Using a Markov reward model to estimate spend-down costs for a geriatric department

A Markov model is used to describe movements of geriatric patients within a hospital system where the states of the Markov chain are acute/rehabilitative, long-stay care, discharge or death. By assigning costs to the states of this model, we can estimate the spend-down costs of running down services given that there are no more admissions and different costs are assigned to acute/rehabilitative and long-stay care. The model is used to estimate the spend-down costs using data previously validated for three Departments of Geriatric Medicine in the South West Thames Region of England. Our approach allows hospital planners to identify cost-effective strategies which take into account the fact that some geriatric patients remain in long-stay care for very long periods of time.     

Continuous-time Markov model for geriatric patients behavior. Optimization of the bed occupancy and computer simulation

Previous research has shown that the flow of patients around departments of geriatric medicine and ex-patients in the community may be modelled by the application of a mixed-exponential distribution. In this paper we considered a five-compartment model using a continuous-time Markov process to describe the flow of patients. Using a M/Ph/c queuing model, we present a way of optimizing the number of beds in order to maintain an acceptable delay probability at a sufficiently low level. Finally, we constructed a Java computer simulation, using data from St George’s Hospital, London.     

Estimating bed requirements for an intermediate care facility

Concern has been expressed in the United Kingdom regarding the proportion of beds intended for acute care that are occupied by patients who do not require acute care. One solution to this problem that is being investigated by some hospitals is the establishment of an intermediate care facility devoted to non-acute care. A key question faced by hospital planners is how many beds such an intermediate care facility should have. We report on a study consisting of a bed use survey and a stochastic analysis exercise that was conducted at the Whittington Hospital NHS Trust in London in order to address this question. Rather than focus on the whether patients in acute beds required acute care throughout their stay in hospital, the study concentrated on identifying periods in patients’ stays when they would have been transferred to an intermediate care facility if one had been available.     

Where to treat the older patient? Can Markov models help us better understand the relationship between hospital and community care?

We have previously used Markov models to describe movements of patients between hospital states; these may be actual or virtual and described by a phase-type distribution. Here we extend this approach to a Markov reward model for a healthcare system with Poisson admissions and an absorbing state, typically death. The distribution of costs is evaluated for any time and expressions derived for the mean and variances of costs. The average cost at any time is then determined for two scenarios: the Therapeutic and Prosthetic models, respectively. This example is used to illustrate the idea that keeping acute patients longer in hospital to ensure fitness for discharge, may reduce costs by decreasing the number of patients that become long-stay. In addition we develop a Markov Reward Model for a healthcare system including states, where the patient is in hospital, and states, where the patient is in the community. In each case, the length of stay is described by a phase-type distribution, thus enabling the representation of durations and costs in each phase within a Markov framework. The model can be used to determine costs for the entire system thus facilitating a systems approach to the planning of healthcare and a holistic approach to costing. Such models help us to assess the complex relationship between hospital and community care.     

An exact approach for relating recovering surgical patient workload to the master surgical schedule

No other department influences the workload of a hospital more than the Department of Surgery and in particular, the activities in the operating room. These activities are governed by the master surgical schedule (MSS), which states which patient types receive surgery on which day. In this paper, we describe an analytical approach to project the workload for downstream departments based on this MSS. Specifically, the ward occupancy distributions, patient admission/discharge distributions and the distributions for ongoing interventions/treatments are computed. Recovering after surgery requires the support of multiple departments, such as nursing, physiotherapy, rehabilitation and long-term care. With our model, managers from these departments can determine their workload by aggregating tasks associated with recovering surgical patients. The model, which supported the development of a new MSS at the Netherlands Cancer Institute–Antoni van Leeuwenhoek Hospital, provides the foundation for a decision support tool to relate downstream hospital departments to the operating room.     

Using phase-type models to cost stroke patient care across health,social and community services

Stroke disease places a heavy burden on society, incurring long periods of time in hospital and community care, and associated costs. Also stroke is a highly complex disease with diverse outcomes and multiple strategies for therapy and care. Previously a modeling framework has been developed which clusters patients into classes with respect to their length of stay (LOS) in hospital. Phase-type models were then used to describe patient flows for each cluster. Also multiple outcomes, such as discharge to normal residence, nursing home, or death can be permitted. We here add costs to this model and obtain the Moment Generating Function for the total cost of a system consisting of multiple transient phase-type classes with multiple absorbing states. This system represents different classes of patients in different hospital and community services states. Based on stroke patients’ data from the Belfast City Hospital, various scenarios are explored with a focus on comparing the cost of thrombolysis treatment under different regimes. The overall modeling framework characterizes the behavior of stroke patient populations, with a focus on integrated system-wide costing and planning, encompassing hospital and community services. Within this general framework we have developed models which take account of patient heterogeneity and multiple care options. Such complex strategies depend crucially on developing a deep engagement with the health care professionals and underpinning the models with detailed patient-specific data.     

The development and use of tools for monitoring the occurrence of surgical wound infections

Infections acquired during patients' hospital stays are a major health care concern in the UK. They can be fatal, lead to excess morbidity and lengthen hospital stay. There is therefore considerable interest in using analytical tools for monitoring the occurrence of infections so that any problems with the quality of patient care can be quickly identified and rectified. The development and implementation of such tools are complicated as some infections can be difficult to diagnose and it can take several weeks before an infection manifests itself. Another important issue is that some patients are more likely to contract an infection than others, regardless of the standard of care they receive. This paper describes work that has been undertaken in collaboration with University College London Hospitals (UCLH) to develop appropriate outcome monitoring tools for surgical wound infections that are easy for hospital staff to use and interpret. The underlying risk model has been developed and validated locally at UCLH, and for more widespread implementation it would require revalidation for new centres.     

Modelling the flow of congestive heart failure patients through a hospital system

The number of hospital admissions in England due to heart failure is projected to increase by over 50% during the next 25 years. This will incur greater pressures on hospital managers to allocate resources in an effective manner. A reliable indicator for measuring the quantity of resources consumed by hospital patients is their length of stay (LOS) in care. This paper proposes modelling the length of time heart failure patients spend in hospital using a special type of Markov model, where the flow of patients through hospital can be thought of as consisting of three stages of care—short-, medium- and longer-term care. If it is assumed that new admissions into the ward are replacements for discharges, such a model may be used to investigate the case-mix of patients in hospital and the expected patient turnover during some specified period of time. An example is illustrated by considering hospital admissions to a Belfast hospital in Northern Ireland, between 2000 and 2004.     

A queueing model for bed-occupancy management and planning of hospitals

The aim of this paper is, on the one hand, to describe the movement of patients through a hospital department by using classical queueing theory and, on the other hand, to present a way of optimising the use of hospital resources in order to improve hospital care. A queueing model is used to determine the main characteristics of the access of patients to hospital, such as mean bed occupancy and the probability that a demand for hospital care is lost because all beds are occupied. Moreover, we present a technique for optimising the number of beds in order to maintain an acceptable delay probability at a sufficiently low level and, finally, a way of optimising the average cost per day by balancing costs of empty beds against costs of delayed patients.     

Administrative days in acute care facilities: a queueing-analytic approach.

After acute care services are no longer required, a patient in an acute care hospital often must remain there while he or she awaits the provision of extended care services by a nursing home, through social support services, or by a home health care service. This waiting period is often referred to as "administrative days" because the time is spent in the acute facility not for medical reasons, but rather for administrative reasons. In this paper we use a queueing-analytic approach to describe the process by which patients await placement. We model the situation using a state-dependent placement rate for patients backed up in the acute care facility. We compare our model results with data collected from a convenience sample of 7 hospitals in New York State. We conclude with a discussion of the policy implications of our models.     

An integrated queuing and multi-objective bed allocation model with application to a hospital in China

In this paper, a multi-objective decision aiding model is introduced for allocation of beds in a hospital. The model is based on queuing theory and goal programming (GP). Queuing theory is used to obtain some essential characteristics of access to various departments (or specialities) within the hospital. Results from the queuing models are used to construct a multi-objective decision aiding model within a GP framework, taking account of targets and objectives related to customer service and profits from the hospital manager and all department heads. The paper describes an application of the model, dealing with a public hospital in China that had serious problems with loss of potential patients in some departments and a waste of hospital beds in others. The performance of the model and implications for hospital management are presented.     

A system dynamics-based simulation study for managing clinical governance and pathways in a hospital

This paper examines the development of clinical pathways (CP) in a hospital in Australia based on empirical clinical data of patient episodes. A system dynamics (SD)-based decision support system is developed and analysed for this purpose. The study highlights the scenarios that will help hospital administrators to redistribute caseloads among admitting clinicians with a focus on multiple diagnostic-related groups (DRGs) as the means to improve the patient turnaround and hospital throughput without compromising quality patient care. DRGs are the best known classification system used in a casemix funding model. Casemix is a DRG-based government funding model for hospitals with a mix of performance measures aiming to reward initiatives that increase efficiencies in hospitals. The classification system groups inpatient stays into clinically meaningful categories of similar levels of complexity that consume similar amounts of resources. Policy explorations reveal various combinations of the dominant policies that hospital management can adopt. With the use of visual interfaces, executives can manipulate the DSS to test various scenarios. Experimental evidence based on focus groups demonstrated that it can enhance group learning processes and improve decision making. The findings are supported by other recent studies of CP implementation on various DRGs. These showed substantial reduction in length of stay, costs and resource utilization.     

Incorporating health outcomes in Pennsylvania hospital efficiency: an additive super-efficiency DEA approach

The health care sector is one of the fastest growing sectors in the United States. Researchers are interested in conducting studies in the area of health economics in order to propose solutions to curb the rapid increase in health care spending and to improve the efficiency of the health care system in the United States. Specifically, hospital efficiency is one important research area in health economics. In this paper, data envelopment analysis (DEA) is used to assess hospital efficiency. An additive super-efficiency model is presented and applied to a sample of general acute care hospitals in Pennsylvania. In addition to the conventional choice of input and output variables, we include the survival rate as a quality measure of health outcome in the set of output variables. Thus our model takes both the quantity and the quality of the output into account. With the results obtained from our proposed DEA model, inefficiencies can be identified for hospitals to address without sacrificing the quality of care.     

Combining Data Mining and Discrete Event Simulation for a value-added view of a hospital emergency department

While simulation models have furthered understanding of the operations of emergency departments (EDs) and the dynamics of the ED within the healthcare system, they only model patient treatment implicitly, tracing the paths patients follow through the ED. By identifying the core patient treatments provided by the ED and incorporating them into a Discrete Event Simulation model, this paper provides insight into the complex relationship between patient urgency, treatment and disposal, and the occurrence of queues for treatment. The essential characteristics of the presented model are used to indicate a generally applicable methodology for identifying bottlenecks in the interface between an ED and a hospital ward.     

Integral resource capacity planning for inpatient care services based on bed census predictions by hour

The design and operations of inpatient care facilities are typically largely historically shaped. A better match with the changing environment is often possible, and even inevitable due to the pressure on hospital budgets. Effectively organizing inpatient care requires simultaneous consideration of several interrelated planning issues. Also, coordination with upstream departments like the operating theatre and the emergency department is much-needed. We present a generic analytical approach to predict bed census on nursing wards by hour, as a function of the Master Surgical Schedule and arrival patterns of emergency patients. Along these predictions, insight is gained on the impact of strategic (ie, case mix, care unit size, care unit partitioning), tactical (ie, allocation of operating room time, misplacement rules), and operational decisions (ie, time of admission/discharge). The method is used in the Academic Medical Center Amsterdam as a decision-support tool in a complete redesign of the inpatient care operations.     

Analysis of capacity management of the intensive care unit in a hospital

A hospital's intensive care unit (ICU) is a limited and critical resource. The efficient utilization of ICU capacity impacts on both the welfare of patients and the hospital's cost effectiveness. Decisions made in the ICU affect the operations of other departments. Yet, decision making in an ICU tends to be mainly subjective and lacking in clear criteria upon which to base any given decision. This study analyzes the admission-and-discharge processes of one particular ICU, that of a public hospital in Hong Kong, by using queuing and computer simulation models built with actual data from the ICU. The results provide insights into the operations management issues of an ICU facility to help improve both the unit's capacity utilization and the quality of care provided to its patients.     

Dimensioning hospital wards using the Erlang loss model

How many beds must be allocated to a specific clinical ward to meet production targets? When budgets get tight, what are the effects of downsizing a nursing unit? These questions are often discussed by medical professionals, hospital consultants, and managers. In these discussions the occupancy rate is of great importance and often used as an input parameter. Most hospitals use the same target occupancy rate for all wards, often 85%. Sometimes an exception is made for critical care and intensive care units. In this paper we demonstrate that this equity assumption is unrealistic and that it might result in an excessive number of refused admissions, particularly for smaller units. Queuing theory is used to quantify this impact. We developed a decision support system, based on the Erlang loss model, which can be used to evaluate the current size of nursing units. We validated this model with hospital data over the years 2004–2006. Finally, we demonstrate the efficiency of merging departments.     

Clinical and Operational Risk: A Bayesian Approach

In health care organizations (HCOs) adverse events may provoke dangerous consequences on patients, such as death, a longer hospital stay, and morbidity. As a consequence, HCO’s department needs to manage legal issues and economic reimbursements. Governances and physicians are interested in operational (OR) and clinical risk (CR) assessment, mainly for forecasting and managing losses and for a correct decision making. Currently, scientific researches, which are objected to a quantification of CR and OR in HCO, are scarce; absence of regulatory constraints and limited awareness of benefits due to risk management do not provide incentives to elaborate on how risks can be quantified. This paper is aimed at proposing Bayesian methods to manage operational and clinical adverse events in health care. Bayesian Networks (BNs) are useful for assessing risks given end stage renal disease (ESRD) as a context of application; some prior probability distributions are advised for representing knowledge before experimental results and Bayesian utility functions for making the optimal decision. The method is described as from the theoretical as from the empirical point of view, thanks to the health care and haemodialysis department, for this application. The ultimate goal is to introduce a methodology useful for managing operational and clinical risk for haemodialysis patients and departments.      

Capacity planning of a perinatal network with generalised loss network model with overflow

Recent literature shows that the arrival and discharge processes in hospital intensive care units do not satisfy the Markovian property, that is, interarrival times and length of stay tend to have a long tail. In this paper we develop a generalised loss network framework for capacity planning of a perinatal network in the UK. Decomposing the network by hospitals, each unit is analysed with a GI/G/c/0 overflow loss network model. A two-moment approximation is performed to obtain the steady state solution of the GI/G/c/0 loss systems, and expressions for rejection probability and overflow probability have been derived. Using the model framework, the number of required cots can be estimated based on the rejection probability at each level of care of the neonatal units in a network. The generalisation ensures that the model can be applied to any perinatal network for renewal arrival and discharge processes.     

Mathematical Models in Medicine

Several mathematical models are discussed to illustrate their applications in medicine. Examples chosen are from different areas of medicine and they utilize various degrees of mathematical sophistication. Problem areas in distribution of drugs in the body, diets for hospital patients, differential diagnosis, body burden in environmental health and patient care in the operating room are developed. The major aim of this survey is to bring to the attention of biomedical scientists and educators the power of mathematical concepts and techniques.