Efficient numerical approximation of a non-regular Fokker-Planck equation associated with first-passage time distributions.

In neuroscience, the distribution of a decision time is modelled by means of a one-dimensional Fokker-Planck equation with time-dependent boundaries and space-time-dependent drift. Efficient approximation of the solution to this equation is required, e.g., for model evaluation and parameter fitting. However, the prescribed boundary conditions lead to a strong singularity and thus to slow convergence of numerical approximations. In this article we demonstrate that the solution can be related to the solution of a parabolic PDE on a rectangular space-time domain with homogeneous initial and boundary conditions by transformation and subtraction of a known function. We verify that the solution of the new PDE is indeed more regular than the solution of the original PDE and proceed to discretize the new PDE using a space-time minimal residual method. We also demonstrate that the solution depends analytically on the parameters determining the boundaries as well as the drift. This justifies the use of a sparse tensor product interpolation method to approximate the PDE solution for various parameter ranges. The predicted convergence rates of the minimal residual method and that of the interpolation method are supported by numerical simulations.

Calculation of the 3-D femoral component's orientation in total hip arthroplasty using a trigonometric algorithm.

Femoral component orientation plays a key role in implant stability and therefore the success rate of total hip arthroplasty. To date, this topic has been studied using various definitions and a variety of imaging modalities and protocols. The aim of this study is a proof of concept that a new algorithm can be used to describe the femoral component's 3D orientation on the three orthogonal anatomical planes and relative to its mechanical axis using input from two orthogonal planes. CT scans of 18 patients with a total of 22 hip arthroplasties were collected. From these, orthogonal coronal and sagittal projections of the complete femur were acquired in the scanning position (MIPs) and relative to the femoral mechanical axis (corrected MIPs). On these images, the orientation of the neck of the femoral component in space and relative to the femoral axis, respectively, was measured by coronal inclination (CIF), sagittal inclination (SIF) and transverse version (TVF). With the algorithm, TVF was also calculated based on CIF and SIF. Differences between measured and calculated TVF and intra- and inter-observer reliability were evaluated using intra-class correlation coefficients (ICC). The error of non-orthogonal imaging (85° angle between the sagittal and coronal reconstructions) was tested on a third series of MIPs. The ICC between the calculated TVF and manually measured TVF, in space and relative to the femoral axis, was 0.98 for both with median absolute differences of 1.3 and 1.5°. For non-orthogonal images this was 0.70 with a median absolute difference of 5°. ICCs for intra-observer and inter-observer reliability for the calculated TVF values were 0.98 and 0.88, respectively. With this algorithm the transverse orientation of the neck of the femoral component can be assessed in space and relative to the mechanical femoral axis by combining its sagittal and coronal orientation. As long as the imaging visualizes two orthogonal planes, the orientation of an implant can be assessed in 3-D, regardless of the imaging modality.

The Effect of Functional Pelvic Tilt on the Three-Dimensional Acetabular Cup Orientation in Total Hip Arthroplasty Dislocations.

BACKGROUND: Anterior and posterior pelvic tilt appears to play a role in total hip arthroplasty (THA) stability. When changing from the standing to the sitting position, the pelvis typically rotates posteriorly while the hips flex and this affects the femoro-acetabular positions. This case-control study compares changes in 3-D acetabular cup orientation during functional pelvic tilt between posterior THA dislocations vs stable THAs. METHODS: Standing and sitting 3-D cup orientation was compared between fifteen posterior dislocations vs 233 prospectively followed stable THAs. 3-D cup orientation was calculated using previously validated trigonometric algorithms on biplanar radiographs. Those algorithms combine the angles in the three anatomical planes (coronal inclination, transverse version, and sagittal ante-inclination) in the standing position with the change in sagittal pelvic tilt from standing to sitting to calculate the 3-D orientation in the sitting position. RESULTS: The standing cup orientation of the dislocated THAs was only characterized by a lower coronal inclination (P = .039). Compared with the controls, from standing to sitting, they showed less posterior pelvic tilt (P < .001). This led to a significant lower coronal inclination (P < .001) and sagittal ante-inclination (P < .001) in the sitting position but similar transverse version (P = .366). CONCLUSIONS: Comparing posterior THA dislocations to stable THAs, there is a lower increase of all three orientation angles from standing to sitting. This leads to a decreased sitting coronal inclination and sagittal ante-inclination which may lead to an increased risk of impingement ensued by THA instability. By contrast, the transverse version was not significantly different in both positions. This confirms the importance of biplanar data on functional cup orientation. LEVEL OF EVIDENCE: Diagnostic, Level III.

The Effect of Postural Pelvic Dynamics on the Three-dimensional Orientation of the Acetabular Cup in THA Is Patient Specific.

BACKGROUND: Sagittal pelvic dynamics mainly consist of the pelvis rotating anteriorly or posteriorly while the hips flexes, and this affects the femoroacetabular or THA configuration. Thus far, it is unknown how the acetabular cup of the THA in the individual patient reorients with changing sagittal pelvic dynamics. QUESTIONS/PURPOSES: The aim of this study was to validate a method that establishes the three-dimensional (3-D) acetabular cup orientation with changing sagittal pelvic dynamics and describe these changes during functional pelvic dynamics. METHODS: A novel trigonometric mathematical model, which was incorporated into an easy-to-use tool, was tested. The model connected sagittal tilt, transverse version, and coronal inclination of the acetabular cup during sagittal pelvic tilt. Furthermore, the effect of sagittal pelvic tilt on the 3-D reorientation of acetabular cups was simulated for cups with different initial positions. Twelve pelvic CT images of patients who underwent THA were taken and rotated around the hip axis to different degrees of anterior and posterior sagittal pelvic tilt (± 30°) to simulate functional pelvic tilt in various body positions. For each simulated pelvic tilt, the transverse version and coronal inclination of the cup were manually measured and compared with those measured in a mathematical model in which the 3-D cup positions were calculated. Next, this model was applied to different acetabular cup positions to simulate the effect of sagittal pelvic dynamics on the 3-D orientation of the acetabular cup in the coronal and transverse plane. After pelvic tilt was applied, the intraclass correlation coefficients of 108 measured and calculated coronal and transverse cup orientation angles were 0.963 and 0.990, respectively, validating the clinical use of the mathematical model. RESULTS: The changes in 3-D acetabular cup orientation by functional pelvic tilt differed substantially between cups with different initial positions; the change in transverse version was much more pronounced in cups with low coronal inclination (from 50° to -29°) during functional pelvic tilt than in cups with a normal coronal inclination (from 39° to -11°) or high coronal inclination (from 31° to 2°). However, changes in coronal inclination were more pronounced in acetabular cups with high transverse version. CONCLUSION: Using a simple algorithm to determine the dynamic 3-D reorientation of the acetabular cup during functional sagittal pelvic tilt, we demonstrated that the 3-D effect of functional pelvic tilt is specific to the initial acetabular cup orientation and thus per THA patient. CLINICAL RELEVANCE: Future studies concerning THA (in)stability should not only include the initial acetabular cup orientation, but also they need to incorporate the effect of sagittal pelvic dynamics on the individual 3-D acetabular cup orientation. Clinicians can also use the developed tool, www.3d-hip.com, to calculate the acetabular cup's orientation in other instances, such as for patients with spinopelvic imbalance.

An optimal adaptive wavelet method for first order system least squares.

In this paper, it is shown that any well-posed 2nd order PDE can be reformulated as a well-posed first order least squares system. This system will be solved by an adaptive wavelet solver in optimal computational complexity. The applications that are considered are second order elliptic PDEs with general inhomogeneous boundary conditions, and the stationary Navier-Stokes equations.