An exact general remeshing scheme applied to physically conservative voxelization

We present an exact general remeshing scheme to compute analytic integrals of polynomial functions over the intersections between convex polyhedral cells of old and new meshes. In physics applications this allows one to ensure global mass, momentum, and energy conservation while applying higher-order polynomial interpolation. We elaborate on applications of our algorithm arising in the analysis of cosmological N-body data, computer graphics, and continuum mechanics problems. We focus on the particular case of remeshing tetrahedral cells onto a Cartesian grid such that the volume integral of the polynomial density function given on the input mesh is guaranteed to equal the corresponding integral over the output mesh. We refer to this as "physically conservative voxelization". At the core of our method is an algorithm for intersecting two convex polyhedra by successively clipping one against the faces of the other. This algorithm is an implementation of the ideas presented abstractly by Sugihara (1994), who suggests using the planar graph representations of convex polyhedra to ensure topological consistency of the output. This makes our implementation robust to geometric degeneracy in the input. We employ a simplicial decomposition to calculate moment integrals up to quadratic order over the resulting intersection domain. We also address practical issues arising in a software implementation, including numerical stability in geometric calculations, management of cancellation errors, and extension to two dimensions. In a comparison to recent work, we show substantial performance gains. We provide a C implementation intended to be a fast, accurate, and robust tool for geometric calculations on polyhedral mesh elements.Comment: Code implementation available at https://github.com/devonmpowell/r3

A Patient-Specific Fracture Risk Assessment Tool for Femoral Bone Metastases:Using the Bone Strength (BOS) Score in Clinical Practice

Patients with femoral metastases are at risk of fracturing bones. It is important to prevent fractures in order to maintain mobility and quality of life. The BOne Strength (BOS) score is based on a computed tomography (CT)-based patient-specific finite element (FE) computer model that objectively calculates bone strength. In this pilot study, the added clinical value of the BOS score towards treatment-related decision making was assessed. In December 2019, the BOS score was implemented in four radiotherapy centers. The BOS scores and fracture risks of individual patients were calculated and returned to the physician to assist in treatment decisions. The physicians filled out a questionnaire, which was qualitatively analyzed. A follow-up to identify fractures and/or death was performed after six months. Until June 2021, 42 BOS scores were delivered (20 high, 9 moderate, and 13 low fracture risk). In 48%, the BOS score led to an adaptation of treatment plans. Physicians indicated that the BOS score provided objective insight into fracture risk, was reassuring for physicians and patients, and improved multidisciplinary discussions and shared decision making. In conclusion, the BOS score is an objective tool to assess fracture risk in femoral bone metastases and aids physicians and patients in making a more informed decision regarding the most appropriate treatment.</p

Quantitative Bildgebung magnetischer Nanopartikel mittels magnetrelaxometrischer Tomographie für biomedizinische Anwendungen

Die Entwicklung neuartiger medizinischer Verfahren, in denen magnetische Nanopartikel (MNP) als Arzneimittelträger oder zur Wärmeinduktion für lokale Krebstherapien eingesetzt werden, ist derzeit Gegenstand intensiver Forschung. Sowohl für die Steuerung, als auch für die Bewertung dieser Therapien werden Messverfahren zum ortsaufgelösten und quantitativen Nachweis der MNP im Körper benötigt. Mit der Magnetrelaxometrie (MRX) steht ein solches Verfahren zur Verfügung, bei dem das abklingende magnetische Moment der MNP nach dem Abschalten eines angelegten Magnetfeldes zur Quantifizierung genutzt wird. Auch ein ortsaufgelöster Nachweis von MNP ist mit diesem Verfahren möglich. Dafür wird die Nanopartikelverteilung mit einem homogenen Magnetfeld zur Relaxation angeregt und die MRX-Signale gleichzeitig von mehreren Sensoren detektiert. Zur Rekonstruktion der Verteilung aus den Messdaten muss ein schlecht-gestelltes, inverses Problem gelöst werden. Dabei muss die Rekonstruktion über zusätzliches a-priori Wissen, wie z.B. der vertikalen Lage der Verteilung in einem 2D Rekonstruktionsgitter oder Annahmen über die Anzahl und Geometrie der MNP-Anreicherungen im Körper, stabilisiert werden. Einen neuartigen Ansatz zur dreidimensionalen Bildgebung der MNP stellt die MRX-Tomographie dar, in der die Rekonstruktion über zusätzliche MRX-Messungen stabilisiert wird. Dafür werden nacheinander unterschiedliche Teilbereiche der Nanopartikelverteilung mit inhomogenen Magnetfeldern zur Relaxation angeregt und die jeweiligen Relaxationssignale wiederum gleichzeitig von mehreren Sensoren aufgezeichnet. In dieser Arbeit wurde erstmals ein experimenteller Aufbau für die MRX-Tomographie konzipiert und realisiert, um damit die Rekonstruktionsqualität dieses Verfahrens zu untersuchen. Zur Detektion der MRX-Signale wurde ein bestehendes Sensorsystem mit 304 der derzeit empfindlichsten Magnetfeldsensoren (SQUIDs) verwendet. Zur selektiven Magnetisierung der MNP-Verteilung wurde ein mehrkanaliges Magnetisierungssystem entwickelt. Dieses stellt in einer wählbaren Abfolge präzise Magnetfelder bereit, die in unmittelbarer Nähe zum Sensorsystem eine ortskodierte Relaxationsantwort der MNP erzeugen. Zur Untersuchung der Bildgebungseigenschaften des MRX-Tomographieaufbaus wurden MNP-Phantome konzipiert und entwickelt, die die spezifischen Gegebenheiten präklinischer Therapiestudien mit MNP nachbilden. Die von den Phantomen bereitgestellten Nanopartikelverteilungen mit klinischen Dosierungen im Milligrammbereich konnten mittels MRX-Tomographie dreidimensional rekonstruiert werden. Bei einer räumlichen Auflösung von wenigen Kubikzentimetern und einem Messvolumen von bis zu 600 cm^3 wurde dabei eine Quantifizierungsunsicherheit von unter 10% erreicht. Die erreichte Gesamtmessdauer einer kompletten MRX-Tomographiesequenz von etwa zwei Minuten lag dabei unterhalb der typischen Narkosedauer in Kleintierstudien. Durch die Verwendung alternativer Anregungssequenzen im MRX-Tomographieaufbau konnte die Gesamtmessdauer ohne wesentlichen Verlust an Rekonstruktionsqualität auf unter 30 Sekunden reduziert werden. Schließlich wurde das Verfahren der MRX-Tomographie unter Berücksichtigung des zeitlichen Relaxationsverlaufes der MNP erweitert. Mit diesem Ansatz konnte auch der Bindungszustand der MNP an das umgebende Medium quantitativ und dreidimensional rekonstruiert werden. Die dazu durchgeführten Versuche belegen das Potential der MRX-Tomographie, den Einfluss der biologischen Umgebung auf die physikalischen Eigenschaften der MNP quantitativ und ortsaufgelöst nachzuweisen. Der in dieser Arbeit entstandene MRX-Tomographieaufbau erlaubt den sicheren quantitativen und ortsaufgelösten Nachweis von MNP-Verteilungen in Kleintieren bis zur Kaninchengröße. Durch eine moderate Anpassung der Anregungsspulen wird somit eine Humananwendung des Verfahrens denkbar. Damit wurde ein wichtiger Schritt in der Entwicklung einer therapiebegleitenden Bildgebung zur Steuerung und quantitativen Bewertung MNP basierter Krebsbehandlungen erreicht.Current biomedical research focuses on the development of novel biomedical applications based on magnetic nanoparticles (MNPs), e.g. for local cancer treatment. These therapy approaches employ MNPs as remotely controlled drug carriers or local heat generators. Since location and quantity of MNPs determine drug enrichment and heat production, quantitative knowledge of the MNP distribution inside a body is essential for the development and success of these therapies. Magnetorelaxometry (MRX) is capable to provide such quantitative information based on the specific response of the MNPs after switching-off an applied magnetic field. Applying a uniform (homogeneous) magnetic field to a MNP distribution and measuring the MNP response by multiple sensors at different locations allows for spatially resolved MNP quantification. However, to reconstruct the MNP distribution from this spatially resolved MRX data, an ill posed inverse problem has to be solved. So far, the solution of this problem was stabilized incorporating a-priori knowledge in the forward model, e.g. by setting priors on the vertical position of the distribution using a 2D reconstruction grid or setting priors on the number and geometry of the MNP sources inside the body. MRX tomography represents a novel approach for quantitative 3D imaging of MNPs, where the inverse solution is stabilized by a series of MRX measurements. In MRX tomography, only parts of the MNP distribution are sequentially magnetized by the use of inhomogeneous magnetic fields. Each magnetizing is followed by detection of the response of the corresponding part of the distribution by multiple sensors. The 3D reconstruction of the MNP distribution is then accomplished by a common evaluation of the distinct MRX measurement series. In this thesis the first experimental setup for MRX tomography was developed for quantitative 3D imaging of biomedical MNP distributions. It is based on a multi-channel magnetizing unit which has been engineered to generate a time-multiplexed sequence of precise magnetic fields for spatially constrained magnetizing of the MNP distribution. The unit has been integrated into a sensor system containing 304 superconducting quantum interference devices (SQUIDs) used for the spatially resolved detection of the MNP responses after each magnetizing. Furthermore, for evaluation of MRX tomography MNP phantoms reflecting the MNP distribution after magnetic drug targeting therapy in animals were designed and implemented. Using these phantoms, MNP distributions with clinical MNP doses in the milligram range could be quantitatively reconstructed by MRX tomography within a field of view up to 600 cm³ and a spatial resolution of a few cubic centimeters. The deviation between the quantified and nominal MNP amount was found to be below 10%. With the present experimental setup MRX tomography measurements of a complete MNP distribution were performed within the typical anesthesia time interval of a few minutes prevailing in preclinical animal studies. By implementing advanced magnetizing sequences this measurement time of the MRX tomography setup could be reduced to below 30 s. Finally, using the same MRX tomography setup a binding state specific quantitative imaging of MNP distributions was achieved by incorporating the temporal MNP relaxation behavior into the reconstruction. Hence, MRX tomography has the potential to image the influence of the local biological environment on the physical properties of the MNPs. The presented MRX tomography setup allows for sensitive and specific spatially resolved 3D quantification of MNPs in small animals. This represents an important step towards the development of a clinical imaging tool for the control and assessment of MNP based cancer treatments. Moreover, by adjusting the excitation coils the field of view could be easily enlarged making MRX tomography quite conceivable for human application

A variational multiscale computational framework for reaction-dominated thermo-chemo-mechanical process modeling in multi-constituent material systems

This dissertation develops a computational framework for modeling multi-constituent material systems characterized by the transport of reacting fluids through deformable solids, and their coupled, nonlinear, thermo-chemo-mechanical response in the reaction-dominated regime. This is accomplished through two major components of the work: (i) new robust variational multi-scale numerical methods that are consistently derived, and (ii) models for multi-physics processes in multi-constituent materials. New robust numerical methods are developed via the variational multiscale (VMS) framework. Through the concept of fine scales in VMS, unresolved physics are recovered and embedded at the coarse scale level, improving stability and accuracy of the method. Focus is placed on fine scales that do not vanish at element boundaries (so-called “edge bubbles”). Using edge bubbles and an explicit time integration algorithm, a VMS Discontinuous Galerkin (VMDG) method is derived for multi-domain problems in elastodynamics where different subdomains can be solved synchronously and concurrently with minimal sharing of information. In addition, a new VMS method is introduced for the reaction-dominated regime of the diffusion–reaction equation. The proposed fine-scale basis consists of enrichment functions that may be nonzero at element edges. The method captures sharp boundary and internal layers, suppresses spurious oscillations, and better satisfies the maximum principle as compared to other existing methods. A priori mathematical analysis of the stability and convergence of the method is presented, and optimal rates of convergence are verified numerically. The numerical methods developed in this work may be applied to many reaction-diffusion systems in mathematical models for coupled thermo-chemo-mechanical phenomena arising from different theoretical frameworks. Here, a model for thermo-chemo-mechanical response of open solid-fluid systems is presented in the context of mixture theory. Derivation starts from constituent-wise equations for balance of mass, momentum, and energy, accounting for energy in formation and breaking of chemical bonds. Interactions between different constituents are captured through interaction terms as per locally homogenized mixture theory. Satisfaction of the second law of thermodynamics is achieved by providing constitutive equations that guarantee non-negative entropy production. Resulting mathematical models yield transient diffusion-advection-reaction problems posed by systems of coupled, nonlinear, second-order partial differential equations (PDEs), whose solution require stable numerical methods. Several numerical studies are presented to highlight stability, accuracy, and other features of the newly developed variational multiscale methods and thermo-chemo-mechanical models. Tests involve hypothetical as well as realistic materials with boundary layers, advancing reaction fronts, chemical swelling, and fingering phenomena