オルソパントモグラフィにおける画像形成の理論解析

抄録

Orthopantomography which is currently used extensively in clinical practice developmented by Paatero in 1961.In 1939 Heckmann first described slit X-ray beam techniqe for radiography curved surface. It is suggested that the principle of the currently used orthopantomography might come from this curved surface tomography method.A number of papers have been published which attempted to elucidate orthopantomography theoretically.
Welander derived a theoretical formula in 1974. His formula agrees with that given by the present authors as well as that given by Tammisalo in respect to mathematical expression.
The size of the blurring image can be estimated by these formulae when the size of an object can be described by rectangle function, although the size of output image can't be calculated when the size of an object is the function of continuous distribution. By theoretical analyses on orthopantomography we have succeeded in deriving a theoretical formura which is almost satisfactory, and in demonstrating it in a experimental study.
Welander derived the following equation on the relation between the real size of an object(d_o) and that of image of the object(d_f) reflected on the film.
Here each element stands for:
R; the radius of the film, r; the radius of the object,
w_o; angular velocity of the X-ray beam or the angular velocity of the object
w_f; the angular velocity of the film, b'_f; the width of beam on the film
D; the distance from the tube target to the object, A; the distance from the target to the film.
Welander's equation (1) can be expressed as follows
Here W is the angle formed by the rotating center and the points where the X-ray beam width crosses the circumference of a circle on which the object is placed.
According to the equation (2), when the slit width becomes closer to infinitely small one, namely b'_f→0, the output image size can be expressed as a multiple of d_f by the factor of Rw_f/rw_o without any blurring components. When the size of an object becomes infinitely small, namely d_o→0, the equation (2) corresponds to Line Spread Function (LSF).
In other words, according to the equation proposed by Welander as well as by authors, the size of the blurring image of an object is expressed as the sum of LSF and the enlarged components which are determined by the rotating radius of an object and film. Mathematically this is a theoretical approximaoe equation although incomplete.
By theoretical analysis of image production of orthopantomography, we have succeeded in deriving our theoretical equation as described below.
Here, each element stand for; d_f(x); X-ray intensity distribution function of the output image,
d_o(x); X-ray intensity distribution function of the object,
R and r; radius of the film and that of the object respectively, w_f and w_o; angular velocity of the film and that of the object (or X-ray beam)
W; angle formed by the points where the X-ray crosses the circumference of the radius r and the rotating center,
b'_f; slit width on the film,
*; convolution integral,
II; rectangle function,