API

struct Isochromat{T<:Real} <: FieldVector{3,T}
    x::T
    y::T
    z::T
end

Holds the x,y,z components of a spin isochromat in a FieldVector, which is a StaticVector (from the package StaticArrays) with custom fieldnames.

decay(m::Isochromat{T}, E₁, E₂) where T

Apply T₂ decay to transverse component and T₁ decay to longitudinal component of Isochromat.

initialize_states(::AbstractResource, ::IsochromatSimulator{T}) where T

Initialize a spin isochromat to be used throughout a simulation of the sequence.

This may seem redundant but to is necessary to share the same programming interface with EPGSimulators.

invert(m::Isochromat{T}, p::AbstractTissueProperties) where T

Invert Isochromat with B₁ insenstive (i.e. adiabatic) inversion pulse

invert(m::Isochromat{T}, p::AbstractTissueProperties) where T

Invert z-component of Isochromat (assuming spoiled transverse magnetization so xy-component zero).

regrowth(m::Isochromat{T}, E₁) where T

Apply T₁ regrowth to longitudinal component of Isochromat.

rotate(m::Isochromat, γΔtGRz, z, Δt, p::AbstractTissueProperties)

Rotation of Isochromat without RF (so around z-axis only) due to gradients and B0 (i.e. refocussing slice select gradient).

rotate(m::Isochromat{T}, γΔtRF::Complex, γΔtGR::Tuple, (x,y,z), Δt, p::AbstractTissueProperties, Δω = zero(T)) where T

RF, gradient and/or ΔB₀ induced rotation of Isochromat computed using Rodrigues rotation formula (https://en.wikipedia.org/wiki/Rodrigues%27rotationformula).

sample!(output, index::Union{Integer,CartesianIndex}, m::Isochromat)

Sample transverse magnetization from Isochromat. The "+=" is needed for 2D sequences where slice profile is taken into account.

sample_xyz!(output, index::Union{Integer,CartesianIndex}, m::Isochromat)

Sample m.x, m.y and m.z components from Isochromat. The "+=" is needed for 2D sequences where slice profile is taken into account.

F̄₋(Ω)

View into the second row of the configuration state matrix Ω, corresponding to the F̄₋ states.

F₊(Ω)

View into the first row of the configuration state matrix Ω, corresponding to the F₊ states.

Z(Ω)

View into the third row of the configuration state matrix Ω, corresponding to the Z states.

decay!(Ω::AbstractConfigurationStates, E₁, E₂)

T₂ decay for F-components, T₁ decay for Z-component of each state.

dephasing!(Ω::AbstractConfigurationStates)

Shift states around due to dephasing gradient: The F₊ go up one, the F̄₋ go down one and Z do not change

excite!(Ω::AbstractConfigurationStates, RF::Complex, p::AbstractTissueProperties)

Mixing of states due to RF pulse. Magnitude of RF is the flip angle in degrees. Phase of RF is the phase of the pulse. If RF is real, the computations simplify a little bit.

excite!(Ω::AbstractConfigurationStates, RF::T, p::AbstractTissueProperties) where T<:Union{Real, Quantity{<:Real}}

If RF is real, the calculations simplify (and probably Ω is real too, reducing memory (access) requirements).

initial_conditions!(Ω::AbstractConfigurationStates)

Set all components of all states to 0, except the Z-component of the 0th state which is set to 1.

initialize_states(::AbstractResource, sequence::EPGSimulator{T,Ns}) where {T,Ns}

Initialize an MMatrix of EPG states on CPU to be used throughout the simulation.

initialize_states(::CUDALibs, sequence::EPGSimulator{T,Ns}) where {T,Ns}

Initialize an array of EPG states on a CUDA GPU to be used throughout the simulation.

invert!(Ω::AbstractConfigurationStates, p::AbstractTissueProperties)

Invert Z-component of states of all orders. Assumes fully spoiled transverse magnetization.

invert!(Ω::AbstractConfigurationStates)

Invert with B₁ insenstive (i.e. adiabatic) inversion pulse

regrowth!(Ω::AbstractConfigurationStates, E₁)

T₁ regrowth for Z-component of 0th order state.

rotate!(Ω::AbstractConfigurationStates, eⁱᶿ::T) where T

Rotate F₊ and F̄₋ states under the influence of eⁱᶿ = exp(i * ΔB₀ * Δt)

rotate_decay!(Ω::AbstractConfigurationStates, E₁, E₂, eⁱᶿ)

Rotate and decay combined

sample_transverse!(output, index::Union{Integer,CartesianIndex}, Ω::AbstractConfigurationStates)

Sample the measurable transverse magnetization, that is, the F₊ component of the 0th state. The += is needed for 2D sequences where slice profile is taken into account.

sample_Ω!(output, index::Union{Integer,CartesianIndex}, Ω::AbstractConfigurationStates)

Sample the entire configuration state matrix Ω. The += is needed for 2D sequences where slice profile is taken into account.

spoil!(Ω::AbstractConfigurationStates)

Perfectly spoil the transverse components of all states.

Ω_eltype(sequence::EPGSimulator{T,Ns}) where {T,Ns} = Complex{T}

By default, configuration states are complex. For some sequences, they will only ever be real (no RF phase, no complex slice profile correction) and for these sequences a method needs to be added to this function.

AbstractTissueProperties{N,T} <: FieldVector{N,T}

Abstract type for custom structs that hold tissue properties used for a simulation within one voxel. For simulations, SimulationParameterss are used that can be assembled with the @parameters macro.

Possible fields:

• T₁::T: T₁ relaxation parameters of a voxel
• T₂::T: T₂ relaxation parameters of a voxel
• B₁::T: Scaling factor for effective B₁ excitation field within a voxel
• B₀::T: Off-resonance with respect to main magnetic field within a voxel
• ρˣ::T: Real part of proton density within a voxel
• ρʸ::T: Imaginary part of proton density within a voxel

Implementation details:

The structs are subtypes of FieldVector, which is a StaticVector with named fields (see the documentation of StaticArrays.jl). There are three reasons for letting the structs be subtypes of FieldVector:

1. FieldVectors/StaticVectors have sizes that are known at compile time. This is beneficial for performance reasons
2. The named fields improve readability of the code (e.g. p.B₁ vs p[3])
3. Linear algebra operations can be performed on instances of the structs. This allows, for example, subtraction (without having to manually define methods) and that is useful for comparing parameter maps.

BlochSimulator{T}

The abstract type of which all sequence simulators will be a subtype. The parameter T should be a number type (e.g. Float64, Float32) and the tissueparameters that are used as input to the simulator should have the same number type. By convention, a BlochSimulator will be used to simulate magnetization at echo times only without taking into account spatial encoding gradients (i.e. readout or phase encoding gradients). To simulate the magnetization at other readout times, including phase from spatial encoding gradients, an AbstractTrajectory will be needed as well.

To make a simulator for a particular pulse sequence:

1. Make a struct that's a subtype of either IsochromatSimulator or EPGSimulator. The struct will hold parameters that are necessary for performing the simulations.

2. Add a method to simulate_magnetization! that implements the pulse sequence. For both performance and GPU compatibility, make sure that simulate_magnetization! does not do any heap allocations. Examples for pSSFP and FISP sequences are found in src/sequences.

3. Add methods to output_eltype and output_size that are used to allocate an output array within the simulate function.

4. [Optional] Add a method to show for nicer printing of the sequence in the REPL

5. [Optional] Add a method to getindex to easily reduce the length of the sequence

6. [Optional] Add a constructor for the struct that takes in data from Matlab or something else and assembles the struct

IMPORTANT

The simulate_magnetization! functions (which dispatch on the provided sequence) are assumed to be type-stable and non-allocating Should be possible to achieve when using functions from operators/epg.jlandoperators/isochromat.jl` and a properly parametrized sequence struct.

IsochromatSimulator{T} <: BlochSimulator{T}

Abstract type of which all sequence simulators that are based on the isochromat model will be a subtype. The parameter T should be a number type (e.g. Float64, Float32) and the tissueparameters that are used as input to the simulator should have the same number type.

EPGSimulator{T,Ns} <: BlochSimulator{T}

Abstract type of which all sequence simulators that are based on the EPG model will be a subtype. The parameter T should be a number type (e.g. Float64, Float32) and the tissueparameters that are used as input to the simulator should have the same number type. The parameter Ns corresponds to the maximum order of configuration states that are tracked in the simulations.

Generic2D{T,V,M,S} where {T<:AbstractFloat, V<:AbstractVector, M<:AbstractMatrix, S} <: IsochromatSimulator{T}

Simulate a generic 2D sequence defined by arrays containing RF and gradient waveforms. Contains a loop over z locations to take into account slice profile effects. The Δt vector stores the time intervals for the waveforms.

Fields

• RF::V{Complex{T}}: Vector with (complex) RF values during each time interval
• GR::M{T}: Matrix with GRx, GRy and GRz values during each time interval
• sample::S: Vector with Bool's to indicate the sample points
• Δt::V{T}: Vector with time intervals
• z::V{T}: Vector with different positions along the slice direction

Generic3D{T,V<:AbstractVector{Complex{T}},W<:AbstractVector{T},M<:AbstractMatrix{T},S} <: IsochromatSimulator{T}

Simulate a generic sequence defined by arrays containing RF and gradient waveforms. Unlike the Generic2D sequence, it is assumed that the excitation is homogenous over the voxel and therefore no summation over a slice direction is applied. The Δt vector stores the time intervals for the waveforms.

Fields

• RF::V{Complex{T}}: Vector with (complex) RF values during each time interval
• GR::M{T}: Matrix with GRx, GRy and GRz values during each time interval
• sample::S: Vector with Bool's to indicate the sample points
• Δt::V{T}: Vector with time intervals

AbstractTrajectory{T}

The abstract type of which all gradient trajectories will be a subtype. The subtypes should contain fields that can describe the full trajectory during a sequence. The type T refers to the precision of the floating point values within the trajectory struct.

SpokesTrajectory{T} <: AbstractTrajectory{T}

Typical Cartesian and radial trajectories have a lot in common: a readout can be described by a starting point in k-space and a Δk per sample point. To avoid code repetition, both type of trajectories are made a subtype of SpokesTrajectory such that some methods that would be the same for both trajectories otherwise are written for SpokesTrajectory instead.

CartesianTrajectory2D{T,I,U,V} <: CartesianTrajectory{T}

Struct that is used to implement a typical Cartesian 2D gradient trajectory. The trajectory is described in a compact fashion by only storing the starting position in k-space (k_start_readout) for each readout as well as the step in k-space per readout point Δk_adc.

Note that CartesianTrajectory2D and RadialTrajectory2D are essentially the same when using this compact description. A SpokesTrajectory struct is therefore defined as a supertype of both and methods are defined for SpokesTrajectory instead to avoid code repetition.

The type parameters are intentionally left vague. The J, for example, may be an integer for sequences where each readout has the same number of samples, but for sequences with different numbers of samples per readout it may be a vector of integers.

Fields

• nreadouts::I: The total number of readouts for this trajectory
• nsamplesperreadout::I: The total number of samples per readout
• Δt::T: Time between sample points
• k_start_readout::U: Starting position in k-space for each readout
• Δk_adc::U: k-space step Δkₓ per sample point (same for all readouts)
• py::V: Phase encoding index for each readout
• readout_oversampling::I: Readout oversampling factor

kspace_coordinates(tr::CartesianTrajectory2D)

Return matrix (nrsamplesperreadout, nrreadouts) with kspace coordinates for the trajectory. Needed for nuFFT reconstructions.

magnetization_to_signal(::Union{CPU1,CPUThreads,CUDALibs}, magnetization, parameters, trajectory::CartesianTrajectory2D, coordinates, coil_sensitivities)

Arguments

• magnetization: Matrix{Complex} of size (# readouts, # voxels) with phase-encoded magnetization at echo times.
• parameters: Tissue parameters of all voxels, including spatial coordinates.
• trajectory: Cartesian trajectory struct.
• coordinates: Vector{Coordinates} with spatial coordinates for each voxel.
• coil_sensitivities: Matrix{Complex} of size (# voxels, # coils) with coil sensitivities.

Returns

• signal: Vector of length (# coils) with each element a Matrix{Complex} of size (# readouts, # samples per readout)

Extended help

As noted in the description of the simulate_signal function (see src/simulate/signal.jl), we simulate the MR signal at timepoint t from coil i as: signalᵢ[t] = sum(m[t,v] * cᵢ[v] * ρ[v] for v in 1:(# voxels)), where cᵢis the coil sensitivity profile of coil i, ρ is the proton density map and m the matrix with the magnetization at all timepoints for each voxel obtained through Bloch simulations.

The output (signalᵢ) for each coil is in principle a Vector{Complex}of length (# samples per readout) * (# readouts). If we reshape the output into aMatrix{Complex}of size (# samples per readout, # readouts) instead, and do something similar form`, then the signal value associated with the s-th sample point of the r-th readout can be expressed as signalᵢ[r,s] = sum( m[r,s,v]] * cᵢ[v] * ρ[v] for v in 1:(# voxels)).

The problem here is that we typically cannot store the full m. Instead, we compute the magnetization at echo times only. The reason is that, if mᵣ is the magnetization at the r-th echo time in some voxel, and E = exp(-ΔtR₂[v]) * exp(im(Δkₓ*x[v])) is the change per sample point (WHICH FOR CARTESIAN SEQUENCES IS THE SAME FOR ALL READOUTS AND SAMPLES), then the magnetization at the s-th sample relative the the echo time can can be computed as mₛ = mᵣ * E[v]^s

Therefore we can write

signalⱼ[r,s] = sum( magnetization[r,v] * E[v]^s * ρ[v] * cⱼ[v] for v in 1:(# voxels)) signalⱼ[r,s] = magnetization[r,:] * (E.^s .* ρ .* cⱼ)

Because the (E.^s .* ρ .* cⱼ)-part is the same for all readouts, we can simply perform this computation for all readouts simultaneously as signalⱼ[:,s] = magnetization * (E.^s .* ρ .* cⱼ)

If we define the matrix Eˢ as E .^ (-(ns÷2):(ns÷2)-1), then we can do the computation for all different sample points at the same time as well using a single matrix-matrix multiplication: signalⱼ = magnetization * (Eˢ .* (ρ .* cⱼ))

The signalⱼ array is of size (# readouts, # samples per readout). We prefer to have it transposed, therefore we compute signalⱼ = transpose(Eˢ .* (ρ .* cⱼ)) * transpose(magnetization) instead.

For the final output, we do this calculation for each coil j and get a vector of signal matrices (one matrix for each coil) as a result.

Note that this implementation relies entirely on vectorized code and works on both CPU and GPU. The matrix-matrix multiplications are - I think - already multi-threaded so a separate multi-threaded implementation is not needed.

sampling_mask(tr::CartesianTrajectory2D)

For undersampled Cartesian trajectories, the gradient trajectory can also be described by a sampling mask.

RadialTrajectory2D{T,I,U,V} <: SpokesTrajectory{T}

Struct that is used to implement a typical radial gradient trajectory. The trajectory can is described in a compact fashion by only storing the starting position in k-space (k_start_readout) for each readout as well as the step in k-space per readout point Δk_adc.

Note that CartesianTrajectory2D and RadialTrajectory2D are essentially the same when using this compact description. A SpokesTrajectory struct is therefore defined as a supertype of both and methods are defined for SpokesTrajectory instead to avoid code repetition.

The type parameters are intentionally left vague. The J, for example, may be an integer for sequences where each readout has the same number of samples, but for sequences with different numbers of samples per readout it may be a vector of integers.

Fields

• nreadouts::I: The total number of readouts for this trajectory
• nsamplesperreadout::I: The total number of samples per readout
• Δt::T: Time between sample points
• k_start_readout::U: Starting position in k-space for each readout
• Δk_adc::U: k-space step Δk between each readout
• φ::V: Radial angle for each readout
• readout_oversampling::I: Readout oversampling factor

add_gradient_delay!(tr::RadialTrajectory2D, S)

Apply gradient delay to radial trajectory in in-place fashion. The delay is described by the 2x2 matrix S and is assumed to influence the start of the readout only, not the readout direction.

kspace_coordinates(tr::RadialTrajectory2D)

Return matrix (nrsamplesperreadout, nrreadouts) with kspace coordinates for the trajectory. Needed for nuFFT reconstructions.

_allocate_array_on_resource(resource, _eltype, _size)

Allocate an array on the specified resource with the given element type _eltype and size _size. If resource is CPU1() or CPUThreads(), the array is allocated on the CPU. If resource is CUDALibs(), the array is allocated on the GPU. For CPUProcesses(), the array is distributed in the "voxel"-dimension over multiple CPU workers.

This function is called by _allocate_magnetization_array and is not intended considered part of te public API.

_allocate_magnetization_array(resource, sequence, parameters)

Allocate an array to store the output of the Bloch simulations (per voxel, echo times only) to be performed with the sequence. For each BlochSimulator, methods should have been added to output_eltype and output_size for this function to work properly.

This function is called by simulate_magnetization and is not intended considered part of te public API.

Returns

• magnetization_array: An array allocated on the specified resource, formatted to store the simulation results for each voxel across the specified echo times.

simulate_magnetization!(magnetization, resource, sequence, parameters)

Simulate the magnetization response for all combinations of tissue properties contained in parameters and stores the results in the pre-allocated magnetization array. The actual implementation depends on the computational resource specified in resource.

This function is called by simulate_magnetization and is not intended considered part of te public API.

If the tissue properties for a single voxel are provided only, the simulation is performed on the CPU in a single-threaded fashion.

If no resource is provided, the simulation is performed on the CPU in a multi-threaded fashion by default.

However, if the parameters are a CuArray, the simulation is performed on the GPU.

If the parameters are a DArray, the simulation is performed on the multiple workers.

function simulate_magnetization(sequence, parameters)

Convenience function to simulate magnetization without specifying the computational resource. The function automatically selects the appropriate resource based on the type of the sequence and parameters. The fallback case is to use multi-threaded CPU computations.

simulate_magnetization(resource, sequence, parameters)

Simulate the magnetization response (typically the transverse magnetization at echo times without any spatial encoding gradients applied) for all combinations of tissue properties contained in parameters.

This function can also be used to generate dictionaries for MR Fingerprinting purposes.

Arguments

• resource::AbstractResource: Either CPU1(), CPUThreads(), CPUProcesses() or CUDALibs()
• sequence::BlochSimulator: Custom sequence struct
• parameters::SimulationParameters: Array with different combinations of tissue properties for each voxel.

Note

• If resource == CUDALibs(), the sequence and parameters must have been moved to the GPU using gpu(sequence) and gpu(parameters) prior to calling this function.
• If resource == CPUProcesses(), the parameters must be a DArray with the first dimension corresponding to the number of workers. The function will distribute the simulation across the workers in the first dimension of the DArray.

Returns

• magnetization::AbstractArray: Array of size (output_size(sequence), length(parameters)) containing the magnetization response of the sequence for all combinations of input tissue properties.

_allocate_signal_array(resource, trajectory::AbstractTrajectory, coil_sensitivities)

Allocate an array to store the output of the signal simulation (all readout points, integrated over all voxels).

_signal_per_coil!(signal, resource, magnetization, parameters, trajectory, coordinates, coil_sensitivities)

Compute the signal for a given coil by calculating a volume integral of the transverse magnetization in each voxel for each time point separately (using the signal_at_time_point! function). Each time point is computed in parallel for multi-threaded CPU computation and on the GPU for CUDA computation.

magnetization_to_signal(resource, magnetization, parameters, trajectory, coordinates, coil_sensitivities)

Allocates memory for the signal and computes the signal for each coil separately using the _signal_per_coil! function.

Implementation details

The _signal_per_coil! function has different implementations depending on the computational resources (i.e. the type of resource). The default implementations loop over all time points and compute the volume integral of the transverse magnetization in each voxel for each time point separately. This loop order is not necessarily optimal (and performance may be) across all trajectories and computational resources. If a better implementation is available, add new methods to this function for those specific combinations of resources and trajectories.

The "voxels" are assumed to be distributed over the workers. Each worker computes performs a volume integral over the voxels that it owns only (for all time points) using the CPU1() code. The results are then summed up across all workers.

Note

When using multiple CPU's, the "voxels" are distributed over the workers. Each worker computes the signal for its own voxels in parallel and the results are summed up across all workers.

simulate_signal(sequence, partitioned_parameters::AbstractVector{<:SimulationParameters})

In situations where the number of voxels is too large to store the intermediate magnetization array, the signal can be calculated in batches: the voxels are divided (by the user) into partitions and the signal is calculated for each partition separately. The final signal is the sum of the signals from all partitions.

When coil sensitivities are not provided, use a single coil with sensitivity = 1 everywhere

simulate_signal(resource, sequence, parameters, trajectory, coil_sensitivities)

Simulate the MR signal at timepoint t from coil i as: sᵢ(t) = ∑ⱼ cᵢⱼρⱼmⱼ(t), where cᵢⱼis the coil sensitivity of coil i at position of voxel j, ρⱼ is the proton density of voxel j and mⱼ(t) the (normalized) transverse magnetization in voxel j obtained through Bloch simulations.

Arguments

• resource::AbstractResource: Either CPU1(), CPUThreads(), CPUProcesses() or CUDALibs()
• sequence::BlochSimulator: Custom sequence struct
• parameters::SimulationParameters: Array with tissue properties for each voxel
• trajectory::AbstractTrajectory: Custom trajectory struct
• coordinates::StructArray{<:Coordinates}: Array with spatial coordinates for each voxel
• coil_sensitivities::AbstractMatrix: Sensitivity of coil j in voxel v is given by coil_sensitivities[v,j]

Returns

• signal::AbstractArray{<:Complex}: Simulated MR signal for the sequence and trajectory. The array is of size (# samples per readout, # readouts, # coils).

f32(x)

Change precision of x to Float32. It uses Functors.fmap to recursively traverse the fields of the struct x. For custom structs (e.g. <:BlochSimulator or <:AbstractTrajectory), it is required that typeof(x) be made a Functors.@functor (e.g. @functor FISP).

It may be necessary to add new adapt rules (by adding new methods to adapt_storage) if new structs with complicated nested fields are introduced.

f64(x)

Change precision of x to Float64. It uses Functors.fmap to recursively traverse the fields of the struct x. For custom structs (e.g. <:BlochSimulator or <:AbstractTrajectory), it is required that typeof(x) be made a Functors.@functor (e.g. @functor FISP).

It may be necessary to add new adapt rules (by adding new methods to adapt_storage) if new structs with complicated nested fields are introduced.

_all_arrays_are_cuarrays(x)

Returns true if all AbstractArray fields in x are CuArrays and false otherwise. Will also return if x does not have any AbstractArray fields.

gpu(x)

Move x to CUDA device. It uses Functors.fmap to recursively traverse the fields of the struct x, converting <:AbstractArrays to CuArrays, and ignoring isbitsarrays. For custom structs (e.g. <:BlochSimulator or <:AbstractTrajectory), it is required that typeof(x) be made a Functors.@functor (e.g. @functor FISP).