The approach rests on a piece of mathematics borrowed from an unlikely field: the analysis of wireless communication channels. When a multi-element ultrasound array fires a pulse from each of its elements in turn and records the echoes on every element, the result is a data structure called the Inter-element Response Matrix, or IRM. Each entry of this matrix describes the signal that travels from one transducer element into the tissue and back to another element. Because the probe both emits and receives, the configuration is purely backscattering, meaning all the information comes from waves that have interacted with the medium and returned. Once the IRM is formed, a mathematical operation known as singular value decomposition, or SVD, breaks it down into a set of singular values that summarize how the medium redistributes acoustic energy among the array elements.

Earlier work by Prada and Fink showed that when point-like scatterers are randomly distributed in a homogeneous medium, each scatterer corresponds to a non-zero singular value of the propagation matrix in the frequency domain. Building on this, Aubry and Derode applied results from Random Matrix Theory, a framework originally developed to describe the statistical properties of large random matrices in physics and telecommunications, to the singular values of acoustic propagation matrices. Their central insight was that when multiple scattering dominates a medium, the distribution of singular values converges to a characteristic shape known as the Quarter Circle Law. The closer the measured distribution hugs this quarter circle, the more the wave field has been randomized by repeated scattering events rather than by simple single reflections.

The North Carolina team exploited this principle by computing a probability density function of the singular values from each IRM and tracking two parameters of that distribution. The first, called E(X), is the expected value of the singular value distribution. The second, called lambda-max, is the peak of the probability density function, that is, the most frequently represented singular value. Theory predicts that as multiple scattering becomes more dominant, E(X) should increase while lambda-max should decrease, because the distribution spreads out and flattens toward the quarter-circle shape. Changes in the microstructure of the medium, such as the density of scatterers, should therefore leave a measurable fingerprint in these two numbers.

To test this idea, the researchers began with computer simulations. They generated two-dimensional numerical structures containing monodisperse circular scatterers made of either plastic or aluminum, randomly distributed in water, with the scatterer area fraction varied from 5 to 25 percent in steps of 5 percent. A simulated 128-element array transducer emitting a 5 MHz Gaussian pulse probed each structure, and the finite-difference time-domain solver SimSonic computed the full wave physics. The results showed that as scatterer density increased, E(X) rose and lambda-max fell, with both metrics reaching a plateau once the area fraction exceeded roughly 15 percent. Beyond that density, multiple scattering had become so dominant that the singular value distribution no longer changed appreciably, a limitation the authors attribute to the saturation predicted by the Quarter Circle Law. Notably, the two metrics could also distinguish between plastic and aluminum scatterers, indicating sensitivity to material properties as well as density.

The team then moved from simulation to laboratory phantoms. Gelatin-glass bead phantoms were fabricated with glass beads embedded at volume fractions of 3 and 10 percent, mimicking highly scattering media in which the scatterer size is comparable to the ultrasound wavelength. Two linear array transducers with central frequencies of 5.1 and 7.8 MHz interrogated these samples, and the singular value distributions were compared against the theoretical quarter circle. At the lower frequency, both E(X) and lambda-max clearly separated the low-density from the high-density phantoms. At the higher frequency, multiple scattering dominated for both densities and the sensitivity of the metrics diminished, reinforcing the simulation finding that the method works best before full scattering saturation sets in.

A second family of phantoms probed a different question: whether the metrics respond to changes in the speed of sound rather than to scattering strength. Polydimethylsiloxane, a soft silicone matrix, was loaded with barium titanate particles at five different mass ratios, producing composites in which the scatterers were much smaller than the wavelength and therefore only weakly scattering, but in which the acoustic velocity varied substantially with particle concentration. The measurements showed that E(X) and lambda-max could differentiate between these weakly scattering media only at the lower particle concentrations. At higher concentrations the trends became difficult to estimate, and conventional speed-of-sound measurements tracked the composition more reliably. The authors note that speed of sound is often hard to measure in biological settings, where through-transmission geometries are rarely practical, which is precisely why backscattering-based metrics could be valuable.

The decisive test came in living animals. The researchers studied thirty-two Sprague Dawley rats: six with healthy lungs serving as controls, eighteen with pulmonary fibrosis of varying severity induced by instilling bleomycin into the airway, and eight fibrotic rats treated with the antifibrotic drug Nintedanib. Under anesthesia, a sternotomy exposed both lungs, and a 128-element array operating at 7.8 MHz recorded five IRMs per animal to ensure reproducibility. After the ultrasound sessions, the lungs were excised, inflation-fixed, sectioned, and stained, and a veterinary pathologist blinded to the animal groups scored fibrosis severity on the modified Ashcroft scale, which ranges from 0 for healthy tissue to 8 for the most severe fibrosis, though no slide in this study scored above 4.

The in vivo results showed statistically significant correlations between both metrics and the histology scores. E(X) decreased with increasing fibrosis severity, while lambda-max increased, a pattern consistent with the underlying physics: fibrotic thickening of the alveolar walls reduces the density of healthy air-filled alveoli, which are the strongest scatterers in lung tissue. Fewer strong scatterers means less multiple scattering, shifting the singular value distribution away from the quarter circle in the direction of lower E(X) and higher lambda-max. The correlations were moderate rather than strong, and the authors examined time windows of 2, 4, 6, and 8 microseconds without finding significant differences, suggesting the result is robust to that processing choice.

The study positions singular value statistics as a potential complement to existing quantitative ultrasound tools such as the backscatter coefficient and envelope statistics, which the same group has previously used to stage interstitial lung disease in rodents. Because the IRM can be acquired with standard multi-element arrays in a backscattering geometry, the approach does not require access to the far side of the organ, a major practical advantage over transmission-based speed-of-sound measurements. The authors caution that only two scatterer volume fractions were tested in the highly scattering phantoms and that the plateau behavior limits sensitivity once multiple scattering fully dominates. They also emphasize that true validation must come from biological tissue, ideally in vivo, which the rat study provides a first step toward. If the moderate correlations seen here can be strengthened and translated to larger animals and eventually patients, the chaotic echoes that once defeated ultrasound imaging of the lung may become a quantitative window into its microstructure.

Subject of Research: Quantifying ultrasound multiple scattering to characterize complex biological media and assess pulmonary fibrosis

Article Title: Characterizing random complex biological media by quantifying ultrasound multiple scattering

Article References: Yousefian, O., Dashti, A., Geithner, H., Karbalaeisadegh, Y., Yao, S., Blackwell, J., Ali, M., Montgomery, S., Zhu, Y., Egan, T., & Muller, M. (2025). Characterizing random complex biological media by quantifying ultrasound multiple scattering. Frontiers in Acoustics, 3, Article 1545057. https://doi.org/10.3389/facou.2025.1545057

Image Credits: AI Generated

DOI: 10.3389/facou.2025.1545057

Keywords: ultrasound, multiple scattering, random matrix theory, singular value decomposition, pulmonary fibrosis, quantitative ultrasound, tissue characterization, lung microstructure, biomedical acoustics, phantoms, in vivo rodent study, Inter-element Response Matrix

The core idea behind US-TSI is elegantly simple. When tissue warms even slightly, the speed of sound within it changes. Because the imaging system assumes a fixed sound speed when reconstructing images, this temperature-induced change causes echoes to shift slightly in time, producing what looks like a tiny apparent displacement of the tissue. The spatial gradient of that displacement is called thermal strain. Crucially, water-rich tissues and lipid-rich tissues respond in opposite ways: sound speeds up with temperature in water-bearing material but slows down in fat. That sign difference is what allows US-TSI to distinguish lipid from water — and, in principle, to quantify how much of each is present.

Despite decades of development, most previous studies relied on assumed or literature-based sound speed–temperature relationships and treated thermal strain mostly as a qualitative contrast signal. What has been missing is a unified framework that connects the actual measured acoustic properties of a material, a physics-based mixture model, realistic ultrasound simulations, and controlled experiments — all in one place. That is precisely what the new study delivers, using oil–gelatin phantoms as controlled surrogates for lipid-bearing tissue.

The researchers began by measuring the speed of sound in pure castor oil and in a 5 percent gelatin solution across a temperature range of 24 to 40 degrees Celsius, spanning both room-temperature phantom conditions and physiological temperatures near 37 degrees. The two materials showed opposite temperature dependence: the gelatin solution’s sound speed increased with temperature, while the oil’s decreased. These measured relationships then fed into Wood’s equation, a classical acoustic mixture model that predicts the effective sound speed of a two-component medium from the densities and sound speeds of its constituents.

With the mixture model in hand, the team turned to Field II, a widely used ultrasound simulation platform, to generate synthetic radiofrequency signals for phantoms containing oil mass fractions from 0 to 40 percent. In each simulation, the scatterer positions were held perfectly fixed while only the sound speed changed between a baseline state and a heated state. Beamforming with an assumed constant sound speed of 1,540 meters per second then produced the characteristic apparent displacement, which was tracked using the same Loupas phase-based algorithm applied to the experimental data. The result was an idealized upper-bound estimate of what US-TSI can detect under perfectly controlled conditions.

The experimental side of the study used a Verasonics Vantage 256 system with a focused heating beam delivering 120-cycle pulses over 50 milliseconds at varying transmit voltages from 10 to 40 volts. Imaging was performed with a five-angle compounded plane-wave sequence at 7.8 megahertz and a frame rate of 500 hertz, capturing enough frames for stable displacement estimation. A calibrated thermocouple measured the actual temperature rise, which was cross-validated against a heat-diffusion cooling model fitted to the post-heating temperature decay. That model showed excellent agreement with direct measurements, with a root-mean-square error of just 0.24 degrees Celsius.

The results are striking in their consistency. Thermal strain varied linearly with temperature rise across the tested range of roughly 1 to 4 degrees Celsius, with coefficients of determination reaching 0.975 to 0.996 in simulation and 0.868 to 0.968 in experiment. Even more importantly for tissue characterization, thermal strain varied linearly with oil mass fraction at each fixed temperature rise, with simulation R² values of 0.998 to 0.999 and experimental R² values of 0.925 to 0.993. Across all 35 matched simulation–experiment conditions, the overall root-mean-square error was 0.016 percent strain, corresponding to just 7.8 percent of the total experimental strain range, and the simulated and experimental values correlated at r = 0.95.

The physical mechanism underlying these trends is the opposite temperature dependence of sound speed in the two components. As temperature rises, sound travels faster through the gelatin matrix but slower through the oil droplets, causing the two materials to diverge acoustically. This divergence amplifies the propagation-time mismatch under constant-speed beamforming, producing larger apparent displacements and stronger strain signals. The result is that heating more — pushing the temperature rise higher — effectively amplifies the compositional contrast without changing the underlying linear relationship, a finding that has direct implications for how clinicians might optimize heating protocols in future applications.

The study also quantified the practical detection limits. Using baseline noise measurements from repeated pre-heating acquisitions, the team estimated an empirical strain detection threshold of approximately 1 × 10⁻⁴, or 0.01 percent strain. Low- and intermediate-oil compositions produced strain signals well above this threshold, while compositions near the high-oil zero-crossing — around 35 percent oil mass fraction — approached the noise floor. This means the technique is most reliable for detecting composition-dependent differences rather than pinpointing an exact oil fraction near the point where the strain signal vanishes. The researchers also note that their simulations represent an idealized upper bound, since they excluded motion, signal decorrelation, and electronic noise that would inevitably degrade performance in living tissue.

The implications extend well beyond phantom experiments. Previous work by overlapping members of this team has already demonstrated US-TSI’s potential for detecting lipids in atherosclerotic plaques in rabbits and pigs and for assessing hepatic fat in obese mice. More recently, a millisecond-level transient heating probe achieved approximately 2 degrees Celsius of temperature rise in just 50 milliseconds in a porcine model, opening the door to real-time, clinically compatible thermal strain imaging. By establishing a rigorous quantitative framework that links composition, temperature, and strain through measured material properties, validated mixture modeling, and matched simulation and experiment, this new study provides the numerical foundation that could help translate US-TSI from a promising research tool into a practical method for noninvasive fat quantification in cardiovascular disease, fatty liver disease, and beyond.

Subject of Research: Quantitative validation of ultrasound thermal strain imaging sensitivity for temperature and lipid composition detection

Article Title: Quantitative validation of ultrasound thermal strain imaging sensitivity using simulation and oil–gelatin phantom experiment

Article References: Wei, R., Sheng, Z., Chen, M., Wu, H., Tzeng, E., Geng, X., Jiang, X., & Kim, K. (2026). Quantitative validation of ultrasound thermal strain imaging sensitivity using simulation and oil–gelatin phantom experiment. Frontiers in Acoustics, 4, Article 1861076. https://doi.org/10.3389/facou.2026.1861076

Image Credits: AI Generated

DOI: 10.3389/facou.2026.1861076

Keywords: ultrasound thermal strain imaging, speed of sound, tissue characterization, lipid detection, Wood's equation, oil–gelatin phantom, Field II simulation, noninvasive thermometry, atherosclerotic plaque, fatty liver, biomedical ultrasound, quantitative imaging

The key to the approach lies in a piece of physics that dates back to 1976, when Robert Alvarez and Albert Macovski showed that X-ray attenuation in the diagnostic energy range can be described as the sum of just two effects: the photoelectric effect and Compton scattering. The photoelectric contribution depends strongly on the atomic number of the material, while the Compton contribution depends mainly on electron density. If attenuation can be measured at several different X-ray energies, the two contributions can be mathematically separated, and from them the material’s mass density and its effective atomic number — a weighted average of the atomic numbers of the elements it contains — can be calculated. This formalism underlies virtually every spectral CT technique in use today, from virtual non-contrast imaging to material decomposition.

What has historically made such calculations difficult in the clinic is that the raw projection data needed for them are locked away inside proprietary scanner software. Manufacturers guard the sinograms, the scanner geometry, the beam-hardening corrections and the spectral detector response functions that a from-scratch reconstruction would require. The German team sidestepped this obstacle elegantly: instead of raw data, they used virtual monoenergetic images, or VMIs, which the scanner’s own software already produces. VMIs simulate what a CT image would look like if the X-ray beam consisted of photons at a single energy, and because the manufacturer has already corrected them for beam hardening, beam quality and detector efficiency, they arrive ready for quantitative analysis. No knowledge of the bow-tie filter or the tube spectrum is needed.

The scanner in question was the Naeotom Alpha from Siemens Healthineers, the first clinically approved photon-counting CT system, released in 2021. Unlike conventional detectors that merely count photons in aggregate, photon-counting detectors register each X-ray photon individually and sort them by energy, providing spectral data intrinsically with every scan and without any additional radiation dose to the patient. From a single acquisition, the researchers generated virtual monoenergetic reconstructions at six energies — 40, 50, 60, 80, 100 and 150 kiloelectronvolts — using the scanner’s Syngo.Via software, and then imported these image sets into their own Python-based analysis pipeline.

The mathematics at the heart of the method is disarmingly compact. Following the Alvarez-Macovski model, the linear attenuation coefficient at each energy is written as a photoelectric term, scaling with density, effective atomic number and an inverse power of the energy, plus a Compton term described by the Klein-Nishina function. Substituting two composite coefficients reduces the problem to a linear system: with attenuation measured at two or more energies, the coefficients can be solved by linear regression, and from them density and effective atomic number follow directly. Using six energies rather than the minimum of two gives the regression extra robustness against the noise that inevitably contaminates real CT data, producing more stable values in the final maps.

Before the algorithm could touch a scanner, its four free parameters had to be calibrated. The team fitted the model to reference attenuation data from the NIST XCOM database for sixteen elements spanning effective atomic numbers from 5, boron, to 20, calcium — precisely the range occupied by the elements that make up human tissue, and a range conveniently free of the abrupt K-edge absorption features that would complicate the fit. The calibration, performed over 40 to 200 keV, yielded parameters remarkably close to those reported by earlier groups, with a coefficient of determination of 0.9996. A verification step then confirmed that the calibrated model reproduces the literature values for density and atomic number with slopes of almost exactly one, small residual deviations being folded in as correction factors.

To test the method on real scanner output, the researchers prepared a series of alcohol-water mixtures — ethanol at 0, 25, 50, 75 and 95 percent by volume, and isopropanol at 0, 10, 35, 60 and 70 percent. These humble mixtures were chosen deliberately: their densities and effective atomic numbers fall squarely in the soft-tissue range of the human body, they are cheap, safe and easy to prepare, and their expected properties can be calculated from published reference data. The tubes were mounted in a custom epoxy-resin phantom nested inside a larger PMMA ring to mimic the scattering conditions of a human torso, and scanned five times at 140 kilovolts so that image noise could be reduced by averaging.

The results were strikingly accurate. Across both mixture series, the measured mass densities deviated from the expected values by an average of just 1.0 percent, while the effective atomic numbers deviated by 0.84 percent. Pure water yielded an effective atomic number of 7.49 against an expected 7.47, and even the most concentrated alcohol solutions tracked their reference values almost perfectly. The maps themselves visually distinguished every concentration in both series. The accuracy is comparable to that reported by earlier image-domain approaches, such as the work of Heismann and colleagues, which required far more complex preprocessing of raw data to achieve deviations of similar magnitude.

The method is not without limits. The Zeff maps proved noisier than the density maps, because Compton scattering depends only weakly on attenuation in this energy range, leaving the atomic-number estimate more vulnerable to image noise, and residual beam-hardening effects produced a gentle gradient across the phantom. The algorithm is also valid only for materials with effective atomic numbers between 5 and 20 and densities up to about 2.0 grams per cubic centimeter — a range that comfortably covers biological tissue, including cortical bone at roughly 1.8 to 2.0, but excludes denser materials such as metal implants. Strong artifacts that distort Hounsfield units, such as severe beam hardening, will propagate into erroneous map values, since the entire method rests on the fidelity of the underlying image data.

Even so, the study lands at a moment of rapid momentum in quantitative spectral imaging. Recent work by Zimmerman and Poludniowski and by Lustermans and colleagues has demonstrated photon-counting CT’s potential for radiotherapy-related material characterization, while other groups are pursuing physics-informed deep-learning approaches to the same problem. Against those data-hungry machine-learning methods, the new algorithm offers a complementary virtue: it is analytically derived, transparently calibrated against tabulated reference data, computationally light, and runs entirely on image data that any clinical workstation can export. The authors suggest it could support applications from radiation therapy planning, where density and atomic number inform dose calculations, to oncology, where tissue characterization may distinguish tumor from benign lesions, or the identification of uric acid stones. Future studies, they note, must validate the approach across different phantoms, protocols, dose levels and realistic tissue compositions — but the demonstration that hospital-ready images alone can yield quantitative chemical maps marks a meaningful step toward making spectral material analysis a routine part of clinical CT.

Subject of Research: Calculation of mass density and effective atomic number maps from virtual monoenergetic photon-counting CT images

Article Title: Computation of the effective atomic number and mass density from virtual monoenergetic photon-counting CT reconstructions

Article References: Horstmeier, S., Thomsen, F. S., & Borggrefe, J. (2026). Computation of the effective atomic number and mass density from virtual monoenergetic photon-counting CT reconstructions. Heliyon, 12(14), Article e45404. https://doi.org/10.1016/j.heliyon.2026.e45404

Image Credits: AI Generated

DOI: 10.1016/j.heliyon.2026.e45404

Keywords: photon-counting CT, effective atomic number, mass density, virtual monoenergetic imaging, spectral CT, Alvarez-Macovski model, material decomposition, radiation therapy, tissue characterization, Heliyon, X-ray attenuation, quantitative imaging