Inducing and imaging thermal strain using a single ultrasound linear array

Photoacoustic thermal-strain measurement towards noninvasive and accurate temperature mapping in photothermal therapy

Photothermal therapy is a promising tumor treatment approach that selectively eliminates cancer cells while assuring the survival of normal cells. It transforms light energy into thermal energy, making it gentle, targeted, and devoid of radiation. However, the efficacy of treatment is hampered by the absence of accurate and noninvasive temperature measurement method in the therapy. Therefore, there is a pressing demand for a noninvasive temperature measurement method that is real-time and accurate. This article presents one such attempt based on thermal strain photoacoustic (PA) temperature measurement. The method was first modelled, and a circular array-based photoacoustic photothermal system was developed. Experiments with Indian ink as tumor simulants suggest that the temperature monitoring in this work achieves a precision of down to 0.3 °C. Furthermore, it is possible to accomplish real-time temperature imaging, providing accurate two-dimensional temperature mapping for photothermal therapy. Experiments were also conducted on human fingers and nude mice, validating promising potentials of the proposed method for practical implementations.

Multifocus Thermal Strain Imaging Using a Curved Linear Array Transducer for Identification of Lipids in Deep Tissue

Thermal strain imaging (TSI) is an ultrasound-based imaging technique intended primarily for diseases in which lipid accumulation is the main biomarker. The goal of the research described here was to successfully implement TSI on a single, commercially available curved linear array transducer for heating and imaging of organs at a deeper depth. For an effective temperature rise of the tissue over a large area, which is key to TSI performance, an innovative multifocus beamforming approach was applied. This yielded a heating area from 32 to 96 mm in the axial direction and -7 to +7 mm in the lateral direction. The pressure fields generated from simulation were in agreement with pressure fields measured with the hydrophone. TSI with safe acoustic power identified with high contrast a rubber inclusion and liposuction fat tissue embedded in a gelatin block.

Evaluating HIFU-mediated local drug release using thermal strain imaging: Phantom and preliminary in-vivo studies

PURPOSE

High-intensity focused ultrasound (HIFU)-mediated drug release becomes a promising therapeutic technique for treatment of cancer, which has merits of deep penetration, noninvasive approach and nonionizing radiation. However, conventional thermocouple-based approach for treatment monitoring would encounter big challenges such as the viscous heating artifact and difficulty in monitoring in the deep region. In this study, we develop an effective method based on thermal strain imaging (TSI) for the evaluation of HIFU-mediated drug release.

METHODS

Both phantom experiments and preliminary animal experiments were performed to investigate the feasibility of the proposed approach. Doxorubicin (DOX)-loaded cerasomes (HIFU and temperature-sensitive cerasomes, HTSCs) were prepared. In the phantom experiments, the HTSC solution is contained inside a cylindrical chamber within a tissue-mimicking phantom. In the animal experiments, the HTSCs are intravenously injected into tumor-bearing mice. An HIFU transducer is used to trigger DOX release from the HTSCs within the phantom or mice, and TSI is performed during HIFU heating. In the phantom experiments, the accuracy of temperature estimation using TSI is validated by measuring with a thermocouple. In animal experiments, the spatial consistency between the distribution of DOX released within the tumor and the location of the heating region estimated by TSI is validated using a spectrofluorophotometer.

RESULTS

In the phantom experiments, the HTSCs show a burst release of DOX when the temperature of the HTSC solution estimated by TSI reaches about 42°C, which is in agreement with the condition for drug release from the HTSCs. The temperature estimation using TSI has high accuracy with error below 2.5%. In animal experiments, fluorescence imaging of the tumor validates that the heating region of HIFU could be localized by the low-strain region of TSI.

CONCLUSION

The present framework demonstrates a reliable and effective solution to the evaluation of HIFU-mediated local drug delivery.

Multi-Focus Beamforming for Thermal Strain Imaging Using a Single Ultrasound Linear Array Transducer

Ultrasound-induced thermal strain imaging (TSI) has been used successfully to identify lipid- and water-based tissues in atherosclerotic plaques in some research settings. However, TSI faces several challenges to be realized in clinics. These challenges include motion artifacts and displacement tracking accuracy, as well as limited heating capability, which contributes to low thermal strain signal-to-noise ratio, and a limited field of view. Our goal was to address the challenge in heating tissue in TSI. Current TSI systems use separate heating and imaging transducers, which require physical alignment of the heating and imaging beams and result in a bulky setup that limits in vivo operation. We evaluated a new design for heating beams that can be implemented on a linear array imaging transducer and can provide improved heating area and efficiency as compared with previous implementations. The heating beams designed were implemented with a clinical linear array imaging transducer connected to a research ultrasound platform. In vitro experiments using tissue-mimicking phantoms with no blood flow revealed that the new design resulted in an effective heating area of approximately 0.85 cm2 and a 0.3°C temperature rise in 2 s of heating, which compared well with in silico finite-element simulations. With the new heating beams, TSI was found to be able to detect a lipid-mimicking rubber inclusion with a diameter of 1 cm from the water-based gelatin background, with a strain contrast of 2.3 (+0.14% strain in the rubber inclusion and -0.06% strain in the gelatin background). Lastly, lipid-based tissue in a 1-cm-diameter human carotid endarterectomy (CEA) sample was identified in good agreement with histology.

Robust Tracking of Small Displacements With a Bayesian Estimator

Radiation-force-based elasticity imaging describes a group of techniques that use acoustic radiation force (ARF) to displace tissue to obtain qualitative or quantitative measurements of tissue properties. Because ARF-induced displacements are on the order of micrometers, tracking these displacements in vivo can be challenging. Previously, it has been shown that Bayesian-based estimation can overcome some of the limitations of a traditional displacement estimator such as normalized cross-correlation (NCC). In this work, we describe a Bayesian framework that combines a generalized Gaussian-Markov random field (GGMRF) prior with an automated method for selecting the prior's width. We then evaluate its performance in the context of tracking the micrometer-order displacements encountered in an ARF-based method such as ARF impulse (ARFI) imaging. The results show that bias, variance, and mean-square error (MSE) performance vary with prior shape and width, and that an almost one order-of-magnitude reduction in MSE can be achieved by the estimator at the automatically selected prior width. Lesion simulations show that the proposed estimator has a higher contrast-to-noise ratio but lower contrast than NCC, median-filtered NCC, and the previous Bayesian estimator, with a non-Gaussian prior shape having better lesion-edge resolution than a Gaussian prior. In vivo results from a cardiac, radio-frequency ablation ARFI imaging dataset show quantitative improvements in lesion contrast-to-noise ratio over NCC as well as the previous Bayesian estimator.

Thermometry and ablation monitoring with ultrasound

In this review we present the current status of ultrasound thermometry and ablation monitoring, with emphasis on the diverse approaches published in the literature and with an eye on which methods are closest to clinical reality. It is hoped that this review will serve as a guide to the expansion of sonographic methods for treatment monitoring and thermometry since the last brief review in 2007.

An adaptive displacement estimation algorithm for improved reconstruction of thermal strain

Thermal strain imaging (TSI) can be used to differentiate between lipid and water-based tissues in atherosclerotic arteries. However, detecting small lipid pools in vivo requires accurate and robust displacement estimation over a wide range of displacement magnitudes. Phase-shift estimators such as Loupas' estimator and time-shift estimators such as normalized cross-correlation (NXcorr) are commonly used to track tissue displacements. However, Loupas' estimator is limited by phase-wrapping and NXcorr performs poorly when the SNR is low. In this paper, we present an adaptive displacement estimation algorithm that combines both Loupas' estimator and NXcorr. We evaluated this algorithm using computer simulations and an ex vivo human tissue sample. Using 1-D simulation studies, we showed that when the displacement magnitude induced by thermal strain was >λ/8 and the electronic system SNR was >25.5 dB, the NXcorr displacement estimate was less biased than the estimate found using Loupas' estimator. On the other hand, when the displacement magnitude was ≤λ/4 and the electronic system SNR was ≤25.5 dB, Loupas' estimator had less variance than NXcorr. We used these findings to design an adaptive displacement estimation algorithm. Computer simulations of TSI showed that the adaptive displacement estimator was less biased than either Loupas' estimator or NXcorr. Strain reconstructed from the adaptive displacement estimates improved the strain SNR by 43.7 to 350% and the spatial accuracy by 1.2 to 23.0% (P < 0.001). An ex vivo human tissue study provided results that were comparable to computer simulations. The results of this study showed that a novel displacement estimation algorithm, which combines two different displacement estimators, yielded improved displacement estimation and resulted in improved strain reconstruction.

Detecting hepatic steatosis using ultrasound-induced thermal strain imaging: an ex vivo animal study

Hepatic steatosis or fatty liver disease occurs when lipids accumulate within the liver and can lead to steatohepatitis, cirrhosis, liver cancer and eventual liver failure requiring liver transplant. Conventional brightness mode (B-mode) ultrasound (US) is the most common noninvasive diagnostic imaging modality used to diagnose hepatic steatosis in clinics. However, it is mostly subjective or requires a reference organ such as the kidney or spleen with which to compare. This comparison can be problematic when the reference organ is diseased or absent. The current work presents an alternative approach to noninvasively detecting liver fat content using US-induced thermal strain imaging (US-TSI). This technique is based on the difference in the change in the speed of sound as a function of temperature between water- and lipid-based tissues. US-TSI was conducted using two system configurations including a mid-frequency scanner with a single linear array transducer (5-14 MHz) for both imaging and heating and a high-frequency (13-24 MHz) small animal imaging system combined with a separate custom-designed US heating transducer array. Fatty livers (n = 10) with high fat content (45.6 ± 11.7%) from an obese mouse model and control livers (n = 10) with low fat content (4.8 ± 2.9%) from wild-type mice were embedded in gelatin. Then, US imaging was performed before and after US induced heating. Heating time periods of ∼ 3 s and ∼ 9.2 s were used for the mid-frequency imaging and high-frequency imaging systems, respectively, to induce temperature changes of approximately 1.5 °C. The apparent echo shifts that were induced as a result of sound speed change were estimated using 2D phase-sensitive speckle tracking. Following US-TSI, histology was performed to stain lipids and measure percentage fat in the mouse livers. Thermal strain measurements in fatty livers (-0.065 ± 0.079%) were significantly (p < 0.05) higher than those measured in control livers (-0.124 ± 0.037%). Using histology as a gold standard to classify mouse livers, US-TSI had a sensitivity and specificity of 70% and 90%, respectively. The area under the receiver operating characteristic curve was 0.775. This ex vivo study demonstrates the feasibility of using US-TSI to detect fatty livers and warrants further investigation of US-TSI as a diagnostic tool for hepatic steatosis.

Flexible integration of high-imaging-resolution and high-power arrays for ultrasound-induced thermal strain imaging (US-TSI)

Ultrasound-induced thermal strain imaging (USTSI) for carotid artery plaque detection requires both high imaging resolution (<100 μm) and sufficient US-induced heating to elevate the tissue temperature (~1°C to 3°C within 1 to 3 cardiac cycles) to produce a noticeable change in sound speed in the targeted tissues. Because the optimization of both imaging and heating in a monolithic array design is particularly expensive and inflexible, a new integrated approach is presented which utilizes independent ultrasound arrays to meet the requirements for this particular application. This work demonstrates a new approach in dual-array construction. A 3-D printed manifold was built to support both a high-resolution 20 MHz commercial imaging array and 6 custom heating elements operating in the 3.5 to 4 MHz range. For the application of US-TSI in carotid plaque characterization, the tissue target site is 20 to 30 mm deep, with a typical target volume of 2 mm (elevation) × 8 mm (azimuthal) × 5 mm (depth). The custom heating array performance was fully characterized for two design variants (flat and spherical apertures), and can easily deliver 30 W of total acoustic power to produce intensities greater than 15 W/cm(2) in the tissue target region.

Creation and characterization of an ultrasound and CT phantom for noninvasive ultrasound thermometry calibration

Ultrasound thermometry provides noninvasive 2-D temperature monitoring, and in this paper, we have investigated the use of computed tomography (CT) radiodensity to characterize tissues to improve the accuracy of ultrasound thermometry. Agarose-based tissue-mimicking phantoms were created with glyceryl trioleate (a fat-mimicking material) concentration of 0%, 10%, 20%, 30%, 40%, and 50%. The speed of sound (SOS) of the phantoms was measured over a temperature range of 22.1-41.1 °C. CT images of the phantoms were acquired by a clinical dedicated breast CT scanner, followed by calculation of the Hounsfield units (HU). The phantom was heated with a therapeutic acoustic pulse (1.54 MHz), while RF data were acquired with a 10-MHz linear-array transducer. Two-dimensional speckle tracking was used to calculate the thermal strain offline. The tissue-dependent thermal strain parameter required for ultrasound thermometry was analyzed and correlated with CT radiodensity, followed by the validation of the temperature prediction. Results showed that the change in SOS with the temperature increase was opposite in sign between the 0%-10% and 20%-50% trioleate phantoms. The inverse of the tissue-dependent thermal strain parameter of the phantoms was correlated with the CT radiodensity (R(2) = 0.99). A blinded ultrasound thermometry study on phantoms with a trioleate range of 5%-35% demonstrated the capability to estimate the tissue-dependent thermal strain parameter and estimate temperature with error less than ~1 °C. In conclusion, CT radiodensity may provide a method for improving ultrasound thermometry in heterogeneous tissues.

Motion Artifact Reduction in Ultrasound Based Thermal Strain Imaging of Atherosclerotic Plaques Using Time Series Analysis

Large lipid pools in vulnerable plaques, in principle, can be detected using US based thermal strain imaging (US-TSI). One practical challenge for in vivo cardiovascular application of US-TSI is that the thermal strain is masked by the mechanical strain caused by cardiac pulsation. ECG gating is a widely adopted method for cardiac motion compensation, but it is often susceptible to electrical and physiological noise. In this paper, we present an alternative time series analysis approach to separate thermal strain from the mechanical strain without using ECG. The performance and feasibility of the time-series analysis technique was tested via numerical simulation as well as in vitro water tank experiments using a vessel mimicking phantom and an excised human atherosclerotic artery where the cardiac pulsation is simulated by a pulsatile pump.

Noninvasive detection of lipids in atherosclerotic plaque using ultrasound thermal strain imaging: in vivo animal study

OBJECTIVES

This study sought to examine the feasibility of in vivo detection of lipids in atherosclerotic plaque (AP) by ultrasound (US) thermal (or temporal) strain imaging (TSI).

BACKGROUND

Intraplaque lipid content is thought to contribute to plaque stability. Lipid exhibits a distinctive physical characteristic of temperature-dependent US speed compared with water-bearing tissues. As tissue temperature changes, US radiofrequency (RF) echoes shift in time of flight, which produces an apparent strain (thermal or temporal strain [TS]).

METHODS

US heating-imaging pulse sequences and transducers were designed and integrated into commercial US scanners for US-TSI of arterial segments. US-RF data were collected while gradually increasing tissue temperature. Phase-sensitive speckle tracking was applied to reconstruct TS maps coregistered to B-scans. Segments from injured atherosclerotic and uninjured nonatherosclerotic common femoral arteries (CFA) in cholesterol-fed New Zealand rabbits, and segments from control normal diet-fed rabbits (N =14) were scanned in vivo at different time points up to 12 weeks.

RESULTS

Lipid-rich atherosclerotic lesions exhibited distinct positive TS (+0.19 ± 0.08%) compared with that in nonatherosclerotic (-0.10 ± 0.13%) and control (-0.09 ± 0.09%) segments (p < 0.001). US-TSI enabled serial monitoring of lipids during atherosclerosis development. The coregistered set of morphological and compositional information of US-TSI showed good agreement with histology.

CONCLUSIONS

US-TSI successfully detected and longitudinally monitored lipid progression in atherosclerotic CFA. US-TSI of relatively superficial arteries may be a modality that could be integrated into a commercial US system for noninvasive lipid detection in AP.

Tissue classification using ultrasound-induced variations in acoustic backscattering features

Ultrasound (US) radio-frequency (RF) time series is an effective tissue classification method that enables accurate cancer diagnosis, but the mechanisms underlying this method are not completely understood. This paper presents a model to describe the variations in tissue temperature and sound speed that take place during the RF time series scanning procedures and relate these variations to US backscattering. The model was used to derive four novel characterization features. These features were used to classify three animal tissues, and they obtained accuracies as high as 88.01%. The performance of the proposed features was compared with RF time series features proposed in a previous study. The results indicated that the US-induced variations in tissue temperature and sound speed, which were used to derive the proposed features, were important contributors to the tissue typing capabilities of the RF time series. Simulations carried out to estimate the heating induced during the scanning procedure employed in this study showed temperature rises lower than 2 °C. The model and results presented in this paper can be used to improve the RF time series.

The effects of transducer geometry on artifacts common to diagnostic bone imaging with conventional medical ultrasound

The portability, low cost, and non-ionizing radiation associated with medical ultrasound suggest that it has potential as a superior alternative to X-ray for bone imaging. However, when conventional ultrasound imaging systems are used for bone imaging, clinical acceptance is frequently limited by artifacts derived from reflections occurring away from the main axis of the acoustic beam. In this paper, the physical source of off-axis artifacts and the effect of transducer geometry on these artifacts are investigated in simulation and experimental studies. In agreement with diffraction theory, the sampled linear-array geometry possessed increased off-axis energy compared with single-element piston geometry, and therefore, exhibited greater levels of artifact signal. Simulation and experimental results demonstrated that the linear-array geometry exhibited increased artifact signal when the center frequency increased, when energy off-axis to the main acoustic beam (i.e., grating lobes) was perpendicularly incident upon off-axis surfaces, and when off-axis surfaces were specular rather than diffusive. The simulation model used to simulate specular reflections was validated experimentally and a correlation coefficient of 0.97 between experimental and simulated peak reflection contrast was observed. In ex vivo experiments, the piston geometry yielded 4 and 6.2 dB average contrast improvement compared with the linear array when imaging the spinous process and interlaminar space of an animal spine, respectively. This work indicates that off-axis reflections are a major source of ultrasound image artifacts, particularly in environments comprising specular reflecting (i.e., bone or bone-like) objects. Transducer geometries with reduced sensitivity to off-axis surface reflections, such as a piston transducer geometry, yield significant reductions in image artifact.

First in vivo use of a capacitive micromachined ultrasound transducer array-based imaging and ablation catheter

OBJECTIVES

The primary objective was to test in vivo for the first time the general operation of a new multifunctional intracardiac echocardiography (ICE) catheter constructed with a microlinear capacitive micromachined ultrasound transducer (ML-CMUT) imaging array. Secondarily, we examined the compatibility of this catheter with electroanatomic mapping (EAM) guidance and also as a radiofrequency ablation (RFA) catheter. Preliminary thermal strain imaging (TSI)-derived temperature data were obtained from within the endocardium simultaneously during RFA to show the feasibility of direct ablation guidance procedures.

METHODS

The new 9F forward-looking ICE catheter was constructed with 3 complementary technologies: a CMUT imaging array with a custom electronic array buffer, catheter surface electrodes for EAM guidance, and a special ablation tip, that permits simultaneous TSI and RFA. In vivo imaging studies of 5 anesthetized porcine models with 5 CMUT catheters were performed.

RESULTS

The ML-CMUT ICE catheter provided high-resolution real-time wideband 2-dimensional (2D) images at greater than 8 MHz and is capable of both RFA and EAM guidance. Although the 24-element array aperture dimension is only 1.5 mm, the imaging depth of penetration is greater than 30 mm. The specially designed ultrasound-compatible metalized plastic tip allowed simultaneous imaging during ablation and direct acquisition of TSI data for tissue ablation temperatures. Postprocessing analysis showed a first-order correlation between TSI and temperature, permitting early development temperature-time relationships at specific myocardial ablation sites.

CONCLUSIONS

Multifunctional forward-looking ML-CMUT ICE catheters, with simultaneous intracardiac guidance, ultrasound imaging, and RFA, may offer a new means to improve interventional ablation procedures.

Thermal strain imaging: a review

Thermal strain imaging (TSI) or temporal strain imaging is an ultrasound application that exploits the temperature dependence of sound speed to create thermal (temporal) strain images. This article provides an overview of the field of TSI for biomedical applications that have appeared in the literature over the past several years. Basic theory in thermal strain is introduced. Two major energy sources appropriate for clinical applications are discussed. Promising biomedical applications are presented throughout the paper, including non-invasive thermometry and tissue characterization. We present some of the limitations and complications of the method. The paper concludes with a discussion of competing technologies.

3D ultrasound reconstruction algorithms from analog and digital data

Freehand 3D ultrasound is increasingly being introduced in the clinic for diagnostics and image-assisted interventions. Various algorithms exist for combining 2D images of regular ultrasound probes to 3D volumes, being either voxel-, pixel- or function-based. Previously, the most commonly used input to 3D ultrasound reconstruction has been digitized analog video. However, recent scanners that offer access to digital image frames exist, either as processed or unprocessed data. To our knowledge, no comparison has been performed to determine which data source gives the best reconstruction quality. In the present study we compared both reconstruction algorithms and data sources using novel comparison methods for detecting potential differences in image quality of the reconstructed volumes. The ultrasound scanner used in this study was the Sonix RP from Ultrasonix Medical Corp (Richmond, Canada), a scanner that allow third party access to unprocessed and processed digital data. The ultrasound probe used was the L14-5/38 linear probe. The assessment is based on a number of image criteria: detectability of wire targets, spatial resolution, detectability of small barely visible structures, subjective tissue image quality, and volume geometry. In addition we have also performed the more "traditional" comparison of reconstructed volumes by removing a percentage of the input data. By using these evaluation methods and data from the specific scanner, the results showed that the processed video performed better than the digital scan-line data, digital video being better than analog video. Furthermore, the results showed that the choice of video source was more important than the choice of tested reconstruction algorithms.

Noninvasive thermometry assisted by a dual-function ultrasound transducer for mild hyperthermia

Mild hyperthermia is increasingly important for the activation of temperature-sensitive drug delivery vehicles. Noninvasive ultrasound thermometry based on a 2-D speckle tracking algorithm was examined in this study. Here, a commercial ultrasound scanner, a customized co-linear array transducer, and a controlling PC system were used to generate mild hyperthermia. Because the co-linear array transducer is capable of both therapy and imaging at widely separated frequencies, RF image frames were acquired during therapeutic insonation and then exported for off-line analysis. For in vivo studies in a mouse model, before temperature estimation, motion correction was applied between a reference RF frame and subsequent RF frames. Both in vitro and in vivo experiments were examined; in the in vitro and in vivo studies, the average temperature error had a standard deviation of 0.7°C and 0.8°C, respectively. The application of motion correction improved the accuracy of temperature estimation, where the error range was 1.9 to 4.5°C without correction compared with 1.1 to 1.0°C following correction. This study demonstrates the feasibility of combining therapy and monitoring using a commercial system. In the future, real-time temperature estimation will be incorporated into this system.

A method for accurate in silico modeling of ultrasound transducer arrays

This paper presents a new approach to improve the in silico modeling of ultrasound transducer arrays. While current simulation tools accurately predict the theoretical element spatio-temporal pressure response, transducers do not always behave as theorized. In practice, using the probe's physical dimensions and published specifications in silico, often results in unsatisfactory agreement between simulation and experiment. We describe a general optimization procedure used to maximize the correlation between the observed and simulated spatio-temporal response of a pulsed single element in a commercial ultrasound probe. A linear systems approach is employed to model element angular sensitivity, lens effects, and diffraction phenomena. A numerical deconvolution method is described to characterize the intrinsic electro-mechanical impulse response of the element. Once the response of the element and optimal element characteristics are known, prediction of the pressure response for arbitrary apertures and excitation signals is performed through direct convolution using available tools. We achieve a correlation of 0.846 between the experimental emitted waveform and simulated waveform when using the probe's physical specifications in silico. A far superior correlation of 0.988 is achieved when using the optimized in silico model. Electronic noise appears to be the main effect preventing the realization of higher correlation coefficients. More accurate in silico modeling will improve the evaluation and design of ultrasound transducers as well as aid in the development of sophisticated beamforming strategies.

Molecular imaging as a tool for translating breast cancer science

The ability to measure biochemical and molecular processes underlies progress in breast cancer biology and treatment. These assays have traditionally been performed by analysis of cell culture or tissue samples. More recently, functional and molecular imaging has allowed the in vivo assay of biochemistry and molecular biology, which is highly complementary to tissue-based assays. This review briefly describes different imaging modalities used in molecular imaging and then reviews applications of molecular imaging to breast cancer, with a focus on translational work. It includes sections describing work in functional and physiological tumor imaging, imaging gene product expression, imaging the tumor microenvironment, reporter gene imaging, and cell labeling. Work in both animal models and human is discussed with an eye towards studies that have relevance to breast cancer treatment in patients.

Analysis of correlation coefficient filtering in elasticity imaging

Correlation-based speckle tracking methods are commonly used in elasticity imaging to estimate displacements. In the presence of local strain, a larger window size results in larger displacement error. To reduce tracking error, we proposed a short correlation window followed by a correlation coefficient filter. Although simulation and experimental results demonstrated the efficacy of the method, it was not clear why correlation coefficient filtering reduces tracking error since tracking error increases if normalization before filtering is not applied. In this paper, we analyzed tracking errors by estimating phase variances of the cross-correlation function and the correlation coefficient at the true time lag based on statistical properties of these functions' real and imaginary parts. The role of normalization is clarified by identifying the effect of the cross-correlation function's amplitude fluctuation on the function's imaginary part. Furthermore, we present analytic forms for predicting axial displacement error as a function of strain, system parameters (signal-to-noise ratio, center frequency, and signal and noise bandwidths), and tracking parameters (window and filter sizes) for cases with and without normalization before filtering. Simulation results correspond to theory well for both noise-free cases and general cases with an empirical correction term included for strains up to 4%.