In our proposed method, the tomosynthesis method plays a very important role. To measure the shift-amount required for a “shift-and-add” operation in the tomosynthesis method, we used a calibration phantom [8]. In the phantom, several thin wires are located at specified angular positions on the standard imaging plane of the dental arch, and the shift-amount at each position is measured as the distance needed to match the positions of a thin wire between the adjacent frame images. Using this measurement, we can obtain a table of the shift-amount at each specified angular position. The details are described in the literature [8]. Figure 1 shows an example of the shift-amount table of an actual dental panoramic apparatus. During the rotation of the x-ray source and detector around an object, several thousands strip images of the object are acquired sequentially. And thus, the

angular position of the detector corresponds to the frame number of a series of the strip images. If the distance between the x-ray source and dental arch is always the same for any angular position, the shift-amount becomes a constant value. However, the rotation center of the x-ray source and detector moves on a specified orbit in the conventional panoramic apparatus, as a result of which the distance between the x-ray source and dental arch changes, and the shift-amount table becomes a cosine-shaped curve as shown in Figure 1.

The cervical vertebrae are located on a curve that is almost symmetrical to the dental arch, and so we assumed an imaginary curve on which the cervical vertebrae are located. We named this curve “a cervical curve” as shown in Figure 2. To reconstruct the cervical vertebrae, a new shift-amount table is required. If the distance between rotation center C and point P on the dental arch is large (Figure 3(a)), the shift-amount, which corresponds to the moved distance (dS_d) of point P on the dental arch for small angular movement (dq), becomes larger. If we exchange the positions of the x-ray source and detector as shown in Figure 3(c), the distance between rotation center C and point Q on the cervical curve becomes smaller, and the shift-amount, which corresponds to the moved distance (dS_c) of point Q on the cervical curve for small angular movement (dq), becomes smaller. As a result, the shift amount table for the cervical curve will be nearly symmetrical (upside down) to that of the dental arch (Figures 3(b) and (d)). The theoretical background to calculate the shift-amount table is shown in the Appendix. An example of the shiftamount table for reconstructing the cervical vertebrae is shown in Figure 4.

Next, we will consider the relationship between the dental arch and cervical curve on a tomosynthesis image. For the illustration, we simplify the imaging objects, in which only two objects are located along the dental arch and cervical curve. Suppose the object distribution on the dental arch f₁(α,β) and that of the cervical curve f₂(α,β) as shown in Figure 5(a), where α indicates the position

along the dental arch or cervical curve, and β is the vertical position to the α-axis. The α- and β-axes correspond to the horizontal and vertical axes of a panoramic image, and α is the position on the curve of the dental arch for the case of the panoramic image of the tooth and jaw bones, and also α is the position on the cervical curve for the case of the panoramic image of the cervical vertebrae.

And we also assume the blurring functions h₁(α;x) and h₂(α;x). h₁(α;x) is a blurring function at position α on the dental arch when the focal point is located on the cervical curve, and h₂(α;x) is a blurring function at position α on the cervical curve when the focal point is located on the dental arch. x is a dummy variable in the α-axis. These functions are shift-variant for position α, that is, a blurring region depends on the shift-amount at position α. These functions are modeled with a Gaussian function whose full width at half maximum (FWHM) corresponds to the shift-amount at each angular position. The shape of the blurring function does not change markedly in the neighborhood of α, and so we assume that these functions are regionally shift-invariant in the small area for x = α.

A given tomosynthesis image can be represented by a

summation of two images: one is an image focused on the teeth and jaw bones or on the cervical vertebrae, and the other is that defocused using one of the above-mentioned blurring functions. That is, tomosynthesis image g₁(α,β) that is focused on the dental arch is represented as follows (see Figure 5(b)):

where means a one dimensional convolution operator for variable α. This equation means that the resultant image of the shift-and-add operation is a summation of the focused image of the teeth and jaw bones and the defocused image of the cervical vertebrae. In actual cases, a cross section of the maxillofacial region includes many organs, and we cannot represent the cross section with this simple model. However, the major attenuating objects are high density organs such as teeth and bones, and so the model can be acceptable as the first approximation.

In the same manner, tomosynthesis image g₂(α,β) that is focused on the cervical curve is represented as follows (see Figure 5(c)):

Using the above two equations, we can remove the shadow of the cervical vertebrae that overlaps on the image of the dental arch, as follows:

(3)

Figure 6 show the case in which rotation center C is close to the dental arch, and in this case, the shift-amount required for reconstructing the cervical curve becomes larger as shown in Figure 6(a).

In this study, we used clinical data acquired with QRMasterP (Telesystems, Co. Ltd., Osaka, Japan). The number of strip images used for reconstructing a tomosynthesis image was 3600, and the size of a strip image was 50 × 1500 pixels. The size of a pixel was 0.2 × 0.2 mm². The x-ray tube voltage was 90 kV and tube current was 10 mA. The data acquisition time was 12 sec. Using the above conditions, we acquired data of two patients (male, ages 45 and 52). This study received the approval of the ethics committee of Asahi University, School of Dentistry.