Data analytics in the practice of Emergency Medicine

Transcription

1 Data analytics in the practice of Emergency Medicine Devashish Das Industrial and Management Systems Engineering University of South Florida November 16, 2018

2 Outline Healthcare systems engineering in ED Emergency medicine at Mayo Clinic Functional regression based monitoring of service systems Conclusions and future work Healthcare systems engineering in ED 2

3 Increasing healthcare cost in the US % of GDP year Healthcare systems engineering in ED 3

4 Overcrowding in the Emergency Department (ED) Emergency Departments are the front-door to healthcare systems, especially for low-income and uninsured population. Over half of the 35.4 million annual inpatient admissions the US begin in the emergency department. 1 Yet, yet more than 5 times as many ED visits are treated and released Healthcare systems engineering in ED 4

5 Outline Healthcare systems engineering in ED Emergency medicine at Mayo Clinic Functional regression based monitoring of service systems Conclusions and future work Emergency medicine at Mayo Clinic 5

6 Mayo Clinic emergency department Level I trauma center in Rochester, Minnesota About 50,000 patients per year in 51 bed facility Emergency medicine at Mayo Clinic 6

7 New technologies to monitor ED performance Electronic dashboard throughout the ED: A multi-patient management system Active RFID tags Emergency medicine at Mayo Clinic 7

8 Optimizing staffing Matching arrival to staffing levels 3 3 Optimization of Multidisciplinary Staffing Improves Patient Experiences at the Mayo Clinic MY Sir, D Nestler, T Hellmich, D Das, MJ Laughlin Jr, MC Dohlman,... Interfaces 47 (5), Wagner prize finalist Emergency medicine at Mayo Clinic 8

9 Improvement in key performance metrics Emergency medicine at Mayo Clinic 9

10 Next steps How to determine if the system is still performing as expected? How to improve the experience individual patient of individual patients? Emergency medicine at Mayo Clinic 10

11 Outline Healthcare systems engineering in ED Emergency medicine at Mayo Clinic Functional regression based monitoring of service systems Conclusions and future work Functional regression based monitoring of service systems 11

12 Simulation and queuing theory Approximate and asymptotic analysis Optimizing the number of servers 4 Non-stationary queues 5 Small ED Pods Discharge/Admit 4 W. A. Massey, The analysis of queues with time-varying rates for telecommunication models, Telecommunication Systems, vol. 21, no. 2-4, pp , O. B. Jennings, A. Mandelbaum, W. A. Massey, and W. Whitt, Server staffing to meet time-varying demand, Management Science, vol. 42, no. 10, pp , Functional regression based monitoring of service systems 12

13 Integrating large scale data Service systems collect vast amounts of performance metric data Stochastic modeling and simulation can be viewed as a forward model, i.e. the intent is to come up with a model that approximates the real data Lack of an inverse modeling approach - Statistically infer the stochastic models from data Disciplined use of available data General purpose - Models can be used in different settings Functional regression based monitoring of service systems 13

14 Existing work on monitoring service systems Limited to M/M/n queues Focus on queue length monitoring 6 No known work on monitoring non-stationary queues 7,8 6 U. N. Bhat and S. S. Rao, A statistical technique for the control of traffic intensity in the queuing systems m/g/1 and g/m/1, Operations Research, vol. 20, no. 5, pp , N. Chen, Y. Yuan, and S. Zhou, Performance analysis of queue length monitoring of m/g/1 systems, Naval Research Logistics (NRL), vol. 58, no. 8, pp , N. Chen and S. Zhou, Cusum statistical monitoring of m/m/1 queues and extensions, Technometrics, vol. 57, no. 2, pp , Functional regression based monitoring of service systems 14

15 Statistical learning framework Predictor: Arrival intensity function λ(t) Response: Departure intensity function δ(t) Arrival rate arrival rate t Departure rate departure rate t Figure: Historical arrival and departure rates Functional regression based monitoring of service systems 15

16 Statistical learning framework Predictor: Arrival intensity function λ(t) Response: Departure intensity function δ(t) Arrival rate arrival rate λ(t) departure rate t Departure rate t Figure: Historical arrival and departure rates Black box: functional regression E (δ(t) λ(t)) Functional regression based monitoring of service systems 15

17 Statistical monitoring framework 9 ( 2 Generate alarm when ˆδ E δ ˆλ) threshold expected actual δ t 8 Das, Devashish and Pasupathy, Kalyan S. and Storlie, Curtis B. and Sir, Mustafa Y. (2018) Functional regression-based monitoring of quality of service in hospital emergency departments, IISE Transactions Functional regression based monitoring of service systems 16

18 Estimating intensity of inhomogeneous Poisson processes 10 τ i,1, τ i,2,... τ i,ni and ν i,1, ν i,2,... ν i,mi arrival and departure time stamps on day i ( N i ) T L(λ i ) λ i (τ i,j ) exp λ(t) dt, j=1 λ(t) = Cf(t), such that f(t)dt = 1 Maximum likelihood estimate of C is N i. Estimation of f(t) is equivalent to kernel density estimation 9 Wu et. al. (2013) ˆλ i (t) = N i j=1 1 w s K s 0 ( ) t τi,j Functional regression based monitoring of service systems 17 w s

19 Estimation of coefficient function Coefficient function β(s, t) is estimated by minimizing difference between expected and observed departure rate from in-control data. min β = min β T i I 0 0 T i I 0 0 ) 2 (ˆδi (t) E(ˆδ i (t) ˆλ i (t)) dt ( 2 t ˆδ i (t) β(s, t)ˆλ i (s)ds) dt. 0 (t T ) Functional regression based monitoring of service systems 18

20 Estimation of coefficient function Coefficient function β(s, t) is estimated by minimizing difference between expected and observed departure rate from in-control data. Challenges: min β = min β Computation T i I 0 0 T i I 0 Nonnegative intensity function Avoid over-fitting 0 ) 2 (ˆδi (t) E(ˆδ i (t) ˆλ i (t)) dt ( 2 t ˆδ i (t) β(s, t)ˆλ i (s)ds) dt. 0 (t T ) Functional regression based monitoring of service systems 18

21 Estimation of coefficient function Therefore β(s, t) can be approximated using kernel functions as follows: with b p 0. P p=1 b p 1 ω K ( s sp ω ) 1 ω K ( t tp ω ), (sp, tp) domain of β(s, t) T T t T s T Functional regression based monitoring of service systems 19

22 Estimation of coefficient function Penalize curvature min b T i I 0 0 φ ( 2 t ˆδ i(t) β(s, t)ˆλ i(s)ds) dt+ 0 (t T ) ( ) 2 2 ( ) β(s, t) 2 2 β(s, t) + dsdt s 2 Proposition 1 For K(u) = 3 4 (1 u2 )I u 1, the least square minimization problem reduces to a quadratic program t 2 min b b T (X T X + Φ)b 2y T Xb + y T y, such that b 0. Functional regression based monitoring of service systems 20

23 Estimation of coefficient function Proposition 2 For proper choice of (s p s q)/ω, optimization problem in Result 1 is strictly convex minimization problem. For Epanechikov s kernel (s p s q)/ω > 0.85 (s4, t4) (s5, t5) (s4, t4) (s5, t5) t (s2, t2) (s3, t3) t (s2, t2) (s3, t3) (s1, t1) (s1, t1) s s (a) (b) Functional regression based monitoring of service systems 21

24 Relationship with infinite server queues Lemma 1 Then, Let λ(t) be the arrival rate to a M t /G t / queue, g(s, t) be the density function of the service time for arrival occurring at time t. and β(s, t) = g(t s, s). Eδ(t) λ(t) = t β(s, t)λ(s)ds Functional regression based monitoring of service systems 22

25 Functional regression based monitoring scheme 1: procedure Phase I of SPC scheme 2: Collect the arrival and departure time instance, τ i,1, τ i,2,... τ i,ni and ν i,1, ν i,2,... ν i,mi, when the system was in-control. 3: Estimate β(s, t). 4: Monitoring statistic e i = T (ˆδi (t) ) 2 ˆβ(s, t)ˆλi (t)ds 0 dt for in-control sample 5: Either use simulation or bootstrapping to estimate h such that P(e i > h) = α 6: procedure Phase II of SPC scheme 7: Collect test sample i I 1. 8: if e i h then 9: sample i is out of control Functional regression based monitoring of service systems 23

26 Simulation model λ(t) t (hour of the day) Figure: Arrival intensity function for simulation case studies Simulate infinite server queues with time varying arrival Arriving entities experience time varying service time distributions Functional regression based monitoring of service systems 24

27 Performance comparison - I Figure: Type II error rate - M t/m/ (µ 0 = 1 hours) FR: Functional regression based monitoring scheme IS: exact calculation based on infinite server lemma Shewhart: x chart for mean service times Functional regression based monitoring of service systems 25

28 Performance comparison - II Figure: Type II error rate - M t/m/ mean service time in control out of contorl D time of the day Figure: Illustration of service time distribution Functional regression based monitoring of service systems 26

29 Performance comparison - III Figure: Type II error rate - M t/m t/ mean service time 1.0 D in control out of contorl D 12 time of the day Figure: Illustration of service time distribution Functional regression based monitoring of service systems 27

30 Infinite server for determining staffing levels number of server D t Figure: Shift in optimal staffing Functional regression based monitoring of service systems 28

31 Detecting changes in staffing Table: Type II error rate for detecting shift in staffing D (hours) FR chart Shewhart Chart Functional regression based monitoring of service systems 29

32 Case Study: Mayo Clinic s ED training - data from 2013, test data - March 2014 Test statistic Test statistic day day (a) (b) Figure: Data from March Monitoring statistic and signaling threshold for (a) functional regression based SPC scheme and (b) Shewhart chart. Functional regression based monitoring of service systems 30

33 Discussion on out of control days March 8, 12, 31 - out-of-control by SH chart. On March 8 and March 12 the length of stay was 2.82 hours and 4.45 hours. On March 21, LOS was 3.8 hours But FR chart signals March 14, 21, 22, 23 and 31 as out-of-control. δ(t) actual δ(t) expected δ(t) t Figure: Departure intensity on March 21 Functional regression based monitoring of service systems 31

34 Outline Healthcare systems engineering in ED Emergency medicine at Mayo Clinic Functional regression based monitoring of service systems Conclusions and future work Conclusions and future work 32

35 Sequence of event occurring during a patient s visit Problem: Define a timeliness of care metric based on a sequence of events occurring during a patient s visit to the ED. Nurse Arrival Doctor Lab values Discharge / Admit time Conclusions and future work 33

36 Real-time monitoring Monitoring Statistic time SPC Test data In control data set Figure: Monitoring CTMCs Conclusions and future work 34

37 Stochastic filtering and smoothing queuing processes Smoothing Problem: For given arrival time stamps τ i,1, τ i,2,... τ i,ni, simulate departure time stamps ν i,1, ν i,2,... ν i,mi. Filtering Problem: For given arrival time stamps τ i,1, τ i,2,... τ i,ni t, simulate departure time stamps ν i,1, ν i,2,... ν i,mi > t. Model free optimal co Problem: For given arrival time stamps τ i,1, τ i,2,... τ i,ni t, simulate departure time stamps ν i,1, ν i,2,... ν i,mi > t. Conclusions and future work 35

38 Thank you Conclusions and future work 36