MRI guided and monitored focused ultrasound thermal ablation methods: a review of progress

Numerical Investigation of Layered Homogeneous Skull Model for Simulations of Transcranial Focused Ultrasound

BACKGROUND AND OBJECTIVES

The influence of the intracranial pressure field must be discussed with the development of a single-element transducer for low-intensity transcranial focused ultrasound because the skull plays a significant role in blocking and dispersing ultrasound wave propagation. Ultrasound propagation is mainly affected by the structure and acoustic properties of the skull; thus, we aimed to investigate the impact of simplifying the acoustic properties of the skull on the simulation of the transcranial pressure field to present guidance for efficient skull modeling in full-wave simulations.

MATERIALS AND METHODS

We constructed a three-dimensional computational model for ultrasound transmission with the same structure but varying acoustic properties of the skull. The structural information and heterogeneous acoustic properties of the skull were acquired from computed tomography images, and we segmented the skull into three layers (3 L), including spongy and compact bones. We then assigned homogeneous acoustic properties to a single layer (1 L) or 3 L of the skull. In addition, we investigated the influence of different types of transducers and different ultrasound frequencies (1.1 MHz, 0.5 MHz, and 0.25 MHz) on the intracranial pressure field to provide a comparison of the heterogenous and homogeneous models.

RESULTS

We indicated the importance of numerical simulations in estimating the intracranial pressure field of the skull owing to beam distortions. When we simplified the skull model, both the 1 L and 3 L models showed contours of the acoustic focus comparable to those of the heterogeneous model. When we evaluated the peak pressure and volume of the acoustic focus, the 1 L model produced a better estimation of peak pressure with a difference <10%, and the 3 L model is suitable to obtain smaller errors in the volume of the acoustic focus.

CONCLUSIONS

In conclusion, we examined the possibility of simplification of skull models using 1 L and 3 L homogeneous properties in the numerical simulation for focused ultrasound. The results show that the layered homogeneous model can provide characteristics comparable to those of the acoustic focus in heterogeneous models.

Noninvasive ultrasonic induction of cerebrospinal fluid flow enhances intrathecal drug delivery

Intrathecal drug delivery is routinely used in the treatment and prophylaxis of varied central nervous system conditions, as doing so allows drugs to directly bypass the blood-brain barrier. However, the utility of this route of administration is limited by poor brain and spinal cord parenchymal drug uptake from the cerebrospinal fluid. We demonstrate that a simple noninvasive transcranial ultrasound protocol can significantly increase influx of cerebrospinal fluid into the perivascular spaces of the brain, to enhance the uptake of intrathecally administered drugs. Specifically, we administered small (~1 kDa) and large (~155 kDa) molecule agents into the cisterna magna of rats and then applied low, diagnostic-intensity focused ultrasound in a scanning protocol throughout the brain. Using real-time magnetic resonance imaging and ex vivo histologic analyses, we observed significantly increased uptake of small molecule agents into the brain parenchyma, and of both small and large molecule agents into the perivascular space from the cerebrospinal fluid. Notably, there was no evidence of brain parenchymal damage following this intervention. The low intensity and noninvasive approach of transcranial ultrasound in this protocol underscores the ready path to clinical translation of this technique. In this manner, this protocol can be used to directly bypass the blood-brain barrier for whole-brain delivery of a variety of agents. Additionally, this technique can potentially be used as a means to probe the causal role of the glymphatic system in the variety of disease and physiologic processes to which it has been correlated.

Numerical Evaluation of the Effects of Transducer Displacement on Transcranial Focused Ultrasound in the Rat Brain

Focused ultrasound is a promising therapeutic technique, as it involves the focusing of an ultrasonic beam with sufficient acoustic energy into a target brain region with high precision. Low-intensity ultrasound transmission by a single-element transducer is mostly established for neuromodulation applications and blood-brain barrier disruption for drug delivery. However, transducer positioning errors can occur without fine control over the sonication, which can affect repeatability and lead to reliability problems. The objective of this study was to determine whether the target brain region would be stable under small displacement (0.5 mm) of the transducer based on numerical simulations. Computed-tomography-derived three-dimensional models of a rat head were constructed to investigate the effects of transducer displacement in the caudate putamen (CP) and thalamus (TH). Using three different frequencies (1.1, 0.69, and 0.25 MHz), the transducer was displaced by 0.5 mm in each of the following six directions: superior, interior, anterior, posterior, left, and right. The maximum value of the intracranial pressure field was calculated, and the targeting errors were determined by the full-width-at-half-maximum (FWHM) overlap between the free water space (FWHMwater) and transcranial transmission (FWHMbase). When the transducer was positioned directly above the target region, a clear distinction between the target regions was observed, resulting in 88.3%, 81.5%, and 84.5% FWHMwater for the CP and 65.6%, 76.3%, and 64.4% FWHMwater for the TH at 1.1, 0.69, and 0.25 MHz, respectively. Small transducer displacements induced both enhancement and reduction of the peak pressure and targeting errors, compared with when the transducer was displaced in water. Small transducer displacement to the left resulted in the lowest stability, with 34.8% and 55.0% targeting accuracy (FWHMwater) at 1.1 and 0.69 MHz in the TH, respectively. In addition, the maximum pressure was reduced by up to 11% by the transducer displacement. This work provides the targeting errors induced by transducer displacements through a preclinical study and recommends that attention be paid to determining the initial sonication foci in the transverse plane in the cases of small animals.

Lesion location and lesion creation affect outcomes after focused ultrasound thalamotomy

MRI-guided focused ultrasound thalamotomy has been shown to be an effective treatment for medication refractory essential tremor. Here, we report a clinical-radiological analysis of 123 cases of MRI-guided focused ultrasound thalamotomy, and explore the relationships between treatment parameters, lesion characteristics and outcomes. All patients undergoing focused ultrasound thalamotomy by a single surgeon were included. The procedure was performed as previously described, and patients were followed for up to 1 year. MRI was performed 24 h post-treatment, and lesion locations and volumes were calculated. We retrospectively evaluated 118 essential tremor patients and five tremor-dominant Parkinson's disease patients who underwent thalamotomy. At 24 h post-procedure, tremor abated completely in the treated hand in 81 essential tremor patients. Imbalance, sensory disturbances and dysarthria were the most frequent acute adverse events. Patients with any adverse event had significantly larger lesions, while inferolateral lesion margins were associated with a higher incidence of motor-related adverse events. Twenty-three lesions were identified with irregular tails, often extending into the internal capsule; 22 of these patients experienced at least one adverse event. Treatment parameters and lesion characteristics changed with increasing surgeon experience. In later cases, treatments used higher maximum power (normalized to skull density ratio), accelerated more quickly to high power, and delivered energy over fewer sonications. Larger lesions were correlated with a rapid rise in both power delivery and temperature, while increased oedema was associated with rapid rise in temperature and the maximum power delivered. Total energy and total power did not significantly affect lesion size. A support vector regression was trained to predict lesion size and confirmed the most valuable predictors of increased lesion size as higher maximum power, rapid rise to high-power delivery, and rapid rise to high tissue temperatures. These findings may relate to a decrease in the energy efficiency of the treatment, potentially due to changes in acoustic properties of skull and tissue at higher powers and temperatures. We report the largest single surgeon series of focused ultrasound thalamotomy to date, demonstrating tremor relief and adverse events consistent with reported literature. Lesion location and volume impacted adverse events, and an irregular lesion tail was strongly associated with adverse events. High-power delivery early in the treatment course, rapid temperature rise, and maximum power were dominant predictors of lesion volume, while total power, total energy, maximum energy and maximum temperature did not improve prediction of lesion volume. These findings have critical implications for treatment planning in future patients.

Anatomical Phase Extraction (APE) Method: A Novel Method to Correct Detrimental Effects of Tissue-Inhomogeneity in Referenceless MR Thermometry-Preliminary Ex Vivo Investigation

Purpose

We present a novel background tissue phase removing method, called anatomical phase extraction (APE), and to investigate the accuracy of temperature estimation and capability of reducing background artifacts compared with the conventional referenceless methods.

Methods

Susceptibility variance was acquired by subtracting pretreatment baseline images taken at different locations (nine pretreatment baselines are acquired and called φ₁ to φ₉). The susceptibility phase data φ_S was obtained using the Wiener deconvolution algorithm. The background phase data φ_T was isolated by subtracting φ_S from the whole phase data. Finally, φ_T was subtracted from the whole phase data before applying the referenceless method. As a proof of concept, the proposed APE method was performed on ex vivo pork tenderloin and compared with other two referenceless temperature estimation approaches, including reweighted ℓ1 referenceless (RW- ℓ1) and ℓ2 referenceless methods. The proposed APE method was performed with four different baselines combination, namely, (φ₁, φ₅, φ₂, φ₄), (φ₃, φ₅, φ₂, φ₆), (φ₇, φ₅, φ₈, φ₄), and (φ₉, φ₅, φ₈, φ₆), and called APE experiment 1 to 4, respectively. The multibaseline method was used as a standard reference. The mean absolute error (MAE) and two-sample t-test analysis in temperature estimation of three regions of interest (ROI) between the multibaseline method and the other three methods, i.e., APE, RW- ℓ1, and ℓ2, were calculated and compared.

Results

Our results show that the mean temperature errors of the APE method-experiment 1, APE method-experiment 2, APE method-experiment 3, APE method-experiment 4, and RW- ℓ1 and ℓ2 referenceless method are 1.02°C, 1.04°C, 1.00°C, 1.00°C, 4.75°C, and 13.65°C, respectively. The MAEs of the RW- ℓ1 and ℓ2 referenceless methods were higher than that of APE method. The APE method showed no significant difference (p > 0.05), compared with the multibaseline method.

Conclusion

The present work demonstrates the use of the APE method on referenceless MR thermometry to improve the accuracy of temperature estimation during MRI guided high-intensity focused ultrasound for ablation treatment.

Emerging Therapeutic Strategies for Brain Tumors

Nearly thirty thousand incidences of primary and 300 thousand incidences of metastatic brain cancer are diagnosed in the USA each year. It has a high mortality rate and is often unresponsive to the standard of care, which includes surgical resection, radiation, and chemotherapy. These treatment strategies are also hindered by their invasiveness and toxic effects on healthy cells and tissues. Furthermore, the blood-brain/tumor barrier severely limits delivery of anti-cancer therapeutics administered intravenously to brain tumors, resulting in poor tumor response to the treatment. There is a critical need to develop new approaches to brain cancer therapy that can overcome these limitations. Focused ultrasound has emerged as a modality that addresses many of these limitations and has the potential to alter the treatment paradigm for brain cancer. Ultrasound transmitted through the skull can be focused on tumors and used for targeted ablation or opening the vascular barriers for drug delivery. This review provides insight on the current status of these unique ultrasound techniques, different strategies of using this technique for brain cancer, experience in preclinical models, and potential for clinical translation. We also debate the safety perspective of these techniques and discuss potential avenues for future work in noninvasive planning, monitoring, and evaluation of the ultrasonic neurointervention.

AAPM Task Group 241: A medical physicist's guide to MRI-guided focused ultrasound body systems

Magnetic resonance-guided focused ultrasound (MRgFUS) is a completely non-invasive technology that has been approved by FDA to treat several diseases. This report, prepared by the American Association of Physicist in Medicine (AAPM) Task Group 241, provides background on MRgFUS technology with a focus on clinical body MRgFUS systems. The report addresses the issues of interest to the medical physics community, specific to the body MRgFUS system configuration, and provides recommendations on how to successfully implement and maintain a clinical MRgFUS program. The following sections describe the key features of typical MRgFUS systems and clinical workflow and provide key points and best practices for the medical physicist. Commonly used terms, metrics and physics are defined and sources of uncertainty that affect MRgFUS procedures are described. Finally, safety and quality assurance procedures are explained, the recommended role of the medical physicist in MRgFUS procedures is described, and regulatory requirements for planning clinical trials are detailed. Although this report is limited in scope to clinical body MRgFUS systems that are approved or currently undergoing clinical trials in the United States, much of the material presented is also applicable to systems designed for other applications.

Focused ultrasound radiosensitizes human cancer cells by enhancement of DNA damage

Purpose

High-intensity focused ultrasound (HIFU/FUS) has expanded as a noninvasive quantifiable option for hyperthermia (HT). HT in a temperature range of 40–47 °C (thermal dose CEM43 ≥ 25) could work as a sensitizer to radiation therapy (RT). Here, we attempted to understand the tumor radiosensitization effect at the cellular level after a combination treatment of FUS+RT.

Methods

An in vitro FUS system was developed to induce HT at frequencies of 1.147 and 1.467 MHz. Human head and neck cancer (FaDU), glioblastoma (T98G), and prostate cancer (PC-3) cells were exposed to FUS in ultrasound-penetrable 96-well plates followed by single-dose X‑ray irradiation (10 Gy). Radiosensitizing effects of FUS were investigated by cell metabolic activity (WST‑1 assay), apoptosis (annexin V assay, sub-G1 assay), cell cycle phases (propidium iodide staining), and DNA double-strand breaks (γH2A.X assay).

Results

The FUS intensities of 213 (1.147 MHz) and 225 W/cm² (1.467 MHz) induced HT for 30 min at mean temperatures of 45.20 ± 2.29 °C (CEM43 = 436 ± 88) and 45.59 ± 1.65 °C (CEM43 = 447 ± 79), respectively. FUS improves the effect of RT significantly by reducing metabolic activity in T98G cells 48 h (RT: 96.47 ± 8.29%; FUS+RT: 79.38 ± 14.93%; p = 0.012) and in PC-3 cells 72 h (54.20 ± 10.85%; 41.01 ± 11.17%; p = 0.016) after therapy, but not in FaDu cells. Mechanistically, FUS+RT leads to increased apoptosis and enhancement of DNA double-strand breaks compared to RT alone in T98G and PC-3 cells.

Conclusion

Our in vitro findings demonstrate that FUS has good potential to sensitize glioblastoma and prostate cancer cells to RT by mainly enhancing DNA damage.

Supplementary Information

The online version of this article (10.1007/s00066-021-01774-5) contains supplementary material, which is available to authorized users.

Computationally Efficient Transcranial Ultrasonic Focusing: Taking Advantage of the High Correlation Length of the Human Skull

The phase correction necessary for transcranial ultrasound therapy requires numerical simulation to noninvasively assess the phase shift induced by the skull bone. Ideally, the numerical simulations need to be fast enough for clinical implementation in a brain therapy protocol and to provide accurate estimation of the phase shift to optimize the refocusing through the skull. In this article, we experimentally performed transcranial ultrasound focusing at 900 kHz on N = 5 human skulls. To reduce the computation time, we propose here to perform the numerical simulation at 450 kHz and use the corresponding phase shifts experimentally at 900 kHz. We demonstrate that a 450-kHz simulation restores 94.2% of the pressure when compared with a simulation performed at 900 kHz and 85.0% of the gold standard pressure obtained by an invasive time reversal procedure based on the signal recorded by a hydrophone placed at the target. From a 900- to 450-kHz simulation, the grid size is divided by 8, and the computation time is divided by 10.

In vivo ultrasound thermal ablation control using echo decorrelation imaging in rabbit liver and VX2 tumor

The utility of echo decorrelation imaging feedback for real-time control of in vivo ultrasound thermal ablation was assessed in rabbit liver with VX2 tumor. High-intensity focused ultrasound (HIFU) and unfocused (bulk) ablation were performed using 5 MHz linear image-ablate arrays. Treatments comprised up to nine lower-power sonications, followed by up to nine higher-power sonications, ceasing when the average cumulative echo decorrelation within a control region of interest exceeded a predefined threshold (- 2.3, log10-scaled echo decorrelation per millisecond, corresponding to 90% specificity for tumor ablation prediction in previous in vivo experiments). This threshold was exceeded in all cases for both HIFU (N = 12) and bulk (N = 8) ablation. Controlled HIFU trials achieved a significantly higher average ablation rate compared to comparable ablation trials without image-based control, reported previously. Both controlled HIFU and bulk ablation trials required significantly less treatment time than these previous uncontrolled trials. Prediction of local liver and VX2 tumor ablation using echo decorrelation was tested using receiver operator characteristic curve analysis, showing prediction capability statistically equivalent to uncontrolled trials. Compared to uncontrolled trials, controlled trials resulted in smaller thermal ablation regions and higher contrast between echo decorrelation in treated vs. untreated regions. These results indicate that control using echo decorrelation imaging may reduce treatment duration and increase treatment reliability for in vivo thermal ablation.

Acoustic radiation force imaging using a single-shot spiral readout

The purpose of this study is to develop and validate rapid magnetic resonance acoustic radiation force imaging (MR-ARFI) using a single shot spiral readout for focused ultrasound (FUS) guidance and for local tissue displacement measurements. A magnetic resonance guided FUS system was used to focus a 3 MHz ultrasound beam to a predetermined position. MR-ARFI was performed with a Bruker 7 T MRI using a modified single-shot spiral readout, with additional motion encoding gradients that convert local displacement into the phase image. Post processing was then used to analyze the resulting displacement and to evaluate the method's performance for the detection of tissue changes resulting from thermal ablation. The single-shot spiral readout acquires a single MR-ARFI image in one second, which is up to two orders of magnitude faster than conventional 2D spin-warp spin echo that acquires the k-space data line by line. The ARFI displacement in tissue mimicking phantoms was detected and localized with less than 5% geometric distortion. The ARFI displacement was also measured pre and post thermal ablation in an ex vivo chicken breast. For transmitted peak negative pressure of 8.6 MPa, the maximum displacement of the tissue that was ablated to 70 °C was 78% lower than the pre-ablated tissue. Since spiral readout is not prone to geometrical distortion, it is well-suited for FUS guidance, without generating undesired temperature elevation. Additionally, local displacement measurements of tissues can be performed rapidly during thermal ablation procedures and may help to assess the success of the treatment.

Establishing sheep as an experimental species to validate ultrasound-mediated blood-brain barrier opening for potential therapeutic interventions

Rationale: Treating diseases of the brain such as Alzheimer's disease (AD) is challenging as the blood-brain barrier (BBB) effectively restricts access of a large number of potentially useful drugs. A potential solution to this problem is presented by therapeutic ultrasound, a novel treatment modality that can achieve transient BBB opening in species including rodents, facilitated by biologically inert microbubbles that are routinely used in a clinical setting for contrast enhancement. However, in translating rodent studies to the human brain, the presence of a thick cancellous skull that both absorbs and distorts ultrasound presents a challenge. A larger animal model that is more similar to humans is therefore required in order to establish a suitable protocol and to test devices. Here we investigated whether sheep provide such a model.

Methods: In a stepwise manner, we used a total of 12 sheep to establish a sonication protocol using a spherically focused transducer. This was assisted by ex vivo simulations based on CT scans to establish suitable sonication parameters. BBB opening was assessed by Evans blue staining and a range of histological tests.

Results: Here we demonstrate noninvasive microbubble-mediated BBB opening through the intact sheep skull. Our non-recovery protocol allowed for BBB opening at the base of the brain, and in areas relevant for AD, including the cortex and hippocampus. Linear time-shift invariant analysis and finite element analysis simulations were used to optimize the position of the transducer and to predict the acoustic pressure and location of the focus.

Conclusion: Our study establishes sheep as a novel animal model for ultrasound-mediated BBB opening and highlights opportunities and challenges in using this model. Moreover, as sheep develop an AD-like pathology with aging, they represent a large animal model that could potentially complement the use of non-human primates.

A dual-mode hemispherical sparse array for 3D passive acoustic mapping and skull localization within a clinical MRI guided focused ultrasound device

Previous work has demonstrated that passive acoustic imaging may be used alongside MRI for monitoring of focused ultrasound therapy. However, past implementations have generally made use of either linear arrays originally designed for diagnostic imaging or custom narrowband arrays specific to in-house therapeutic transducer designs, neither of which is fully compatible with clinical MR-guided focused ultrasound (MRgFUS) devices. Here we have designed an array which is suitable for use within an FDA-approved MR-guided transcranial focused ultrasound device, within the bore of a 3 Tesla clinical MRI scanner. The array is constructed from 5  ×  0.4 mm piezoceramic disc elements arranged in pseudorandom fashion on a low-profile laser-cut acrylic frame designed to fit between the therapeutic elements of a 230 kHz InSightec ExAblate 4000 transducer. By exploiting thickness and radial resonance modes of the piezo discs the array is capable of both B-mode imaging at 5 MHz for skull localization, as well as passive reception at the second harmonic of the therapy array for detection of cavitation and 3D passive acoustic imaging. In active mode, the array was able to perform B-mode imaging of a human skull, showing the outer skull surface with good qualitative agreement with MR imaging. Extension to 3D showed the array was able to locate the skull within  ±2 mm/2° of reference points derived from MRI, which could potentially allow registration of a patient to the therapy system without the expense of real-time MRI. In passive mode, the array was able to resolve a point source in 3D within a  ±10 mm region about each axis from the focus, detect cavitation (SNR ~ 12 dB) at burst lengths from 10 cycles to continuous wave, and produce 3D acoustic maps in a flow phantom. Finally, the array was used to detect and map cavitation associated with microbubble activity in the brain in nonhuman primates.

3D-printed adaptive acoustic lens as a disruptive technology for transcranial ultrasound therapy using single-element transducers

The development of multi-element arrays for better control of the shape of ultrasonic beams has opened the way for focusing through highly aberrating media, such as the human skull. As a result, the use of brain therapy with transcranial-focused ultrasound has rapidly grown. Although effective, such technology is expensive. We propose a disruptive, low-cost approach that consists of focusing a 1 MHz ultrasound beam through a human skull with a single-element transducer coupled with a tailored silicone acoustic lens cast in a 3D-printed mold and designed using computed tomography-based numerical acoustic simulation. We demonstrate on N  =  3 human skulls that adding lens-based aberration correction to a single-element transducer increases the deposited energy on the target 10 fold.

Real-Time Spatiotemporal Control of High-Intensity Focused Ultrasound Thermal Ablation Using Echo Decorrelation Imaging in ex Vivo Bovine Liver

The ability to control high-intensity focused ultrasound (HIFU) thermal ablation using echo decorrelation imaging feedback was evaluated in ex vivo bovine liver. Sonications were automatically ceased when the minimum cumulative echo decorrelation within the region of interest exceeded an ablation control threshold, determined from preliminary experiments as -2.7 (log-scaled decorrelation per millisecond), corresponding to 90% specificity for local ablation prediction. Controlled HIFU thermal ablation experiments were compared with uncontrolled experiments employing two, five or nine sonication cycles. Means and standard errors of the lesion width, area and depth, as well as receiver operating characteristic curves testing ablation prediction performance, were computed for each group. Controlled trials exhibited significantly smaller average lesion area, width and treatment time than five-cycle or nine-cycle uncontrolled trials and also had significantly greater prediction capability than two-cycle uncontrolled trials. These results suggest echo decorrelation imaging is an effective approach to real-time HIFU ablation control.

Post Hoc Analysis of Passive Cavitation Imaging for Classification of Histotripsy-Induced Liquefaction in Vitro

Histotripsy utilizes focused ultrasound to generate bubble clouds for transcutaneous tissue liquefaction. Bubble activity maps are under development to provide image guidance and monitor treatment progress. The aim of this paper was to investigate the feasibility of using plane wave B-mode and passive cavitation images to be used as binary classifiers of histotripsy-induced liquefaction. Prostate tissue phantoms were exposed to histotripsy pulses over a range of pulse durations (5- ) and peak negative pressures (12-23 MPa). Acoustic emissions were recorded during the insonation and beamformed to form passive cavitation images. Plane wave B-mode images were acquired following the insonation to detect the hyperechoic bubble cloud. Phantom samples were sectioned and stained to delineate the liquefaction zone. Correlation between passive cavitation and plane wave B-mode images and the liquefaction zone was assessed using receiver operating characteristic (ROC) curve analysis. Liquefaction of the phantom was observed for all the insonation conditions. The area under the ROC (0.94 versus 0.82), accuracy (0.90 versus 0.83), and sensitivity (0.81 versus 0.49) was greater for passive cavitation images relative to B-mode images ( ) along the azimuth of the liquefaction zone. The specificity was greater than 0.9 for both imaging modalities. These results demonstrate a stronger correlation between histotripsy-induced liquefaction and passive cavitation imaging compared with the plane wave B-mode imaging, albeit with limited passive cavitation image range resolution.

Optical flow and image segmentation analysis for noninvasive precise mapping of microwave thermal ablation in X-ray CT scans - ex vivo study

PURPOSE

To develop image processing algorithms for noninvasive mapping of microwave thermal ablation using X-ray CT.

METHODS

Ten specimens of bovine liver were subjected to microwave ablation (20-80 W, 8 min) while scanned by X-ray CT at 5 s intervals. Specimens were cut and manually traced by two observers. Two algorithms were developed and implemented to map the ablation zone. The first algorithm utilises images segmentation of Hounsfield units changes (ISHU). The second algorithm utilises radial optical flow (ROF). Algorithm sensitivity to spatiotemporal under-sampling was assessed by decreasing the acquisition rate and reducing the number of acquired projections used for image reconstruction in order to evaluate the feasibility of implementing radiation reduction techniques.

RESULTS

The average radial discrepancy between the ISHU and ROF contours and the manual tracing were 1.04±0.74 and 1.16±0.79mm, respectively. When diluting the input data, the ISHU algorithm retained its accuracy, ranging from 1.04 to 1.79mm. By contrast, the ROF algorithm performance became inconsistent at low acquisition rates. Both algorithms were not sensitive to projections reduction, (ISHU: 1.24±0.83mm, ROF: 1.53±1.15mm, for reduction by eight fold). Ablations near large blood vessels affected the ROF algorithm performance (1.83±1.30mm; p < 0.01), whereas ISHU performance remained the same.

CONCLUSION

The two suggested noninvasive ablation mapping algorithms can provide highly accurate contouring of the ablation zone at low scan rates. The ISHU algorithm may be more suitable for clinical practice as it appears more robust when radiation dose reduction strategies are employed and when the ablation zone is near large blood vessels.

Ultrasound-mediated drug delivery to the brain: principles, progress and prospects

The blood-brain barrier (BBB) limits drug delivery to the central nervous system. When combined with microbubbles, ultrasound can transiently permeate blood vessels in the brain. This approach, which can be referred to as sonoporation or sonopermeabilization, holds significant promise for shuttling large therapeutic molecules, such as antibodies, growth factors and nanomedicine formulations, across the BBB. We here describe the basic principles of BBB permeation using ultrasound and microbubbles, and we summarize several (pre-) clinical studies showing the potential of BBB opening for improving the treatment of cancer and neurodegenerative disorders.

In-bore MRI interventions: current status and future applications

PURPOSE OF REVIEW

This review discusses the feasibility, recent advances and current status of in-bore MRI-guided interventional techniques for diagnosis and treatment of focal prostate cancer (PCa) and also explores the future applications, highlighting the emerging strategies for the treatment of PCa.

RECENT FINDINGS

Multiparametric MRI has opened up opportunities for diagnosis and targeted therapeutics to the site of disease within the organ wherein minimizing the incidence of treatment-related toxicity of whole gland therapy. MRI-guided targeted biopsy has a higher detection rate for significant cancer and lower rate of detection of insignificant cancer. In comparison to ultrasound-guided focal therapy, in-bore treatment provides the advantage of real time thermal monitoring during treatment and assessment of treatment coverage by an enhanced scan immediately post-treatment. Preliminary results of ongoing phase I and II in-bore focal PCa treatment trials via transperineal, transrectal and transurethral routes, using different energy modalities for the ablation, have shown promising results.

SUMMARY

Advances in multiparametric-MRI has opened up opportunities for in-bore targeted focal treatment of PCa in the correctly selected patient.

Real-Time MRI-Guided Focused Ultrasound for Focal Therapy of Locally Confined Low-Risk Prostate Cancer: Feasibility and Preliminary Outcomes

OBJECTIVE

Focal therapy is an emerging approach to the treatment of localized prostate cancer. The purpose of this study was to report the 6-month follow-up oncologic and functional data of the initial phase 1 trial of patients treated with focal transrectal MRI-guided focused ultrasound in North America.

SUBJECTS AND METHODS

Four patients with a prostate-specific antigen (PSA) level of 10 ng/mL or less, tumor classification cT2a or less, and a Gleason score of 6 (3 + 3) were prospectively enrolled in the study and underwent multiparametric MRI and transrectal ultrasound-guided prostate systematic biopsy. Under MRI guidance and real-time monitoring with MR thermography, focused high-frequency ultrasound energy was delivered to ablate the target tissue. The incidence and severity of treatment-related adverse events were recorded along with responses to serial quality-of-life questionnaires for 6 months after treatment. Oncologic outcomes were evaluated with multiparametric MRI and repeat transrectal ultrasound-guided biopsy 6 months after treatment.

RESULTS

Four patients with a total of six target lesions were treated and had complications graded Clavien-Dindo I or less. Quality-of-life parameters were similar between baseline and 6-months. All four patients had normal MRI findings in the treated regions (100%), biopsy showed that three patients (75%) were clear of disease in the treated regions, representing complete ablation of five target lesions (83%). All patients had at least one Gleason 6-positive core outside of the treated zone.

CONCLUSION

MRI-guided focused ultrasound is a feasible method of noninvasively ablating low-risk prostate cancers with low morbidity. Further investigation and follow-up are warranted in a larger patient series with appropriate statistical analysis of oncologic and functional outcome measures.

Facilitation of Drug Transport across the Blood-Brain Barrier with Ultrasound and Microbubbles

Medical treatment options for central nervous system (CNS) diseases are limited due to the inability of most therapeutic agents to penetrate the blood–brain barrier (BBB). Although a variety of approaches have been investigated to open the BBB for facilitation of drug delivery, none has achieved clinical applicability. Mounting evidence suggests that ultrasound in combination with microbubbles might be useful for delivery of drugs to the brain through transient opening of the BBB. This technique offers a unique non-invasive avenue to deliver a wide range of drugs to the brain and promises to provide treatments for CNS disorders with the advantage of being able to target specific brain regions without unnecessary drug exposure. If this method could be applied for a range of different drugs, new CNS therapeutic strategies could emerge at an accelerated pace that is not currently possible in the field of drug discovery and development. This article reviews both the merits and potential risks of this new approach. It assesses methods used to verify disruption of the BBB with MRI and examines the results of studies aimed at elucidating the mechanisms of opening the BBB with ultrasound and microbubbles. Possible interactions of this novel delivery method with brain disease, as well as safety aspects of BBB disruption with ultrasound and microbubbles are addressed. Initial translational research for treatment of brain tumors and Alzheimer’s disease is presented.

MRI-guided biopsies and minimally invasive therapy for prostate cancer

Recent advances in multiparametric magnetic resonance imaging (mp-MRI) have led to a paradigm shift in the diagnosis and management of prostate cancer (PCa). Its sensitivity in detecting clinically significant cancer and the ability to localize the tumor within the prostate gland has opened up discussion on targeted diagnosis and therapy in PCa. Use of mp-MRI in conjunction with prostate-specific antigen followed by targeted biopsy allows for a better diagnostic pathway than transrectal ultrasound (TRUS) biopsy and improves the diagnosis of PCa. Improved detection of PCa by mp-MRI has also opened up opportunities for focal therapy within the organ while reducing the incidence of side-effects associated with the radical treatment methods for PCa. This review discusses the evidence and techniques for in-bore MRI-guided prostate biopsy and provides an update on the status of MRI-guided targeted focal therapy in PCa.

Real-time microwave imaging of differential temperature for thermal therapy monitoring

A microwave imaging system for real-time 3-D imaging of differential temperature has been developed for the monitoring and feedback of thermal therapy systems. Design parameters are constrained by features of a prototype-focused microwave thermal therapy system for the breast, operating at 915 MHz. Real-time imaging is accomplished with a precomputed linear inverse scattering solution combined with continuous vector network analyzer (VNA) measurements of a 36-antenna, HFSS-modeled, cylindrical cavity. Volumetric images of differential change of dielectric constant due to temperature are formed with a refresh rate as fast as 1 frame/s and 1 (°)C resolution. Procedures for data segmentation and postprocessed S-parameter error-correction are developed. Antenna pair VNA calibration is accelerated by using the cavity as the unknown thru standard. The device is tested on water targets and a simple breast phantom. Differentially heated targets are successfully imaged in cluttered environments. The rate of change of scattering contrast magnitude correlates 1:1 with target temperature.

Non-invasive temperature monitoring and hyperthermic injury onset detection using X-ray CT during HIFU thermal treatment in ex vivo fatty tissue

PURPOSE

This paper examines X-ray CT, to serve as an image-guiding thermal monitoring modality for high intensity focused ultrasound (HIFU) treatment of fatty tissues.

MATERIALS AND METHODS

Six ex vivo porcine fat tissue specimens were scanned by X-ray CT simultaneously with the application of HIFU. Images were acquired during both heating and post-ablation stages. The temperature at the focal zone was measured simultaneously using a thermocouple. The mean values of the Hounsfield units (HU) at the focal zone were registered and plotted as a function of temperature.

RESULTS

In all specimens studied, the HU versus temperature curves measured during the heating stage depicted a characteristic non-linear parabolic trajectory (R(2) > 0.87). The HU-temperature trajectory initially decreased to a minimum value at about 44.5 °C and then increased substantially as the heating progressed. The occurrence of this nadir point during the heating stage was clearly detectable. During post-ablation cooling, on the other hand, the HU increased monotonically with the decreasing temperature and depicted a clearly linear trajectory (R(2) ≥ 0.9).

CONCLUSIONS

Our results demonstrate that the HU-temperature curve during HIFU treatment has a characteristic parabolic trajectory for fat tissue that might potentially be utilised for thermal monitoring during HIFU ablation treatments. The clear detection of 44.5 °C, presumably marking the onset of hyperthermic injury, can be detected non-invasively as an occurrence of a minimum on the HU-time curve without any need to relate the HU directly to temperature. Such features may be helpful in monitoring and optimising HIFU thermal treatment for clinically applicable indications such as in the breast by providing a non-invasive monitoring of tissue damage.

Ultrasound-mediated blood-brain barrier disruption for targeted drug delivery in the central nervous system

The physiology of the vasculature in the central nervous system (CNS), which includes the blood-brain barrier (BBB) and other factors, complicates the delivery of most drugs to the brain. Different methods have been used to bypass the BBB, but they have limitations such as being invasive, non-targeted or requiring the formulation of new drugs. Focused ultrasound (FUS), when combined with circulating microbubbles, is a noninvasive method to locally and transiently disrupt the BBB at discrete targets. This review provides insight on the current status of this unique drug delivery technique, experience in preclinical models, and potential for clinical translation. If translated to humans, this method would offer a flexible means to target therapeutics to desired points or volumes in the brain, and enable the whole arsenal of drugs in the CNS that are currently prevented by the BBB.

In vivo thermal ablation monitoring using ultrasound echo decorrelation imaging

Previous work indicated that ultrasound echo decorrelation imaging can track and quantify changes in echo signals to predict thermal damage during in vitro radiofrequency ablation (RFA). In the in vivo studies reported here, the feasibility of using echo decorrelation imaging as a treatment monitoring tool was assessed. RFA was performed on normal swine liver (N = 5), and ultrasound ablation using image-ablate arrays was performed on rabbit liver implanted with VX2 tumors (N = 2). Echo decorrelation and integrated backscatter were computed from Hilbert transformed pulse-echo data acquired during RFA and ultrasound ablation treatments. Receiver operating characteristic (ROC) curves were employed to assess the ability of echo decorrelation imaging and integrated backscatter to predict ablation. Area under the ROC curves (AUROC) was determined for RFA and ultrasound ablation using echo decorrelation imaging. Ablation was predicted more accurately using echo decorrelation imaging (AUROC = 0.832 and 0.776 for RFA and ultrasound ablation, respectively) than using integrated backscatter (AUROC = 0.734 and 0.494).

Leveraging the power of ultrasound for therapeutic design and optimization

Contrast agent-enhanced ultrasound can facilitate personalized therapeutic strategies by providing the technology to measure local blood flow rate, to selectively image receptors on the vascular endothelium, and to enhance localized drug delivery. Ultrasound contrast agents are micron-diameter encapsulated bubbles that circulate within the vascular compartment and can be selectively imaged with ultrasound. Microbubble transport-based estimates of local blood flow can quantify changes resulting from anti-angiogenic therapies and facilitate differentiation of angiogenic mechanisms. Microbubbles that are conjugated with targeting ligands attach to endothelial surface receptors that are upregulated in disease, providing high signal-to-noise ratio images of pathological vasculature. In addition to imaging applications, microbubbles can be used to enhance localized gene and drug delivery, either by changing membrane and vascular permeability or by carrying and locally releasing cargo. Our goal in this review is to provide an overview of the use of contrast-enhanced ultrasound methodologies in the design and evaluation of therapeutic strategies with emphases on quantitative blood flow mapping, molecular imaging, and enhanced drug delivery.

See, reach, treat: ultrasound-triggered image-guided drug delivery

The integration of therapeutic interventions with diagnostic imaging has been recognized as one of the next technological developments that will have a major impact on medical treatments. Therapeutic applications using ultrasound, for example thermal ablation, hyperthermia or ultrasound-induced drug delivery, are examples for image-guided interventions that are currently being investigated. While thermal ablation using magnetic resonance-guided high-intensity focused ultrasound is entering the clinic, ultrasound-mediated drug delivery is still in a research phase, but holds promise to enable new applications in localized treatments. The use of ultrasound for the delivery of drugs has been demonstrated, particularly in the field of cardiology and oncology for a variety of therapeutics ranging from small-molecule drugs to biologics and nucleic acids exploiting temperature- or pressure-mediated delivery schemes.

Real-time passive acoustic monitoring of HIFU-induced tissue damage

Thermal ablation by high-intensity focused ultrasound (HIFU) shows great promise as a noninvasive cancer therapy. This work proposes a novel method of real-time HIFU treatment monitoring that uses the passively monitored acoustic signal emanating from the focus during HIFU exposure. We performed 212 exposures in seven freshly excised ox livers using 1.067-MHz HIFU at a 95% duty cycle for a range of insonation durations and acoustic intensities. Acoustic emissions were recorded using a 15-MHz passive detector aligned confocally and coaxially with the HIFU transducer. Lesion presence and size were ascertained by slicing the tissue in the transverse and axial focal planes post exposure. Our results demonstrate that successful formation of HIFU lesions in ex vivo ox liver is highly correlated with the presence of pronounced dips in the magnitude of the received signal at integer harmonics of the insonation frequency. A detector based on this observation predicted lesioning with >80% accuracy in regimes that were very likely to create lesions (≥60 J of energy) and had an error rate of <6% for exposures that were too short to cause lesioning (≤1 s long). The overall sensitivity and specificity of the detector were 75.6% and 74.2%, respectively. The proposed detector could therefore provide a low-cost means of effectively monitoring clinical HIFU treatments passively and in real time.

Conformal microwave array (CMA) applicators for hyperthermia of diffuse chest wall recurrence

PURPOSE

This article summarises the evolution of microwave array applicators for heating large area chest wall disease as an adjuvant to external beam radiation, systemic chemotherapy, and potentially simultaneous brachytherapy.

METHODS

Current devices used for thermotherapy of chest wall recurrence are reviewed. The largest conformal array applicator to date is evaluated in four studies: (1) ability to conform to the torso is demonstrated with a CT scan of a torso phantom and MR scan of the conformal water bolus component on a mastectomy patient; (2) specific absorption rate (SAR) and temperature distributions are calculated with electromagnetic and thermal simulation software for a mastectomy patient; (3) SAR patterns are measured with a scanning SAR probe in liquid muscle phantom for a buried coplanar waveguide CMA; and (4) heating patterns and patient tolerance of CMA applicators are characterised in a clinical pilot study with 13 patients.

RESULTS

CT and MR scans demonstrate excellent conformity of CMA applicators to contoured anatomy. Simulations demonstrate effective control of heating over contoured anatomy. Measurements confirm effective coverage of large treatment areas with no gaps. In 42 hyperthermia treatments, CMA applicators provided well-tolerated effective heating of up to 500 cm(2) regions, achieving target temperatures of T(min) = 41.4 ± 0.7°C, T(90) = 42.1 ± 0.6°C, T(ave) = 42.8 ± 0.6°C, and T(max) = 44.3 ± 0.8°C as measured in an average of 90 points per treatment.

CONCLUSION

The CMA applicator is an effective thermal therapy device for heating large-area superficial disease such as diffuse chest wall recurrence. It is able to cover over three times the treatment area of conventional hyperthermia devices while conforming to typical body contours.

A unified approach to combine temperature estimation and elastography for thermal lesion determination in focused ultrasound thermal therapy

Sonogram-based temperature estimation and elastography have both shown promise as methods of monitoring focused ultrasound (FUS) treatments to induce thermal ablation in tissue. However, each method has important limitations. Temperature estimates based on echo delays become invalid when the relationship between sound speed and temperature is nonlinear, and are further complicated by thermal expansion and other changes in tissue. Elastography can track thermal lesion formation over a wider range of elasticity, but with low specificity and high noise. Furthermore, this method is poor at small lesion detection. This study proposes integrating the two estimates to improve the quality of monitoring FUS-induced thermal lesions. Our unified computational kernel is tested on three types of phantoms. Experiments with type I and type II phantoms were conducted to calibrate the thermal mapping and elastography methods, respectively. The optimal settings were then used in experiments with the type III phantom, which contains ex vivo swine liver tissue. Three different spatial-peak temporal-average intensities (I(spta); 35, 133 and 240 W cm(-2)) were delivered with a sonication time of 60 s. The new procedure can closely monitor heating while identifying the dimensions of the thermal lesion, and is significantly better at the latter task than either approach alone. This work may help improve the current clinical practice, which employs sonograms to guide the FUS-induced thermal ablation procedure.

Model predictive filtering for improved temporal resolution in MRI temperature imaging

A novel method for reconstructing MRI temperature maps from undersampled data is presented. The method, model predictive filtering, combines temperature predictions from a preidentified thermal model with undersampled k-space data to create temperature maps in near real time. The model predictive filtering algorithm was implemented in three ways: using retrospectively undersampled k-space data from a fully sampled two-dimensional gradient echo (GRE) sequence (reduction factors R = 2.7 to R = 7.1), using actually undersampled data from a two-dimensional GRE sequence (R = 4.8), and using actually undersampled data from a three-dimensional GRE sequence (R = 12.1). Thirty-nine high-intensity focused ultrasound heating experiments were performed under MRI monitoring to test the model predictive filtering technique against the current gold standard for MR temperature mapping, the proton resonance frequency shift method. For both of the two-dimensional implementations, the average error over the five hottest voxels from the hottest time frame remained between +/-0.8 degrees C and the temperature root mean square error over a 24 x 7 x 3 x 25-voxel region of interest remained below 0.35 degrees C. The largest errors for the three-dimensional implementation were slightly worse: -1.4 degrees C for the mean error of the five hottest voxels and 0.61 degrees C for the temperature root mean square error.

Non‐invasive thermometry for monitoring hepatic radiofrequency ablation

This paper reviews a selection of methods for non-invasive thermometry with special attention to limitations of possible relevance for hepatic radiofrequency ablation.

High intensity focused ultrasound in the treatment of abdominal and gynaecological diseases

In recent years high intensity focused ultrasound (HIFU) has received increasing interest as a non-invasive modality for the treatment of tumours of solid organs. Surgeons continue their quest to find the optimal technique whereby a diseased organ can be treated with a minimum of damage to the patient, while providing a comprehensive treatment to produce either cure or resolution of symptoms. Two of the areas in which HIFU is beginning to establish itself as a real therapeutic alternative, are in the treatment of abdominal and gynaecological disease. In this paper, we will review the literature available regarding the use of HIFU in the treatment of various organs: liver, kidney, pancreas, bladder, uterus and vulva.

MRI-based thermal dosimetry and diffusion-weighted imaging of MRI-guided focused ultrasound thermal ablation of uterine fibroids

PURPOSE

To investigate tissue changes observed in diffusion-weighted imaging (DWI) and its relation to contrast imaging, thermal dosimetry, and changes in the apparent diffusion coefficient (ADC) after MRI-guided focused ultrasound surgery (MRgFUS) of uterine fibroids.

MATERIALS AND METHODS

Imaging data were analyzed from 45 fibroids in 42 women treated with MRgFUS. The areas of the hyperintense regions in DWI and of nonperfused regions in T1-weighted contrast enhanced imaging (both acquired immediately after treatment) were compared with each other and to thermal dosimetry based estimates. Changes in ADC were also calculated.

RESULTS

Hyperintense regions were observed in 35/45 fibroids in DWI. When present, the areas of these regions were comparable on average to the thermal dose estimates and to the nonperfused regions, except for in several large treatments in which the nonperfused region extended beyond the treated area. ADC increased in 19 fibroids and decreased in the others.

CONCLUSION

DWI changes, which includes changes in both in T2 and ADC, may be useful in many cases to delineate the treated region resulting from MRgFUS. However, clear DWI changes were not always observed, and in some large treatments, the extent of the nonperfused region was under estimated. ADC changes immediately after MRgFUS were unpredictable.

Short-duration-focused ultrasound stimulation of Hsp70 expression in vivo

The development of transgenic reporter mice and advances in in vivo optical imaging have created unique opportunities to assess and analyze biological responses to thermal therapy directly in living tissues. Reporter mice incorporating the regulatory regions from the genes encoding the 70 kDa heat-shock proteins (Hsp70) and firefly luciferase (luc) as reporter genes can be used to non-invasively reveal gene activation in living tissues in response to thermal stress. High-intensity-focused ultrasound (HIFU) can deliver measured doses of acoustic energy to highly localized regions of tissue at intensities that are sufficient to stimulate Hsp70 expression. We report activation of Hsp70-luc expression using 1 s duration HIFU heating to stimulate gene expression in the skin of the transgenic reporter mouse. Hsp70 expression was tracked for 96 h following the application of 1.5 MHz continuous-wave ultrasound with spatial peak intensities ranging from 53 W cm(-2) up to 352 W cm(-2). The results indicated that peak Hsp70 expression is observed 6-48 h post-heating, with significant activity remaining at 96 h. Exposure durations were simulated using a finite-element model, and the predicted temperatures were found to be consistent with the observed Hsp70 expression patterns. Histological evaluation revealed that the thermal damage starts at the stratum corneum and extends deeper with increasing intensity. These results indicated that short-duration HIFU may be useful for inducing heat-shock expression, and that the period between treatments needs to be greater than 96 h due to the protective properties of Hsp70.

Driving delivery vehicles with ultrasound

Therapeutic applications of ultrasound have been considered for over 40 years, with the mild hyperthermia and associated increases in perfusion produced by ultrasound harnessed in many of the earliest treatments. More recently, new mechanisms for ultrasound-based or ultrasound-enhanced therapies have been described, and there is now great momentum and enthusiasm for the clinical translation of these techniques. This dedicated issue of Advanced Drug Delivery Reviews, entitled "Ultrasound for Drug and Gene Delivery," addresses the mechanisms by which ultrasound can enhance local drug and gene delivery and the applications that have been demonstrated at this time. In this commentary, the identified mechanisms, delivery vehicles, applications and current bottlenecks for translation of these techniques are summarized.

Fast temperature optimization of multi-source hyperthermia applicators with reduced-order modeling of 'virtual sources'

The goal of this work is to build the foundation for facilitating real-time magnetic resonance image guided patient treatment for heating systems with a large number of physical sources (e.g. antennas). Achieving this goal requires knowledge of how the temperature distribution will be affected by changing each source individually, which requires time expenditure on the order of the square of the number of sources. To reduce computation time, we propose a model reduction approach that combines a smaller number of predefined source configurations (fewer than the number of actual sources) that are most likely to heat tumor. The source configurations consist of magnitude and phase source excitation values for each actual source and may be computed from a CT scan based plan or a simplified generic model of the corresponding patient anatomy. Each pre-calculated source configuration is considered a 'virtual source'. We assume that the actual best source settings can be represented effectively as weighted combinations of the virtual sources. In the context of optimization, each source configuration is treated equivalently to one physical source. This model reduction approach is tested on a patient upper-leg tumor model (with and without temperature-dependent perfusion), heated using a 140 MHz ten-antenna cylindrical mini-annular phased array. Numerical simulations demonstrate that using only a few pre-defined source configurations can achieve temperature distributions that are comparable to those from full optimizations using all physical sources. The method yields close to optimal temperature distributions when using source configurations determined from a simplified model of the tumor, even when tumor position is erroneously assumed to be approximately 2.0 cm away from the actual position as often happens in practical clinical application of pre-treatment planning. The method also appears to be robust under conditions of changing, nonlinear, temperature-dependent perfusion. The proposed approach of using virtual sources reduces the number of variables that must be optimized to achieve a tumor-focused temperature distribution, thereby reducing the calculation time required in real-time control applications to about 1/3 to 1/4 of that required for full optimization.

Gaussian representation of high-intensity focused ultrasound beams

A method for fast numerical simulation of high-intensity focused ultrasound beams is derived. The method is based on the frequency-domain representation of the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation, and assumes for each harmonic a Gaussian transverse pressure distribution at all distances from the transducer face. The beamwidths of the harmonics are constrained to vary inversely with the square root of the harmonic number, and as such this method may be viewed as an extension of a quasilinear approximation. The technique is capable of determining pressure or intensity fields of moderately nonlinear high-intensity focused ultrasound beams in water or biological tissue, usually requiring less than a minute of computer time on a modern workstation. Moreover, this method is particularly well suited to high-gain simulations since, unlike traditional finite-difference methods, it is not subject to resolution limitations in the transverse direction. Results are shown to be in reasonable agreement with numerical solutions of the full KZK equation in both tissue and water for moderately nonlinear beams.