Organ motion induced by respiration may cause clinically significant targeting errors and greatly degrade the effectiveness of conformal radiotherapy. It is therefore crucial to be able to model respiratory motion accurately. A recently proposed lung motion model based on principal component analysis (PCA) has been shown to be promising on a few patients. However, there is still a need to understand the underlying reason why it works. In this paper, we present a much deeper and detailed analysis of the PCA-based lung motion model. We provide the theoretical justification of the effectiveness of PCA in modeling lung motion. We also prove that under certain conditions, the PCA motion model is equivalent to 5D motion model, which is based on physiology and anatomy of the lung. The modeling power of PCA model was tested on clinical data and the average 3D error was found to be below 1 mm.
Deep Dive into PCA-based lung motion model.
Organ motion induced by respiration may cause clinically significant targeting errors and greatly degrade the effectiveness of conformal radiotherapy. It is therefore crucial to be able to model respiratory motion accurately. A recently proposed lung motion model based on principal component analysis (PCA) has been shown to be promising on a few patients. However, there is still a need to understand the underlying reason why it works. In this paper, we present a much deeper and detailed analysis of the PCA-based lung motion model. We provide the theoretical justification of the effectiveness of PCA in modeling lung motion. We also prove that under certain conditions, the PCA motion model is equivalent to 5D motion model, which is based on physiology and anatomy of the lung. The modeling power of PCA model was tested on clinical data and the average 3D error was found to be below 1 mm.
Respiration-induced organ motion is one of the major uncertainties in lung cancer radiotherapy, which may cause clinically significant targeting errors and greatly degrade the effectiveness of conformal radiotherapy. It is therefore crucial to be able to accurately model the respiratory motion. We distinguish between two main categories of respiratory motion models: one that devotes solely to the study of the motion of a single point (usually tumor center of mass); and the other one that attempts to model the motion of the entire lung by employing the spatial relations among different regions of the lung. This work will focus on the latter. Most works in the literature [1] rely more or less on the assumption of regular breathing and can yield suboptimal solutions if irregular breathing occurs.
As far as we know, there exist two spatiotemporal models which do not assume regular breathing patterns. Low et al [2] described a 5D lung motion model parameterized by tidal volume and airflow measured with spirometry, which allows characterization of hysteresis and irregular breathing patterns. More recently, Zhang et al [3] applied principal component analysis (PCA) to the 3D deformation field derived from a deformable image registration between a reference phase and other phases in a four-dimensional computed tomography (4DCT) data set. Although the PCA motion model in [3] seems promising for a small number of patients, there is still a need to understand the underlying reason why it works and whether there is any connection between the two lung motion models. In this paper, we will present a much deeper and detailed analysis of the PCA-based lung motion model. We provide the theoretical justification of the effectiveness of PCA in modeling lung motion. We shall see that it is closely related to Low’s physiological 5D lung motion model and that under certain conditions, these two models are actually equivalent.
We first briefly describe how PCA may be used to construct a lung motion model. We form a matrix X , where each row represents the displacement vectors of a certain voxel in the lung along one of the three coordinates in space at all time points. If we perform PCA on the covariance matrix of X , we will get a set of eigenvectors
, it is the motion of the corresponding voxel along one direction over time. We can rewrite as:
. If we look at the difference between any two rows (i.e., motion of two voxels along one direction), we can see that, ( )
where i w are the column vectors of W . ∆ ≤ ∆ y x ɶ ɶ , we have:
The implication is that if two voxels move similarly, then their motion represented by PCA will also be similar, provided that the principal components kept in the model do not have vanishing eigenvalues associated with them. Without this property, the magnitude of the difference between two eigenvectors is not limited by the corresponding difference between the real motion and can be arbitrarily large, then the motion of two voxels reconstructed by PCA can be wildly different, even if they move very similarly, which is not desirable.
The main feature of respiration is that it is somewhat (though not perfectly) periodic. The simplest function that captures this feature is a cosine function. In the cosine respiratory phantom, the motion of each voxel along each of three coordinates in space is in the form of cosine functions. We allow arbitrary amplitude and arbitrary phase for each cosine function. In matrix form:
where X is an N by M matrix; N is the number of voxels in the lung times 3, and M is the number of samples in time. 1 ,…, N A A and 1 ,…, N ϕ ϕ are amplitude and phase; θ is the time interval between successive samples.
It may seem to be an idealistic respiratory phantom at first. However, notice that any 3 rows in the above matrix is exactly the parametric form of an ellipse in 3D, so the 3D trajectory of each voxel follows an ellipse, and since we allow arbitrary amplitude and phase for each spatial coordinate, this ellipse (i.e., the 3D trajectory of a voxel) can have arbitrary shape, size and orientation in space. Therefore, this respiratory phantom can be seen as a coarse approximation to regular breathing.
In this respiratory phantom, we replace the cosine functions in the above phantom with even power of cosine functions, i.e.,
- Φ . We still allow arbitrary amplitude and arbitrary phase for each function. This formula has been used by Lujan et al [4] to model lung motion. The bias term in this function does not matter for PCA because of centering and is set to zero without loss of generality.
In this study, 2 patients were enrolled under an IRBapproved protocol and scanned using a 64-slice CT scanner (Philips 64-slice Brilliance CT) operating in ciné mode with a slice-thickness of 0.625 mm. Each contiguous set of the simultaneously acquired 64 CT slices was called a couch position, which covers 4 cm in the longitudinal axis of the patient. The scanner was operated to acq
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