## View-angle tilting MRI (part 3): Fewer approximations

This is a third (and presumably final) text about view-angle tilting (or VAT) MRI concluding the explanations posted on 2020-09-10 and on 2020-09-15. I don’t really like the mathematical approximations that I used in my first text and, therefore, I would like to try again; the simple graphical visualization should correspond to an equally simple mathematical derivation …

Let’s start as before with the MRI signal equation for an excited slice without any distortions

`S`(

`t`) = ∫d

`x`

`ρ`(

`x`,

`ẑ`) exp (–i

`γG`

_{x}

`x`

`t`)

with frequency-encoding direction `x` and frequency-encoding gradient `G`_{x} (phase encoding is generously omitted). The `x` integral is over the complete imaged object (but for a compact object that fits into our field of view, we can integrate from – ∞ to ∞). The slice-selection direction is denoted by `ẑ`, where I use the hatted (“^”) variable to differentiate the undistorted (original) slice position `ẑ` from the distorted position `z` = `z`(`x`) analyzed below.

I will ignore the slice thickness, since the essence of the VAT technique can pretty well be explained without it. And everything gets really complicated if we consider varying slice thicknesses – or even slices split in several parts as illustrated in the following figure.

It’s useful to keep in mind the resonance relation between an excitation pulse with frequency `ω`_{rf} applied after switching on the slice-selection gradient `G`_{z,slc} and the slice position `ẑ` (without field inhomogeneities):

`ω`

_{rf}=

`γG`

_{z,slc}

`ẑ`or

`ẑ`= (

`ω`

_{rf})/(

`γG`

_{z,slc}).

If we now consider inhomogeneity-induced distortions of a slice (due to a field inhomogeneity, `ΔB`(`x`,`z`)), then each original slice position `ẑ` (corresponding to the excitation frequency `ω`_{rf}) is “moved” to a new position, `z`(`x`), which is, in general, different for each `x`. The new position, `z`(`x`), is the solution of the equation

`ω`

_{rf}=

`γ`(

`G`

_{z,slc}

`z`(

`x`) +

`ΔB`(

`x`,

`z`(

`x`)))

for each (fixed) value of `x`. The relation to the original slice position, `ẑ`, can be expressed (after division by `γG`_{z,slc}) as

`ω`

_{rf})/(

`γG`

_{z,slc}) =

`ẑ`=

`z`(

`x`) + (

`ΔB`(

`x`,

`z`(

`x`)))/(

`G`

_{z,slc}),

which can be re-arranged to

`z`(

`x`) =

`ẑ`– (

`ΔB`(

`x`,

`z`(

`x`)))/(

`G`

_{z,slc}).

(Of course, the previous equation is not an explicit solution describing `z`(`x`), since `z`(`x`) still appears also on the right-hand side of the equation.)

If we go back to the MRI signal equation and include the distorted slice geometry, `z`(`x`), the integral changes to

`S`(

`t`) = ∫d

`x`

`ρ`(

`x`,

`z`(

`x`)) exp(–i

`ω`(

`x`,

`z`(

`x`))

`t`),

where, `ω`(`x`,`z`) is used to describe the Larmor frequencies during readout. Without field inhomogeneities, this is simply (and independent of `z`) `ω`(`x`,`z`) = `γG`_{x} `x`. The inhomogeneity, `ΔB`(`x`,`z`) changes the Larmor frequencies to `ω`(`x`,`z`(`x`)) = `γ` (`G`_{x} `x` + `ΔB`(`x`, `z`(`x`))). If we also include the additional VAT readout gradient, `G`_{z,VAT}, we get

`ω`(

`x`,

`z`(

`x`)) =

`γ`(

`G`

_{x}

`x`+

`ΔB`(

`x`,

`z`(

`x`)) +

`G`

_{z,VAT}

`z`(

`x`)).

We can now insert the expression for `z`(`x`) derived above into the third addend of this last expression, which gives

`ω`(

`x`,

`z`(

`x`)) =

`γ`(

`G`

_{x}

`x`+

`G`

_{z,VAT}

`ẑ`+ (1 – (

`G`

_{z,VAT})/(

`G`

_{z,slc}))

`ΔB`(

`x`,

`z`(

`x`))).

So, by setting `G`_{z,VAT} = `G`_{z,slc} (as proposed for VAT MRI), the inner parenthesis vanishes and neither `ΔB` nor `z`(`x`) (which implicitly contains `ΔB` as well) appears in the expression for the Larmor frequency. This means that the artifacts in readout direction (due to changed readout frequencies) are removed:

`S`(

`t`) = ∫d

`x`

`ρ`(

`x`,

`z`(

`x`)) exp(–i

`γ`(

`G`

_{x}

`x`+

`G`

_{z,VAT}

`ẑ`)).

Other artifacts caused by the changed slice geometry, `z`(`x`), obviously remain (as indicated by the term `ρ`(`x`, `z`(`x`)) in the integral) and are *not* corrected by the VAT approach.

Since I’ve omitted the integration (∫d`z` …) over the finite slice thickness, the meaning of the slice coordinate `ẑ` in the final results is not completely obvious. Actually, the appearance of `ẑ` during readout corresponds to a tilted readout (that’s why it’s called *view-angle tilting*), which becomes clearer if one includes the omitted integration over the slice direction in the resulting MRI signal equation.

pdf version available

URL: <https://dtrx.de/od/mmm/mmm_20200929.html>