M3 – Monologues, MRI, Miscellanea

What’s special about 1.256 431 208 626… in MRI?

posted on 2018-06-28 (by Olaf Dietrich)

Consider the following simple question in magnetic resonance imaging (MRI) or spectroscopy (MRS): Given a fixed total measurement time, Ttotal, (e. g., a typical breath-hold duration of 16 s or a maximum accepted sequence duration of 10 min) and the possibility to fit into this duration an acquisition with several repeated read-outs (that are to be averaged to increase the data quality), what is the optimum balance between the repetition time (TR or TR) and the number of averaged signals (also called simply the number of averages, N, or the number of excitations, NEX or Nex)? (For simplicity, let’s consider only pulse sequences with 90 ° excitations (e. g., spin-echo sequences), in which all available magnetization is flipped to the transverse plain in each repetition.)

The solution (for 90° excitations)

Dependence of the signal-to-noise ratio $Ψ$ on the repetition time $T_{	ext{R}}$ at fixed total measurement time with a maximum at $T_{	ext{R}} / T_1 = 1.256 431 208 626…$
Dependence of the signal-to-noise ratio Ψ on the repetition time TR at fixed total measurement time with a maximum at TR / T1 = 1.256 431 208 626…

In general, a single MRI acquisition (without averaging) requires M repeated excitations separated by TR (e. g., M could be the number of phase-encoding steps for conventional spin-echo acquisitions or M = 1 for single-shot EPI or one-dimensional MR spectroscopy). Hence, the acquisition time without data averaging is Tacq = MTR, and, neglecting for now that the number of averages must be integer, this number of averages is N = Ttotal / Tacq = Ttotal / (MTR). In the following, the actual relevant time parameter is, thus, Tavail = Ttotal / M, the time available for each required excitation, and the number of averages is N = Tavail / TR. We can also express the repetition time by these parameters as:

TR = Tacq / M = Ttotal / (M N) = Tavail / N.

On the one hand, averaging data increases the obtainable signal-to-noise ratio (SNR or Ψ) proportional to the square root of the number of averages:

Ψ ∝ √(N) = √(Tavail / TR).

Dependence of the SNR $Ψ$ on the number $N$ of averages (left) and the $T_{	ext{R}}$ (right, at fixed total measurement time) due to averaging neglecting the influence of $T_{	ext{R}}$ on $T_1$ relaxation
Dependence of the SNR Ψ on the number N of averages (left) and the TR (right, at fixed total measurement time) due to averaging neglecting the influence of TR on T1 relaxation

On the other hand, increasing the number of averages (at fixed total measurement duration!) results in a shortened time for longitudinal relaxation: TR = Tavail / N and, thus, in less SNR:

Ψ ∝ 1 – exp(–TR / T1) = 1 – exp (Tavail)/(NT1).

Dependence of the SNR $Ψ$ on the number $N$ of averages (left, at fixed total measurement time) and the $T_{	ext{R}}$ (right) due to $T_1$ relaxation neglecting the influence of signal averaging
Dependence of the SNR Ψ on the number N of averages (left, at fixed total measurement time) and the TR (right) due to T1 relaxation neglecting the influence of signal averaging

The resulting total SNR is proportional to the product of both factors, i. e.,

Ψ ∝ √(Tavail / TR)(1 – exp(–TR / T1)):

Resulting dependence of the SNR $Ψ$ on $T_{	ext{R}}$ at constant total measurement time (and thus, implicitly, on the number of averages) considering both averaging and relaxation effects
Resulting dependence of the SNR Ψ on TR at constant total measurement time (and thus, implicitly, on the number of averages) considering both averaging and relaxation effects

The SNR (as a function of TR at constant total measurement time) has, thus, a maximum in an intermediate range; for very short TR (and, consequently, many averages), the longitudinal magnetization cannot relax sufficiently, which reduces the available signal; for very long TR, the relaxation is approximately complete anyway, but the number of averages decreases.

To calculate the optimum TR (or N), TR and Tavail are best expressed in terms of T1 using τ = TR / T1 and T = Tavail / T1, which gives

Ψ ∝ √((T)/(τ))(1 – exp(–τ))

The maximum of this expression with respect to τ is obtained by setting its derivative to zero, i. e.

0 = ()/(τ) [√(T / τ)(1 – exp(–τ))] = (exp(–τ))/(2)√((T)/(τ3))(1 + 2τ – exp(τ))

which gives

1 + 2τ – exp(τ) = 0.

The solution of this equation is

τ =  – (1)/(2) – W–1(–1 / (2 √(e))) = 1.256 431 208 626…

where W–1 is the lower branch of the Lambert W-function (or product-log or Omega function, which gives the solution for W in z = Wexp(W), i. e., it is the inverse function of f(W) = Wexp(W)). Hence, the optimal choice for TR is (at least theoretically):

TR ≈ 1.256 T1.

Fortunately, the function to be maximized has a rather broad maximum, and one obtains values between 99 % and 100 % of the maximum SNR for TR between about 0.9937 T1 and 1.5773 T1 and values between 95 % and 100 % of the maximum SNR for TR between about 0.7293 T1 and 2.0882 T1 (values determined by numerical evaluation and illustrated in the first figure at the top).

So, practically, choosing TR to be T1 gives still a nearly optimal SNR; if a larger number of signal acquisitions is preferred, e. g., for statistical evaluation, then TR may even be shortened to 0.75 T1, and if – on the other hand – the base SNR of every single acquisition becomes very low (but is to be reconstructed before signal averaging) then TR may be chosen to be 2 T1 to increase the quality of each single (not averaged) data set.

The presented result is not new; in fact, it has been derived before (at least numerically) in several publications. An early example is, e. g., a publication by R. R. Ernst and R. E. Morgan (1973).

The presented analysis is valid only for simple 90° excitations; FLASH sequences or steady-state pulse sequences which refocus the transverse magnetization as well will exhibit a different behavior.

Of course, optimizing TR as described above is not an option if T1-weighted or T2-weighted images are to be acquired. In these cases, the optimal value of TR depends on the desired contrast and not on the total SNR.

(This is a shortened version of a slightly longer pdf document that contains some more details and discusses also a few special cases.)

Categorized: MRI, NMR, SNR, physics
pdf version available
URL: <https://dtrx.de/od/mmm/mmm_20180628.html>

Looking for reference values of water diffusion?

posted on 2018-04-25 (by Olaf Dietrich)

This is a short advertisement for my web document “Diffusion Coefficients of Water”. If you have ever looked for reference values of the self-diffusion coefficient of water at different temperatures, then chances are that you might like to bookmark the link above.

Screenshot from \url{https://dtrx.de/od/diff/}
Screenshot from <https://dtrx.de/od/diff/>

(If you have never heard of molecular (self-)diffusion, then it’s probably not so relevant for you. As an ultra-short explanation: Self-diffusion is a term for the thermal motion of molecules. If we could look at the individual molecules of a liquid such as water (say, with a really big microscope), then we would see all molecules in constant random motion. This molecular motion is known as the cause of the Brownian motion, which has been observed already in 1827 by the botanist Robert Brown. The extent of this motion is directly related to the temperature of the liquid or – which is essentially the same – to the thermal kinetic energy of the molecules. This motion can be described by a physical quantity called the (self-)diffusion coefficient D, which has units of m2 / s. The meaning of a diffusion coefficient of, e. g., D = 2 × 10–3 mm2 / s is that after a time t = 1 s the space “visited” by an average diffusing molecule (in a certain statistical sense) has a size s (i. e., a radius) of s = √( 6Dt) = √( 12 × 10–3 mm2) ≈ 0.1 mm.)

Why is it good to know the exact relationship between the temperature and the diffusion coefficient of water? Typical applications are:

The web document introduced above provides an interactive interface to calculate self-diffusion coefficients of water at different temperatures (or, alternatively, to calculate the temperature corresponding to a given diffusion coefficient). This calculation is based on the results taken from several published articles with exactly this kind of data (namely, measured diffusion coefficients of water at different temperatures).

The web page has actually existed for quite some time (since about 2002), but it was never made public. No (hyper)links to it existed and it was used only by me (and by some colleagues and our Master or PhD students). Thus, I was somewhat surprised when I recently found out that the page is indexed by google and has already been cited in several reports and theses. Consequently, I decided to slightly update its content (and to pretty up its appearance). The most relevant change is that I added a non-linear (quadratic) fit that describes the data in the Arrhenius plot much more accurately (and can be used for data over a greater temperature range) than the previously used linear fit. Note that using the (recommended) quadratic fit, calculated diffusion coefficients for intermediate temperatures around 15 to 30 °C will differ from those of the earlier (linear) version by a few tenths of a percent.

Categorized: diffusion, MRI, NMR, physics, reference values
pdf version available
URL: <https://dtrx.de/od/mmm/mmm_20180425.html>

Double field strength gives double SNR – or not?

posted on 2018-02-22 (by Olaf Dietrich)

Some time ago at lunch, we had a discussion about the advantages of high magnetic field strengths B0 in MRI. We happily agreed that higher field strengths result in higher signal-to-noise ratios (SNR). But then several opinions surfaced about the exact quantitative relation between the SNR and B0 – ranging from linear to quadratic and including some very specific exponents in between such as 7/4. It turns out that more than one correct answer exists … and there are some surprising technical subtleties.

Discussed relationships of signal-to-noise ratio (SNR) and magnetic field strength $B_0$
Discussed relationships of signal-to-noise ratio (SNR) and magnetic field strength B0

As starting point of a quantitative discussion, we have to define what we consider as SNR in MRI: The SNR (sometimes given the symbol Ψ) is defined as the amplitude of the image signal (at some point or region of interest) divided by the standard deviation of the noise signal. (There are other SNR definitions used in other disciplines that involve squared amplitudes (signal power) and logarithms (resulting in decibels), but in MRI we generally use the very simple ratio of the signal amplitude and the noise standard deviation.)

To simplify things, we can divide the analysis of field strength and SNR into two parts: the discussion of the signal and of the noise. The signal part is rather straight forward: If the field strength B0 increases, then

The measured signal S is the voltage induced in the radio-frequency (rf) receiver coil by the precessing nuclear magnetization. As known from the theory of electromagnetic induction, this induced voltage is proportional to both the (angular) frequency ω = γB0 and the magnetization MN = χB0, and taking both factors together, we find S ∝ B02.

So, if we stop at this point, we may hope for four times the SNR at double field strength. But we haven’t considered the noise yet. And there the trouble begins …

There are several sources of noise in MRI and depending on the setup, different sources can dominate the noise generation. Generally, the thermal noise voltage Unoise can be expressed as Unoise = √( 4kTRΔf), where Δf is the signal bandwidth and R is a resistance associated to the rf receiver coil. For very small samples (milliliters), this resistance R is dominated by the actual coil resistance Rcoil. Rcoil is proportional to the square root of the Larmor frequency Rcoil ∝ √(ω) ∝ √(B0) because of the rf skin effect, which reduces the penetration depth and, thus, the conducting area of the coil wires proportional to 1 / √(ω). Therefore, Unoise ∝ Rcoil1 / 2 ∝ ω1 / 4 ∝ B01 / 4 and the resulting SNR is proportional to S / Unoise ∝ B07 / 4 in this case.

However, a different looking result is presented by A. Abragam in his classic monograph “The principles of nuclear magnetism” (1961). There (on p. 83) we find for the SNR Ψ ∝ √(Q)B03 / 2 with the coil quality factor Q. This is derived from the shunt resistance R = QLω of a circuit (here the receive coil) with inductance L. This apparently contradicting result can be explained if the frequency dependence of the quality factor Q ∝ √(ω) (for a Q-optimized solenoid coil) is considered, which then gives the same exponent of 7 / 4 as before.

Finally, for larger samples – such as human subjects in clinical MRI – additional noise is generated because of the inductive (or magnetic) losses associated with the electrical conductivity of the sample or tissue. Electrical power is dissipated in the sample proportional to the squared induced voltage in the sample, i. e., proportional to ω2 and proportional to the squared current I2 in the coil. This dissipation of power can therefore be expressed as an additional apparent coil resistance Rsample that is also proportional to the squared Larmor frequency Rsample ∝ ω2. Consequently, considering only this apparent coil resistance associated to the sample, we find Unoise ∝ Rsample1 / 2 ∝ ω ∝ B0, and now the resulting SNR is proportional to S / Unoise ∝ B0.

These results (and detailed derivations) can be found in two classic MRI papers by D. I. Hoult and R. E. Richards (1976) and by D. I. Hoult and P. C. Lauterbur (1979).

An obvious question now is: Can’t we simply measure the SNR dependence on the field strength to find out what’s actually going on in MRI? Well, we can try, but there are several complications. One major problem is that we cannot use the same rf coils at different field strengths. There will generally be differences of the coil design for different field strengths and these must be expected to influence the measured SNRs. Another complication is the influence of the electromagnetic wavelengths in tissue that approach the dimensions of the samples at about 3 to 7 T. Furthermore, the relaxation times (T1, T2, T2*) change with B0 and may influence SNR measurements.

Nevertheless, there are some publications reporting SNRs at different field strengths, e. g. by D. I. Hoult et al. (1986), J. T. Vaughan et al. (2001), C. Triantafyllou et al. (2005), and recently by R. Pohmann et al. (2016). They all show a clear increase of SNR with B0, but there are still some discrepancies with respect to the exact dependency. Particularly the latest paper by R. Pohmann et al. demonstrates and discusses a slightly better than linear increase of SNR at high fields. But in conclusion and as rule of thumb, assuming an approximately linear relationship of SNR and B0 appears still justified for large (clinical) MRI systems.

Categorized: MRI, NMR, physics, signal-to-noise ratio, SNR
pdf version available
URL: <https://dtrx.de/od/mmm/mmm_20180222.html>

Is T2 necessarily shorter than T1?

posted on 2018-01-17 (by Olaf Dietrich)

If you are familiar with MRI (or NMR in general), then probably also with the relaxation time constants T1 and T2. These tissue-specific (or substance-specific) constants describe how fast the nuclear magnetization returns to its equilibrium value, M0, after excitation by a pulsed radio-frequency (rf) field. Shortly summarized, T1 describes the exponential recovery of the longitudinal magnetization ML (i. e., of the magnetization parallel to the external static magnetic field B0), and T2 describes the exponential decay of the transverse magnetization MT (that is precessing in the plane orthogonal to B0).

Longitudinal (blue) and transverse (red) relaxation of nuclear magnetization after a 90° rf pulse
Longitudinal (blue) and transverse (red) relaxation of nuclear magnetization after a 90° rf pulse

Typically (as shown in the first figure), T2 values of tissue are considerably lower than T1 values, i. e., the transverse magnetization decays quicker than the longitudinal relaxation needs for recovery. For most tissues in vivo, T1 varies between about 300 ms and 3 s, while T2 varies between about 10 ms and 200 ms. Longer T2 relaxation times (up to about 3 s as well) are found for liquids.

So one may ask if there are good physical reasons for T2 values being shorter than or – at most – equal to T1. As a physicist, I’d start with checking some extreme cases, e. g., assuming that T2 is much longer than T1, i. e. T2 ≫ T1. Then the longitudinal magnetization can fully recover while at the same time some transverse magnetization would be preserved. As a result, the magnitude of the total magnetization √(ML2 + MT2) would become greater than the equilibrium value M0 – which is physically impossible.

To analyze these properties of T1 and T2 in more detail, longitudinal and transverse relaxation can also be plotted together in a diagram showing the transverse magnetization on the horizontal axis and the longitudinal magnetization on the vertical axis. These diagrams show the evolution of the magnetization (for experts: in a rotating frame of reference) after a 90° rf pulse (all trajectories start at the lower right corner of the diagram). The left-hand side of the following figure shows three cases T2 = T1 / 2 (blue), T2 = T1 (green), and T2 = 2 T1 [sic!] (cyan); and all three curves show a “benign” behavior in that they lie in the shaded area below the black circle segment ML2 + MT2 = M02. This means that the magnitude of the total magnetization vector is always smaller than M0.

Evolution of the magnetization after a 90° rf pulse for $T_2 ≤2 T_1$ (left hand side) and $T_2 > 2 T_1$ (red curve on right-hand side)
Evolution of the magnetization after a 90° rf pulse for T2 ≤ 2 T1 (left hand side) and T2 > 2 T1 (red curve on right-hand side)

However, the right-hand side of this figure shows what’s happening if T2 becomes greater than 2 T1 – in this case, T2 = 3 T1 (red curve): Now the magnitude of the total magnetization vector increases above the physical limit of M0, i. e., the red line crosses the black border of physically benign behavior!

In fact, it can be shown (see appendix if interested) that the maximum T2 value, for which the red curve stays always below the black line, is exactly T2 = 2 T1. And as so often, almost everything that is physically possible is also realized in nature (although the case T1 < T2 < 2 T1 is really extremely rare), as described by Malcolm H. Levitt in his highly recommendable NMR text book “Spin dynamics” (2nd ed., section 11.9.2, note 13):

The case where T2 > T1 is encountered when the spin relaxation is caused by fluctuating microscopic fields that are predominantly transverse rather than longitudinal. One mechanism which gives rise to fields of this form involves the antisymmetric component of the chemical shift tensor (not to be confused with the CSA). […] Molecular systems in which this mechanims is dominant are exceedingly rare (see F. A. L. Anet, D. J. O’Leary, C. G. Wade and R. D. Johnson, Chem. Phys. Lett., 171, 401 (1990)).

So, the answer to the title question is: No, T2 can in fact be greater than T1 in very special circumstances, but it can never be greater than 2 T1.

Appendix

The maximum T2 value, for which the red curve stays always below the black line, is exactly T2 = 2 T1. This can be seen by analyzing the inequality

MT2 + ML2 = (M0 exp(–t / T2))2 + (M0 (1 – exp(–t / T1))2 ≤ M02.

First, we divide by M02 and set T2 = αT1 as well as β = exp(–t / T1), yielding

(β1 / α)2 + (1 – β)2 = β2 / α + 1 – 2β + β2 ≤ 1

which is (after subtraction of 1 and division by β)

β2 / α – 1 – 2 + β ≤ 0  or  β2 / α – 1 ≤ 2 – β.

β is by definition (for positive t) between 0 and 1, so the right-hand side of the last inequality is a linear function descending from 2 to 1 (i. e. always  ≤ 2). Its left-hand side has very different shapes depending on α: it is increasing from 0 to 1 for 0 < α ≤ 2 (since then the exponent 2 / α – 1 ≥ 0); but it is going to infinity for β → 0 if α > 2 (since then the exponent 2 / α – 1 < 0). So, the last inequality will not hold in the latter case for sufficiently small values of β, which means that non-physical behavior occurs if α > 2 or, using the definition from above, if T2 > 2 T1.

Categorized: MRI, physics, relaxation
pdf version available
URL: <https://dtrx.de/od/mmm/mmm_20180117.html>

Gadolinium does not increase the signal, but the speed of MRI

posted on 2017-12-22 (by Olaf Dietrich)

You may have heard that gadolinium-based MRI contrast agents can enhance or increase the signal of tissue. This is generally a good description of what’s going on. Here, however, I would like to argue why this is – strictly speaking – not true: contrast agents cannot really increase the signal available for MRI.

Some basic facts first: gadolinium-based contrast agents are very frequently used in clinical MRI to improve the image contrast as illustrated in the following example.

$T_1$-weighted MRI without (left) and with (right) contrast agent
T1-weighted MRI without (left) and with (right) contrast agent

An important fact about (conventional) MRI contrast agents is that it’s never the contrast agent itself that is visible in MR images. Instead, the contrast agent changes the behavior of the atomic nuclei in its neighborhood – in clinical MRI, these are the nuclei of hydrogen, i. e. the protons. As a consequence, these protons now appear brighter in T1-weighted MRI than protons which are not influenced by the contrast agent.

So, apparently the proton signal is increased by gadolinium? Yes, apparently … Actually, there is always a maximum signal that is available for MRI and that depends on three major factors:

But the presence of a contrast agent does not increase this maximum signal.

Instead, we are cheating: First, we artificially decrease the MRI signal by choosing short repetition times (TR). And only afterwards, a certain part of this suppressed signal is recovered due to the influence of the contrast agent! This is illustrated in the following diagram:

Effect of contrast agent on MRI signal
Effect of contrast agent on MRI signal

Obviously, the maximum signal, Smax, with contrast agent is exactly the same as the maximum signal without contrast agent – but we can obtain this maximum signal considerably faster (i. e, at shorter TRs). That’s why gadolinium can be described to increase the speed, but not the MRI signal. In agreement with this observation, no additional gadolinium-induced signal enhancement can be found in proton-density-weighted MR images (with very long TRs). But, hypothetically, if contrast agent could be distributed homogeneously in the tissue (which in reality is not possible), then PD-weighted MRI could be accelerated by using shorter TRs without changing the contrast.

Categorized: MRI, physics, contrast
pdf version available
URL: <https://dtrx.de/od/mmm/mmm_20171222.html>

Simultaneous multi-slice excitation done differently

posted on 2017-12-15 (by Olaf Dietrich)

Last week, Koray Ertan and colleagues published an MRM paper (doi: 10.1002/mrm.27031) (expanding on an ISMRM abstract presented earlier this year) with an interesting approach for simultaneous multi-slice excitation. So, what’s it about?

In MRI, we are frequently interested in data from a single two-dimensional slice of the imaged subject or object – or actually in data from several such slices. These slices are then displayed as conventional 2D MR images. A procedure called slice selection is used to restrict our data acquisition to each single slice. To understand slice selection, one has to know that MRI is based on the resonant excitation of spins in a static magnetic field B0. Spins have a characteristic, so-called Larmor frequency ω = γB depending on the magnetic field B (the constant of proportionality γ is the gyromagnetic ratio). By applying a radio-frequency (rf) field with exactly this Larmor frequency, spins can be excited (i. e., they can be made to generate a measurable signal).

The basic idea of slice selection is to excite only spins in a single slice by (first) superposing a linear magnetic field gradient gz e. g. in z direction, resulting in the spatial field distribution (shown in the first figure):

B(x,y,z) = B0 + gz z.

(These linear magnetic fields or gradient fields are one of the basic ingredients of MR imaging. Each MR imager comes with built-in coils to apply gradient fields in all possible spatial orientations.)

Then, by choosing an rf frequency ωslc = 2πfslc, we can select a plane in space where

ωslc = γB(x,y,z) = γB0 + γgz zslc,

and only spins in this plane centered around zslc = (ωslc / γ – B0) / gz are excited (see first figure).

Slice selection of a single slice in $z$ direction
Slice selection of a single slice in z direction

More advanced MRI techniques can excite several spatially separated slices at once by applying rf fields with more than one frequency – the difficult part is then to separate data acquired from these slices for reconstruction.

Simultaneous slice selection of two slices in $z$ direction
Simultaneous slice selection of two slices in z direction

The nice idea realized in the paper by Koray Ertan et al. is to excite multiple slices not by applying rf fields with a mixture of several frequencies, but instead by modifying the gradient field to a spatially non-linear magnetic field. Consequently the mapping between frequencies ω and spatial positions (z) is no longer one-to-one, but several positions can correspond to a single frequency:

Simultaneous slice selection of two slices in $z$ direction using non-linear magnetic fields
Simultaneous slice selection of two slices in z direction using non-linear magnetic fields

So, depending on the shape of the magnetic field variation two or more slices can be excited using a single excitation frequency. This has the advantage that we can use short and simple standard rf pulses for slice excitation. The obvious disadvantage is that it requires additional gradient hardware providing the non-linear magnetic fields, which is currently not available at existing MRI systems.

However, this may change in future, since there are currently some promising approaches for non-linear encoding fields. In particular, I’m thinking about an impressive MRM paper (doi: 10.1002/mrm.26700) by Sebastian Littin and colleagues published also this year, in which an 84-channel matrix gradient coil is presented, which is capable of providing very flexible linear or non-linear field configurations.

Categorized: MRI, physics, publications
pdf version available
URL: <https://dtrx.de/od/mmm/mmm_20171215.html>