Fig 1.
(a) Semi-confined compression of a cylindrical sample (diameter of 30 mm, height of 17 mm) of Sylgard 527 gel for determining the gel material parameters. Technical specification of the apparatus is available at http://isml.ecm.uwa.edu.au/ISML/. We used Bestech KD40S-5N tension-compression load-cell with 5N force range (www.bestech.com.au). (b) Example of force measured in the experiments. We conducted the experiments for three gel samples from a given batch of gel.
Fig 2.
X-ray images of the cylindrical sample (diameter of 30 mm, height of 17 mm) of Sylgard 527 gel during needle insertion.
The images were acquired using X-ray C-arm General Electric 9900 apparatus located at The University of Western Australia Clinical Training and Evaluation Centre CTEC. (a) Calibration of the X-ray apparatus image acquisition system and image distortion evaluation using the X-ray opaque chessboard-like calibration pattern (the pattern was machined from a printed circuit board PCB). (b) X-ray image of the sample at the start of needle insertion. (c) X-ray image of the sample after the needle insertion to the depth of 8 mm. (d) Locally enlarged X-ray image after the needle insertion to the depth of 8 mm. Gel deformation in the area adjacent to the needle is clearly visible. U1 is the needle insertion depth (i.e. the needle tip displacement in relation to top sample surface) and U2 is the maximum deflection of the sample surface along the needle shaft. The deformation coefficient (see section 3.1 Kinematic approach) CD = U2/U1. For the experiment shown in this figure CD = U2/U1≈0.4.
Fig 3.
Geometry of the needle used in this study.
Fig 4.
Experimental set-up for the needle insertion into silicone gel cylindrical (diameter of 30 mm and height of 17 mm) sample and needle force measurement.
The needle insertion was conducted using the specialized apparatus we also applied in compression tests to determine Sylgard 527 gel material properties (see Fig 1). The needle force was measured using Bestech KD40S-5N tension-compression load-cell with 5N force range (www.bestech.com.au).
Fig 5.
Meshless computational grids used when verifying convergence of the method for modeling of needle insertion proposed in this study.
(a) Coarse grid (7,480 nodes); (b) Moderate grid (17,730 nodes); and (c) Refined grid (30,294 nodes). Convergence analysis was conducted through modeling of needle insertion into a cylindrical sample according to the experimental set-up shown in Fig 4.
Table 1.
Parameters of the Ogden and neo-Hookean material models when evaluating the sensitivity of our method for needle insertion modeling to the material properties of the analyzed continuum.
Fig 6.
(a) Top and (b) side view (X-ray image) of the three-layered non-homogenous (each layer has different material properties) cylindrical Sylgard 527 silicone gel sample with the layers of steel beads (black dots) embedded within the sample; (c) computational grid /nodal distribution (32,276 nodes and 178,993 integration cells) we apply to represent the spatial domain when modeling needle insertion into the sample.
Table 2.
Ogden material model parameters for the three layers of the non-homogenous cylindrical Sylgard 527 gel sample.
Fig 7.
Experimental set-up for conducting the experiments on needle insertion within a CT scanner.
The experiments were conducted to obtain quantitative information about the deformation field induced by needle insertion. We used an in-house CT-compatible test apparatus built from hard plastic. The needle is attached to a roller bearing located beneath the actuating screw to prevent transmission of the rotary motion of the screw to the needle (i.e. the needle undergoes only translational/linear motion). For CT image acquisition, we used Siemens SOMATOM XCT scanner located at Medical XCT Facility of Commonwealth Scientific and Industrial Research Organization (CSIRO) in Kensington, Western Australia.
Fig 8.
Complex geometry (brain phantom made from Sylgard 527 silicone gel) we used in this study to evaluate the performance of our algorithm for needle insertion simulation.
(a) Photograph of the anatomically accurate human skull cast by 3B Scientific (Hamburg, Germany; https://www.3bscientific.com) we used to manufacture the phantom (inside the skull). (b) Computed tomography (CT) image (sagittal section) of the brain phantom. To extract information about the phantom geometry from the images, we used 3D Slicer—an open source software platform for image processing and three-dimensional visualization [61].
Fig 9.
Meshless discretization (using the MTLED algorithm) for simulation of the needle insertion into the brain phantom geometry extracted from the phantom radiographic images.
The geometry was discretized using 73,926 nodes (blue and red dots) and 417,790 background tetrahedral integration cells with one Gauss point per cell. Red dots indicate the nodes that are rigidly constrained.
Fig 10.
(a) Histograms displaying the differences in displacement field components (ux -top, uy -middle, uz -bottom) between (a) the coarse (7,480 nodes) and refined (30,294 nodes) computational grids; and (b) the moderate (17,730 nodes) and refined (30,294 nodes) grids. The comparison was done node-by-node for the refined grid. Interpolation was applied to compute the displacements at the locations of nodes of the refined grid using coarse and moderate density grids. The needle is inserted in the z-direction. Note practically negligible differences (under 0.1 mm for all the nodes) between the displacement field components obtained using moderate and refined grids. As the differences between the displacements obtained using coarse and refined grids were up to 0.6 mm, we used the 0.6 mm as our axial scale so that all the displacement differences (coarse to refined grids and moderate to refined grids) are on the same scale.
Table 3.
Normalized Root Mean Square Error (NRMSE) for displacement components (ux, uy, uz) for successively denser grids obtained when modeling needle insertion into a cylindrical sample (diameter 30 mm; height 17 mm) of silicone gel.
Fig 11.
Measured (red solid line) and computed (black solid line) force acting on the needle during insertion into small cylindrical gel sample (diameter of 30 mm, height of 17 mm).
The sample is shown in Fig 4.
Fig 12.
Modeling needle insertion into a small cylindrical sample (diameter of 30 mm and height of 17 mm; the sample is shown in Fig 4) when varying the sample material properties (shear modukus) and material model (Ogden and neo-Hookean) as described in Table 1.
The insertion depth is 15 mm. The computational grid consists of 17,730 nodes and 96,038 integration cells with one integration point per cell. Histograms show the node-by-node differences (in millimeters) in displacement field components (ux -top, uy -middle, uz -bottom) between (a) Materials 1 and 2 and (b) between Materials 1 and 3 and (c) between Materials 2 and 4. Materials 1, 2 and 3 are Ogden, and Material 4 is neo-Hookean. The shear moduli are 72.0 Pa for Material 1, 722.0 Pa for Material 2, 7222.0 Pa for Material 3, and 1000.0 Pa for Material 4. See Table 1 for more information about Material 1, Material 2, Material 3 and Material 4.
Table 4.
Modeling needle insertion into a cylindrical sample (diameter of 30 mm and height of 17 mm) when varying the sample material model and material properties as described in Table 1.
Fig 13.
Needle insertion (to depth of 5 mm) into the non-homogenous cylindrical sample (diameter of 65 mm and height of 34 mm) of Sylgard 527 gel (Figs 6 and 7).
The needle diameter is 1.6 mm. The predicted contours of the sample (blue lines) are overlaid on the CT images acquired during the needle insertion. The needle is indicated in figure (a), and its outline can be distinguished in figures (a)-(d). (a) Sections through the planes located at (a) 0 mm, (b) 0.16 mm, (c) 0.32 mm, (d) 0.48 mm, (e) 0.64 mm, (f) 0.80 mm, (g) 0.96 mm, and (h) 1.12 mm anteriorly from the central plane of the needle shaft.
Fig 14.
Needle insertion (to depth of 5 mm) into the non-homogenous cylindrical sample (diameter of 65 mm and height of 34 mm) of Sylgard 527 gel (Figs 6 and 7)—close-up view of the insertion area.
The needle diameter is 1.6 mm. The predicted contours of the sample (blue lines) are overlaid on the CT images acquired during the needle insertion. The needle is indicated in figure (a), and its outline can be distinguished in figures (a)-(d). Sections through the planes located at (a) 0 mm, (b) 0.16 mm, (c) 0.32 mm, (d) 0.48 mm, (e) 0.64 mm, (f) 0.80 mm, (g) 0.96 mm, and (h) 1.12 mm anteriorly from the central plane of the needle shaft.
Fig 15.
Needle insertion (to a depth of 15 mm) into the non-homogenous cylindrical sample (diameter of 65 mm and height of 34 mm) of Sylgard 527 gel (Figs 6 and 7).
The needle diameter is 1.6 mm. The predicted contours of the sample (blue lines) are overlaid on the CT images acquired during the needle insertion. The maximum principal Green strain predicted in the simulation is over 70%. The needle is indicated in figure (a), and its outline can be distinguished in figures (a)-(f). Sections through the planes located at (a) 0 mm, (b) 0.16 mm, (c) 0.32 mm, (d) 0.48 mm, (e) 0.64 mm, (f) 0.80 mm, (g) 0.96 mm, and (h) 1.12 mm anteriorly from the central plane of the needle shaft.
Fig 16.
Needle insertion (to a depth of 15 mm) into the non-homogenous cylindrical sample (diameter of 65 mm and height of 34 mm) of Sylgard 527 gel (Figs 6 and 7)—close-up view of needle insertion area.
The needle diameter is 1.6 mm. The predicted contours of the sample (blue lines) are overlaid on the CT images acquired during the needle insertion. The maximum principal Green strain predicted in the simulation is over 70%. The needle is indicated in figure (a), and its outline can be distinguished in figures (a)-(f). Sections through the planes located at (a) 0 mm, (b) 0.16 mm, (c) 0.32 mm, (d) 0.48 mm, (e) 0.64 mm, (f) 0.80 mm, (g) 0.96 mm, and (h) 1.12 mm anteriorly from the central plane of the needle shaft.
Fig 17.
Needle insertion to the depth of 5 mm into the non-homogenous cylindrical sample (diameter of 65 mm and height of 34 mm) of Sylgard 527 gel (Figs 6 and 7).
Comparison of the displacement field components at the beads location predicted using the MTLED algorithm with the kinematic approach for needle insertion modeling we introduced in this study and experimentally determined from the CT images. Histograms of the differences in the (a) x-direction, (b) y-direction and (c) z-direction. The displacements are in millimeters. The predicted and experimentally determined displacement field magnitudes for all beads are listed in S1 Table.
Fig 18.
Needle insertion to the depth of 5 mm into the non-homogenous cylindrical sample (diameter of 65 mm and height of 34 mm) of Sylgard 527 gel (Figs 6 and 7).
(a) Top view (X-Y plane) and (b) Transverse view (X-Z plane) of the steel beads (black and red solid circles) embedded in the gel sample as shown in Fig 5A and 5B. Black circles: the difference between the bead displacements predicted using the MTLED algorithm with the kinematic approach for needle insertion modeling and experimentally determined from the CT images does not exceed 0.32 mm (twice the in-plane-image resolution). Red circles with numbers (bead ID): the difference between the bead displacements predicted using the MTLED algorithm with the kinematic approach for needle insertion modeling and experimentally determined from the CT images exceeds 0.32 mm (twice the in-plane-image resolution).
Fig 19.
Needle insertion to the depth of 15 mm into the non-homogenous cylindrical sample (diameter of 65 mm and height of 34 mm) of Sylgard 527 gel (Figs 6 and 7).
Comparison of the displacement field components at the beads location predicted using the MTLED algorithm with the kinematic approach for needle insertion modeling we introduced in this study and experimentally determined from the CT images. Histograms of the differences in the (a) x-direction, (b) y-direction, and (c) z-direction. The displacements are in millimeters. Values of the predicted and experimentally determined displacement field magnitude for all beads are listed in S2 Table.
Fig 20.
Needle insertion to the depth of 15 mm into the non-homogenous cylindrical sample (diameter of 65 mm and height of 34 mm) of Sylgard 527 gel (Figs 6 and 7).
(a) Top view (X-Y plane) and (b) Transverse view (X-Z plane) of the steel beads (black and red solid circles) embedded in the gel sample as shown in Fig 5A and 5B. Black circles: the difference between the bead displacements predicted using the MTLED algorithm with the kinematic approach for needle insertion modeling and experimentally determined from the CT images does not exceed 0.32 mm (twice the in-plane-image resolution). Red circles with numbers (bead ID): the difference between the bead displacements predicted using the MTLED algorithm with the kinematic approach for needle insertion modeling and experimentally determined from the CT images exceeds 0.32 mm (twice the in-plane-image resolution).
Fig 21.
Modeling of needle insertion (insertion depth of 15 mm) into the non-homogenous cylindrical (diameter of 65 mm and height of 34 mm) sample of Sylgard 527 gel.
Vector plot of the predicted (red vectors) using our model and experimentally determined from the CT image analysis (blue vectors) displacement field at the beads’ location. The gel sample is shown in Fig 6, and the experimental set-up—in Fig 7.
Fig 22.
Measured (red solid line) and predicted (black solid dashed) force acting on the needle during insertion into the non-homogenous cylindrical sample (diameter of 65 mm and height of 34 mm) of Sylgard 527 gel. The sample and the model are shown in Fig 6. The experimental set-up is shown in Fig 4.
Fig 23.
Modeling needle insertion into a cylindrical sample (diameter of 65 mm and height of 34 mm) using uniformly the simplest neo-Hookean model with μ = 1,000 Pa.
The insertion depth is 15 mm. Histograms displaying the node-by-node difference (in mm) for the (a) ux (b) uy and (c) uz displacement field components between Ogden and neo-Hookean material models. In the simulations using the Ogden material model, three layers with different material properties were distinguished in the sample as listed in Table 2. For the neo-Hookean material model, the sample was modeled as a homogenous continuum (uniform material properties for the entire sample). Note practically negligible differences (up to 0.05 mm for all the nodes) between the displacements obtained using the two material models.
Fig 24.
(a) Initial configuration of the brain phantom geometry (the geometry is shown in Figs 8 and 9) and (b) Magnitude of the displacement field (in m) when modeling needle insertion by 0.10 m in z-direction.
Fig 25.
Modeling of needle insertion into a continuum with complex geometry (the brain phantom–see Fig 8) using Ogden (with μ = 722 Pa and α = -1.3) and the simplest hyperelastic neo-Hookean material models with μ = 1,000 Pa.
The insertion depth is 100 mm. Histograms displaying the node-by-node difference (in mm) for the (a) ux (b) uy and (c) uz displacement field components between Ogden and neo-Hookean material models. The brain phantom geometry was discretized using 73,926 nodes and 417,790 background tetrahedral integration cells with one integration point per cell as shown in Fig 9.