Free Water and MRI Part 1
[MRI Writing in progress…
Magnetic resonance imaging (MRI) is a medical imaging technique which uses magnetic fields to look at organs in a living body. This post describes the math behind a particular MRI imaging technique called Diffusion Tensor Imaging (DTI). DTI is used to map the structure of the brain by measuring how restricted the water molecules are in their local environment. It is a clever technique as it relies on taking a bunch of overexposed images from many different directions and then using a statistical model to try and recover what the water molecules are doing. If we can figure out how the water molecules move, we get an idea of the local environment that the water molecules are in and thus build a map of the brain.
Magnetic Resonance Imaging (MRI)
Magnetic resonance imaging (MRI) machines are huge and expensive but they work on a simple principle - when certain atoms are inside a large and varying magnetic field they absorb and emit radio frequency energy which can be detected.
Most MRI scans try to excite the hydrogen atoms in the water and fat molecules
in your body with magnetic fields. They vary the duration and also the
direction of the magnetic fields in what is known as an MRI sequence. There
are many different types of sequences and they give us different images for the
same object.
The image above has different weighting schemes (T1, T2, and PD) which show the same region of the brain. See how the images look very different as the different MRI sequences bring different structures of the brain into focus.
Diffusion Weighting
Diffusion weighting differs the weighting schemes above to give images that are overexposed and then tries to use statistics to reconstruct an image.
Here is an analogy. Suppose we have to design an algorithm to look at a picture
of a soccer player playing a soccer game. We want the algorithm to look at this
single picture and tell us whether the player runs around alot or doesn’t run
around much. This would be almost impossible to do using a single picture -
wouldn’t it? If we were able to take multiple pictures in sequence of the same
player in action we could get something similar to video and then it would be
an easier task. But what if you were forced to only use a single picture to
make this decision. One thing you could do would be to lengthen the exposure
the single picture you take. If the player is running around alot then you
should see a streak across your overexposed image. If the player is mostly
standing in a single spot then the overexposed image will appear blurry but you
should mostly be able to detect the player at a single location. 
This is quite a clever idea and using statistics
we can make guesses about how the player is moving. This is analogous to what
Diffusion imaging is doing to figure out how the water molecules are moving in
the brain.
The math here depends on complicated physics (which I do not fully understand)
however the final formulas are easy to understand and interpret. For a given
voxel (3d version of a pixel), we have the following formula that relates the
signal for a short exposure \(S_0\) to a signal with a longer exposure
\(S_t\).
Here the \(b\) term depends on the direction of the pulse, the time duration of the pulse, the time duration between two pulses, the strength of the pulse etc. All these can be fixed and so for a given sequence we can gather all those terms into a constant \(b\). The \(ADC\) term is also a constant and is called the apparent diffusion coefficient. It can be estimated if we know the values of \(S_t\), \(S_0\) and \(b\).
There is greater loss of signal with the long exposures \( S_t < S_0 \) and so the left hand side of the equation above is between 0 and 1. This gives us a map of ADC values at every voxel and we can plot that to get a diffusion weighted image. (see the image on the right) This is something new and interesting and gives us different information from just the short exposures.
The diffusion coefficient of water at body temperature is \( d = 3 \times 10^{-3}
mm^2/s \). Free water is water that is not restricted by surrounding
tissue. If we have a voxel that contains free water then we will probably get
an ADC value for that voxel that is close to the diffusion coefficient of
water.
Diffusion Tensor Imaging (DTI)
Diffusion tensor imaging extends this concept of an apparent diffusion coefficient to three dimensions. This will tell us how the diffusion is happening in 3D space. Here we let \(b\) be a constant that depends on the pulse strength, duration and timing between pulses but we do not let it depend on the gradient direction. It is a constant for a given pulse sequence and it can be calculated quite precisely. We now have a slightly different formula for the signal decay.
Here, just like before, \( S_t \) and \( S_0 \) are the signal intensities for the long exposure and the short exposure respectively. \(q\) is now a 3x1 vector that represents the direction of the pulse (gradient). \( D \) is now a 3x3 positive definite symmetric matrix that generalizes the apparent diffusion coefficient.
Now, instead of trying to estimate one ADC coefficient per voxel we need to estimate a whole 3x3 matrix \(D\) for each voxel.
Since the 3x3 matrix is symmetric we only need to estimate 6 values instead of 9. Also, we have the additional problem of making sure that the values we estimate form a positive definite matrix i.e. \(q^T D q > 0 \) for all non-zero values of q.
We know what \(b\) is and also the value of \(S_0\). If we have atleast 6 different values of \(q\) and have the corresponding values of \(S_t \), then we can estimate \(D\) via regression and a system of equations. Making sure that the answer is positive definite is a harder and will require constraints on the solutions.
If we have a voxel with water unconstrained by surrounding tissue then we expect where \(d = 3 \times 10^{-3} \) and I is an identity 3x3 matrix.
Free water Elimination
One problem we have is that voxels are quite large in practice (2mm x 2mm x 2mm) isn’t uncommon. This is large enough that the diffusion coefficient inside the voxel isn’t homogenous. There might be a mixture of free water and tissue in that voxel and we need to figure out how to estimate how much free water and how much tissue.
Here all the symbols mean the same as above and the only new symbol is \(f\) which measure the volume fraction of this voxel that is made up of tissue. Now if we can find out what the value of $f$ is and also find the value of \(D\) then we can figure out how much free water \( 1-f \) is contained in each voxel.
If you have just one value of \(b\) then it is hard to estimate \(f\) and \(D\) as many solutions are likely to fit the equations. Choosing among them is hard. There are two approaches.
- Multi-shell diffusion tensor imaging - where we vary the sequences of pulses and so have different values of b. This reduces the number of solutions down to 1 again. However this is more expensive and takes more time in the MRI machine.
- Single-shell diffusion tensor imaging with constraints - where we put constraints on the possible solutions and reduce it down to one again. Of course, we have the additional problem of trying to figure out whether the solution we have is accurate or not as in most cases there is no ground truth.
In the next post we will look at some of the math behind a single shell free water elimination model.