Applying mathematical methods to problems associated with health care

Steve GallivanDirector

Clinical Operational Research Unit

University College London,UKwww.ucl.ac.uk/operational-research

(CORU)

Talk outline

• Overview of CORU

• Discuss methods used in CORU • Examples of studies that rely on

“back of envelope” analysis.

Health warning

4-dimensional cube

WARNINGHARD SUMS

SG 2000

Major funding reviewsMathematics Department

83 90 00 059585

CORUset up at

UCL

M.Utleybecomes

Deputy Director

Ray Jackson Director of CORU

Statistical Science Department

CORU’s History

S.GallivanbecomesDirector

Cervical cancer screeningAudit systemsMedical decision making Down syndromePatient progress modellingCardiac surgery outcomesRisk modellingPharmacy errorsSafety in surgeryAdvising public inquiriesGenetic screening

Infection dynamicsBed demand modellingEmergency service rotasAortic Abdominal AneurismsAudit of cardiac bypass surgerySeverely injured patientsCancer chemotherapyRoad traffic safetyCancer data sourcesMental healthEvaluation of treatment centres

What does CORU do?

What does CORU do?

Whatever it is, there’s a lot of it!

Funding Review Summary: 1998 – 2004

- 92 publications- 65 conference presentations- 25 official reports- 91 unpublished papers

Very cost-effective research

Knowledge/Belief

SystemImprovements

Data and observation

Knowledge

Inductive reasoning

Deductive reasoning

The two principle modes of scientific enquiry

What does CORU do?

Inductive Reasoning

Deductive Reasoning

The spectrum of health care research activity

Randomised controlled trials

Modelling

Laboratorystudies

Observationalstudies

Consensus groups

CORU methodological profile

Inductive Reasoning

Deductive Reasoning

CORU

Most healthcare research

Methods used in CORU

OR – a filing cabinet of mathematical modelling methods

Decision analysisStatistical modelling

Optimisation theory

Data envelopment analysis

Stochastic analysis

Systems dynamics

SimulationSoftware development

IF ALL ELSE FAILSINVENT SOMETHING

Deriving equations

Health OR – trend towards specialisation

Decision analysis (mostly health economics)

Statistical modelling

Optimisation theory

Data envelopment analysis

Stochastic analysis

Systems dynamics

SimulationSoftware development

Deriving equations

IF ALL ELSE FAILSINVENT SOMETHING

Diversity of methods used within CORU

Methodology Example projectsDeriving equations Bed demand estimation;

Decision analysis Cardiac surgery;Genetic testing.

Statistical modelling Monitoring outcomes in surgery;Development and testing of risk models.

Optimisation and game theory Modelling competition between Trusts;Admissions planning;

Data envelopment analysis

Stochastic analysis Evaluation of screening programmes;Capacity needs; infection dynamics

Systems dynamics

Simulation Prioritised booking systems.

Software development Clinical decision support systems;Clinical audit systems.

When all else fails, invent something

Patient choice;Assessment of runs of poor outcome.

Applying mathematical methods to problems associated with health care

Can we audit surgical outcomes taking case-mix into account?

Step 1: Talk to doctors and find out what they think

Example 1

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

Operation number

0

2

4

6

8

10Cumulative perioperative mortalities

Use of charts to monitor surgical outcomes

MR de Leval, K Francois, C Bull, W Brawn, D SpiegelhalterAnalysis of a cluster of surgical failures.

Application to a series of neonatal arterial switch operationsJ. Thoracic and Cardiovascular Surgery, 107(3): 914 – 924, 1994.

Use pre-operative factors to model surgical risk

0 10 20 30 40 50 60 70 80 90 100 110120 130140150

Operation number

0

2

4

6

8

10Cumulative perioperative mortalities

Expected mortality (from risk model)

Actual mortalityPar for the course

Net life gain

Calculating Net Life Gain based on pre-operative risks

0 10 20 30 40 50 60 70 80 90 100110120130140150Operation number

012345

-1-2-3-4

Net life gain

VLAD THE IMPALER

THE VENERABLE BLEED

HAWKEYE PIERCE

Survivoragainstthe odds

Unexpecteddeath

Comparing 3 fictitious surgeons

J. Lovegrove, O.Valencia, T.Treasure, C.Sherlaw-Johnson, S.Gallivan Monitoring the result of cardiac surgery by variable life adjusted display (VLAD),Lancet; 350:1128-1130, 1997

Comparing several surgeons at a single hospital

4

3 3

2 2 2

1 1 1 1

0 0 0 0 0

Successive cases

Num

ber

of d

eath

s

Move up with probability equal to the risk of death

Otherwise move horizontally

Key

Model outcomes as a random walk

Bad runs of outcome can happen by chance

-15

-10

-5

0

5

10

15

0 50 100 150 200 250 300 350

Operation number

Net

life

gai

n

Is poor overall outcome a coincidence?

Unlikely to be achance coincidence

Sherlaw-Johnson C, Gallivan S, Treasure T, Nashef S. (2004)Computer tools to assist the monitoring of outcomes in surgery.European Journal of Cardio-thoracic Surgery. 26: 1032-1036.

Example 2

Giving patients the right to choose

Mathematical methods to assist with hospital operation and planning

Devolution of decision making can degrade system

performance

To be able to offer choice, the system needs more capacity

Cautionary Tale

Patient choice analogy - aircraft catering

14 first class passengers will be offered meat or fish dinners. Both are equally popular. How many of each should be stocked?

7 meat + 7 fish10 meat + 10 fish14 meat + 14 fish

Stock Probability of being understocked

80%5%0%

PatientPool

Trust A

Trust B

Patients’ preferences for two equally attractiveoptions

Total number of patients offered choice

1000900800700600500400300200100

Ext

ra c

apac

ity r

equi

red

to o

ffer

choi

ce (

%)

30

25

20

15

10

5

0

-5

Ext

ra c

apac

ity s

uppl

ied

abo

ve t

hat e

xpec

ted

(%

)

Total number of patients in programme

Percentage chance of Unitbeing oversubscribed

1%

5%

10%50%

KEY

Capacity requirements for a Unit catering for patientschoosing between two equally attractive sites

MORAL:Choice requires substantial unused capacity (which has to be funded)

Bed needs estimation

Applying mathematical methods to problems associated with health care

Example 3

AdmissionsLength of stay

Capacityrequired =

Averagedaily number of

admissions

Lengthof

stay

VARIABIL

ITY

How many beds are needed ?

A simple extension to the conveyor belt model

How many beds are needed to honour commitments ?

Length of stayvariable

Booked admissions

N per day

Full attendanceNo emergency admissions

DELIBERATELY SPARSEASSUMPIONS

pi - probability that a patient is still resident

i days after admission;

Probability theory gives a ‘simple’ analysis

The steady state probability that k beds are occupied on

a given day is the coefficient of xk in the power series:

N

iii xppxQ

0

)1()(

WARNINGHARD SUMS

Variable length of stay causes variation in bed demand

beds available

Conveyor belt estimateof bed needs

13121110987654

100

80

60

40

20

03210

% Days overloaded

Gallivan, Utley, et al, ‘Booked inpatient admissions and hospital capacity: a mathematical modelling study’

BMJ 324:280- 282, 2002.

Pro’s and con’s of variability model

• Starkly illustrates that variability affects capacity needs;

• conveyed this to clinicians;

• established potential dangers of booking policy;

• BUT, not intended to assist real planning;

• for that, one needs added realism.

Booked admissionsFriThuWedTuesMonSunSat

Attendance?Length of stay?

?Emergencyadmissions

Adding other source of variability

Shorter stays Longer stays

Utley M., Gallivan S. et al. ‘Analytical Methods for Calculating the Capacity Required to Operate an Effective Booked Admissions Policy for Elective Inpatient Services’. Health Care Manag. Sci. 6:97-104, 2003.

Goodness me,can use linearprogramming

Cyclic demand during planning cycle

Gallivan S, Utley M, ‘Modelling admissions booking of elective in-patients into a treatment centre’IMA J. Management Math, 16, 305-315, 2005

hjdwCc

H

h w

hcdwCc

C

cchd ppn )(,

1 0)(,

1,

2 1

0

)(,0 1

,w

hcdwCc

H

h

C

cchd pnMean bed

demand

Variance ofbed demand

(Both cyclic)

0

)(,1

,0w

hcdwCc

C

cc pn

Length of stay distributions

Decision variables: numbers of bookings

Optimisation to assist capacity and admissions planning

Optimum admissions plan Weekly pattern of bed needsMaximised reserve capacity

Typeof case

Length of stayDistributions Emergency

AdmissionRates

DNARates

Contractualobligations

Input data

M T W T F S S

Reserve capacity (%)

0

70

Optimised admissions

FIFO

Example 4

Great Ormond Street HospitalPaediatric Intensive Care Unit (PICU)

Mathematical methods to assist with hospital operation and planning

Variable LOS

FriThuWedTuesMonSunSat

Unplanned admissions

FriThuWedTuesMonSunSat

FriThuWedTuesMonSunSat

Elective

Emergency

Planned admission days

Optimisation analysis

DECISION VARIABLES

Mean and varianceof bed demand by

day of week

FriThuWedTuesMonSunSat

FriThuWedTuesMonSunSat

Elective

Emergency

Planned admission days

FriThuWedTuesMonSunSat

FriThuWedTuesMonSunSat

Elective

Emergency

Planned admission days

Paediatric Intensive Care Analysis using Stochastic System Optimisation

Analysis implemented in simple EXCEL system

Inventing acronyms: Waste Of Money Brains And Time

Example 5

Multistage health care processes

Mathematical methods to assist with hospital operation and planning

GP Referral

Initial assessment

Low intensity therapy Psychological therapy

Leave system

Stepped care for depression and anxiety: Example pathways

Initial Assessment

Low Intensive Therapies

Psychologicaltherapy

End of treatment

Time

Await Assessment

Assessment

Low intensitytherapy

Psycologicaltherapy

Left system

Time

Time

Time

Time

Time

Stage probability distributionsp (t)

1

p (t)2

p (t)3

p (t)4

p (t)5

Using stage probabilities we can• Set up an optimisation problem that...

• Identifies system bottlenecks;

• Smoothes out overload in system;

• Identifies optimal admission pattern;

• Estimates resource requirements

WARNINGHARD SUMS

Prototype planning tool has been programmed

So new, we haven’t got an acronym and haven’tgot any data

Conclusions• Clinical OR is very cost effective health research;

• Using many modelling approaches is sensible;

• ‘Back of envelope’ modelling is very powerful;

• Can link stochastic demand models to other OR techniques such as optimisation to give practical tools to assist with health planning and operation.

That’s all folks