Review: Prediction of Respiratory Motion - Prediction and Classification of Respiratory Motion - page 15

Current data point x act ( t )

Predicted position
x pred ( t+

Δ )

Respiratory motion trace

1417.4

1417.2

Prediction error
( t +) Actual Position

x act ( t +)

Δ ε

1417

Δ

1416.8

1416.6

1416.4

1416.2

1416

1415.8

SHL

1415.6

2.751 2.7511 2.7512 2.7513 2.7514 2.7515 2.7516 2.7517 2.7518

x 10 4

Time Stamp (sec)

SHL = Signal History Length Δ = Response time

Fig. 2.7 Explanation of signal history length (SHL) explanation of SHL, response time (D) and
prediction error with respect to the current data point. Let x(t) denote the actual respiratory
motion curve at time t after SHL. The predicted position x pred (t ? D) can be calculated based on
the sinusoidal curve fit model over SHL

In KF, the predicted position x(t) can be derived from the previous state x(t - 1)
and the current measurement z(t)[ 82 , 84 ]. Sharp et al. showed that RMSE for the
prediction accuracy is around 2.5 mm with 200 ms latency [ 75 ]. Because of state
update process with new data, KF is effective for linear dynamic systems, but
prediction accuracy is degraded when breathing patterns change from one linear
state to another [ 73 ]. KF was enhanced to interactive multiple model (IMM) filter
with constant velocity (CV) and constant acceleration (CA) based on KF by Putra
et al. in Fig. 2.4 [ 81 , 83 ]. Hong et al. also suggested the first-order extended Kalman
filter (EKF) can be used to process and update the state estimate [ 72 ].

2.3.1.3 Sinusoidal Model

Regular respiratory motion shows a continuous sinusoidal pattern with respect to
the time sequence. This sinusoidal curve can be adjusted to respiratory motion
over signal history length (SHL). We show Fig. 2.7 to clarify the ideas of SHL,
response time (D), and prediction error for a single point of respiratory motion
trace. Let x(t) denote the actual respiratory motion curve at time t after SHL.
Vedam et al. represented a sinusoidal wave model to estimate the predicted
position for a given response time (D), as follows [ 74 ]: