Sverre Holm

University of Oslo · Department of Physics

This book integrates concepts from physical acoustics with those from linear viscoelasticity and fractional linear viscoelasticity. Compressional waves and shear waves in applications such as medical ultrasound, elastography, and sediment acoustics often follow power law attenuation and dispersion laws that cannot be described with classical viscou...

The physics of shear waves traveling through matter carries fundamental insights into its structure, for instance, quantifying stiffness for disease characterization. However, the origin of shear wave attenuation in tissue is currently not properly understood. Attenuation is caused by two phenomena: absorption due to energy dissipation and scatteri...

Absorption of elastic waves in complex media often depends on frequency in a linear way for both longitudinal and shear waves. This universal property occurs in media such as rocks, unconsolidated sediments, and human tissue. Absorption is due to relaxation processes and we argue that these processes are thermally activated. Unusual for ultrasonics...

Acoustic propagation through a random distribution of 1 m ice cubes, from 100 to 1000 Hz, was simulated in a 3D finite element model. The effective sound speed and attenuation as functions of frequency were calculated from the simulated signals. Attempts were made to fit a number of models to the wave speed and attenuation, including single scatter...

Numerical emulations of the piano have been a subject of study since the early days of sound synthesis. High-accuracy sound synthesis of acoustic instruments employs physical modeling techniques which aim to describe the system’s internal mechanism using mathematical formulations. Such physical approaches are system-specific and present significant...

The amateur radio community is a global, highly engaged, and technical community with an intense interest in space weather, its underlying physics, and how it impacts radio communications. The large-scale observational capabilities of distributed instrumentation fielded by amateur radio operators and radio science enthusiasts offers a tremendous op...

Two popular theories for wave propagation in sediments, the Biot poroelastic theory and the Viscous Grain Shearing (VGS) theory, have been formulated in terms of dispersion relations. The lack of explicit wave equations for these models could be considered to be a weakness. Hence, providing wave equations that are fully consistent with these theori...

In an ordinary time-varying capacitor, there is debate on whether a time-domain multiplication or a time-domain convolution of capacitance and voltage determines charge. The objective of this work is to resolve this question by experiments on a time-varying capacitor in parallel with a resistor. It was implemented by a motor-driven potentiometer an...

“In the science, Evolution is a theory about changes: in the Myth, it is a fact about improvements” is what Lewis wrote in the Funeral of a Great Myth in the 1940’s, demonstrating both an acceptance of and a skepticism to the explanatory power of biological evolution. There seems to be an ambiguity in reading Lewis, as Peterson in 2010 called him...

The amateur radio community is a global, highly engaged, and technical community with an intense interest in space weather, its underlying physics, and how it impacts radio communications. The large-scale observational capabilities of distributed instrumentation fielded by amateur radio operators and radio science enthusiasts offers a tremendous op...

Power-law attenuation in elastic wave propagation of both compressional and shear waves can be described with multiple relaxation processes. It may be less physical to describe it with fractional calculus medium models, but this approach is useful for simulation and for parameterization where the underlying relaxation structure is very complex. It...

In an ordinary time-varying capacitor, there is debate whether a time-domain multiplication or a time-domain convolution of capacitance and voltage determines charge. A time-varying capacitor in parallel with a resistor was implemented by a motor-driven potentiometer and op-amps. The response matched a power-law function over about two decades of t...

The two main theories for wave propagation in sediments, the Extended Biot theoryand the Viscous Grain Shearing (VGS) theory, have been formulated in terms of dispersion equations only.Some have considered the lack of a wave equation to be a weakness that may indicate lack of a physical basisfor these theories. In the Biot theory, it is the frequen...

Power laws in time and frequency appear in fields such as linear viscoelasticity and acoustics, viscous boundary layer problems, and dielectrics. This is consistent with fractional derivatives in the fundamental descriptions, since power laws in time and frequency are related by the Fourier transform, and also associated with fractional derivatives...

Background: Biomechanical tissue properties of glioblastoma tumors are heterogeneous, but the molecular mechanisms involved and the biological implications are poorly understood. Here, we combine magnetic resonance elastography (MRE) measurement of tissue stiffness with RNA sequencing of tissue biopsies to explore the molecular characteristics of t...

The two main theories for wave propagation in sediments, the Extended Biot theory and the Viscous Grain Shearing (VGS) theory, have been formulated in terms of dispersion equations only. Some have considered the lack of a wave equation to be a weakness that may indicate lack of a physical basis for these theories. In the Biot theory, it is the freq...

Background: The biomechanical tissue properties of glioblastoma tumors are heterogeneous, but the molecular mechanisms involved and the biological implications are poorly understood. Here, we combine magnetic resonance elastography (MRE) measurement of tissue stiffness with RNA sequencing of tissue biopsies to explore the molecular characteristics...

Positioning systems have become increasingly popular in the last decade for location-based services such as navigation and asset tracking and management. As opposed to outdoor positioning, where the Global Navigation Satellite System became the standard technology, there is no consensus yet for indoor environments despite of the availability of dif...

Single-mode equivalent space-time representations of the acoustic wave propagating in a Biot poroelastic medium have previously been found only for asymptotic cases: In the low frequency regime, where the viscous skin depth is greater than the characteristic pore size, the time domain equivalent is represented with integer order temporal and spatia...

This paper proposes some fractional nonlocal viscoelastic models which are under the framework of both Eringen’s nonlocal theory and gradient elasticity theory. Introducing different combinations of new mechanical elements derived from the spatial nonlocal theory, a set of time-space-fractional constitutive models for nonlocal material, such as the...

Purpose Understanding how mechanical properties relate to functional changes in glioblastomas may help explain different treatment response between patients. The aim of this study was to map differences in biomechanical and functional properties between tumor and healthy tissue, to assess any relationship between them and to study their spatial dis...

The calibration of nodes in a sensor network or in a positioning system is a time-consuming process. This is especially relevant in indoor environments, where GPS signals are not available. Usually, it involves a user making transmissions either at known points, or by transmitting while moving under the area of interest. Some self-calibration metho...

This an English translation of Chapter 1 of a book in Norwegian - "Den innbilte konflikten. Om naturvitenskap og Gud". The first chapter in the original language is here: https://www.lundeforlag.no/lun/content/mma/publish/01/139/13934/Den_innbilte_konf-utdrag.pdf

The automotive industry continuously spends resources to reduce fuel consumption, operating costs, and harmful emissions. Namely, regulations are becoming more restrictive and customers’ expectations are growing. To achieve these goals in compression-ignition engines, an approach employs innovative Common Rail Injection Systems, based on advanced c...

Non-integer derivatives in acoustics and other areas of physics

PURPOSE Understanding how mechanical properties relate to functional changes in glioblastomas may help explain different treatment response between patients. The aim of this study was to map differences in biomechanical and functional properties between tumor and healthy tissue, to assess any relationship between them and to study their spatial dis...

The talk is about the non-integer (fractional) derivative, its mathematical formulation by Abel in 1823, and present day applications in modeling power-law behavior. These applications are in acoustics of complex media like tissue and sediments as well as in rheology, turbulence, and dielectrics. It builds on my book “Waves with Power-Law Attenuati...

The constant phase element (CPE) is a capacitive element with a frequency-independent negative phase between current and voltage which interpolates between a capacitor and a resistor. It is used extensively to model the complexity of the physics in e.g. the bioimpedance and electrochemistry fields. There is also a similar element with a positive ph...

Background Changes in brain stiffness can be an important biomarker for neurological disease. Magnetic resonance elastography (MRE) quantifies tissue stiffness, but the results vary between acquisition and reconstruction methods. Purpose To measure MRE repeatability and estimate the effect of different reconstruction methods and varying data quali...

The Cole-Cole model for a dielectric is a generalization of the Debye relaxation model. The most familiar form is in the frequency domain and this manifests itself in a frequency dependent impedance. Dielectrics may also be characterized in the time domain by means of the current and charge responses to a voltage step, called response and relaxatio...

Invited presentation 9.12.2020. Session: “Fractional Calculus Models of Compressional and Shear Waves for Medical Ultrasound” Abstract: Wave equations with non-integer derivative operators describe attenuation which increases with frequency with other powers than two, unlike ordinary wave equations. It is desirable to try to understand what prope...

Desirable properties for a bandlimited fractional-like operator, more in agreement with measurements of attenuation in real media in medical imaging (ultrasound and elastrography) and sediment acoustics

No PDF available ABSTRACT The imaginary part of the complex shear modulus in tissue is not negligible. In liver the phase angle (ranging between 0 and 1) is about 0.2 while in kidney it is about 0.3. The presence of dispersion can have its origin either in a constitutive loss—i.e., absorption of energy—or in scattering of the wave and hence represe...

No PDF available ABSTRACT Shear wave viscoelastic properties of tissue are now measured by a wide variety of methods including elastography, imaging scanners, rheological shear viscometers, and a variety of calibrated stress-strain analyzers. Because absorption and sound speed can be strong functions of frequency, fitting the data to an viscoelasti...

No PDF available ABSTRACT Wave equations with non-integer derivative operators describe attenuation which increases with frequency with other powers than two, unlike ordinary wave equations. It is desirable to try to understand what properties of, e.g., biological tissue that give rise to this behavior. The main attenuation mechanisms of standard a...

The constant phase element (CPE) with a frequency-independent negative phase between current and voltage is used extensively in e.g. the bioimpedance and electrochemistry fields. Its physical meaning is only partially understood. Here we show that the responses of both the common capacitive CPE as well as the inductive CPE are exactly the same as t...

Time-varying viscosity of viscoelastic materials has been found to induce complex rheology behaviors, which cannot be well characterized by the classical viscoelastic models. In this paper, different types of time-varying viscosity, namely, linearly varying viscosity, exponentially varying viscosity, and the proposed power-law viscosity are introdu...

Several wave equations for power-law attenuation have a spatial fractional derivative in the loss term. Both one-sided and two-sided spatial fractional derivatives can give causal solutions and a phase velocity dispersion which satisfies the Kramers–Kronig relation. The Chen–Holm and the Treeby–Cox equations both have the two-sided fractional Lapla...

A rising wave of technologies and instruments are enabling more labs and clinics to make a variety of measurements related to tissue viscoelastic properties. These instruments include elastography imaging scanners, rheological shear viscometers, and a variety of calibrated stress-strain analyzers. From these many sources of disparate data, a common...

No PDF available ABSTRACT The grain shearing (GS) and viscous grain shearing (VGS) models for saturated, unconsolidated marine sediments fit well to real data. But the models have much further relevance. The inherent convolution is in fact the definition of a non-integer derivative. Therefore, the GS theory is exactly described by the fractional di...

This paper considers the modelling of curing adhesive properties using fractional derivatives. A systematic approach is adopted where results can be related to a physical interpretation of the system rather than relying on a purely data-driven approach. The method relies on selecting standard integer order models based on the pre-cure and post-cure...

The biphasic extended Biot poroviscoelastic model takes into account the squirt flow in grain-grain contacts and introduces the bulk and shear relaxation modes associated with it. This model has been criticized for its empirical approach, but here the constitutive equations and the time domain wave equations of the model are derived. This also make...

A rising wave of technologies and instruments are enabling more labs and clinics to make a variety of measurements related to tissue viscoelastic properties. These instruments include elastography imaging scanners, rheological shear viscometers, and a variety of calibrated stress-strain analyzers. From these many sources of disparate data, a common...

Accurate localization in indoor environments is crucial for the correct operation of location-aware and augmented reality applications, indoor navigation, and inventory management, among others. Magnetic, radiofrequency and inertial navigation systems typically provide room or meter-level accuracy. Despite being the most widely used, they are affec...

Much effort has been expended on finding the relationship between fractional models and fractal geometries beyond the semantic similarity. A general connection has not been found, but a geometrical and physical interpretation of fractional integration and differentiation is in Podlubny (Fractional Calculus and Applied Analysis 5:367–386, 2002).

Poroelasticity or poroviscoelasticity is a rich field with complex physics-based models. It can model both compressional and shear waves in sub-bottom ocean sediments and is also used for modeling shear waves in the human body in elastography.

The spring–damper models for viscosity and relaxation described in Chap. 2 can be justified in processes taking place at length scales which may be much smaller than the wavelength, often at the molecular level. This is especially so in the acoustics case. In Sect. 2.6.3 specific examples of processes were given due to \(H_2O\), \(MgSO_4\), and \(B...

The aim of this chapter is to describe the conventional acoustic models in the framework of linear elasticity. The two main attenuation mechanisms are the viscous and the relaxation ones. It is shown that the viscous model derives from the Kelvin–Voigt spring–damper system, and that the relaxation model is based on the standard linear solid or Zene...

As mentioned in the introduction of Chap. 5, this chapter is concerned with wave equations that are made to fit frequency domain power laws with little consideration for time domain properties. First wave equations which are found by manipulating the common wave equations will be discussed. This is done by replacing ordinary derivatives in time or...

In previous chapters it has been shown that the standard linear viscoelastic models can be generalized to their fractional counterparts. It has also been demonstrated that the fractional Zener, Kelvin–Voigt, Maxwell, and Newtonian models all satisfy physical criteria such as passivity and causality. These models have temporal responses that are cha...

This chapter gives an overview of the main methods for characterizing viscoelastic systems in terms of the relaxation modulus and the creep response. A comparison is also made between linear differential equation descriptions and convolution descriptions, and in particular those with fading convolution kernels

The exponential response has been the typical medium response in the first part of this book. Examples are the creep compliances of the Kelvin–Voigt and Zener models ( 3.30) and ( 3.30) as well as the relaxation responses of the Maxwell and Zener models (( 3.37) and ( 3.45)). But in many complex media this model is too simple, and power laws are ob...

Mapping neuronal activity noninvasively is a key requirement for in vivo human neuroscience. Traditional functional magnetic resonance (MR) imaging, with a temporal response of seconds, cannot measure high-level cognitive processes evolving in tens of milliseconds. To advance neuroscience, imaging of fast neuronal processes is required. Here, we sh...

Ultrasound tissue characterization based on the coefficient of nonlinearity, βn = 1 + B/2A, has been demonstrated to produce added diagnostic value due to its large variation and sensitivity to tissue structure. However, the parameter has been observed to be significantly correlated to the speed of sound and density. These relationships are analyze...

Wave equations with non-integer order derivatives may model power law behavior in medical and sediment acoustics. As experiments only support a finite bandwidth, there is a limit to how much physical insight that can be gained from such models. Other ways to model a power law are with a fractional heat law, hierarchical ladder models for polymer ch...

The extended Biot poro-viscoelastic model [Chotiros and Isakson, JASA (2014)] like the standard Biot poroelastic model is biphasic and predicts two compressional waves (fast and slow) and one shear wave. Taking squirt flow into account, the model introduces two additional relaxation modes apart from the Biot crossover relaxation mode (global flow)....

Abstract Time domain analysis of the intracranial pressure (ICP) waveform provides important information about the intracranial pressure-volume reserve capacity. The aim here was to explore whether the tympanic membrane pressure (TMP) waveform can be used to non-invasively estimate the ICP waveform. Simultaneous invasive ICP and non-invasive TMP si...

The purpose of this study is to investigate the use of fundamental rheological parameters as quantified by MR elastography (MRE) to measure liver fibrosis and inflammation simultaneously in humans. MRE was performed on 45 patients at 3 T using a vibration frequency of 56 Hz. Fibrosis and inflammation scores were obtained from liver biopsies. Biomec...

This paper aims at presenting a survey of the fractional derivative acoustic wave equations, which have been developed in recent decades to describe the observed frequencydependent attenuation and scattering of acoustic wave propagating through complex media. The derivation of these models and their underlying elastoviscous constitutive relationshi...

DOI:https://doi.org/10.1103/PhysRevE.97.059901

This study explored the hypothesis that the central aortic blood pressure (BP) waveform may be used for non-invasive estimation of the intracranial pressure (ICP) waveform. Simultaneous invasive ICP and radial artery BP waveforms were measured in 29 individuals with idiopathic normal pressure hydrocephalus (iNPH). The central aortic BP waveforms we...

Tissue characterisation using the parameter of nonlinearity B/A has shown promising diagnostic value in medical imaging. Its estimation usually infers an assumption of constant sound speed and density. However, an empirical result known as Ballou’s rule shows a significant correlation between the reciprocal sound speed and the parameter of nonlinea...

The viscosity extended Biot theory [Sahay, Geophysics (2008)] incorporates pore fluid viscosity in the constitutive relation of the classical Biot theory. The strain rate term due to pore fluid viscosity turns the zero velocity slow S wave mode into a diffusive process. In this framework, it is claimed that fast P and S waves bleed energy when the...

There are two mechanisms for attenuation of mechanical waves: absorption and multiple scattering. We explore the second mechanism for shear waves in the context of MR elastography. The theory for attenuation was first given in the seismic field (O’Doherty & Anstey, Geophys. Prosp. 1971). Later phase was included, and the mean field concept was appl...

The paper by Scafetta entitled ”High resolution coherence analysis between planetary and climate oscillations”, May 2016 claims coherence between planetary movements and the global temperature anomaly. The claim is based on data analysis using the canonical covariance analysis (CCA) estimator for the magnitude squared coherence (MSC). It assumes a...

Low intensity pulsed ultrasound (LIPUS) is a therapy used clinically for bone fracture and soft tissue healing LIPUS induces functional alterations in cell behavior and has shown potential for cancer therapy. In order to understand the underlying biophysical mechanisms, the ability to control acoustic exposure parameters and elimination of unwanted...

In this work, in vivo ultrasound cardiac images created with Capon's minimum variance adaptive beamformer are compared with images acquired with the conventional delay-and-sum beamformer. Specifically, we provide three views of a human heart imaged through the parasternal short-axis, the parasternal long-axis and the apical four-chamber views. The...

Most derivations of acoustic wave equations involve ensuring that causality is satisfied. Here we explore the consequences of also requiring that the medium should be passive. This is a stricter criterion than causality for a linear system and implies that there are restrictions on the relaxation modulus and its first few derivatives. The viscous a...

Many complex media of great practical interest, such as in medical ultrasound, display an attenuation that increases with a power law as a function of frequency. Usually, measurements can only be taken over a limited frequency range, while wave equations often modelattenuation over all frequencies. There is therefore some freedom in how the models...

In MR elastography, it is common to use an elastic model for the tissue's response in order to interpret the results properly. More complex models, such as viscoelastic, fractional viscoelastic, poroelastic, or poroviscoelastic ones, are also used. These models appear at first sight to be very different, but here it is shown that they may all be ex...

This paper proposes a fractional biharmonic operator equation model in the time-space domain to describe scattering attenuation of acoustic waves in heterogeneous media. Compared with the existing models, the proposed fractional model is able to describe arbitrary frequency-dependent scattering attenuation, which typically obeys an empirical power...

The characteristic time-dependent viscosity of the intergranular pore-fluid in Buckingham's grain-shearing (GS) model [Buckingham, J. Acoust. Soc. Am. 108, 2796–2815 (2000)] is identified as the property of rheopecty. The property corresponds to a rare type of a non-Newtonian fluid in rheology which has largely remained unexplored. The material imp...

Many of the most interesting complex media are non-Newtonian and exhibit time-dependent behavior of thixotropy and rheopecty. They may also have temporal responses described by power laws. The material behavior is represented by the relaxation modulus and the creep compliance. On the one hand, it is shown that in the special case of a Maxwell model...

All adaptive beamformers aim at increased resolution and better contrast ratio. However, stretching the dynamic range (e.g. 40log(x)) may produce the same effects. Adaptive beamformers are often tested on wire targets or uniform cysts, which do not allow the study of dynamic range (DR) alterations. DR stretching may possibly account for some of the...

We propose a framework for extracting the bone surface from B-mode images employing the eigenspace minimum variance (ESMV) beamformer and a ridge detection method. We show that an ESMV beamformer with a rank-1 signal subspace can preserve the bone anatomy and enhance the edges, despite an image which is less visually appealing due to some speckle p...

This work describes the implementation of software plane-wave beamforming performed on the GPU of an Apple iPhone 6s and 6s Plus. The code can run on any current iOS device that supports the Metal API. The implementation is largely written in Swift, with some Objective-C, while the core processing component was written in Metal, Apple's new GPU pro...

This paper is accepted for publishing in Fra