Principles of Doppler Tomography

This paper shows how the Radon Transform can be used to determine vector fields. A scheme to determine the velocity field of a moving fluid by measurements with a continuous Doppler signal is suggested. When the flow is confined to a bounded domain, as is the case in most applications, it can be uniquely decomposed into one gradiental and one rotational part. The former vanishes if the fluid is incompressible and source-free, and the latter can be completely reconstructed by the methods proposed in this paper if the domain is simply connected. Special attention is paid to laminar flow in a long cylindrical vessel with circular cross-section. Under such conditions the flow profile becomes parabolic, which makes the vessel recognizable as a typical "N-shaped" pattern in an image describing the rotation of the velocity field. The vessel yields the same Doppler tomographic pattern, no matter how it is sectioned. The ideas presented should be applicable also when studying the flow in blood vessels, even if the flow profile in these is not quite parabolic. The discrepancies only make the "N-shape" somewhat distorted. The scalar Radon Transform The Radon and related transforms are currently in wide use in the fields of applied mathematics. In particular, during the last decade medical imaging techniques have turned more and more towards tomography. X-ray Computed Tomography (CT), Emission CT, Ultrasound CT and Nuclear Magnetic Resonance Imaging are the dominant areas. The enormous increase in computer power available at a relatively low cost is probably the single most important factor behind this development. Advances in electronics and improved computer algorithms have also contributed. However, the underlying mathematics is much the same as before. Typically, the issue is to measure how a tissue scatters or absorbs the energy in an applied test beam. The equation governing this process can often be written as = /oexp(-jf/i(r)«tf), where / is the intensity of the test beam and L is the path that the beam takes. The object is to determine how the linear attenuation coefficient, (*(x), varies in space. Consider a transverse section of a three dimensional body of finite extent. Introduce an orthogonal coordinate system such that the section lies in the xy plane. Each straight line in this plane is uniquely determined by the pair (s, 9), defined in Figure 1 below. Note that —oo denote the unit vector u> = (cos 9, sin 0,0). Then u> and 9 are interchangeable, and the line L can hr* ./ritten in the form L(s,9) = { r : r e J = 0r r-u; = s}. Figure 1: Definition of the parameters for the Radon transform. Let us now confine ourselves to the plane 2 = 0, and introduce the function (i as fi(sj) = log (y-) = / /i(r)rf/. Vin/ Jr-w=i Then fi is the two dimensional Radon transform (or X-ray transform) of /x which we also write fi(a.O) = TZfi. Measuring how a test beam is attenuated along a straight line, L(s,6), one can evaluate fi(s.O). and as A* and 9 vary to generate any line in the xy plane, the function /i(.s, 6) can be obtained for the entire sO plane. With the aid of the inverse Radon transform, (i = TZ~ft, it is then possible to calculate the linear attenuation coefficient, /i(x,y), at any point in the xy plane. For further background on the scalar Radon transform, the reader is referred to [3] and [4]. Moving Fluids and Doppler Shift If the domain under consideration contains a moving fluid, then the Doppler shift car. be used to measure the velocity of this motion. Disregarding relativistic effects, the Doppler shift, Aw, in an emitted signal with angular frequency w is described by the equation