Compensating for respiratory motion is a key challenge for stereotactic body radiation therapy. To overcome latencies in the systems, prediction of future motion is necessary. This is related to the assumption of a stable correlation between external and internal motion. We present a new application for on-line model checking to introduce fail-safety to respiratory motion prediction and show its relevance by comparing to the widely used nLMS predictor. We demonstrate that the regularity of the external motion can be modeled and tested using OMC and deviations from regular respiratory motion can be detected.
Copyright c by the paper’s authors. Copying permitted for private and academic purposes. course of respiratory surrogate signals and to identify potential failures. We introduce a respiratory model for on-line model checking (OMC) and discuss how fail-safety could be integrated into SBRT. We show the relevance of OMC for the validation of respiratory motion by looking at situations where the prediction error of common predictors would be inconspicuous but the model of respiratory motion does not hold anymore. 2 2.1
The predictor of a non independent and identically distributed (i.i.d) time series Yn = (y1, . . . , yn) is called model-free if no assumption about an underlying statistical model is required. In a nutshell, the basic idea involves transforming the non-i.i.d history Yn to an easy to predict i.i.d dataset and transforming back, deriving the prediction of the time series.
Model-free prediction algorithms include normalized Least Mean Squares filter (nLMS) used in the Synchrony system for the CyberKnife system [SLS+07], Support Vector Regression [ES09] as well as Artifical Neural Networks [MP09].
Performance of a typical model-free prediction is measured by the error between predicted and actual signal. Naturally no model of the underlying signal is available. The additional lack of error estimation and information on likelihoods of errors enable fail-safety only as a reaction to already occurred errors. 2.2
Regarding respiratory motion SBRT represents a closed-loop treatment of patients. The model of the underlying mapping between internal and external surrogates is checked at equally distributed intervals and after huge errors in the prediction, the predictor of the respiratory motion is only validated retrospectively based on the occurred error.
OMC presents a new iterative verification approach [RSG14a, RSG14b]. Properties of the continuously updated model are verified concurrently at run time of the system.
To represent the respiratory motion we choose the model
4 x(t) = b + d · t + X ci cos(i · f · t) + si sin(i · f · t), i=1 (1) which is a combination of a discrete Fourier series with four frequency terms and a linear component. The values x(t) represent the predicted state of the system at time t0 + t where t0 is the time, the model was derived. The parameters b, d, s1, . . . , s4, c1, . . . , c4 and f are computed based on the two last breathing cycles at the time of calculation. A new model is derived at regular time steps of k ∈ R+ seconds.
Typically OMC is used to check the validity of the system and the prediction is used alongside with a relaxation parameter to guarantee the system behaves in the sense of the model. This allows to get some estimate of the probability for errors to happen while a model is active. 3
We compare the model-free predictor nLMS with OMC. We focus on the suitability for fail-safety applications. We choose not to compare the quality of prediction explicitly. While OMC by design has some prediction capability (see Figure 1) a typical prediction horizon of well over one second makes it hard to react to sudden changes in respiratory motion. While the long prediction horizon proves to be unsuited for real-time prediction it is essential for the model checking abilities.
Typical predictors, even when based on a model, will accept and process additional data points as long as the rate of change is not too high. This can result in undesirable behavior: consider a patient holding his breath. The respiratory motion in this case is easy to predict and conventional error measures are of no use. Clearly, for a patient holding his breath the prerequisites of the mapping between internal and external surrogates do not hold, introducing possible extreme but unrecognized errors in SBRT. The long prediction horizon of OMC alongside with the probability measure makes this case easy to detect. The same holds true for similar situation like misplaced or gradually detached tracking markers. Clearly a long prediction horizon is also advantageous when detection of breathing artifacts like coughing, yawning or sneezing is of interest. For an example see Figure 1. Between approximately 21566 and 21570 seconds regular breathing is disturbed by an irregularity. While the model does not hold, the probability is low and returns to normal values once the irregularity comes to an end.
Taking into account the history of error probabilities introduces additional applications. For sequences of reliable breathing as reported by the OMC the beam width of the linear accelerator in SBRT could be reduced and vice versa. For patients that show continuously irregular respiratory motion it may be advantageous to disable Synchrony motion tracking all together. This idea could also be expanded to test whether a patient is suited for motion correction before hand. 4
In our comparison we lined out, that OMC could become a useful algorithm in validating the quality of respiratory motion correction. Alongside the ability of model-free algorithms to work on any data and thus being easily expandable to multivariate setups [DWE+14] using classic algorithms for prediction is a better choice at the moment and OMC should only be used in addition to common predictors. OMC enables on-line validation of a given model and when compared to traditional model-based predictors [EDSS13] allows for more complete models, also accounting for deformations, e.g., in combination with 4D imaging. This not only introduces the possibility to improve prediction but also provides error probabilities and thus allows for more advanced applications.