Description of the proposed protocol
Our proposed protocol utilizes the AM algorithm[2] included in SimVascular[3], an open source software package. The output of this protocol is several adapted meshes and the corresponding FE simulation solutions. Since the overall goal of FE simulations is to achieve sufficient resolution for the parameters of interest (PoI), which are quantities derived from the simulated flow physics important to the research question at hand, the user selects the adapted mesh which adequately resolves the PoI with a minimum number of elements. The protocol consists of an iterative procedure with several operational recommendations. This iterative procedure performs serial adaptations of an initial input mesh using SimVascular’s AM algorithm. In each iteration, an adapted mesh is produced which is then supplied to the subsequent iteration as the input mesh. In each iteration, the input parameters to the AM algorithm are gradually refined to ensure progressive improvements in the mesh. The operational recommendations are a set of suggestions regarding the initial mesh and the settings prescribed to the iterative procedure that can help prevent the occurrence of the aforementioned undesirable outcomes.
The iterative procedure
The iterative procedure (Figure 1) is governed by parameters prescribed to it, which may be broadly divided into two categories:
- Iteration parameters: Number of iterations (n), Error threshold (Emin), Initial mesh edge size (H0), Refinement Factors (F1 and F2), Preliminary maximum edge size (Hpmax), Preliminary minimum edge size (Hpmin).
- AM algorithm parameters:
- Mesh adaptation strategy: Isotropic versus anisotropic. Our protocol exclusively uses the anisotropic setting as recommended by the authors of the AM algorithm[2].
- MIE reduction factor (R): The AM algorithm attempts to reduce the MIE through each iteration by this factor. It should always be less than 1.
- Maximum edge size (Hmax) and Minimum edge size (Hmin): The maximum and minimum edge size parameters constrain the size of the elements produced in the adapted mesh.
The iterative procedure begins with a uniform mesh where the global edge size for the bulk volume of the geometry is set to H0. In order to refine the mesh progressively, the protocol reduces Hmin and Hmax by refinement factors F1 and F2, respectively, in each iteration. For the first iteration, Hmax and Hmin are set equal to Hpmax and Hpmin, respectively. The error threshold, Emin, is the user-defined value of MIE at which the protocol will stop operation irrespective of the number of completed iterations. Since the MIE of the initial mesh may be unknown when the protocol is initiated, the initial value of Emin should be set to an arbitrarily large value. Once the MIE of the initial mesh is determined after the post-processing step of the first iteration, Emin is updated to the desired value, i.e., a certain proportion (less than 1) of the MIE of the initial mesh.
Operational Recommendations
In addition to the iterative procedure described above, the following operational recommendations are critical for the proper working of the protocol:
- The value of Hmax prescribed to the AM algorithm should always be lesser than H0, to prevent mesh coarsening.
- We have observed that the AM algorithm does not affect faces to which a boundary layer mesh is prescribed. It is our suggestion that this behavior is utilized to preserve the geometric fidelity of “wall” faces and other complex faces by prescribing a fine face mesh and a boundary layer mesh. The fine face mesh and boundary layer mesh prevent changes to those faces, thereby preserving geometric fidelity without significantly increasing the total mesh size.
- In the initial mesh, the mesh for the gross volume can be quite coarse. The value of H0 can be one order of magnitude larger than the edge size prescribed to the wall face.
An example application of the protocol
To demonstrate the working of the protocol, we consider a scenario modelling a Fontan surgical junction with physiologic flow and a patient-specific geometry derived from the MRI scan of a patient with a 19mm-diameter extracardiac conduit (Figure 2A). The initial mesh is identical in all three trials and was generated using MeshSim (Simmetrix Inc., NY). In the subsequent section, we present three trial runs of the protocol designated Trials 1, 2 and 3, which represent conservative, moderate and aggressive approaches to MIE reduction, respectively. In all trials Emin was set to the value of 30. For this example, we consider the PoI to be the pressure developed at the IVC and SVC faces. A detailed description of the physiologically realistic boundary conditions, initial mesh, simulation and protocol settings are provided in Additional File 1.