Chapter 25: Phase-Contrast CT

Every CT so far has measured X-ray absorption. The difference in attenuation coefficient $\mu$ from material to material becomes the image contrast directly. But soft tissues (muscle, fat, tumor, cartilage) all have densities close to water, so their $\mu$ differs by very little. Relying on absorption alone, they come out as nearly the same shade of gray. To distinguish soft tissue without contrast agents, phase-contrast CT measures a different physical quantity. A bright, coherent X-ray source, the synchrotron, made this measurement practical.

The other part of the refractive index

The refractive index of matter for X-rays can be written as a complex number:

$$n \;=\; 1 - \delta + i\beta$$

The imaginary part $\beta$ corresponds to absorption (attenuation $\mu = 4\pi\beta/\lambda$), the quantity we have been measuring all along. The real-part term $\delta$ describes how much a material shifts the phase of the transmitted X-ray (refraction). The important point is that in soft tissue $\delta$ is two to three orders of magnitude larger than $\beta$. Tissues that absorption can barely tell apart can carry large contrast as a phase shift. Phase-contrast imaging images this $\delta$.

Turning phase into intensity

A detector can only measure intensity; phase itself is invisible directly. The simplest, apparatus-free approach is the propagation-based method. Instead of recording intensity right at the sample, we record it at a distance $z$. The phase shift tilts the wavefront leaving the sample slightly (refraction), and as it propagates through free space, light converges or diverges at boundaries, forming bright/dark fringes (edge enhancement). In the near field, this intensity change is proportional to the Laplacian of the phase (the transport-of-intensity equation, TIE):

$$I_z \;\approx\; I_0\left(1 - \frac{z}{k_0}\nabla^2\varphi\right)$$

At $z=0$ (flush against the sample) this is just $I_0$, the absorption image. The farther $z$, the stronger the fringes proportional to $\nabla^2\varphi$. Only the boundaries stand out while the interior stays flat, so this fringe image looks as though an edge detector had been applied.

Paganin phase retrieval

The boundary fringes catch the eye, but they cannot be used directly for quantitative CT reconstruction. What we want is not fringes but a filled-in image of the phase (the projected thickness). If the sample can be assumed to be a single material (a constant ratio of $\delta$ to $\beta$), the single-shot inverse filter of Paganin (2002) recovers this from the intensity at a single distance.

$$T \;=\; -\frac{1}{\mu}\ln\!\left(\mathcal{F}^{-1}\!\left[\frac{\mathcal{F}(I_z)}{1 + \dfrac{z\,\delta}{\mu}\,|\mathbf{k}|^2}\right]\right)$$

The denominator is a low-pass filter in Fourier space. Where propagation enhanced the boundaries (lifted the high frequencies), it suppresses those high frequencies to roll the effect back, recovering the low-frequency contrast that the absorption image had collapsed. We handled the ramp filter of FBP in Chapter 7 and MRI's $k$-space in Chapter 17; here again, the heart of reconstruction reduces to weighting in Fourier space.

Simulation: seeing with phase what absorption cannot

We set up a phantom mimicking water-like soft tissue: a large organ with a few structures of very small absorption difference inside it. On the left is the contact absorption image, nearly flat (note how low the contrast number is). In the middle is the propagated intensity at distance $z$; raising $z$ raises bright/dark fringes at the boundaries. On the right is the thickness image recovered by the Paganin inverse filter, where the fringes roll back into a filled-in, high-contrast image. With the same sample and the same dose, switching the measured quantity from absorption to phase changes what can be seen.

From reconstruction to image processing

Once phase contrast has furnished directed projections, the tomographic reconstruction that follows is the same as always. Line up the phase-retrieved projections at each angle into a sinogram, and return to the volume with FBP or an iterative method. As with SPECT, what is new is only what and how it measures; the skeleton of the inverse problem is shared.

That completes the story of how images are made from measurements. It set out from a single X-ray in Chapter 1 (the line integral, the Radon transform), assembled the tools of reconstruction (Fourier methods, iteration, compressed sensing, deep learning), and covered industrial CT, MRI's Fourier measurement, nuclear medicine's emission counting and collimators, and in the Frontiers part, tomosynthesis, photoacoustics, electron tomography, and phase contrast. The physical quantities measured are all different, but the framework of the inverse problem (recovering a volume from limited measurements) and the tools that solve it (the Fourier transform, iteration, regularization, statistical models, learning) were shared throughout. The image-processing part that follows turns to using the images so made: display, denoising, segmentation, registration, and quantification.

References

• Paganin D, Mayo SC, Gureyev TE, Miller PR, Wilkins SW. Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object. Journal of Microscopy 206, 33–40 (2002) — single-distance, single-material phase retrieval.
• Wilkins SW, Gureyev TE, Gao D, Pogany A, Stevenson AW. Phase-contrast imaging using polychromatic hard X-rays. Nature 384, 335–338 (1996) — a pioneer of propagation-based phase contrast.
• Momose A. Recent Advances in X-ray Phase Imaging. Japanese Journal of Applied Physics 44, 6355–6367 (2005) — an overview including grating interferometry.
• Bravin A, Coan P, Suortti P. X-ray phase-contrast imaging: from pre-clinical applications towards clinics. Physics in Medicine and Biology 58, R1–R35 (2013).