CT versus MRI: Exploring the Similarities and Distinctions

In this blog post, we take a brief dive into the fundamental principles of Magnetic Resonance Imaging (MRI) and Computerized Tomography (CT), highlighting the inherent differences between the two modalities. While recent strides in AI-driven techniques have enabled simulations of CT from MRI and vice versa, it's essential to recognize the distinct nature of each imaging method. Given their distinct characteristics, one cannot simply replace the other. CT scans offer tomographic (cross-sectional) images based on material linear attenuation coefficients, by directing X-rays through the body from various angles and measuring the decrease in X-ray intensity along linear paths. CT scans outperform MRI in their ability to visualize bone fractures and dense tissues, rendering them indispensable for evaluating traumatic injuries. Despite the ionizing radiation exposure associated with CT scans, they are still utilized for guiding procedures such as biopsies and surgeries, which may not be feasible with MRI. In contrast, MRI operates by manipulating hydrogen atoms within biological tissue. When subjected to an external magnetic field, these hydrogen atoms become magnetized, producing detectable magnetic fields within the MRI scanner's coil. MRI offers better differentiation between different types of soft tissues compared to CT, providing visualization of ligaments, tendons, and muscles. Due to this capability, MRI scans are widely used for diagnosing neurological disorders, musculoskeletal injuries, and soft tissue abnormalities. MRI have limitations, including its high cost, time-intensive data processing requirements, and limited availability in emergency situations. Given that both CT and MRI scans visualize the internal structures of the body, numerous advancements have been made in translating MRI images into CT-like representations and vice versa using neural networks such as GANs and Diffusion models. However, I want to emphasize that deep learning is not akin to magic. Therefore, if certain details aren't discernible in an MRI image, they can't be magically made visible in a CT-like image translated by AI, and vice versa. Keeping this in mind, let's delve briefly into the fundamental physical principles of CT and MRI.

MRI Basics 

This part is written based on the book "Electro-Magnetic tissue properties MRI", Jin Keun Seo, Eung Je Woo, Ulrich Katscher and Yi Wang, Imperial College Press, 2014. 

The human body is placed within the MRI scanner, where it is exposed to a primary magnetic field $\mathbf{B}_0$. This field is assumed to be homogeneous, represented as $\mathbf{B}_0 = B_0\hat{\mathbf{z}}$, with $\hat{\mathbf{z}}=(0,0,1)$ indicating the direction of the main magnetic field of the MRI scanner, where $B_0>0$ is a constant. Within an external magnetic field, described as $\mathbf{B} = \mathbf{B}_0 + [\mathbf{G} \cdot \mathbf{r}] \hat{\mathbf{z}}$, the magnetization vector $\underline{\mathbf{M}}$ is induced to precess around the $\hat{\mathbf{z}}$-axis. Here, $\underline{\mathbf{M}}(\mathbf{r},t)=(\underline{M}_x, \underline{M}_y, \underline{M}_z)$ depends on time $t$, position $\mathbf{r}$, and the initial magnetization $\underline{\mathbf{M}}(\mathbf{r},0)$. The interaction of $\underline{\mathbf{M}}$ with the external magnetic field $\mathbf{B}(\mathbf{r})=(0,0, B(\mathbf{r}))$ is governed by the Bloch equation: $$\frac{\partial}{\partial t} \underline{\mathbf{M}}=-\underbrace{\gamma \mathbf{B}}_{\omega \hat{\mathbf{z}}} \times \underline{\mathbf{M}}$$where $\omega(\mathbf{r}) =\gamma B(\mathbf{r})$ and $\gamma$ represents the gyromagnetic ratio.

To extract a signal of $\underline{\mathbf{M}}$ within a specific slice of the body, it's necessary to flip $\underline{\mathbf{M}}$ towards the $xy$-plane, generating its $xy$-component. The flipping requires the application of a second magnetic field, $\underline{\mathbf{B}}_1$, which is perpendicular to to the primary magnetic field $\mathbf{B}_0$. The process involves utilizing an RF magnetic field $\underline{\mathbf{B}}_1$, produced by an RF coil into which a sinusoidal current of I mA is introduced. This current oscillates at the Larmor frequency, defined by $\omega_0 = \gamma B_0$, to effectively rotate $\underline{\mathbf{M}}$ towards the $xy$-plane.  Consequently, the evolution of $\underline{\mathbf{M}}(\mathbf{r},t)$ satisfies the Bloch equation $$\frac{\partial}{\partial t} \underline{\mathbf{M}}=-\gamma \underline{\mathbf{B}}\times \underline{\mathbf{M}}-\frac{1}{T_2} \underline{\mathbf{M}}_{xy} +\mbox{relaxation effects},$$ where $\underline{\mathbf{M}}_{xy} = \underline{\mathbf{M}}_x\hat{\mathbf{x}} + \underline{\mathbf{M}}_y\hat{\mathbf{y}}$, and the total magnetic field $\underline{\mathbf{B}}(\mathbf{r},t)$ combines the steady and gradient fields with the RF field, represented as:$$\underline{\mathbf{B}}(\mathbf{r},t)= (B_0+G\cdot\mathbf{r} ) \hat{\mathbf{z}}+\underline{\mathbf{B}}_1(\mathbf{r},t).$$

Once the RF field is turning off, the magnetization vector $\underline{\mathbf{M}}$ begins to realign with the primary magnetic field $\mathbf{B}_0$, moving back towards its equilibrium state. As the magnetization vector realigns with the primary magnetic field, the k-space signals (that is linked to the Fourier transform of the MR image) are recorded. MR signal detection relies on Faraday's law and the principle of reciprocity. An external receiver coil picks up MR signals emitted from the body.

To interpret these MR signals accurately, it's essential to comprehend two fundamental relaxation processes: spin-lattice relaxation (known as $T_1$ relaxation) and spin-spin relaxation (known as $T_2$ relaxation). $T_1$ relaxation refers to the time it takes for spins to return to their equilibrium distribution along the magnetic field direction after being disturbed by RF excitation. This is a measure of how quickly they release energy to their surroundings, or "lattice". On the other hand, $T_2$ relaxation is the time it takes for spins to lose phase coherence among each other in the transverse plane, independent of the magnetic field, indicating how quickly they dephase due to interactions with neighboring spins.  MRI data can be adjusted to emphasize different characteristics of tissues, resulting in images with varying contrast styles, such as T1-weighted and T2-weighted images.

Therefore, the MR signals used to make MR images are quite different from the projection data used in CT scans due to their reliance on distinct tissue physical properties.  The MR images are sensitive to variations in the water content and the molecular structure of tissues. This means that MRI has the ability to detect details and structures that may be overlooked by CT scans. 

CT Basics

CT scans are known to be faster than MRI scans, with most procedures completed within a few minutes. The CT scan operates by emitting X-ray beams from an anode within an X-ray tube, directing these beams toward a series of detectors. The core principle of CT involves utilizing these X-rays to traverse the body, capturing projection data from various angles to construct the CT image. Specifically, images are reconstructed using X-ray data collected from different angles as the X-ray tube and detectors rotate around the patient.

To simplify the explanation, let's consider the basic principles of CT imaging using a two-dimensional parallel-beam CT model, although it is not used in clinical environment. CT is to visualized  the attenuation coefficient $\mu(\mathbf{x})$ at position $\mathbf{x}=(x_1,x_2)$, that measures how much the X-ray beam is weakened or absorbed as it passes through the medium.  Projection data, represented as $P(\varphi,s)$, is collected at a specific projection angle $\varphi$, ranging from $0$ to $2\pi$ radians, and at a detector position $s$. Assuming that the X-ray source emits monochromatic (single-energy) X-rays, the relationship between the measured projection data $P$ and the attenuation distribution $\mu$ along a beam's path $\ell_{\varphi,s}= \{\mathbf{x}\in \Bbb R^2~:~\mathbf{x}\cdot(\cos\varphi,\sin\varphi)=s\}$  can be described using the Beer-Lambert law: $$P (\varphi,s)= \int_{\ell_{\varphi,s}}\mu(\mathbf{x})dl_{\mathbf{x}}$$ where  $dl_{\mathbf{x}}$ is the line element. This integral computes the total attenuation experienced by the X-ray beam as it travels through the body, providing the projection data $P$  for reconstructing the CT image.  CT scans work by measuring how different tissues in the body absorb X-rays differently. Dense materials like bone soak up more X-rays, making them show up white on the CT images. Softer tissues, on the other hand, don't absorb as much, so they appear in various shades of gray.

About AI-based Image Translation between MRI and CT Scans

A thousand years ago, China and Korea, despite their vastly different spoken languages, managed to communicate effectively using Chinese characters. These characters served as a bridge by conveying meanings and intentions, bypassing the need for exact linguistic equivalence. In this ancient system, a spoken language, such as Korean,  was encoded into Chinese characters, which were then decoded back into another spoken language, like Chinese.

In a similar vein, the realm of artificial intelligence leverages this core principle of concentrating on the underlying essence (namely, the meaning and intent) to facilitate language translation across diverse linguistic landscapes. This approach is paralleled in the domain of medical imaging, wherein AI-based methodologies are capable of translating between CT and MRI scans. This translation works well for images in which CT and MRI both provide common details on anatomical structures.

However, it's crucial to acknowledge that specific details, especially within soft tissues, are exclusively visible through MRI and may not be discernable with CT. Conversely, certain aspects are uniquely detectable by CT scans and might be difficult to visualize with MRI. AI cannot transfer specific details from MRI to CT images when these details are missing in the MRI data. Pushing AI excessively for image translation may lead it to focus on broad trends within the images rather than capturing the distinctive features of individual patients. This limitation is intrinsically tied to the nature of AI technology, with its reliance on training data. Consequently, accomplishing a precise translation between CT and MRI images may not be achievable.

Concluding this blog, I'd like to emphasize the fundamental differences between MRI and CT imaging: MRI relies on the behavior of hydrogen protons, whereas CT depends on electron density. This implies that because they leverage different physical properties of tissues, achieving a flawless translation from one to the other is intrinsically difficult.