BLAKJac

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Introduction

BLAKJac is implemented in the Julia language. It relates to the MR-STAT functionality and aims to predict or minimize the noise levels in the reconstructed quantitative maps.

Two of its main functions are

• Analyzer: given a sequence of encoding values and of RF pulse angles, it predicts the resulting noise level.
• Optimizer: given a sequence of encoding values, it generates a sequence of RF pulse angles that is presumed to be optimal in terms of noise level. The Optimizer makes use of the Analyzer.

Inputs and outputs

Analyzer

In essence, the analyzer has as its input

• The RF sequence: a vector of $N{TR}$ complex values, with e.g. $N{TR}=5\cdot 224=1120$. The magnitude of the complex values expresses the flip angle in degrees.
• The trajectorySet. See dedicated section below.
• A large bunch of options, in the form of a dictionary. For details, see BLAKJac_analysis and BLAKJac_defaults

There is one additional parameter:

• [Actually an output] Optionally, a dictionary of $I^{-1}$ matrices, where $I$ is the diagonalized Fisher Information Matrix. Such a dictionary may be useful if the optimizer is run a multiplicity of times.

As an Output, the Analyzer returns

• A vector of noise levels of length $p$, with $p$ being the number of properties. For the time being, $p$ is always 3.
• An estimate of the information content that can be expected from the scan; for the time being, a pretty immature metric.
• b1factorRMS: an indication on how sensitive the sequence is to deviations of B1+.

Mostly, only the first output is used, the vector of noise levels. These have a somewhat quantitative meaning: these are scaled such that if $T_1$ and $T_2$ were to be known beforehand, an ideal sequence would result in the vector [1.0, 0.0, 0.0]. Typically, the results are on the order of 3; a value of 2 is “good”, a value over 4 is generally bad and if it results in values below 1.0, then something is fishy – unless strong regularization has been applied (see “options”).

The interface, in full:

function BLAKJac_analysis!(
    RFdeg::Vector{ComplexF64}, 
    trajectorySet::Vector{<:Vector{<:TrajectoryElement}}, 
    options::Dict, 
    saved_H::Dict=Dict())

Optimizer

In brief, the optimizer has a trajectorySet as input and it spits out an RF sequence. And it needs a lot of options.

The input:

• A trajectorySet. See dedicated section below.
• A large bunch of options, in the form of a dictionary.
• (Optionally) a random-value seed. Relevant if the option opt_initialize (see below) is set to cRandom30. By default, every run of the optimizer will always return the same pattern. If different realizations are needed, each call to the optimizer must be done with a different seed (Int).

Output

• It returns an RF sequence: a vector of $N{TR}$ complex values, with e.g. $N{TR}=5\cdot 224=1120$. The magnitude of the complex values expresses the flip angle in degrees.

The interface, in full:

function BLAKJac_optimize(trajectorySet, options::Dict, mersenneTwisterSeed=1)

TrajectorySet

The trajectorySet. This is a vector of $N_{TR}$ vectors (in Cartesian applications, these are vectors of length 1) of TrajectoryElement. And a TrajectoryElement is a tuple, mostly used for $(k_y,k_z)$ – and then, in 2D-cases, the value of $k_z$ is fixed (typically, always 0).

Note that in the 2D case, $k_x$ is actually ignored, assuming that all samples of the readout share the same history – which ignores T2-decay during readout.

The reason of making the trajectorySet a vector of vectors is for non-Cartesian: then, it can be e.g. a vector of 20 spirals of 1000 samples each and “$(k_y,k_z)$” has to be interpreted as $(k_x,k_y)$.

The function TrajectorySet(kyVector, kzVector) may be useful before invoking the Analyzer: it converts an $N_{TR}$-vector of $ky$ values and an $N{TR}$-vector of $kz$ values into an $N{TR}$-vector of singleton-vectors of tuples $(k_y,k_z)$. That function has the following (tricky?) logic: if any of the provided $k_y$ or $k_z$ values is negative, then it assumes $(k_y,k_z)=(0,0)$ corresponds to zero spatial frequency. If the provided values are all nonnegative, it assumes the center of the range to correspond to zero spatial frequency. This logic is due to legacy reasons, where typically $k_y=1:224$ and $k_z=1$ would be provided, so TrajectorySet shifts the values by (112,1) in that example.

Options

There are lots of option parameters, some of them are deprecated and some have defaults that are very rarely adapted. For that purpose, the following function is available:

BLAKJac_defaults!(trajectorySet::Vector{<:Vector{<:TrajectoryElement}}, options::Dict)

It does not return anything, but the “!” indicates that options will be modified by it. Note: it only assigns uninitialized options to a default value. Options that are already filled in will be left unmodified. To set all options to default, use

recon_options = Dict() # erase all existing settings

BLAKJac.BLAKJac_defaults!(trajectorySet, recon_options)

The trajectorySet is a parameter, since some defaults are set accordingly to the encoding sequence.

The list of options:

Remarks:

(R1) : maxMeas: Maximum expected number of measurements per phase-encoding value. A ToDo item: this should be automatic, but for the time being, it has to be provided by the user. If there is no variable density, then the default value has to be set to 4 times the number of k-spaces, i.e. 4nTR/nky in 2D sequences and to 8nTR/(nky*nkz) if two phase-encoding directions are applied (which points to a flaw in the default, in the case of 3D).

Why 4 or 8 and not 1? One factor of 2 is needed because of assumed positivity of T1 and T2, for which the signal at $(k_y,k_z)$ and at $(-k_y,-k_z)$ has to be considered jointly. Another factor of 2 (per dimension) is needed because the entered $k_y$ or $k_z$ values are allowed to be fractional. In the case of a fractional value of $k_y$, e.g. at $k_y=n+f$ with $f\in [0,1)$ and $n$ an integer, part of the signal is assigned to the integer location $n$ and another part to integer location $n+1$. To be precise, a fraction $\cos({\pi f}/{2})$ is assigned to $n$ and $\sin({\pi f}/{2})$ to $(n+1)$. This implies that we have some, albeit minimalistic, multi-coil capacity in our system. The cos() and sin() functions have been chosen in order to make this “splitting” noise-neutral, since it preserves the total noise power.

(R2) : invregval: Although the actual reconstruction does not apply explicit regularization, the Analyzer assumes that it does. With the assumed default of $200^{-2}$, the Analyzer acts as if regularization would bias the reconstructed maps towards $(\rho,T_1,T2)=(0,T{1,ref},T{2,ref})$, by striving for a minimal squared error in the case that the SNR is around 200. In our case, “SNR” has to be seen as the expected value of ${\sqrt{(1/N{voxels})\sum_{voxels}(T1(r)-T{1,ref})^2}}/{\sigma_{T_1}}$. The parameter can also be used to indicate that a given parameter is “known”. E.g. by setting invregval to $[10^{+6}.10^{+6},200^{-2}]$, we indicate that and T1 are known and that all data are assumed to be used to estimate T2.

(R3) : The current default of the T1T2set is set to [(0.8183, 0.0509), (0.67, 0.2066), (1.2208, 0.1253), (0.2465, 0.0461), (2.2245, 0.3082), (0.3677, 0.0461), (0.3677, 0.1253)]

(R4) : The options sizeStep, portion_size, portion_step and opt_iterations_limit are Opimizer options. Following approach is used: in a first stage the Vector sizeStep is relevant; let it be $[n_1,n_2,n_3]$ as an example. In that first stage, the Optimizer internally runs a series of optimizations. The first one is to optimize the flip angle on $n_1$ equidistant points, the other flip angles following by cubic-spline interpolation. Subsequently, the result is cubic-spline interpolated to $n_2$ equidistant points to act as a starting point on an optimization on $n_2$ points, etc. Once this is done, the result is used as a starting point for the second stage of optimizations (deprecated by now): the Optimizer will then optimize individual RF pulses. Since the optimization process cannot reasonably handle a large number (over 50 or so) of parameters simultaneously, it only addresses a portion of size portion_size at a time, taking all other RF pulses as fixed. Once such a portion is considered optimized, the portion advances by portion_step pulses. In some cases, this may result in sequences with pulses around 180 degrees, but not if a sar_limit has been imposed to e.g. 40 degrees RMS. The total number of internal runs should therefore be

length(sizeStep)+ceil(nTR/portion_step).

This can be reduced by the option parameter opt_iterations_limit. When omitting the second stage (which is advised), the default opt_iterations_limit=length(sizeStep) is appropriate.

(R5) : Internal Optimizer options. Default is Optim.Options(time_limit = 3000.0, iterations = 100000, f_tol=1.0e-2, g_tol = 1.0e-3) The first two options are useful to stop the prevent “the optimizer being stuck, either by non-convergence or due to a parameter choice that would require ages to converge. On f_tol and g_tol, the latest insights are that these should be set stricter in order to avoid grossly-suboptimal false minima. A value of around $10^{-8}$ should be default.

(R6) : Valid values are:

• "noise level" (not accounting for SAR or high flip angles),
• "low-flip corrected noise" (penalty on the maximum flip angle in the sequence, on the assumption that, on a fixed TR, it will thereby reduce the sampling-time and consequently increase the noise by a certain factor),
• "information content", "information content penalized" (immature – do not use),
• (Legacy )"sar-limited noise" (a heavy penalty term is added if the root-mean-square of the RF pulses exceeds a limit). Deprecated. Instead, combine "noise level" with account_SAR=true