Ultrasonic Therapy Research Articles

Realization of Airy pattern acoustic bullets by depth of focus synthesis.

Energy concentration is a key parameter for the characterization of light or acoustic bullets. To the best of our knowledge, the highest encircled-energy ratio in the mainlobe reported is merely 25% for the solitary bullet waves and 30% for the nonsolitary bullet waves. This letter reports experimentally and theoretically on a new family of acoustic bullets, namely, Airy pattern ultrasonic bullets, realized by the depth of focus synthesis. The Airy pattern ultrasonic bullets, regarded as a beam with an ultra-long depth of focus, exhibit Airy pattern energy distribution and contain 83.8% of total energy in their mainlobe. In the depth of focus synthesis proposed, an ultra-long depth of focus could be longitudinally synthesized by successive acoustic foci by using an acoustic reflection cavity (ARC). In contrast to the well-known aperture synthesis which utilizes a platform motion to extend the aperture, the depth of focus synthesis employs multiple reflections in ARC to extend its physical dimensions. The Airy pattern ultrasonic bullets presented would be potentially useful in long-range ultrasound imaging, ultrasonic therapy, and acoustic wireless energy transmission.

Effect of Hamstring Active Release Technique in Cervicogenic Headache

Objectives: To investigate the effect of Hamstring active release technique on pain intensity level, headache intensity, pressure pain threshold, cervical flexion rotation range of motion and Hamstring flexibility in patients with cervicogenic headache. Methods: Fifty-six patients with cervicogenic headache participated in the study, aged 25-45 years. Patients were randomized into two groups of 28 individuals each. Group A was given conventional physiotherapy only in the form of ultrasonic therapy, stretching as well as strengthening exercises. Group B was given conventional physiotherapy and active release technique for Hamstring. Each group was treated three times a week for one month. Pain intensity level, headache intensity, pressure pain threshold of Sub-occipital and Hamstring muscles trigger points, cervical flexion rotation range of motion and Hamstring flexibility were measured by visual analogue scale, the Arabic version of 6-item headache impact test, digital pressure algometry, CROM device and straight leg raising test respectively, pre and post-treatment. Results: Within group comparison showed significant difference in pain intensity level, headache intensity, pressure pain threshold of Sub-occipital and Hamstring muscles, cervical flexion rotation range of motion and Hamstring flexibility. Between groups comparison showed no significant difference in all variables, pre-study. Where a significant difference in all variables, post study, in favor of group B except right and left cervical flexion rotation range of motion. Conclusion: Hamstring active release technique improves pain intensity level, headache intensity, pressure pain threshold and Hamstring flexibility, in patients with cervicogenic headache.

Study on high-order-mode oscillation and migration characteristics of single bubble in the ultrasonic field

Abstract Investigating the motion characteristics of cavitation bubbles in an ultrasonic field is important for industries such as ultrasonic cleaning and ultrasonic therapy. This paper investigated the oscillation and migration characteristics of a single cavitation bubble in a 20 kHz ultrasonic field. Peristaltic pump syringe injection tests were performed at different locations in a 20 kHz ultrasonic field, and a high-speed camera was used to observe the bubble motion characteristics. The bubble was found to resonate with the ultrasonic waves at a higher order mode, changing from spherical shape to honeycomb and golf ball shapes. Also, we found that the ultrasonic waves may prevent the buoyancy of the bubble. Through revealing the motion characteristics of the bubble in the ultrasonic field, we may control the bubble motion by ultrasonic, and strengthen the bubble cavitation effect.

Effects of extracorporeal shockwave therapy versus ultrasonic therapy and deep friction massage in the management of lateral epicondylitis: a randomized clinical trial

The study's goal was to compare and evaluate the benefits of deep friction massage and ultrasonic therapy (US) vs extracorporeal shockwave therapy (ESWT) for people with lateral epicondylitis. This double-blind, parallel-arm randomized clinical trial was conducted after ethical approval on a sample of 80 subjects with lateral epicondylitis. Participants were enrolled based on predefined eligibility criteria. They were randomly allocated to groups A and B. Group A received ESWT, while Group B received the US combined with deep friction massage. Data was collected using the Numeric Pain Rating Score (NPRS) and Patient-rated tennis elbow evaluation questionnaire (PRTEE) at baseline, at 3rd, and at 7th week of treatment. On the basis of the normality of the data, a non-parametric test was applied to evaluate between-group and within-group differences. P value ≤ 0.05 was considered significant. There was a significant difference between groups (p < 0.001). Comparisons of PRTEE scores at 3rd week and 7th week of intervention were found significant for both groups (p < 0.001). While considering between-group comparisons based on percentile scores of PRTEE at baseline, 3rd and 7th week of intervention, in group A Median (IQR) at the baseline was 24.00 (5.00), at 3rd week, 10.00 (5.00) and 7th week was 1.50 (2.50) and in group B Median (IQR) at the baseline was 25.00 (4.00), at 3rd week 19.50 (4.50) and at 7th week was 11.50 (2.50). The results were significant in both groups (p = 0.000), but between-group analysis revealed that ESWT is more effective in patients with lateral epicondylitis.

Combined Effects of Manual Therapy and Functional Electrical Stimulation on Torticollis among Adults

Background: Acquired muscular torticollis is a common musculoskeletal disorder that occurs after birth due to various etiologies. This condition is characterized by the abnormal tilt and twist of the neck muscles, causing significant discomfort and functional impairment. Objective: This study aimed to investigate the combined effects of manual therapy and functional electrical stimulation (FES) on adults with acquired torticollis. Methods: A 45-year-old male patient with acquired torticollis was evaluated using a detailed general movement assessment and an arthrodial goniometer to measure cervical ranges of motion. The patient exhibited tightness of the right sternocleidomastoid (SCM) muscle with limited ranges of motion. The treatment protocol included both manual therapy and electrotherapy for three months, with sessions conducted four days per week, each lasting 40 minutes. Manual therapy consisted of soft tissue mobilization, stretching of the right SCM, and isometric exercises for the left SCM. Electrotherapy included transcutaneous electrical nerve stimulation (TENS), DL2 stimulator, infrared radiation therapy (IRR), and ultrasonic therapy. The patient’s cervical ranges of motion were measured before and after the intervention to assess the effectiveness of the combined therapy. Results: Significant improvements were observed in the patient’s cervical ranges of motion after the 12-week treatment protocol. The mean lateral flexion of the affected right SCM improved from 35 degrees to 20 degrees, and the mean rotation improved from 43 degrees to 28 degrees. These changes indicate a noticeable improvement in both the ranges of motion and posture of the patient. Conclusion: The combined effects of manual therapy and functional electrical stimulation significantly improved the cervical ranges of motion and overall posture in an adult patient with acquired torticollis. These findings suggest that integrating multiple therapeutic modalities can enhance treatment outcomes for musculoskeletal deformities.

Noninvasive release of tendons using MRI guided focused ultrasound: a hybrid therapy using long-pulse focused ultrasound followed by thermal ablation

Introduction Magnetic Resonance-guided Focused Ultrasound (MRgFUS) thermal ablation is an effective noninvasive ultrasonic therapy to disrupt in vivo porcine tendon but is prone to inducing skin burns. We evaluated the safety profile of a novel hybrid protocol that minimizes thermal spread by combining long-pulse focused ultrasound followed by thermal ablation. Methods In-vivo Achilles tendons (hybrid N = 15, thermal ablation alone N = 21) from 15 to 20 kg Yorkshire pigs were randomly assigned to 6 treatment groups in two studies. The first (N = 21) was ablation (600, 900, or 1200 J). The second (N = 15) was hybrid: pulsed FUS (13.5 MPa peak negative pressure) followed by ablation (600, 900, or 1200 J). Measurements of ankle range of motion, tendon temperature, thermal dose (240 CEM43), and assessment of skin burn were performed in both groups. Results Rupture was comparable between the two protocols: 1/5 (20%), 5/5 (100%) and 5/5 (100%) for hybrid protocol, compared to 2/7 (29%), 6/7 (86%) and 7/7 (100%) for the ablation-only protocol with energies of 600, 900, and 1200 J, respectively. The hybrid protocol produced lower maximum temperatures, smaller areas of thermal dose, fewer thermal injuries to the skin, and fewer full-thickness skin burns. The standard deviation for the area of thermal injury was also smaller for the hybrid protocol, suggesting greater predictability. Conclusion This study demonstrated a hybrid MRgFUS protocol combining long-pulse FUS followed by thermal ablation to be noninferior and safer than an ablation-only protocol for extracorporeal in-vivo tendon rupture for future clinical application for noninvasive release of contracted tendon.

Study on the ultrasonic cavitation damage to early atherosclerotic plaque

Ultrasonic cavitation can damage surrounding material and be used for destruction of the target tissue. In this paper, we investigated the interaction between atherosclerotic plaque (AP) and cavitation bubbles to determine whether the mechanical effect of cavitation damage could be potentially useful in therapy for treating atherosclerotic plaques. A two-bubble–fluid–solid model was established to study the dynamic behavior of bubbles near the AP and the AP damage by ultrasound-induced cavitation. A low-intensity focused ultrasound (LIFU) transducer was used for testing cavitation-based AP damage. We found that the nonlinear oscillation of bubbles causes the relative positions of the bubbles to shift, either toward or away from one another, these phenomena lead to changes in the bond failure rate between the fiber bundles, and the value of BRF exhibits an upward trend, this is the reason why the fibers suffered from reversible stretching and compressing. However, the AP damage is irreversible and diminishes as the number of cycles in the ultrasonic burst. It appears that the bigger the radii, regardless of whether the bubble (3 − i)’s and bubble i's radii are equal, the greater the AP damage. Ultrasonic cavitation therapy may not be appropriate for advanced AP patients, and the calcified tissue has a greater impact on the stability of the plaque. The damage area should be strictly selected. Additionally, the tissue damage phenomenon was found in experimental results. This work shows that the severity of AP damage is correlated with acoustic parameters and the surrounding environment from both simulation and experimental perspectives. The results show that ultrasonic cavitation may provide a new choice for the treatment of AP.

Active non-Hermitian complementary acoustic metamaterials for transcranial imaging

High-frequency ultrasound has long been a safe and effective tool for medical imaging, diagnosis, and noninvasive treatments. However, the presence of the porous skull poses a challenge, impeding wave transmission and limiting noninvasive ultrasonic brain imaging due to a significant impedance mismatch and energy attenuation. In this study, we propose an innovative approach by introducing an active non-Hermitian complementary metamaterial (NHCMM) to enhance transcranial imaging. The NHCMM integrates piezoelectric elements, hydrogel materials, and a feedback control circuit. This actively modulates the effective acoustic properties of the metamaterial, ensuring precise tuning to match the impedance of the skull, thereby reducing energy loss and enhancing transmission. This design not only creates a transparent window for high-frequency ultrasound but also significantly improves brain imaging quality. The presented work establishes a foundational framework for non-invasive ultrasound-based brain imaging and high-precision ultrasonic therapies, preserving the structural integrity of the cranial barrier. We anticipate that this research holds immense potential for advancing the fields of brain imaging and ultrasonic treatments, paving the way for future innovations in medical diagnostics and therapeutic interventions.

Comparing Outcomes of Various Non-surgical Treatments, such as Scaling and Root Planing, Ultrasonic Therapy, and Laser Therapy

Abstract Background: Periodontitis is a prevalent inflammatory condition leading to progressive attachment loss and potential tooth loss. Nonsurgical periodontal therapy, including scaling and root planing (SRP), ultrasonic therapy, and laser therapy, is essential in managing periodontitis. This study compares the clinical and patient-reported outcomes of these three treatment modalities to determine their relative effectiveness. Objective: We aimed to compare the clinical outcomes and patient-reported experiences of SRP, ultrasonic therapy, and laser therapy in patients with chronic periodontitis. Materials and Methods: This randomized controlled trial involved 126 participants with moderate-to-severe chronic periodontitis, randomly assigned to one of three groups (42 participants each): Group 1 (SRP), Group 2 (ultrasonic therapy), and Group 3 (laser therapy). Clinical outcomes, including clinical attachment level (CAL), probing depth (PD), and bleeding on probing (BOP), were measured at baseline and 3 months post-treatment. Patient-reported outcomes such as pain, discomfort, and satisfaction were assessed using a visual analog scale. Data were analyzed using analysis of variance, with a P &lt; 0.05 considered statistically significant. Results: All groups showed significant improvements in CAL, PD, and BOP with no significant differences between groups (P &gt; 0.05). Laser therapy resulted in the lowest pain and discomfort scores and the highest satisfaction (P &lt; 0.05). Conclusion: SRP, ultrasonic therapy, and laser therapy are effective nonsurgical treatments for periodontitis, with comparable clinical outcomes. Laser therapy, however, offers superior patient comfort and satisfaction, suggesting that it may be a preferred option for patients sensitive to pain or discomfort.

Phonophoresis in Physiotherapy: Mechanisms, Applications, and Emerging Trends for Enhanced Drug Delivery and Therapeutic Efficacy

Phonophoresis, a widely utilized treatment in physiotherapy, combines topically applied gel or cream-based medications with ultrasonic therapy to enhance percutaneous absorption of pharmacological agents. The procedure employs ultrasound frequencies of 0.7 to 1.1 MHz with intensities ranging from 0.0 to 3.0 Watts per cm². Indications for phonophoresis span various inflammatory, deformative, dermatological, and rheumatic/neurological conditions. Recent studies suggest its clinical efficacy in pain relief and improved function, especially in conditions like lateral epicondylitis and osteoarthritis. The mechanism of action involves both thermal and non-thermal effects, with cavitation playing a key role in enhancing transdermal transport. Phonophoresis offers a non-invasive alternative for drug delivery, bypassing hepatic metabolism and minimizing systemic side effects. Various medicinal gels, particularly diclofenac, are used for transdermal drug delivery through phonophoresis. Low frequency phonophoresis has gained attention for enhancing transdermal transport, especially for macromolecules. Its applications extend beyond physiotherapy to include ocular drug delivery, nail therapy, gene therapy, and sports sciences. Clinical studies demonstrate the effectiveness of phonophoresis in conditions like anterior knee pain, tuberculous lymphadenitis, acute low back pain, and latent myofascial trigger points. Dosage considerations emphasize optimizing thermal effects without causing tissue damage. Studies highlight the potential of phonophoresis in relieving pain and improving function, particularly in knee osteoarthritis. Additionally, phonophoresis with Phyllanthus amarus nanoparticle gel shows promise in reducing pain and enhancing functional capacity in symptomatic knee osteoarthritis. In summary, phonophoresis stands as a valuable modality in physiotherapy, showcasing diverse applications and demonstrating clinical efficacy in various musculoskeletal and inflammatory conditions. Further research is warranted to explore its full potential and optimize treatment protocols.

Exploring the Efficacy of Ultrasonic Therapy for Ankle Sprains: A Narrative Review

Exploring the Efficacy of Ultrasonic Therapy for Ankle Sprains: A Narrative Review

Ultrasonic therapies for seizures and drug-resistant epilepsy.

Ultrasonic therapy is an increasingly promising approach for the treatment of seizures and drug-resistant epilepsy (DRE). Therapeutic focused ultrasound (FUS) uses thermal or nonthermal energy to either ablate neural tissue or modulate neural activity through high- or low-intensity FUS (HIFU, LIFU), respectively. Both HIFU and LIFU approaches have been investigated for reducing seizure activity in DRE, and additional FUS applications include disrupting the blood-brain barrier in the presence of microbubbles for targeted-drug delivery to the seizure foci. Here, we review the preclinical and clinical studies that have used FUS to treat seizures. Additionally, we review effective FUS parameters and consider limitations and future directions of FUS with respect to the treatment of DRE. While detailed studies to optimize FUS applications are ongoing, FUS has established itself as a potential noninvasive alternative for the treatment of DRE and other neurological disorders.

Advances in Brain Tumor Therapy Based on the Magnetic Nanoparticles.

Brain tumors, including primary gliomas and brain metastases, are one of the deadliest tumors because effective macromolecular antitumor drugs cannot easily penetrate the blood-brain barrier (BBB) and blood-brain tumor barrier (BTB). Magnetic nanoparticles (MNPs) are considered the most suitable nanocarriers for the delivery of brain tumor drugs because of their unique properties compared to other nanoparticles. Numerous preclinical and clinical studies have demonstrated the potential of these nanoparticles in magnetic targeting, nuclear magnetic resonance, magnetic thermal therapy, and ultrasonic hyperthermia. To further develop and optimize MNPs for the diagnosis and treatment of brain tumors, we attempt to outline recent advances in the use of MNPs to deliver drugs, with a particular focus on their efficacy in the delivery of anti-brain tumor drugs based on magnetic targeting and low-intensity focused ultrasound, magnetic resonance imaging for surgical real-time guidance, and magnetothermal and ultrasonic hyperthermia therapy. Furthermore, we summarize recent findings on the clinical application of MNPs and the research limitations that need to be addressed in clinical translation.

Advancements in Laser and Ultrasound Therapeutic Strategies for Cancer Cells: Recent Review

Cancer is a disease caused by uncontrollable cell growth and division. Surgery, chemotherapy, radiotherapy, and hormonotherapy are all cancer treatment options. In addition to noninvasive cancer ablative therapy. As an example, ultrasonic therapy, even with low-intensity pulsed ultrasound (LIPUS) or high-intensity focused ultrasound (HIFU), and Laser therapy (photo-biomodulation therapy) in low-level laser therapy (LLLT) with different wavelength ranges from ultraviolet (UV), visible and infrared (IR) that all have demonstrated different results depending on the target of treatment so previous trials therapies are being studied. This paper reviews recent studies on the in vitro treatment effect of ultrasound therapy and laser therapy on normal and cancerous cell lines with specific parameters. The effect of ultrasound results showed a decrease in cell proliferation and an increase in apoptosis in different types of cells, depending especially on sound intensity, known as Special Peak Temporal Average Intensity (ISPTA). While the laser effect is noticed on cell viability, either enhance or inhibit their viability depending upon the dose of exposure and other specific parameters like wavelength, energy density, and power density used in each treatment protocol. The previous studies conclude that each response would have a treatment method with specific parameters, even an increase or decrease in cell viability. Further studies need to be applying these methods in vivo.

Laser-generation of focused acoustic vortex with Fresnel spiral optoacoustic surfaces

We report a laser-based method to generate high central-frequency focused ultrasound vortices by introducing the Fresnel spiral for optoacoustic conversion. The formed broadband optoacoustic waves are coherently superimposed with each other, giving rise to acoustic vortices at different positions along the propagating axis according to their corresponding frequencies, while still maintaining the generation of focused vortex at the predetermined focal length. The laser-generated vortices show the typical vortex features including a helical wavefront and a central null pressure. A semi-parabolic focusing detection system is designed and constructed to scan the generated ultrasound vortices with high spatial resolution. The resulting acoustic vortices with the central frequency of about 3 MHz are presented in three dimensions, and agree well with the predictions. The flexibility of this scheme and its scalability to higher frequencies open tremendous perspectives in particle manipulation, ultrasonic therapy and so on.

Noninvasive magnetic resonance-guided focused ultrasound for tendon disruption: an in vivo Animal study

Purpose Surgical resection of the tendon is an effective treatment for severe contracture. Magnetic Resonance-guided Focused Ultrasound (MRgFUS) is a non-invasive ultrasonic therapy which produces a focal increase in temperature, subsequent tissue ablation and disruption. We evaluated MRgFUS as a clinically translatable treatment modality to non-invasively disrupt in vivo porcine tendons. Material and methods In vivo Achilles tendons (n = 28) from 15–20kg Yorkshire pigs (n = 16) were randomly assigned to 4 treatment groups of 600, 900, 1200 and 1500 J. Pretreatment range of motion (ROM) of the ankle joint was measured with the animal under general anesthesia. Following MRgFUS treatment, success of tendon rupture, ROM increase, temperature, thermal dosage, skin burn, and histology analyses were performed. Results Rupture success was found to be 29%, 86%, 100% and 100% for treatment energies of 600, 900, 1200 and 1500 J respectfully. ROM difference at 90° flexion showed a statistically significant change in ROM between 900 J and 1200 J from 16° to 27°. There was no statistical significance between other groups, but there was an increase in ROM as more energy was delivered in the treatment. For each of the respective treatment groups, the maximal temperatures were 58.4 °C, 63.3 °C, 67.6 °C, and 69.9 °C. The average areas of thermal dose measured were 24.3mm2, 53.2mm2, 77.8mm2 and 91.6mm2. The average areas of skin necrosis were 5.4mm2, 21.8mm2, 37.2mm2, and 91.4mm2. Histologic analysis confirmed tissue ablation and structural collagen fiber disruption. Conclusions This study demonstrated that MRgFUS is able to disrupt porcine tendons in vivo without skin incisions.

EFFECTIVENESS OF COMBINATION THERAPY AND SHOULDER GLIDES ON FROZEN SHOULDER: A CASE REPORT

Frozen shoulder is the gradual loss of movement in the shoulder (glenohumeral) joint. When the shoulder is frozen, the joint has become stiff and its movement is limited. It is also called adhesive capsulitis and commonly treated by manual therapy and exercise, usually delivered together as components of a physical therapy intervention. The differential diagnosis includes biceps tendinopathy, glenohumeral osteoarthritis, neoplasm, rotator cuff tendinopathy or tear (with or without impingement), and subacromial and subdeltoid. The case report is based on use of combination therapy and shoulder glides to reduce the pain and improve the Range of motion of the joint. Combination therapy comprises of transcutaneous electrical nerve stimulation and ultrasonic therapy at different intensities according to different musculoskeletal conditions. Outcome measure used were Numerical Pain Rating Scale for pain and universal goniometer for measuring range of motion. In conclusion, the study supports the use of combination therapy and shoulder glides are beneficial in reducing pain and improve shoulder Range of motion in patients with case of Frozen shoulder. Keywords: Frozen shoulder, transcutaneous electrical nerve stimulation, ultrasonic therapy, range of motion, numerical pain rating scale, pain.

The Effects of Ultrasonic and Gamma Irradiation on the Flavor of Potato Wines Investigated by Sensory Omics.

Aroma is one of the most fascinating and least-known mysteries of Baijiu research. The volatile compounds (VOCs) of potato wine were evaluated by sensory omics techniques in order to comprehend their overall flavor characteristics and investigate the effects of ultrasonic treatment and gamma irradiation therapy on the aroma of the wine. The findings revealed that a total of 14 flavor compounds were identified by GC-MS. Isoamyl alcohol, ethyl octanoate, and 1,1-diethoxyethane were the key aroma components, according to GC-O analysis. A total of 50 volatile substances were identified by GC-IMS. After being subjected to irradiation and ultrasonic treatment, the alcohol level of the potato wine reduced while the esters content increased. By calculating the relative odor activity value, a total of 29 aroma components were classified as key aroma compounds (ROAV > 1). According to the results of the sensory evaluation-fruity, Fen-flavor, and sweet-and the acceptability of the irradiated and ultrasonicated potato wine were improved. Therefore, the use of ultrasonic and irradiation therapy in potato wine, as well as the overall aroma building of potato wine, can be supported theoretically by this study.

Ultrasonic Therapy Device Using Fuzzy-Logic for Clinical Use

In this paper, an ultrasonic therapy device using fuzzy-logic(UTD-FL) has been designed, constructed and tested with phantom materials. Fuzzy-logic rules have been determined using four parameters. In clinical practice, ultrasonic therapy is conducted solely based on subjective evaluation of medical experts, but in UTD-FL, fuzzy-logic rules decide automatically the three critically important characteristic parameters of applicable output power to the patient, namely the power of the signal, the percent of the duty-cycle and the signal frequency. The signal frequency and specifically its amplitude have critical effect on the temperature rise of the tissue test-point or surface. Therefore the intensity of the ultrasonic frequency and the duration of treatment-time are absolutely vital, that is why this instrument has been developed. This instrument is expected to prevent possible side effects, injuries, and potential damages on real tissues due to experts’ uneasiness. The test results of this newly developed medical device have been compared with clinical-practice. The instrument produces optimum output power due to its fuzzy-logic rules based mode design, and IR-temperature sensor based feedback effect; whereas the clinical mode inputs only clinical experience base gained medical data.

Quadratic versus linear models to estimate the mean scattering spacing as a function of temperature in ex-vivo tissue

Previous works have shown the feasibility of temperature estimation during ultrasonic therapy using pulse-echo diagnostic ultrasound. These methods are based on the measurement of thermally induced changes in backscattered RF echoes due to thermal expansion and changes in ultrasonic velocity. They assume a joint contribution of these two parameters and a linear dependence with temperature. In this work, the contributions of velocity changes and thermal expansion to the evolution of the mean scatterer spacing of ex vivo bovine skeletal muscle tissue samples were decoupled. This was achieved by employing an experimental setup which allows measuring the absolute velocity value, using the through-transmission technique in a direct transmission configuration. The mean-scatterer spacing was estimated from spectral analysis of the backscattered signals obtained in pulse-echo mode. We propose a quadratic model of the thermal expansion coefficient to fit the evolution of the mean-scatterer spacing with temperature. The temperature increase estimated by the linear model, in the range of 29.5–47 °C, presents a percentage error (mean square error) of 11 %, while for the quadratic model the error is 4.8 %.