Extended Phase Graphs (EPG)#

Extended Phase Graphs (EPG) are first described in Hennig’s 1988 paper Multiecho imaging sequences with low refocusing flip angles – although the term “Extended Phase Graph” appears in a later paper (Echoes—how to generate, recognize, use or avoid them) – is an invaluable tool for fast simulation of MRI sequences. It allows simulation of all main phenomena, including diffusion, motion, and magnetization transfer.

The EPG model is based on the spatial Fourier transform of the isochromats present in a voxel. Originally developed for sequences with a very regular time course (e.g. RARE or bSSFP), it has been later extended for non-regular sequences

Sycomore provides three EPG models:

  • a one-dimensional regular model where all gradient moments throughout the simulation are assumed equal,

  • a one-dimensional discrete model where gradients moments may differ, their values being discretized (i.e. binned) to avoid numerical instabilities along the simulation,

  • a three-dimensional discrete model, where gradients can be specified in three dimensions.

All three models share a common API, and can handle single-pool or two-pools (exchange and MT) variants.

To create a single-pool model, a single species is sufficient. Additionally, the initial magnetization \(M_0\) can be specified. If missing, it defaults to \([0,0,1]\).

species_a = sycomore.Species(779*ms, 45*ms)
single_pool_model = sycomore.epg.Discrete(species_a)

sycomore::Species species_a(779*ms, 45*ms);
sycomore::epg::Discrete single_pool_model(species_a);

For an exchange model two species, their initial magnetizations, and the exchange rate from pool a to pool b must be provided. An optional frequency offset of pool b relative to pool a may be provided, default to 0 Hz.

species_b = sycomore.Species(100*ms, 20*ms)
M0_a = sycomore.Array[float](0, 0, 0.8)
M0_b = sycomore.Array[float](0, 0, 0.2)
k_a = 2*Hz
exchange_model = sycomore.epg.Discrete(species_a, species_b, M0_a, M0_b, k_a)

sycomore::Species species_b(100*ms, 20*ms);
sycomore::Vector3R M0_a{0, 0, 0.8}, M0_b{0, 0, 0.2};
auto k_a = 2*Hz;
sycomore::epg::Discrete exchange_model(species_a, species_b, M0_a, M0_b, k_a);

In an MT model, pool b is assumed to have a very short \(T_2\) and thus no transversal magnetization: one species, the \(R_1\) or \(T_1\) of pool b, both initial magnetizations, and the exchange rate from pool a to pool b must be provided.

R1_b = 100*ms
mt_model = sycomore.epg.Discrete(species_a, R1_b, M0_a, M0_b, k_a)

auto R1_b = 100*ms;
sycomore::epg::Discrete mt_model(species_a, R1_b, M0_a, M0_b, k_a);

Once the model is created, the main features are:

  • Simulation of an RF pulse using the function apply_pulse. For a single pool model, the flip angle must be provided, and the phase is optional, defaulting to 0°. For an exchange model, a second pair of flip angle and phase may be provided – if missing, the same pulse will be applied to the two pools. For an MT model, the phase and direct saturation are mandatory.

  • Simulation of a time interval (relaxation, diffusion, exchange, etc.) using the function apply_time_interval. Refer to the model-specific pages for details.

  • Accessing the non-dephased magnetization using the echo property

  • Accessing the orders and the states using the orders and states properties.

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