. We derive sufficient conditions for the optimality of control-limit policies with respect to patient's lipid-ratio (LR) levels and age and parameterize our model using clinical data. We compute the optimal treatment policies and illustrate the importance of individualized treatment factors by comparing their performance to those of the guidelines in use in the U.S

ELICITING PATIENT PREFERENCES AND PLACING EXPEDITED ORGANS

Liver transplantation plays a crucial role in saving lives when no other alternatives exist. Each year approximately 5,500 liver transplants are performed in the US. However, annually still 2,000 lives are lost due to lack of livers. Much effort has been spent on improving the organ allocation system. In this dissertation, we focus on patient preference elicitation which is an essential component of medical decision models and expedited organ placement which is relatively unexplored component of the organ allocation system. When livers become available, they are offered to patients according to an order (match list) specified by a set of rules. Each patient can accept/reject the offer. Other researchers have considered this accept/decline decision. Estimating patient preferences over health states is an important component of these decision making models. Direct approaches, which involve asking patients abstract uestions, have significant drawbacks. We propose a new approach that infers patient preferences based on observed decisions via inverse optimization techniques. We illustrate our method on the timing of a living-donor liver transplant. If it appears that the standard allocation procedure will not result in a match before the organ becomes nonviable, the liver’s placement can be expedited, meaning that it is offered to a transplant center instead of an individual patient. We study the subsequent decision problem faced by a transplant center, namely which, if any, of its patients should receive the organ independent of their positions on the match list. We develop a simulation model and compare different policies for expedited liver placement. Our study indicates that a policy which gives higher priorities to patients whose likelihood of death is higher performs the best based on several metrics. We also formulate the transplant center’s decision problems as an average reward Markov Decision Process (MDP). Due to the complexity of the model, traditional methods used to solve MDP problems cannot be utilized for our model. Thus, we approximate the solution via Least Square Policy Iteration (LSPI) method. Despite the extensive search on basis functions, the LSPI method yields promising, yet not better outcomes than the policies found to be the best via simulation

ANTICIPATING U.S. POPULATION-LEVEL HEALTH AND ECONOMIC IMPACTS USING DISCRETE-EVENT SIMULATION TO GUIDE HEALTH POLICY DECISIONS

This dissertation presents two applications of discrete-event simulation (DES) to represent clinical processes: (1) a model to quantify the risk of the maternal obese and diabetic intrauterine environment influence on progression to adult obesity and diabetes, and (2) a model to evaluate health and economic outcomes of different smoking cessation strategies. The first application considers the public health impact of the diabetic and obese intrauterine environment\u27s effect on the prevalence of diabetes and obesity across subsequent generations. We first develop a preliminary DES model to investigate and characterize the epidemiology of diabetes during pregnancy and birth outcomes related to maternal obesity and diabetes. Using data from the San Antonio Heart Study (SAHS), the 1980 Census and the NCHS we are able to verify a simplified initial version of our model. Our methodology allows us to quantify the impact of maternal disparities between different racial/ethnic groups on future health disparities at the generational level and to estimate the extent to which intrauterine exposure to diabetes and obesity could be driving these health disparities. The populace of interest in this model is women of child-bearing age. The preliminary model is next modified to accommodate data and assumptions representing the United States population. We use a mixed-methods approach, incorporating both statistical methods and discrete event simulation, to examine trends in weight-gain over time among white and black women of child-bearing age in the US from 1980 to 2008 using United States Census projections and National Health and Nutrition Examination Survey (NHANES) data. We use BMI as a measure of weight adjusted for height. We establish an underlying population representative of the population prior to the onset of the obesity epidemic. Assessing the rate of change in body mass index (BMI) of the population prior to the obesity epidemic allows us to make \u27unadjusted\u27 projections, assuming that subsequent generations carry the same risk as the initial cohort. Unadjusted projections are compared to actual trends in the US population. This comparison allows us to quantify the trends in weight-gain over time. This model is interesting as a first step in understanding the trans-generational impact of obesity during pregnancy at the population level. The aim of the second application is to understand the impact of different pharmacologic interventions for smoking cessation in achieving long-term abstinence from cigarette smoking is an important health and economic issue. We design and develop a clinically-based DES model to provide predictive estimates of health and economic outcomes associated with different smoking cessation interventions. Interventions assessed included nicotine replacement therapy, oral medications (bupropion and varenicline), and abstinence without pharmacologic assistance. We utilized data from multiple sources to simulate patients\u27 actions and associated responses to different interventions along with co-morbidities associated with smoking. Outcomes of interest included estimates of sustained abstinence from smoking, quality adjusted life years, cost of treatment, and additional health-related costs due to long-term effects of smoking (lung cancer, chronic obstructive pulmonary disease, stroke, coronary heart disease). Understanding the comparative effectiveness and intrinsic value of alternative smoking cessation strategies can improve clinical and patient decision-making and subsequent health and economic outcomes at the population level. This dissertation contributes to the field of industrial engineering in healthcare. US population-level data structures are not always available in the desired format and there is not one method for managing the data. The key element is to be able to link the mathematical model with the available data. We illustrate various methods (i.e. bootstrap techniques, mixed-effects regression, application of probability distributions) for extracting information from different types of data (i.e. longitudinal data, cross-sectional data, incidence rates) to make population-level predictions. Methods used in cost-effectiveness evaluations (i.e. incremental cost-effectiveness ratio, bootstrap confidence intervals, cost-effectiveness plane) are applied to output measures obtained from the simulation to compare alternative smoking cessation strategies to deduce additional information. While the estimates resulting from the two models are topic-specific, many of the modules created for these studies are generic and can easily be transferred to other disease models. It is believed that these two models will aid decision makers in recognizing the impact that preventative-care initiatives will have, and to evaluate possible alternatives

Estimating the price of privacy in liver transplantation

In the United States, patients with end-stage liver disease must join a waiting list to be eligible for cadaveric liver transplantation. However, the details of the composition of this waiting list are only partially available to the patients. Patients currently have the prerogative to reject any offered livers without any penalty. We study the problem of optimally deciding which offers to accept and which to reject. This decision is significantly affected by the patient's health status and progression as well as the composition of the waiting list, as it determines the chances a patient receives offers. We evaluate the value of obtaining the waiting list information through explicitly incorporating this information into the decision making process faced by these patients. We define the concept of the patient's price of privacy, namely the number of expected life days lost due to a lack of perfect waiting list information.We develop Markov decision process models that examine this question. Our first model assumes perfect waiting list information and, when compared to an existing model from the literature, yields upper bounds on the true price of privacy. Our second model relaxes the perfect information assumption and, hence, provides an accurate representation of the partially observable waiting list as in current practice. Comparing the optimal policies associated with these two models provides more accurate estimates for the price of privacy. We derive structural properties of both models, including conditions that guarantee monotone value functions and control-limit policies, and solve both models using clinical data.We also provide an extensive empirical study to test whether patients are actually making their accept/reject decisions so as to maximize their life expectancy, as this is assumed in our previous models. For this purpose, we consider patients transplanted with living-donor livers only, as considering other patients implies a model with enormous data requirements, and compare their actual decisions to the decisions suggested by a nonstationary MDP model that extends an existing model from the literature