As MRI sequences become more complex, the complexity of the dependencies between its components increases likewise. It is desirable to encapsulate parts of the sequence structure to enable reuse, but on the other hand sequence modules are configured in multiple interwoven, partially ordered steps. Established approaches enforce a linear order on the calculation steps. That does not capture this behavior adequately and thereby produces sequence descriptions that are hard to understand, maintain and extend. Those sequence definitions also require additional developer input to efficiently check derived parameters for consistency—such as during binary search of protocol configuration.

This work presents a novel sequence description and sequence traversal strategy which yields a new perspective on MRI sequences, eases encapsulation and development, and is guaranteed to perform calculations at highest efficiency.

Sequence preparation (see [fig1])

A sequence consist of modules that are arranged hierarchically. Each module is defined isolatedly and describes its parameters, which can be used for calculations by other modules, and parameter dependencies, which are resolved in the sequence preparation stage to yield an acyclic directed graph of all parameters and dependencies. Opposed to conventional approaches, the calculation order is not set explicitly.

Sequence traversal (see [fig2])

After the sequence is prepared, it can be traversed as dictated by the sequence structure, i.e. to run the sequence or calculate channel waveforms. However, the traversal does not evoke any calculations directly. Setting parameter nodes of the graph recursively iterates all dependent nodes and invalidates their values.

When a basic sequence module is reached in the traversal process, its defining parameters are calculated through the graph. Each invalid dependency input is tagged recursively to find the minimal calculation graph required for that parameter. Afterwards, those calculations are performed, starting at seed points identified by the tagging process. Once calculated, nodes stay valid unless invalidated by setting a parameter that it depends upon. This allows a reuse of calculated parameters by other sequence modules. Also, parameters that are not needed to parameterize the object are not calculated, neither are parameters that merely depend upon those parameters.

The twice-refocused, spin-echo diffusion MRI scheme [1] is a typical example of a sequence with complex parameter dependencies due to the interactions between the EPI readout and the diffusion preparation. It is defined through multiple parameter conditions that imply a certain calculation and calculation order. The parameter set has to fulfill a number of conditions simultaneously, such as the nulling of eddy currents, the level of diffusion weighting and the scan geometry. An imperative programming paradigm requires the explicit calculation order, and a minor change in the conditions, such as spoiling instead of eddy current nulling, usually has side-effects that are hard to grasp from the sequence code.

The algorithm of this work derives the calculation order automatically and reveals potential side effects that can be expected when exchanging parts of the dependencies. (see [fig3])

Furthermore, the graph yields maximally efficient and non-redundant calculations for any parameter. Therefore, testing protocol parameter ranges for sequence validity can be optimized by prioritizing parameters that are likely provoke sequence failure. This way, calculations that are likely to be non-critical can be postponed to be performed only when all critical parameters are feasible. (see [fig4])The algorithm was used successfully to configure a twice-refocused, spin-echo diffusion prepared EPI sequence. (see [fig5])The algorithm shows great potential in calculation efficiency and simplification of the sequence development process. It removes the need for explicit calculation order definition and adds the benefit of a transparent parameter dependency sequence perspective and maximally efficient calculations. The approach can be extended by graph operations and sequence modules that control the sequence traversal based on graph properties. This enables plug-in mechanisms of sequence modules and the development of generic physical models that are applicable to arbitrarily defined sequences.