Modelling Activities at a Neurological Rehabilitation Unit

Richard WoodJeff 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

Neuro Rehab

Home

LT care

District Hospital Specialist Unit

Rookwood Hospital

• Cardiff• Treatment provided by a multidisciplinary team

• Annual demand: 375• 21 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

Prob

abili

ty d

ensi

ty

Prob

abili

ty d

ensi

ty

Home

Nursing home

Other hospital

Demand / referrals

Queuing System (simple)

Q

µ

µ

µ

EXITDemand / referrals

λ

Length of stay

M | M | 21

Queuing System (bed-blocking)

Q

Coxk

EXITDemand / referrals

λ

Active

M | Ck + M | 21Exp

Coxk Exp

Coxk Exp

Blocked

Queuing System (balking and reneging)

Q

Coxk

EXITDemand

Active

MB+R | Ck + M | 21Exp

Coxk Exp

Coxk Exp

Blocked

Referrals

Balking Reneging

Queuing System (patient groups)

Q

Coxk1

EXIT

Demand

Active MB+R | Ck1 + M | r1

Exp

Coxk1 Exp

Blocked

Referrals

Balking

Reneging

Q

Coxkp

EXITExp

Coxkp ExpReneging

Referrals

MB+R | Ckp + M | rp

EXIT

∑ 𝑟 𝑖=21

Required Output

1. Steady-state results2. Performance measures3. What if? analysis

Activ

e LO

SBl

ocke

d LO

SCART Analysis Queuing

System

1 2 43

1

2

3

4

1 2 3 4

1 2 3 4

149 days

100 days 231 days

86 days 162 days 175 days 255 days

72 days 136 days 122 days 178 days

14 days 26 days 53 days 77 days

6 beds

3 beds

3 beds

9 beds

Solution

Balking Reneging

Results

Probability of reneging 0.62

Mean bed occupancy 20.8 patients

Annual throughput 51 patients/year

Mean queue length 10 referrals

Mean waiting time 29 days

Annual cost £3.64m

Validated against data

What-if AnalysisMeasure Original

modelReduce delays to discharge (50% / 100%)

One-third increase in older patients

Increase/decrease number of beds

Reneging probability0.62 0.58 / 0.45 0.65

Annual throughput51 57 / 60 51

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

Coxk1

EXIT

Demand

Active MB+R | Ck1 + M | r1

Exp

Coxk1 Exp

Blocked

Referrals

Balking

Reneging

Q

Coxkp

EXITExp

Coxkp ExpReneging

Referrals

MB+R | Ckp + M | rp

EXIT

∑ 𝑟 𝑖=21

Active Length of Stay

Active LOS Treatment Intensity

Aver

age

Activ

e LO

S

Scaled Active LOS

Mean + Variance Mean Variance

For a particular patient group:

• Treatment intensity cannot be directly controlled

• Dependent on treatment timetables

Prob

abili

ty d

ensi

ty

Prob

abili

ty d

ensi

ty

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 availabilityon 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-hand Program

Objective function value

(normalised)1 0.54

Demanded sessions scheduled

(avg per patient) 85% 86%

Sessions with neither

primary/secondary therapist

(avg per patient)41% 21%

SchedulingThe 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

ModellingThe 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 flowThese insights are now used on a regular basis in waiting list management and

admissions meetingsThe research work has had a huge impact in how we utilise our resourcesThe investment from the department in support of the research has been

well worth it

rich.wood@lloydsbanking.comrichardwoodgb@hotmail.co.uk

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)