Modelling Activities at a Neurological Rehabilitation Unit Richard Wood Jeff Griffiths Janet Williams

Neurological Injury An injury to the brain that has occurred since birth ABI = TBI + Non-TBI Typical patient pathway: A + E ICU NeuroRehab Home LT care District Hospital Specialist Unit

Rookwood Hospital Cardiff Treatment provided by a multidisciplinary team Annual demand: beds Average LOS: 5 months Annual throughput: 50 Average bed cost per day: £480 Average cost of patient episode: £72,000 Highly sought after Expensive 150 days

Time between arrivals Service time Average LOS = 5 months Queuing System Q Bed 1 Bed 2 Bed 21 EXIT Average time = 2.75 days Probability density Home Nursing home Other hospital Demand / referrals

Queuing System (simple) Q µ µ µ EXIT Demand / referrals λ Length of stay M | M | 21

Queuing System (bed-blocking) Q Cox k EXIT Demand / referrals λ Active M | C k + M | 21 Exp Cox k Exp Cox k Exp Blocked

Queuing System (balking and reneging) Q Cox k EXIT Demand Active M B+R | C k + M | 21 Exp Cox k Exp Cox k Exp Blocked Referrals BalkingReneging

Queuing System (patient groups) Q Cox k 1 EXIT Demand Active M B+R | C k 1 + M | r 1 Exp Cox k 1 Exp Blocked Referrals Balking Reneging Q Cox k p EXIT Exp Cox k p Exp Reneging Referrals M B+R | C k p + M | r p EXIT

Required Output 1.Steady-state results 2.Performance measures 3.What if? analysis Active LOS Blocked LOS CART Analysis Queuing System days 100 days 231 days 86 days 162 days 175 days255 days 72 days 136 days 122 days178 days 14 days 26 days 53 days77 days 6 beds 3 beds 9 beds

Solution BalkingReneging

Results Probability of reneging0.62 Mean bed occupancy20.8 patients Annual throughput51 patients/year Mean queue length10 referrals Mean waiting time29 days Annual cost£3.64m Validated against data

What-if Analysis Measure Original model Reduce delays to discharge (50% / 100%) One-third increase in older patients Increase/decrease number of beds Reneging probability / Annual throughput 5157 / 6051 Annual cost £3.64m£3.62m / £3.57m£3.68m.... can we use the model to assess other meaningful what-if scenarios? Better for patients More costly   

Effect of Treatment Intensity on LOS Length of stay is dependent on the number of hours of therapy each week More therapy = quicker recovery To incorporate this concept within the model: Service rates in queuing system must be dependent on treatment intensity

Queuing System Q Cox k 1 EXIT Demand Active M B+R | C k 1 + M | r 1 Exp Cox k 1 Exp Blocked Referrals Balking Reneging Q Cox k p EXIT Exp Cox k p Exp Reneging Referrals M B+R | C k p + M | r p EXIT

Active Length of Stay Active LOSTreatment Intensity Average Active LOS Scaled Active LOS Mean + Variance MeanVariance For a particular patient group: Treatment intensity cannot be directly controlled Dependent on treatment timetables Probability density

Scheduling Treatment Each week: Demand set for each patient Supply determined by availability of staff Demand fitted to supply (excess demand) Aim: automate scheduling process to rapidly evaluate the effects of changes to…. Staff skill-mix and availability Patient demand and availability on average treatment intensity.... For each patient group Automated scheduling program  Excel/VBA  Multi-objective hierarchical combinatorial optimisation problem  Heuristics  Purpose-built alogrithms to target constraint violations  Meta-heuristics  Simulated annealing  Tabu search

Intensity vs LOS Fit 1-over-x relations to data Constrain to known LOS for typical treatment intensity What-if scenario Change to timetable variables E.g. Increases intensity for PG4 Reduces active LOS Scale service rates in Coxian distribution to reflect this Solve system PG 1 PG 3 PG 2 PG 4

What-if Analysis 1.More group sessions Amend schedule, run program, find avg LOSs from intensities, solve system 3 extra patients per year, 2 days fewer waiting time, reduced reneging 2.Composition of workforce (budget cuts) Retain number of FTEs, but skew towards lower bands Leads to lower treatment intensity (since staff cannot lead sessions) Thus: wasted resources, longer LOS, fewer patients per year 

Automated Scheduling Program Used since January 2011 Before 8 hours each week After More time for clinical work Better solution Performance measures Audit data Results of dry-run (3 trial average) By-handProgram Objective function value (normalised) Demanded sessions scheduled (avg per patient) 85%86% Sessions with neither primary/secondary therapist (avg per patient) 41%21%

Scheduling  The scheduling work has released at least 4 hours a week of qualified physiotherapists who would otherwise be involved in scheduling the patient treatments for the following week  The automated computer scheduling creates a fairer system for patients as it takes into account what treatment the patient received the previous week Modelling  The service modelling work has been a real asset in that it has opened the eyes of the operational service managers to the issues regarding patient flow  These insights are now used on a regular basis in waiting list management and admissions meetings  The research work has had a huge impact in how we utilise our resources  The investment from the department in support of the research has been well worth it

Scheduling physiotherapy treatment in an inpatient setting Operations Research for Health Care (2012) Modelling activities at a neurological rehabilitation unit European Journal of Operational Research (2013) Optimising resource management in neurorehabilitation NeuroRehabilitation (In press)