Magnetic resonance imaging (MRI) and nuclear magnetic resonance (NMR) spectroscopy are both able to determine diffusion coefficients or, in the case of anisotropic media, diffusion tensors. Both technologies do so by the same means: by making the resonance frequency for a nuclear spin dependent on its position in space by the use of pulsed magnetic field gradients. As spins diffuse, their resonance frequencies are random (stochastic) functions of their position. This causes a decoherence of the overall signal as the oscillating signals from each spin, all randomly changing frequency, increasingly interfere destructively as time goes on. Eventually there is no coherence among signal phase, as all spins have sampled a broad distribution of frequencies. How quickly the signal loses coherence and cancels out depends on how quickly a broad range of frequencies can be experienced by the diffusing spins, and thus how rapidly a random distribution of phases can be reached. So the signal dies off quicker for faster diffusion (smaller or less constrained molecules), but also if a more powerful magnetic field gradient is used.

We will provide a quantitative treatment from basic principles, which will be in close analogy with the theory of NMR relaxation. It will be shown that the magnetic resonance signal during a pulse field gradient is equivalent to a correlation function for resonance frequency or signal phase, and to the Fourier transform of the conditional probability for diffusion in the direction of the field gradient. We will derive in full the formulae necessary to fit models of isotropic and anisotropic diffusion to magnetic resonance data.

As a side note, the term "resonance frequency" is a little loosely-used here, but should be interpreted as the energy of the Zeeman interaction between a nuclear spin and the applied magnetic field, which in a quantum theoretical treatment corresponds to the rate of spin state transitions. It is also the "precession frequency", meaning the rate at which the transverse magnetic moment for a nuclear spin oscillates about the axis of the applied magnetic moment, which is its interpretation in the classical vector (Bloch equation) description. It is therefore the oscillation frequency of the signal observed from a given nuclear spin.

## Mathematical Treatment

We will use the simple Bloch equation vector model of magnetic resonance and seek an expression for the magnetic resonance signal in the presence of translational diffusion and a magnetic field gradient. In this theory, each observable nuclear spin is considered independently, then we add up contributions over the ensemble to get the total observed signal. The complex-valued, transverse magnetic moment for a single diffusing spin in a static magnetic field with Larmor frequency ω0, chemical shift ωcs, field gradient vector g , isotropic diffusion coefficient D, position vector X and transverse relaxation rate constant R can be described by a Bloch equation:

We will usually find it convenient to work in a rotating frame of reference, rotating with the chemically shifted Larmor frequency. Without change of notation, in this frame, the relevant Bloch equation is:

We could equivalently write

with

However, in the treatment of diffusion, X is not simply a coordinate, it is the random (stochastic) path of coordinates that the spin samples by Brownian motion. So we will replace X with X_t to indicate this stochastic path. The magnetisation is therefore itself also a stochastic function. To indicate this, we will change the notation of m to m_t, for the stochastic magnetisation from a single spin. Naively, we could then write:

However, stochastic functions are not differentiable (exactly why is out of scope here). Instead, a meaningful stochastic differential equation (SDE) for the magnetisation from a single diffusing spin is obtained by multiplying through by dt:

The method of variation of constants can be used to write a formal solution:

This is sufficient to simulate the signal from diffusing spins in a static field with field gradient. The figures below show such a simulation. The signal's frequency varies as the particle samples different spatial positions with different magnetic fields due to the field gradient.

*The stochastic magnetic resonance signal (top: real part, bottom: phase) from a particle diffusing through a magnetic field gradient, with D=0.7e-9 m^2 / s, and a field gradient magnitude of 0.05 T / m*

To obtain the signal observed in a magnetic resonance experiment over a large ensemble of spins, we must integrate over the ensemble:

This equation introduces two important quantities: the probability

of observing a spin with gradient frequency ωt, and the conditional probability

of observing a spin with gradient frequency ωt at time t given that it was observed to have gradient frequency ωt' at t'. The integrals take the instantaneous magnetisation for a partcular gradient frequency (equivalent to position), scaled by the probability of observing such a frequency, scaled by the probability of a transition to such a frequency from some starting frequency. So the observed signal for a sufficiently large ensemble is simply a weighted sum of all possible signals. From the Bloch equation in the rotating frame, we know that

This leaves us with the problem of finding the two probability functions, which requires examining the diffusion process.

## The stochastic motion of a particle: Langevin equation

The stochastic motion of a particle is described by the Langevin equation:

with v_t and X_t the stochastic velocity and position respectively, γ a friction coefficition and β =k_B T where k_B is Botzmann's constant and T is temperature. The term W(t,n) (sorry about the lack of super/subscripts) refers to the stochastic forces acting on particles in n spatial dimensions. It denotes a standard Wiener process. This is a stochastic process with special properties. In particular, the Wiener process is a non-stationary Gaussian process whose increments W_t - W_t' are stationary. The increments W_t - W_t' have variance t-t', so that the distribution of jumps in the process is broader as the time delay between observations gets larger. In the figures below, a 3-dimensional Brownian motion path is shown, with coordinates sampled every 1 ms and every 10 ms with an isotropic diffusion coefficient of 0.7 x 10-9 m^2 s^-1. We can see that the jumps are larger when the time between samples is longer, which is the behavious of a Wiener process.

The diffusion of particles which can be realistically observed in any magnetic resonance experiment occurs in the so-called overdamped limit in which γ is large, which eqilibrates the system quickly. Consideration of diffusion therefore means we need only consider "long" time-scales. By a descent into stochastic theory that is out of scope for now, in this limit we need only consider position, which furthermore reduces to a scaled Wiener process:

Or as an SDE:

with

In the treatment of diffusion, the Wiener process has the same dimensionality as the problem we are treating, normally 3-dimensional (although 2-dimensional probelms occur, such as diffusion on a membrane or surface of a sphere etc). The quantity σ is in general an n-dimensional tensor, n being the dimensionality of the system. However, it can be treated as a scalar if diffusion is isotropic.

*Brownian motion path of a particle sampled at 1 ms intervals (top) and 10 ms intervals (bottom), D=0.7e-9 m^2 / s.*

## The Fokker-Planck Equation

For any Markov process, of which diffusion is an example, it is possible to associate with it an equation of motion for its conditional and non-conditional probabilities. This is achieved using the Fokker-Planck equation (FPE, also known as the forward Kolmogorov equation in mathematics). If X_t is a random variable driven by n-dimensional Wiener process W(t,n), diffusion σ and drift b described by the SDE:

then the FPE can be written:

Where the diffusion tensor field D(x) is:

In the case of the diffusion of particles in the over-dampled limit and in the absence of any constraining potential, there is no drift (friction) term and the FPE becomes:

Solving this equation yields allows us to obtain the conditional probability P=P(X,t | X,t'), meaning the probability that the system be located in configuration X,t at time t given that it was in configuration X,t' at t'. In magnetic resonance experiments, we shall mainly be interested in cases where the diffusion tensor is (or can be modelled as) constant in space and time:

It is sometimes possible to treat diffusion as isotropic, in which case the tensor reduces to a coefficient:

In the case of n-dimensional isotropic translational diffusion, the solution is:

which can be verified by substitution. In the case of anisotropic diffusion with diffusion tensor **D**, the solution is:

This is harder to prove, but can be verified by substitution with some effort.

## Solution for stochastic gradient frequency

We are interested not in the 3-dimensional problem of particle position but the 1-dimensional problem of a linearly varying magnetic field (thus resonance frequency). Therefore we are seeking the condidtional probability

of observing a spin with gradient frequency ω(t) at time t given that it was observed to have gradient frequency ω(t') at t'. Recall that the gradient frequency (the contribution to resonance frequency due to the magnetic field gradient) is given by:

ow we wish to express ω as a stochastic process. Recalling that position is a Wiener process:

where n is the dimensionality of the system (usually 3 for the problems we are interested in here), the (1-dimensional) stochastic gradient frequency is given by:

Therefore the gradient frequency is a 1-dimensional Wiener process in the direction of g. The final line is simply a change in notation with T indicating the vector transpose. The FPE can be written:

with direction-dependent diffusion coefficient for frequency related to the translational diffusion tensor:

This can be easier to see if the system is first transformed into a coordinate system with g parallel to the x-axis (left to the reader). The fact that D(ω) is scalar has simplified the FPE for gradient frequency. It is projecting out the diffusion parallel to g. We have already seen that the solution can be written:

Examples of the 1-dimensional Wiener process of gradient frequency are provided in the figures below, with corresponding histograms of the jumps in gradient frequency.

*The gradient frequency path of a particle sampled at 1 ms intervals (top) and 10 ms intervals (bottom), D=0.7e-9 m^2 / s , with a field gradient magnitude of 0.05 T / m*

*The gradient frequency increment distribution of a diffusimg particle sampled at various intervals, D=**0.7e-9 m^2 / s , with a field gradient magnitude of 0.05 T / m*

## The ensemble magnetisation

We are now ready to calculate the ensemble magnetisation that we will observe in a magnetic resonance experiment:

With the single-particle maagnetisation function in the rotating frame given by:

The relaxation term R won't affect the integrals since frequency doesn not appear, so this will simply be an exponential damping term, as it is without treating diffusion. Therefore we will ignore it here for clarity, but keep in mind that relaxation damping is present in reality.

In the treatment of a magnetic resonance experiment, be in in NMR or MRI, we normally assume P(ω) constant over the domain being observed. We can also exploit the fact that the diffusion process is stationary, so only the time difference matters between the points at which ω,t and ω',t' are observed, not both t and t'. Therefore the conditional probability simplifies to:

There we have:

The diffusion attenuation of the observed magnetisation is therefore the Fourier transform of the conditional probability for a 1-dimensional diffusion process parallel to the magnetic field gradient. It is a property of Fourier transforms that the narrow the function, the broader its Fourier transform. So if the conditional probability is narrow, attenuation is slow, and vice versa. The conditional probability is narrow if large jumps in frequency, which imply large jumps of a particle in space parallel to the field gradient, are rare. This is the case for slow diffusion. So slowly-diffusing particles should have slow attenuation of the magnetic resonance signal, and vice versa. Completing the calculation by evaluating the Fourier transform:

Yields

Where all constants are absorbed into M_0. We can simplify this by introducing a tensor-valued variable **b** defined by:

to give:

with the circle operator the Hadamard (element-wise) product. Note that in many magnetic resonance pulse sequence the b-value (as this tensor is often known) must also take account of refocussing pulses, but the salient point remains: The observable signal is exponentially attenuated by diffusion. The faster the diffusion in the direction of the magnetic field gradient, the quicker the system loses memory of its initial frequencies and thus phase. It is this function that is often fitted to data in diffusion tensor imaging (A separate post will cover data fitting). Note that if we include relaxation and chemical shift, the solution is:

## Diffusion attenuation as a memory process

Attenuation due to diffusion occurs as the system loses memory of its initial distribution of gradient frequencies. The diffusion attenuation is equivalent to the correlation function for gradient frequency. The stochastic gradient frequency is an ergodic Markov process. As such an ensemble average for some time point is equivalent to a long-time average for a single particle. A correlation function can be defined as:

For a stationary process, of which Brownian motion is an example, the correlation function depends only on the time difference:

The second line is the basic function for diffusion attenuation we have already met, so we can identify it with a correlation function. The function f in our context is the single-particle magnetisation (written without pre-factors and relaxation for clarity):

We can also identify the phase:

such that the diffusion attenuation can be identified as the correlation function for phase:

Diffusion attenuation is conceptually therefore the dissipation of phase coherence, as the system loses memory of its initial distribution of phases. An illustration of this is provided in below, showing magnetisation vectors initially with the same phase eventually becoming random as each spin samples a different range of frequencies due to its unique stochastic path through the field gradient.

*The magnetic moments of 20 spins as they diffuse, sampled every 3 ms, D=0.6e-9 m^2 / s **with a field gradient magnitude of 0.05 T / m*

We determined that this correlation function is proportional to the Fourier transform of the conditional probability of the 1-dimensional Wiener process of Brownian motion parallel to the magnetic field gradient. It is worth noting that the spectral density function is the Fourier transform of the correlation function. Therefore, the conditional probability is proportional to the spectral density function. The conditional probability therefore indicates the density of motion occurring as a function of gradient frequency. Recalling this conditional probability:

A narrow conditional probability means that motion is condensed into low frequencies (velocities), which implies a large particle. Since such a particle moves slowly, it maintains coherence with its initial frequency (or phase) for longer. This is rather similar to the considerations made in the treatment of magnetic resonance relaxation. The system relaxes as it loses memory of its starting configuration due to stochastic motions and becomes random again. So why is diffusion attenuation, or diffusion-mediated decoherence, not considered to be relaxation? The distinction lies not in the nature of the stochastic motions, but in the reasons for which such motions expose the system to a stochastic magnetic field. In the case of nuclear spin relaxation, stochastic motion leads to a stochastic magnetic field because the local magnetic field depends on the spin system itself. For example, chemical shift depends on orientation relative to applied field. The applied field may be constant, but (rotational) diffusion exposes the spin to a random field because of the chemical shift anisotropy interaction. The random field modulation is because of the spin system itself. By contrast, in diffusion-mediated decoherence, the field experienced by spins only changes because a varying field is imposed by the field gradient, which is external to the spin system. The random field modulation is because of the instrumentation.

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