Precisely Forecasting the Location and Motion of Lung Tumors Dan Cervone PQ Talk May 6, 2013 Dan Cervone (PQ Talk) Lung Tumor Forecasting May 6, 2013 1 / 21
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 (PQ Talk) Lung Tumor Forecasting May 6, 2013 2 / 21
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 (PQ Talk) Lung Tumor Forecasting May 6, 2013 3 / 21
Illustration of the process Fiducial External beam radiotherapy Dan Cervone (PQ Talk) Lung Tumor Forecasting May 6, 2013 4 / 21
What we observe 2.80 s 12 y coordinate (mm) 10 Observed tumor 8 position in 3D at 6 30 Hz. 4 1-5 min of 2 observations per ● 0 series. −2 0 2 4 6 8 x coordinate (mm) Dan Cervone (PQ Talk) Lung Tumor Forecasting May 6, 2013 5 / 21
What we predict 7.03 s 10 y coordinate (mm) 8 Current 6 observation 4 predicts future ● 2 observation 0.4s ● 0 ahead. −2 −2 −1 0 1 2 3 x coordinate (mm) Dan Cervone (PQ Talk) Lung Tumor Forecasting May 6, 2013 6 / 21
Modeling approaches First principal component usually has over 99% of the variance, so we work with this alone: position (mm) 10 5 0 10 20 30 40 50 60 time (s) Dan Cervone (PQ Talk) Lung Tumor Forecasting May 6, 2013 7 / 21
Modeling approaches First principal component usually has over 99% of the variance, so we work with this alone: position (mm) 10 5 0 10 20 30 40 50 60 time (s) Statistical methods: State space model [2], Kalman filtering [7], ridge regression [3], wavelets [6]. Machine learning methods: Support Vector Regression [5, 1], Neural Networks [3, 4]. Neural Networks are considered most effective. [3] Dan Cervone (PQ Talk) Lung Tumor Forecasting May 6, 2013 7 / 21
Philosophical musings When predicting in a complex system, we must consider bias/variance tradeoff. Especially in prediction, more structural assumptions tend to lower variance but increase bias. Structure is often articulated in the form of distributional assumptions and dependence. Can also find structure by assuming certain types of invariance. When structure involves paramaters, these are usually explored in the space of statistical models, not always in the space of the data. Dan Cervone (PQ Talk) Lung Tumor Forecasting May 6, 2013 8 / 21
Model overview For a specific patient, we consider: Dictionary of d distinct motifs, Γ = { γ (1) , . . . , γ ( d ) } , characterize different breath cycle shapes. Location of tumor during breath cycle up to time t , µ t , is a sequence of r transformed dictionary elements µ t = ( g φ 1 ( γ 1 ) , g φ 2 ( γ 2 ) , . . . , g φ r ( γ ∗ r )) where φ i parameterizes the transformation of γ i . Dan Cervone (PQ Talk) Lung Tumor Forecasting May 6, 2013 9 / 21
Model overview For a specific patient, we consider: Dictionary of d distinct motifs, Γ = { γ (1) , . . . , γ ( d ) } , characterize different breath cycle shapes. Location of tumor during breath cycle up to time t , µ t , is a sequence of r transformed dictionary elements µ t = ( g φ 1 ( γ 1 ) , g φ 2 ( γ 2 ) , . . . , g φ r ( γ ∗ r )) where φ i parameterizes the transformation of γ i . γ ∗ r denotes a partial observation of γ r ; the time series ends in the middle of a motif. We measure the tumor location with correlated noise: Y t ∼ N ( µ t , K θ ( t )) Dan Cervone (PQ Talk) Lung Tumor Forecasting May 6, 2013 9 / 21
Transformation parameters For each i = 1 , ..., r , φ i is four-dimensional. φ i consists of four parameters: length scaling, position scaling, location drift, and skewness. System constrains µ t to be continuous. We accumulate information for φ i very quickly as we observe the i th motif—this gives us precise predictions. Over time our prior for φ becomes more informative, increasing precision of future predictions both within and across motifs. Dan Cervone (PQ Talk) Lung Tumor Forecasting May 6, 2013 10 / 21
Illustration of (conditional) data-generating process Dan Cervone (PQ Talk) Lung Tumor Forecasting May 6, 2013 11 / 21
Illustration of (conditional) data-generating process Dan Cervone (PQ Talk) Lung Tumor Forecasting May 6, 2013 11 / 21
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