A Closed-Form Differential Formulation for Ultrasound Spatial Calibration: Single Wall Phantom

Abstract

Calibration is essential in freehand 3-D ultrasound to find the spatial transformation from the image coordinates to the sensor coordinate system. Ease of use, simplicity, precision and accuracy are among the most important factors in ultrasound calibration, especially when aiming to make calibration more reliable for day-to-day clinical use. We introduce a new mathematical framework for the simple and popular single-wall calibration phantom with a plane equation pre-determination step and the use of differential measurements to obtain accurate measurements. The proposed method provides a novel solution for ultrasound calibration that is accurate and easy to perform. This method is applicable to both radiofrequency (RF) and B-mode data, and both linear and curvilinear transducers. For a linear L14-5 transducer, the point reconstruction accuracy (PRA) of reconstructing 370 points is 0.73 ± 0.23 mm using 100 RF images, whereas the triple N-wire PRA is 0.67 ± 0.20 mm using 100 B-mode images. For a curvilinear C5-2 transducer, the PRA using the proposed method is 0.86 ± 0.28 mm on 400 points using 100 RF images, whereas N-wire calibration gives a PRA of 0.80 ± 0.46 mm using 100 B-mode images. Therefore, the accuracy of the proposed variation of the single-wall method using RF data is practically similar to the N-wire method while offering a simpler phantom with no need for accurate design and construction.

Introduction

Ultrasound is a tolerable, portable, inexpensive and real-time modality that can produce 2-D and 3-D images, and therefore, it is a valuable intra-operative imaging modality to guide surgeons aiming to achieve higher accuracy in the intervention and improve patient outcomes. Specifically, there has been a surge of interest in integrating ultrasound imaging into a number of clinical procedures, such as laparoscopic procedures (Nakamoto et al. 2008), minimally invasive cardiac surgeries and therapies (Huang et al. 2010), spinal fusion surgeries (Yan et al. 2011), orthopedic surgeries (Paulius et al., 2008, Peters et al., 2010), guidance for breast biopsy (Cosio et al. 2010), tumor resection (Krekel et al. 2011), brain neurosurgery (Unsgård, 2009) and radiation therapy (Chinnaiyan et al. 2003).

In many such clinical procedures, there is a benefit in tracking the spatial location of the transducer while sweeping over the anatomy of interest. This “freehand” 3-D ultrasound imaging approach can be used for visualization and quantitative measurements such as 3-D locations, sizes and volumes of anatomic structures. Also, by tracking an ultrasound transducer, multiple ultrasound data sets can be mapped into the same coordinate system to construct larger volumes with an extended field of view. Ultrasound with positional information also facilitates registration to complementary image modalities such as magnetic resonance imaging (Melvær et al. 2012). In some applications, laparoscopic, biopsy and surgical tools are also tracked, and their positions should be converted to a common coordinate system as the ultrasound images. Augmented reality is yet another application that can benefit from tracked ultrasound transducers.

The accuracy of freehand-tracked ultrasound is an important factor in the overall accuracy of the aforementioned procedures (Peterhans et al. 2010). In many cases, high accuracy results in numerous clinical benefits. For example, intra-operative ultrasound imaging of the vertebrae, combined with automated registration to pre-operative computed tomography, could improve spine surgery by improving accuracy, reducing operative time and decreasing invasiveness. The resulting benefits include lower surgical risk, increased possibility of performing more complex instrumentation, decreased post-operative complications, more confidence in the surgical procedures and better post-operative function (Yan et al. 2011). In performing neuronavigation based on intra-operative 3-D ultrasound, precise surgical planning and intervention are possible, resulting in the reduction of residual tumor volumes, reduced operation times and better patient outcomes (Lindseth et al. 2002). In ultrasound-guided liver tumor resection, the surgeon relies on ultrasound volumes for accurate orientation with respect to the tumor. High accuracy is needed to provide tumor-free resection margins and to preserve vessels close to the tumor (Gulati et al. 2009). For a breast tumor biopsy, the needle tip should be accurately located inside several positions of the tumor (Cosio et al. 2010). In real-time visualization of high-intensity focused ultrasound for prostate cancer treatment with 3-D ultrasound, precise knowledge of the size and location of the tumor and the treated areas can improve the outcome (Rouvire et al. 2007).

In all the clinical applications that use freehand-tracked ultrasound to reconstruct 3-D ultrasound volumes, such as those examples cited above, the challenge is to precisely locate the ultrasound image pixels with respect to a tracking sensor on the transducer. In a process called spatial calibration, the spatial transformation between the ultrasound image coordinates and the transducer's coordinate system is determined.

Many methods have been proposed for ultrasound calibration over the last two decades. Most methods are based on imaging an artificial object with known geometric parameters called a phantom. To calculate the calibration parameters, the phantom geometry, the ultrasound image features and usually a mathematical model are used. Calibration methods can thus be categorized according to the phantom shape.

Point-based phantoms can be constructed as a bead (Amin et al., 2001, Detmer et al., 1994), crossed-wires (Melvær et al., 2012, Trobaugh et al., 1994, Yaniv et al., 2011) or the center of a sphere (Brendel et al. 2004). Wire-based phantoms usually have N- or Z-shaped patterns, but other configurations can also be used (Boctor et al., 2004, Chen et al., 2009, Hsu et al., 2008b, Pagoulatos et al., 2001, Peterhans et al., 2010). The method of Chen et al. (2009) is used in the open-source PLUS ultrasound software employed by several research groups (Lasso et al. 2012). In plane-based methods, the phantom can be a fixed plane, as in the single-wall method (Najafi et al., 2012b, Prager et al., 1998, Yaniv et al., 2011) or its variant the Cambridge phantom (Prager et al. 1998), or multiple planes (Najafi et al. 2012a). Another approach is based on registration of 2-D ultrasound images with the 3-D model of the phantom (Bergmeir et al., 2009, Blackall et al., 2000, Lange et al., 2011). Some calibration methods do not require a phantom and use a calibrated stylus (Hsu et al., 2008a, Khamene and Sauer, 2005, Muratore and Galloway, 2001) or use changes in speckle from transducer movements (Boctor et al. 2006).

Calibration methods can also be categorized according to their mathematical solution technique. Some of the calibration methods solve the calibration parameters by iteratively minimizing a cost function based on the mathematical geometry of the problem (Detmer et al., 1994, Melvær et al., 2012, Prager et al., 1998). Iterative methods are subject to suboptimal local minima and are sensitive to initial estimates; therefore, they are less robust in general than closed-form solutions (Eggert et al. 1997). Some methods use a closed-form solution derived from the geometry of the phantom to determine calibration parameters (Boctor et al., 2004, Chen et al., 2009, Najafi et al., 2012b). Not all methods use a mathematical solver to calculate calibration parameters. For example, there are methods that are based on iterative manual alignment of the ultrasound image with a thin planar phantom (Gee et al., 2005, Lindseth et al., 2003). Detailed reviews, comparison of different calibration methods and a summary of various validation techniques can be found in survey papers (Hsu et al., 2009, Mercier et al., 2005).

Ease of use, simplicity, precision (repeatability) and accuracy are among the most important factors in ultrasound calibration, especially when the aim is to make calibration more reliable for day-to-day clinical use. Phantoms that must be built with a specific geometry, or from specific material, or with a specific scanning or alignment protocol or phantoms that use complicated segmentation or registration algorithms are barriers to simplicity and ease of use for a user. The single-wall method uses perhaps the simplest phantom among other calibration methods. It merely requires a planar object such as the flat bottom surface of the water tank. Such a phantom is part of the popular Stradwin freehand ultrasound system freely available and used by many research groups (Prager et al. 1999).

One of the most important limiting factors in increasing the accuracy of calibration is accurate, absolute localization of phantom features in ultrasound images (Lange et al. 2011). One reason for this is the blurry appearance of features resulting from the finite resolution of the ultrasound images and the presence of noise. Moreover, image formation errors arise from speed of sound variations, refraction and a finite beam width, all of which contribute to distortions in the shape of the depicted features.

Absolute localization of features is even more challenging in point-based and wire-based methods compared with plane-based methods. It is very difficult to accurately localize the actual intersection point of a wire or a point target in an ultrasound image. The reason is that the finite resolution of the ultrasound image and imaging artifacts cause small point-shaped objects to appear as short blurred lines in ultrasound images (Chen et al. 2009), and therefore, they do not appear in the shape of a circle, but rather as an asymmetric cloud with a width of several pixels. The finite beam width also affects the accuracy of feature localization of planar objects (Prager et al. 1998). A plane appears in the ultrasound image as a line with a thickness related to the beam width and plane orientation. Unlike point features, the appearance/shape of the line remains unchanged along the line as long as the plane's inclination relative to the beam direction is not high.

To improve accuracy, the need for absolute localization of the calibration phantom features should be reduced. Therefore in this work, we propose use of the differential measurements of the relative distance between two image features. Advances in differential measurements for ultrasound motion tracking in recent years enable accurate measurements of the relative locations of phantom features. This accuracy can be as high as a few microns when radiofrequency (RF) ultrasound data are used (Walker and Trahey 1995).

The differential measurement can be especially accurate when the shapes of the features are very similar. Because the appearance of a wire varies with depth in the image (Chen et al. 2009), it is more difficult to perform accurate measurements of the distance difference of point features unless only pairs of features with similar shapes are selected. Planar phantoms, on the other hand, are ideal because the plane appears as a line with uniform thickness. In fact, echo RF pulses in all RF scan lines exhibit a similar pattern because they all experience the same physics of reflection except for a variation in the ultrasound focus (and, therefore, beam width), which slightly varies with the depth. The relative axial distance of line features can be measured as the relative shift between echo RF pulses and is therefore a differential measurement. This measurement can be performed very accurately, especially with RF data. Redundancy of measurements, because of the presence of many scan lines, is another advantage of planar objects over wires. Redundancy allows averaging of measurements, which can also reduce error from measurement noise.

Previously, we developed a closed-form differential formulation based on a multiwedge phantom comprising five planes (Najafi et al. 2014). The method achieved very high accuracy using differential measurements on RF data. A 3-D printer was required to precisely manufacture the multiwedge phantom. However, some users without such manufacturing ability would prefer a calibration method that does not require a special phantom.

The popular single-wall method (Prager et al. 1998) is an obvious choice to consider in extending the differential calibration method to improve ease-of-use. The original single-wall method uses an iterative solver to find the calibration parameters as well as the plane equation. In fact, unlike most calibration methods, the wall phantom is not required to be tracked, and the plane’s position and orientation parameters are also determined in the solution. Therefore, although the simplicity of the wall phantom is appealing, it is quite a challenge to achieve high accuracy with a small number of images because of the large number of parameters to be solved.

According to Prager et al. (1998), the single-wall phantom produced slightly ill-conditioned sets of calibration equations because of the limited range of scanning motions that resulted in clear images of the wall. There was significant uncertainty in the calibration parameters that led to relatively poor point reconstruction precision. Approximately 500 images were used in those experiments (Prager et al. 1998). To improve accuracy, the Cambridge phantom (Prager et al. 1998) was proposed, which was more accurate but again required a custom apparatus and accurate mounting on the transducer. The technique was also not fully automatic.

In summary, despite the wealth of calibration methods, including the recent introduction of accurate methods using differential measurements on multiple wedges, there is still a need for improved calibration using the simplest and most popular single-wall method. In this article, we propose an improved variation of the single-wall calibration for both linear and curvilinear transducers and for both RF and B-mode input data. The key is to adapt the differential measurement approach to a single wall. We also propose to simplify the calibration problem by explicitly measuring the position of the plane using a stylus and, thus, reduce the number of unknowns and formulate a closed-form solution. The goal is to obtain good accuracy with a relatively small number of images.

In our preliminary work (Najafi et al. 2012b), we developed a closed-form differential formulation for the traditional untracked single-wall method for linear transducers. An updated and detailed description of that method can be found in the Appendix. Similar to the traditional single-wall method, the limited range of possible rotations caused that method to be sensitive to measurement error. Therefore, a new “pre-determined single-wall” method is proposed with fewer parameters to solve by ultrasound calibration. We describe the new method and compare it with the traditional single-wall and N-wire calibrations because they are the most commonly used calibration methods according to our review of the literature to date.