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A Location-Mixture Autoregressive Model for Online Forecasting of Lung Tumor Motion Dan Cervone , Natesh Pillai, Debdeep Pati, Ross Berbeco, John H. Lewis Harvard Statistics Department and Harvard Medical School August 6, 2014 Dan Cervone (Harvard) Online forecasting of lung tumor motion August 6, 2014 1 / 17

Introduction External beam radiotherapy: Lung tumor patients are given an implant (fiducial) at the location of their tumor. X-ray tomography can reveal location of the fiducial, thus the tumor. Radiotherapy is applied to the tumor location in a narrow beam, minimizing exposure within healthy tissue. Dan Cervone (Harvard) Online forecasting of lung tumor motion August 6, 2014 2 / 17

Introduction Fiducial External beam radiotherapy Dan Cervone (Harvard) Online forecasting of lung tumor motion August 6, 2014 3 / 17

Introduction Our problem: Patient’s respiration means the tumor is in constant motion. Tracking the fidual lags 0.1-2s behind, depending on specific machinery used. We need to forecast the location of the tumor to overcome this latency and ensure concentrated, accurate radiothrapy. Dan Cervone (Harvard) Online forecasting of lung tumor motion August 6, 2014 4 / 17

The data We consider data from 11 patients: multiple days receiving treatment. multiple radiotherapy sessions (“beams”) per treatment. Patient 11, day 6 beam 3 −6.5 −7.0 X position (mm) −7.5 −8.0 −8.5 −9.0 25 Y position (mm) 20 15 10 5 0 −12.0 −12.5 Z position (mm) −13.0 −13.5 −14.0 −14.5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 time (s) Dan Cervone (Harvard) Online forecasting of lung tumor motion August 6, 2014 5 / 17

The data We consider data from 11 patients: multiple days receiving treatment. multiple radiotherapy sessions (“beams”) per treatment. Patient 11, day 6 beam 3 20 PC 1 position (mm) 15 10 5 0 −5 −10 PC 2 position (mm) 1.0 0.5 0.0 −0.5 −1.0 0.6 PC 3 position (mm) 0.4 0.2 0.0 −0.2 −0.4 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 time (s) Dan Cervone (Harvard) Online forecasting of lung tumor motion August 6, 2014 5 / 17

Semiperiodic features 10 First PC position (mm) 5 0 −5 0 10 20 30 40 time (s) Signals at different frequencies. Fluctuations in location/amplitude/periodicity. Repeated motifs. Dan Cervone (Harvard) Online forecasting of lung tumor motion August 6, 2014 6 / 17

Location-mixture autoregressive process Assume Y i , i = 0 , . . . , n is a time series in R , and for some p , d n � α n , j N ( µ n , j , σ 2 ) Y n | Y n − 1 , Y n − 2 , . . . ∼ j =1 p � µ n , j = ˜ µ n , j + γ ℓ Y n − ℓ ℓ =1 Dan Cervone (Harvard) Online forecasting of lung tumor motion August 6, 2014 7 / 17

Location-mixture autoregressive process Assume Y i , i = 0 , . . . , n is a time series in R , and for some p , d n � α n , j N ( µ n , j , σ 2 ) Y n | Y n − 1 , Y n − 2 , . . . ∼ j =1 p � µ n , j = ˜ µ n , j + γ ℓ Y n − ℓ ℓ =1 α n , j are mixture weights: � d n j =1 α n , j = 1. Mixture means have autoregressive component � p ℓ =1 γ ℓ Y n − ℓ . Mixture means have location shift component ˜ µ n , j . Dan Cervone (Harvard) Online forecasting of lung tumor motion August 6, 2014 7 / 17

Location-mixture autoregressive process Let Y i − 1: p be the subseries ( Y i − 1 . . . Y i − p ) ′ , and define mixture components: − 1 2 ( Y i − 1: p − Y i − j − 1: p ) ′ Σ − 1 ( Y i − 1: p − Y i − j − 1: p ) � � exp α i , j = − 1 � 2 ( Y i − 1: p − Y i − ℓ − 1: p ) ′ Σ − 1 ( Y i − 1: p − Y i − ℓ − 1: p ) � � ℓ ≤ i − p exp µ i , j = Y i − j − γ ′ Y i − j − 1: p + γ ′ Y i − 1: p Or, using latent variables, Y i | M i = j , Y i − 1 , . . . , Y 0 ∼ N ( Y i − j + γ ′ ( Y i − 1: p − Y i − j − 1: p ) , σ 2 ) P ( M i = ℓ | Y i − 1 , . . . , Y 0 ) ∝ � − 1 � 2( Y i − 1: p − Y i − ℓ − 1: p ) ′ Σ − 1 ( Y i − 1: p − Y i − ℓ − 1: p ) exp Dan Cervone (Harvard) Online forecasting of lung tumor motion August 6, 2014 8 / 17

Location-mixture autoregressive process Latent variables M i instantiate time series motifs: let � τ 2 γ ′ � Ω = , Σ γ Dan Cervone (Harvard) Online forecasting of lung tumor motion August 6, 2014 9 / 17

Location-mixture autoregressive process Latent variables M i instantiate time series motifs: let � τ 2 γ ′ � Ω = , Σ γ then M i = j implies ( Y i − 0: p − Y i − j − 0: p ) ′ Ω − 1 ( Y i − 0: p − Y i − j − 0: p ) = small. Dan Cervone (Harvard) Online forecasting of lung tumor motion August 6, 2014 9 / 17

Location-mixture autoregressive process Latent variables M i instantiate time series motifs: let � τ 2 γ ′ � Ω = , Σ γ then M i = j implies ( Y i − 0: p − Y i − j − 0: p ) ′ Ω − 1 ( Y i − 0: p − Y i − j − 0: p ) = small. Assuming Y i − 0: p | M i = j , Y i − j − 0: p ∼ N ( Y i − j − 0: p , Ω) yields the LMAR predictive distribution for Y i | M i = j , Y i − 1 , . . . Dan Cervone (Harvard) Online forecasting of lung tumor motion August 6, 2014 9 / 17

Illustration of time series motifs 20 Fist PC position (mm) 15 10 5 0 −10 40 50 60 70 80 90 100 time (s) Y i − 1: p most recent p values of times series. Y i − j − 1: p previous instances of this motif. Dan Cervone (Harvard) Online forecasting of lung tumor motion August 6, 2014 10 / 17

Illustration of time series motifs 15 First PC position (mm) 10 ● 5 Current trajectory Similar past trajectories Future point ● 0 99.6 99.8 100.0 100.2 100.4 time (s) Y i − 1: p most recent p values of times series. Y i − j − 1: p previous instances of this motif. Dan Cervone (Harvard) Online forecasting of lung tumor motion August 6, 2014 10 / 17

Parameter estimation We focus on estimating Ω: 1-1 correspondence between Ω and LMAR parameters γ , Σ , σ 2 . Latent variables M i yield estimating equation: − 1 � h (Ω) = 2 log( | Ω | ) − i ≥ p 1 � 1 [ M i = j ]( Y i − 0: p − Y i − j − 0: p ) ′ Ω − 1 ( Y i − 0: p − Y i − j − 0: p ) 2 j ≤ i − p ˆ Ω = argmax( h (Ω)) computable quickly using EM algorithm. Not loglikelihood, but ˆ Ω enjoys same large-sample properties as MLE. Dan Cervone (Harvard) Online forecasting of lung tumor motion August 6, 2014 11 / 17

k step ahead predictive distributions Given a point estimate ˆ Ω, k -step ahead predictive distributions can be approximated by marginalizing over future steps 1 to ( k − 1): � α k j N ( µ k j , σ 2 Y i + k | Y i , Y i − 1 , . . . ∼ k ) . j Dan Cervone (Harvard) Online forecasting of lung tumor motion August 6, 2014 12 / 17

k step ahead predictive distributions Given a point estimate ˆ Ω, k -step ahead predictive distributions can be approximated by marginalizing over future steps 1 to ( k − 1): � α k j N ( µ k j , σ 2 Y i + k | Y i , Y i − 1 , . . . ∼ k ) . j Mixture components α k j , µ k j , σ 2 k available cheaply, analytically. No need for sampling or Monte Carlo. Dan Cervone (Harvard) Online forecasting of lung tumor motion August 6, 2014 12 / 17

Predictive performance on clinical data We tested predictive performance using the following procedure for all clinical observations in our data set. 1 Train model on first 35s of data 2 Fit prediction model in under 5 seconds 3 Simulate predictions for the next 40 seconds of data using model fit 4 Compare predictions with observed values Dan Cervone (Harvard) Online forecasting of lung tumor motion August 6, 2014 13 / 17

Predictive performance on clinical data We tested predictive performance using the following procedure for all clinical observations in our data set. 1 Train model on first 35s of data 2 Fit prediction model in under 5 seconds 3 Simulate predictions for the next 40 seconds of data using model fit 4 Compare predictions with observed values Predictions must be made in real time (30Hz)! Dan Cervone (Harvard) Online forecasting of lung tumor motion August 6, 2014 13 / 17

Predictive performance on clinical data We tested predictive performance using the following procedure for all clinical observations in our data set. 1 Train model on first 35s of data 2 Fit prediction model in under 5 seconds 3 Simulate predictions for the next 40 seconds of data using model fit 4 Compare predictions with observed values Predictions must be made in real time (30Hz)! Alternative prediction models considered: Our model: LMAR Neural networks Penalized AR model LICORS (Georg and Shalizi, 2012) For all methods, any tuning parameters were set to patient-independent optimal values, using separate data. Dan Cervone (Harvard) Online forecasting of lung tumor motion August 6, 2014 13 / 17

Example 1 ( k = 6) TRAINING DATA 8 4 0 −4 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 LMAR 8 4 0 −4 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 NN 8 4 0 −4 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 PENAL. AR 8 4 0 −4 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 LICORS 8 4 0 −4 Dan Cervone (Harvard) Online forecasting of lung tumor motion August 6, 2014 14 / 17