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Stiffness mapping for early detection of breast cancer: combined force and displacement measurements

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Abstract

Early detection of breast cancer is crucial to patient survival. Our ultimate goal is to automate and refine the manual breast exam process using an electromechanical device that gently indents the tissue at multiple locations and measures the required indentation forces and the resulting displacements of the tissue surface. Our current experiments use a simplified electromechanical device and average-sized (180 cc) silicone breast phantoms to collect the force/displacement data. This data is used with finite element methods and a genetic algorithm to create a three-dimensional map of the stiffness inside the breast—unusually stiff regions are suspected tumors. We tested 14 tumor-free phantoms and 14 tumor-containing phantoms. Using a combination of 10% force data and 90% displacement data we could correctly classify all phantoms as tumor-containing or tumor-free, which was substantially more robust than either measurement modality alone. In addition, the approach is robust to errors in the stiffness assumed in the computations.

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Notes

  1. A-341 from Factor II, Inc.

  2. Dow Xiameter PMX200 50 cs silicone fluid, Ellsworth Adhesives.

  3. Johnson’s Baby Powder, Original.

  4. HP 3D Scanner Pro S3 with dual cameras.

  5. Futek item FSH03871 LSB200 250g JR S-Beam Load Cell with FSH03927 External USB Output Kit.

  6. Again, the error for a tumor-free FE individual remains constant over all generations.

  7. It is 1/6 because there are 6 nodes in each surface element.

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Acknowledgements

This work was supported by generous contributions to the Lawrence J. Giacoletto endowment. Computing support was provided by the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number ACI-1548562. Resources were obtained through the Texas Advanced Computing Center (TACC) at the University of Texas at Austin.

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  1. Mechanical Engineering, Rose-Hulman Institute of Technology, Terre Haute, IN, 47802, USA

    Lorraine G. Olson

  2. Electrical Engineering, Rose-Hulman Institute of Technology, Terre Haute, IN, 47802, USA

    Robert D. Throne

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Correspondence to Lorraine G. Olson.

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This work was supported by generous contributions to the Lawrence J. Giacoletto endowment. This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number ACI-1548562. Resources were obtained through the Texas Advanced Computing Center at the University of Texas at Austin.

Appendices

A displacement data processing

There are two phases to processing the displacement data so that it can be used as part of an error norm comparing finite element normal displacements to experimentally measured displacements: generating finite element normal displacements and converting experimentally measured STL files to corresponding normal displacements.

1.1 A.1 Finite element normal displacements and nodal areas

Let us suppose that we have a finite element mesh such as that shown in the stiffness map in Fig. 1, which is typical for our phantoms. Notice that some elements are small and some elements are large. If we compute the forward solution for an indentation site we will have the force required to impose that indentation along with the x, y, z displacements of the surface nodes. While each element has a well-defined normal direction the nodes do not have a normal as such—they are connected to several elements each of which has a different normal and a different size. In addition, the elements were created by processing an STL of the geometry and this sometimes introduces small peaks that result in widely varying normals in a small region.

To address these issues, we create a transformation matrix that takes the x, y, z displacements at the nodes and converts it into smoothed normal displacements at the nodes:

  1. 1.

    We find the normal to each surface element and also calculate its area. The elements are straight-sided 6-node elements so the normal and area are well-defined.

  2. 2.

    For each node on the surface we find all of the elements to which it is attached.

  3. 3.

    The normal for this node is the area-weighted average of the elements to which it is attached.

  4. 4.

    For each surface node, we replace its normal with the area-weighted average of the normals of all the nodes within a 5 mm radius

  5. 5.

    Finally we use these smoothed normals to create a matrix that transforms the x, y, z nodal displacements to normal nodal displacements

We also need to associate an area with each node in the cost calculation for the displacements—the area for each node is taken to be 1/6 of the sum of the areas of the elements to which it is attachedFootnote 7.

1.2 A.2 Processing the STL files

The STL file processing is repeated independently for each indentation site. Recall from Section 4 that we have two “before indentation” STL files (STL1 and STL2) and two “after indentation” STL files (STL3 and STL4). An STL file consists of triangles and vertices, and for our purposes we are only interested in the vertices which we treat as a cloud of surface xyz data points. We need to turn this cloud of points into normal nodal displacements for comparison with the finite element results. We also need to mark the nodes where there is no data. Figure 18 shows a typical cloud of points from a single STL.

Fig. 18
figure 18

Point cloud from a single STL scan file. (Approximately 450K points. The keyhole is missing data due to the indenter blocking the camera)

Full size image

We rotate and translate the four STL files into the same coordinate system that was used to generate the finite element mesh. This is straightforward.

For each STL file, for each finite element node:

  1. 1.

    Find all of the points in the STL file which are within 2 mm of the smoothed normal vector through the node.

  2. 2.

    If there are at least 10 points, mark that node as having “good” data. (Otherwise it is marked “bad” and not used in the cost calculation.)

  3. 3.

    Average the xyz location of those near points.

  4. 4.

    Find the normal distance from the finite element node to the average xyz location using the smoothed normals.

  5. 5.

    This is the normal distance from the node to the cloud.

Each STL file then produces a set of nodal normal distances and a preliminary list of nodes with good data.

We create an overall list of nodes with good data for this indenter which includes only nodes which have good data from all four STL files. If the difference between the nodal normal distances from the two precompression scans (STL1 and STL2) differ by more than 0.1 mm that node is also marked as bad. Similarly, if the difference between the nodal normal distances for the two indentation scans (STL3 and STL4) differ by more than 0.1 mm that node is marked as bad.

Next we create the final nodal normal displacement and list of nodes with good data by combining the nodal normal distance information from the four scans:

  1. 1.

    Average the normal nodal distance data for the two precompression scans.

  2. 2.

    Average the normal nodal distance data for the two indentation scans.

  3. 3.

    The nodal normal displacement is the difference of the indentation average normal nodal distance and the precompression average normal nodal distance.

  4. 4.

    If the nodal normal displacement at any node is more than 0.4 mm different than the average value of all of its neighbor nodes (within a radius of 2.5 mm) mark that node as bad.

This gives us (for one indentation site) a set of nodal normal displacements and a list of which nodes have good data. The normal displacements can be plotted as shown earlier in Figure 5.

B Additional details on genetic algorithm and computational costs

Here we briefly summarize some additional details on the genetic algorithm and it’s computational cost which may be of particular interest. Complete details about the genetic algorithm are available in our earlier papers. [41, 42].

1.1 B.1 Genetic algorithm details

The genetic algorithm individuals are completely defined by a chromosome which records whether the tissue in each clump is healthy or tumor tissue.

The 288 individuals in the first generation are chosen carefully to speed convergence and to help avoid falling into local minima. The first generation always includes one tumor-free individual. This choice ensures that a tumor-free result is always considered. The first generation also includes 144 individuals which have exactly one tumor clump each. This choice ensures that tumors throughout the tissue are considered. The remaining 143 individuals are generated by randomly assigning tumor or healthy tissue properties to each clump in the individual (equal likelihood of tumor or healthy). This choice ensures that a wide variety of tumor sizes is considered.

We evaluate the cost associated with each of the 288 individuals. The best 16 individuals (16 with lowest cost) are taken as “parents” for the next generation. Parents are retained for the next generation and then mated in pairs. For each pair of parents we generate 17 children. To generate a “child” we compare the genes on the chromosomes for the two parents: if the parents have the same gene (healthy or tumor) in a given location then the child has that gene; if the parents have different genes then the child has an even (random) chance for healthy or tumor tissue. Once all the children are created we choose a random number of children to receive mutations and on those children we switch a random number of genes. These mutations also help to avoid falling into a local minimum. The resulting population of 16 parents and 272 children are used for the next generation.

1.2 B.2 Computational costs

We are using Stampede 2 at the Texas Advanced Computing Center at University of Texas at Austin. Our code runs on 6 nodes with 48 processes per node (total 288 processes). The wall clock time for a full inverse to compute the stiffness map for one experiment is approximately 2 hours. We anticipate that algorithmic enhancements and hardware computing advancements will reduce this time in the future. The run time is essentially unchanged whether our cost function uses forces or displacements or their combination—the majority of the computation time is spent in solving the forward finite element problem which intrinsically produces both force and displacement information.

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Olson, L.G., Throne, R.D. Stiffness mapping for early detection of breast cancer: combined force and displacement measurements. Engineering with Computers 38, 4023–4041 (2022). https://doi.org/10.1007/s00366-022-01741-3

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