15 Advanced topic: Bloch’s model and MRI
We will take a classical modellers perspective to understand how Magnetic Resonance Imaging (MRI) works. This will also enable us to appreciate its potential to experimentally investigate continuum mechanical processes. We will introduce necessary concepts of electromagnetism, quantum mechanics, nuclear magnetic resonance and MRI in a rather straight-foward and ad hoc way inspired by the two books Callaghan (1993) and Dössel (2000). While the idealized classical modellers perspective takes us quite far, there are also limitations. Becoming an expert in the field of nuclear magnetic resonance and MRI certainly requires a further in-depth study of quantum mechanics, electromagnetism and electrical engineering.
15.1 Magnetic field and magnetic dipole
A magnetic field \(\mathbf B\) exerts a torque on a magnetic dipole, which means that the object exhibits a magnetic dipole moment. The magnetic dipole moment can be measured by the torque that a dipole exters on it in a homogeneous magnetic field:
\[ \underbrace{\mathbf \tau}_{\text{torque in }Nm} = \underbrace{\mathbf m}_{\text{mag. dipole moment in }Am^2}\times \underbrace{\mathbf B}_{\text{magnetic field in }\frac{Nm}{Am^2}=:T} \]
The magnetic dipole moment is a first order tensor, hence has a direction and a magnitude. Example for magnetic dipoles are
- compass needle in Earth’s magnetic field
- general permanent bar magnets
- curcuit currents, which act as dipoles (\(\mathbf m = I \cdot F \cdot \mathbf e_n\))
15.2 Magnetization and the idea of MRI
Materials react in different ways to externally applied magnetic fields. On the macro-scale, we distinguish
- diamagnetic materials: extremely weak response, e.g., water, organic material
- paramagnetic materials: weak response, e.g., aluminium, lithium
- ferromagnetic materials: strong response, e.g., iron, cobalt
On the quantum mechanical scale, however, every material responds to an externally applied magnetic field. This is used for MRI.
In order to understand this better, we need to introduce so-called magnetization \(\mathbf M\) of a specific reference volume \(V\). It is defined as the superposition of all magnetic dipole moments in that volume:
\[ \mathbf M = \frac{1}{V} \sum \mathbf m_i, \]
in which index \(i\) denotes the individual dipoles. Quite interestingly, certain atoms also acts as magnetic dipoles, e.g. protons. Ensembles of protons hence give rise to an effective magnetization of a reference volume. The fundamental idea of MRI is hence to utilize the magnetization of ensembles of protons in our body for imaging purposes. The next sections will be devoted to understanding how this can be done.
15.3 Precession of a magnetic gyro
Let’s assume we are give a magnetic dipole that rotates within an external magnetic field \(\mathbf B\). The rotating magnetic dipole then becomes a magnetic gyro. It’s angular momentum is given by
\[ \underbrace{\mathbf L}_{\text{angular momentum}} = \underbrace{\mathbf I}_{\text{momentum of inertia}} \cdot \underbrace{\omega.}_{\text{angular velocity}} \]
From Newton’s laws, we know
\[ \frac{d}{dt}\mathbf L \underbrace{=}_{\text{Newton's law}} \underbrace{\mathbf \tau}_{\text{torque}} = \underbrace{\mathbf m}_{= m \mathbf e_m} \times \underbrace{\mathbf B}_{= B \mathbf e_B}. \]
We can hence conclude that the dipole \(\mathbf m\) is rotating unaffected by the magnetic field \(\mathbf B\) as long as \(\mathbf e_m || \mathbf e_B\), because torque \(\mathbf \tau\) vanishes in this situation. As soon as there is an angle between \(\mathbf m\) and \(\mathbf B\), hence between \(\mathbf e_B\) and \(\mathbf e_m\), the torque will no longer vanish and \(\tfrac{d \mathbf L}{dt}\) will be orthogonal to both \(\mathbf m\) and \(\mathbf B\). This will induce a precession motion at precession angular velocity
\[ \frac{d \phi}{dt} = \omega_0 = - \underbrace{\frac{m}{L}}_{:=\gamma} B. \]
Here, \(\phi\) denotes the angular position of the precession motion, \(\omega_0\) denotes the corresponding angular velocity, and \(\gamma\) stands for the so-called gyromagnetic ratio, also referred to as magnetogyric ratio. Note, that the angular precession speed \(\omega_0\) is independent of the angle \(\alpha\) between \(\mathbf e_m\) and \(\mathbf e_B\).
15.4 Precession of an ensemble of protons
Inspired by our insight into magnetic gyros, we now turn our focus to much smaller scale magnetic dipoles. Certain nuclei, such as protons and phosphor atoms also possess an intrinsic magnetic diploe moment. When placed into a static, external magnetic field, these particle hence share characteristics of a magnetic gyro in the sense that they experience a torque that tends to align the direction of the magnetic dipole moment and the magnetic field. Instead of rotation, we talk about spin and will exploit the analogy between both in the following sections.
Nuclear magnetic resonance has been simultaneously discovered by Purcell, Torrey and Pound in Harvard and by Bloch, Hansen and Packard in Stanford. At the heart of this discovery is the observation that the torque extered by a magnetic field \(\mathbf B = B_0 \mathbf e_B\) to the spinning magnetic dipole moment of a nuclei induces a precession subject to gyromagnetic ratio
\[ \gamma := \frac{\mu}{L}. \]
Here, the magnetic dipole moment is referred to as \(\mu\) to account for the fact that we are on the particle level. \(\gamma\) depends on the type of particle, for example
| type of particle | \(^1\) H | \(^{13}\) C | \(^{19}\) F | \(^{31}\) P |
|---|---|---|---|---|
| \(\gamma\) in \(MHz/T\) | 42,6 | 40 | 10,8 | 17,2 |
The precision frequency - also referred to as the Lamor frequency is then given by
\[ \omega_0 = \gamma B_0, \]
hence depends on the strength of the homogeneous magnetic field \(B_0\). For \(B_0=50 \mu T\), for instance, which roughly corresponds to Earth’s magnetic field, a proton will end up precessing at a frequency of \(2,13 kHz\). The same proton will precess at \(170,4 MHz\) in a \(1T\) MRI scanner.
So far, the analogy seems pretty good. What’s the catch?
In quantum mechanical systems the angular momentum of particles, such as protons, is subject to a so-called spin number. For protons this would be \(\tfrac{1}{2}\). In a static magnetic field aligned with the \(z\)-axis (\(\mathbf e_B = \mathbf e_z\)), we have \(\mathbf B=(0,0,B_0)^T\) and the angular momentum is quantized, namely
\[ L = \sqrt{l(l+1)} \hbar \quad \text{and} \quad L_z = m_l \hbar \]
For spin \(\tfrac{1}{2}\) particles we have \(l=\tfrac{1}{2}\), such that
\[ L = \frac{\sqrt{3}}{2} \hbar \quad L_z = \pm \frac{1}{2} \hbar \]
An important consequence of these laws of quantum mechanics is that the \(z\)-component of the particle’s angular momentum denoted by \(L_z\) can take one of two values, namely \(\pm \tfrac{1}{2} \hbar\). Components that are orthogonal to the \(z\)-axis are uncertain, hence cannot be specified. We refer to countless texts on quantum mechanics for further details.
Now, we assume an ensemble of protons, e.g. a reference volume within in our body. Since we are dealing with a large collection of protons it makes sense to consider the expectancy of \(\mu\). For its \(z\)-component, we find
\[ < \mu_z > = \gamma L_z = \pm \frac{1}{2} \gamma \hbar. \]
This means that the collective magnetization in direction of the magnetic field is composed of particles at a lower energy level \(-\frac{1}{2} \gamma \hbar B_0\) that spin-up and those at a higher energy level \(\frac{1}{2} \gamma \hbar B_0\) that spin-down.
This raises the question: Does the overall magnetization in the direction of the magnetic field simply get cancelled?! Luckily this is not so! The distribution of quantized energy levels follows the Boltzmann statistics, which tells us that the ratio between the number of spin-down protons \(N^{-}\) and the number of spin-up \(N^{+}\) is in thermal equilibrium:
\[ \frac{N^{-}}{N^{+}} = \exp \left( \frac{\Delta E}{k T} \right) \]
The energy difference between the two energy levels is given by \[ \Delta E = \frac{1}{2} \gamma \hbar B_0-(-\frac{1}{2} \gamma \hbar B_0) = \gamma \hbar B. \]
Substitution leads to
\[ \frac{N^{-}}{N^{+}} = \exp \left( \frac{\gamma \hbar B_0}{k T} \right) \]
and can be solved. For protons in a magnetic field of strength \(B=1T\) at a temperature of \(37^\circ\) we for instance find 1,0000066, hence 6,6ppm more spin-ups than spin-downs. The magnetization of \(1mm^3\) of water, which is approximately \(6,7 \cdot 10^{19}\) portons is therefore around \(3\cdot 10^{-3} Am^{-1}\).
The effective magnetization within a reference volume \(\mathbf x\) denoted by \(\mathbf M(\mathbf x) = (M_x(\mathbf x),M_y(\mathbf x),M_z(\mathbf x))^T\) now corresponds to the expectancy of its precessing protons. For \(\mathbf B=(0,0,B_0)^T\) we end up with a collective magnetization that has a non-vanishing \(z\)-component \(M_{z0}\), while the individual \(x\)- and \(y\)-components are not correlated, and lead to \(M_{x0} = M_{y0}=0\). We refer to this is the equilibrium magnetization
\[ \mathbf M_0 = \left( \begin{array}{c} M_{x0} \\ M_{y0} \\ M_{z0} \end{array} \right) = \left( \begin{array}{c} 0 \\ 0 \\ M_{z0} \end{array} \right). \]
15.5 The Bloch equations
A model for the evolution of an ensemble’s magnetization vector \(\mathbf M (\mathbf x,t)\) is given by the Bloch equations:
\[ \frac{d}{dt} M(\mathbf x,t) = \underbrace{\gamma M(\mathbf x,t) \times B(\mathbf x,t)}_{\text{excitation}} + \underbrace{\mathbf R(\mathbf x) \left( M(\mathbf x,t) - \mathbf M_0 \right)}_{\text{relaxation}} \]
with \[ \mathbf R(\mathbf x) = \left( \begin{array}{c c c} -\frac{1}{T_2(\mathbf{x})} & 0 & 0 \\ 0 & -\frac{1}{T_2(\mathbf{x})} & 0 \\ 0 & 0 & -\frac{1}{T_1(\mathbf{x})} \end{array} \right). \]
Considering that \(\gamma \mathbf M\) is nothing but the angular momentum, we see that the excitation part of the Bloch system again mimicks Newton’s law. In a static magnetic field, this would lead to a collective precession as soon as \(\mathbf M\) is out of the \(\mathbf B\) axis. In thermal equilibrium, however, this is not the case, see above.
Let’s focus on the _excitation part: for a start: We superpose the static magnetic field denoted by \(\mathbf B_0=(0,0,B_0)^T\) with an transversal magnetic field \(\mathbf B_1(t)\) that oscillates at Lamor frequency and arrive at
\[ \mathbf B = \mathbf B_0 + \mathbf B_1(t) = B_0 \mathbf e_z + B_1 \cos (\omega_0 t) \mathbf e_x - B_1 \sin (\omega_0 t) \mathbf e_y. \]
In an MRI scanner, this field stems from a radio frequency pulse emitted into the scanned area. The excitation part of Bloch’s equations reads
\[ \begin{align*} \dot{M}_x &= \gamma \left(M_y B_0 + M_z B_1 \sin (\omega_0 t) \right) \\ \dot{M}_y &= \gamma \left(M_z B_1 \cos (\omega_0 t) - M_x B_0 \right)\\ \dot{M}_z &= \gamma \left(-M_x B_1 \sin (\omega_0 t) - M_y B_1 \cos (\omega_0 t) \right). \end{align*} \]
For the initial condition \(\mathbf M(0) = \mathbf M_{z0}\) the solution reads
\[ \begin{align*} M_x(t) &= M_{z0} \sin (\omega_1 t) \sin (\omega_0 t) \\ M_y(t) &= M_{z0} \sin (\omega_1 t) \cos (\omega_0 t)\\ M_z(t) &= M_{z0} \cos (\omega_1 t), \end{align*} \]
in which freuqnecy \(\omega_1 = \gamma B_1\) is determined by the amplitude of the emitted radio frequency pulse. The solution is hence a superposition of a precession motion around the \(z\)-axis and a rotation away from the \(z\)-axis referred to as flipping. The latter is seen best, when \(\mathbf M(t)\) is transformed into a coordinate system that rotates at Lamor frequency \(\omega_0\). The rotating magnetization will be denoted by \(\mathbf M'(t)\). The solution to Bloch’s equations then takes an even simple form in which the superposed rotation away from the \(z\)-axis is directly evident:
\[ \begin{align*} M'_x(t) &= M_{z0} \sin (\omega_1 t) \\ M'_y(t) &= 0 \\ M'_z(t) &= M_{z0} \cos (\omega_1 t). \end{align*} \]
The so-called flipping angle \(\alpha\) defined as
\[ \alpha = \gamma \omega_1 \tau, \]
denotes the angle between \(z\)-axis and magnetization vector after time \(\tau\). Note, how a strong RF pulse (large \(B_1\)) and a short pulse duration can lead to the same flipping angle than a weaker RF pulse applied for a longer duration.
Due to the oscillating nature of the flipping motion, it is possible to induce an exact \(90^\circ\) flip by choosing \(\tau\) just right. We’ll denote this duration as \(\tau_{90}\) and refer to it as a \(90^\circ\)-pulse. It corresponds to the time that we need to apply a RF pulse in oder to turn the magnetization vector completely into the \(x-y\) plane. Twice that duration denoted by \(\tau_{180}\) will flip the magnetization vector by \(180^\circ\) to point in negative \(\mathbf B_0\) direction.
Any type of flipping \(\alpha \neq 0\) constitutes a non-equilibrium situation. The system hence tends to relax back into its thermal equilibrium. The component-wise relaxation part of the Bloch equations in a rotating reference frame reads
\[ \begin{align*} \dot{M'}_x &= - \frac{1}{T^\ast_2} M'_x \\ \dot{M'}_y &= - \frac{1}{T^\ast_2} M'_y \\ \dot{M'}_z &= - \frac{1}{T_1} (M'_z - M_{z0}) \end{align*} \]
Here, \(T_1\) accounts for the so-called spin-lattice relaxation back to thermal equilibrium, whereas \(T^\ast_2\) accounts for the apparent decay (dephasing) of the transverse magnetization, hence the decay of the magnetization in the transverse \(x-y\)-plane. It is composed of the so-called spin-spin relaxation \(T_2\) due to interaction with neighbouring magnetic dipoles as well as relaxation due to inhomogeneities \(T_{2i}\) according to
\[ \frac{1}{T^\ast_2} = \frac{1}{T_2} + \frac{1}{T_{2i}} \]
Assuming as initial condition the magnetization after a \(90^\circ\) pulse, hence \(M(0) = (M_{z0},0,0)\), we result at
\[ \begin{align*} M'_x(t) &= M_{z0} \exp \left(-\frac{1}{T^\ast_2} \right) \\ M'_y(t) &= 0 \\ M'_z(t) &= M_{z0} \left(1-\exp \left(-\frac{1}{T_1} \right) \right). \end{align*} \]
15.6 Free induction decay
Let’s observe what happens, if we at first excite an ensemble of protons, and then switch of the pulse after a duration \(\tau_{90}\). The magnetization vector gets flipped into \(x,y\)-plane as an oscillating transverse magnetization. The magnitude of the correlated \(x\)- and \(y\)-components decays at rate \(\tfrac{1}{T^\ast_2}\), see figure Figure 15.1.
Note, that \(M_T\) denotes the amplitude of the transverse magnetization and is introduce to facilitate a condensed notation. You can think of it as the \(x\)-component the rotating reference frame above. The decay of \(M_T\) is referred to as free induction decay (FID) and due to the previously described spin-spin relaxation caused by de-phasing of the precessing spins and further inhomogeneities in the local magnetic field. At the same time, the magnetization’s \(z\)-component \(M_z\) relaxes back into thermal equilibrium \(M_{z0}\) due to spin-lattice relaxation at rate \(\tfrac{1}{T_1}\). Timescale \(T_1\) is always longer than timescale \(T_2\).
Two questions remain:
- How can we measure the evolving magnetization and its relaxation times \(T_1\) and \(T_2\)/\(T_2^\ast\)?
- How can we spatially resolve the signal to differentiate between neighboring tissue types based on differences in relaxation time scales?
In this lecture, we will discuss the fundamental ideas, while a detailed study requires to consult expert literature.
15.7 Measuring relaxation time scales
The fundamental technological idea is that the oscillating transverse magnetization \(M_T\) can be detected via receiver coils. It is hence possible to capture the FID signal during a so-called read-out. In contrast to this, the \(z\)-component cannot be measured directly. This poses a challenge, as \(T_1\) is typically larger than \(T_2\). Even after signal decay in the receiver coils we cannot immediately excite the ensemble again, as it takes longer to restore the equilibrium magnetization. It is hence crucial to determine \(T_1\) not only to characterize tissue material, yet also to design efficient MRI pulses.
An indirect way to determine \(T_1\) is referred to as inversion recovery sequence.
An initial \(180^\circ\) pulse leads to a complete inversion of \(M_z\). It hence points into negative \(z\)-direction. Consequently, the transverse magnetization will be almost undetectable. This time, however, it is not our goal to capture the FID. We rather wait for some time, assuming that a part of the longitudinal magnetization will have relaxed back into equilibrium. Now, a second pulse is applied, which this time is a \(90^\circ\) pulse. Due to the fact that the thermal equilibrium had not yet been totally restored, we can only flip a fraction of the magnitization into transverse plane. As a result, the amplitude of the following FID will be smaller than what we started with, see figure Figure 15.2. This is exactly what we want, however, as quantifying the amount of amplitude reduction allows us to infer on \(T_1\).
Another essential pulse sequence is referred to as spin echo sequence to measure \(T_2\). Its fundamental idea is to at first induce a FID, which correponds to a dephasing of the oscillating transverse magnetization, and then flip the magnetization by \(180^\circ\) to invert the magnetization. The latter induces a refocussing of the tranvsere magnetization (_back-phasing), which is referred to as a spin echo. Its maximal amplitude is smaller than that of the orginal FID signal. Detecting the decay of maxima corresponding to subsequent echos provides a means to determine \(T_2\).
15.8 Spatial resolution in MRI
Being able to distinguishing between different tissue types based on their relaxation time scales is a good start. In order to utilize this for an MRI scan, however, we have to also be able to spatially resolve the origine of the signal. Only then will we be able to infer on magnetization \(\mathbf M(\mathbf x,t)\) and relaxation times \(T_1(\mathbf x)\) and \(T_2(\mathbf x)\) as functions of space \(\mathbf x\). Three techniques are employed to facilitate \(z\)-, \(y\)- and \(x\)- resolution.
i) Spatial resolution in \(z\)-direction through a gradient field:
Idea: Superpose the background magnetic field with a static gradient field in longitudinal direction:
\[ \mathbf B(\mathbf x) = \mathbf B_0 + G_z \cdot z \]
We learned before that the excitation requires a RF-pulse that matches the Lamor frequency. The gradient \(G_z\) implies that the Lamor frequency changes in longitudinal direction:
\[ \omega_0 (z) = \gamma \left( \mathbf B_0 + G_z \cdot z \right). \]
A given RF-pulse hence excites only the slice whose Lamor frequency corresponds to the chosen excitation frequency. Any detected signal in the receiver coil has consequently been emitted by that slice, which facilitates \(z\)-resolution.
ii) Spatial resolution in \(y\)-direction through phase coding:
Idea: Apply a gradient in \(y\)-direction before signal detection in order to imprint a spatially correlated dephasing of the transverse magnetic field before read out.
\[ G_y = \frac{\partial B_{z0}}{\partial y} \quad \text{for a duration of } T_y \]
iii) Spatial resolution in \(x\)-direction through frequency coding:
Idea: Apply a gradient in \(x\)-direction during signal detection in order to induce a systematic dephasing of the transverse magnetic field during read out.
\[ G_x = \frac{\partial B_{z0}}{\partial x} \quad \text{during signal detection} \]
15.9 MRI imaging
Considering all three spatial resolution techniques, we get an expression for the to-be-expected signal \(S\) within an excited slice:
\[ S(t,T_y) = \int \int M_{z0} \exp \left( - i \gamma G_x x t - i \gamma G_y y T_y\right) dx dy \]
Note, that for convenience, we express the \(x-\) and \(y\)- oscillation in the complex plane with \(i\) being the imaginary unit. The signal depends on both \(t\) and on the duration of the applied gradient during phase coding. With
\[ k_x := \gamma G_x t \quad \text{and} \quad k_y := \gamma G_y t \]
we can furthermore re-write the signal as a 2d Fourier transform of the magnetization:
\[ S(k_x,k_y) = \int \int M_{z0} \exp \left( - i k_x x - i \gamma k_y y \right) dx dy. \]
This means that Fourier inversion of the signal yields a spatially resolved magnetization vector in 2d. This in retur can be interpretated in terms of apparent relaxation times, hence tissues types.
A vast amount of advanced techniques have been developed that build upon the presented fundamental concepts to facilitate efficient imaging as well as to facilitate imaging tailored towards certain functionality of the tissue.
15.10 Bloch-Torrey model
In a continuum mechanical context, magnetization can also be interpreted as a property of an advecting or diffusing substance, which gives rise to the Bloch-Torrey model as an extension to an underlying material model:
\[ \frac{d}{dt} M(\mathbf x,t) = \underbrace{\gamma M(\mathbf x,t) \times B(\mathbf x,t)}_{\text{excitation}} + \underbrace{\mathbf R(\mathbf x) \left( M(\mathbf x,t) - M_0 \right)}_{\text{relaxation}} - \underbrace{\mathbf v(\mathbf x) \cdot \nabla M(\mathbf x,t) }_{\text{advection}} - \underbrace{\nabla \cdot D(\mathbf x) \nabla M(\mathbf x,t) }_{\text{diffusion}} \]