Gelatin-Based Liver Phantoms for Training Purposes: A Cookbook Approach.

Background: Patients with liver pathology benefit from image-guided interventions. Training for interventional procedures is recommended to be performed on liver phantoms until a basic proficiency is reached. In the last 40 years, several attempts have been made to develop materials to mimic the imaging characteristics of the human liver in order to create liver phantoms. There is still a lack of accessible, reproducible and cost-effective soft liver phantoms for image-guided procedure training. Methods: Starting from a CT-scan DICOM file, we created a 3D-printed liver mold using InVesalius (Centro de Tecnologia da informação Renato Archer CTI, InVesalius 3 open-source software, Campinas, Brazil) for segmentation, Autodesk Fusion 360 with Netfabb (Autodesk software company, Fusion 360 2.0.19426 with Autodesk Netfabb Premium 2023.0 64-Bit Edition, San Francisco, CA, USA) for 3D modeling and Stratasys Fortus 380 mc 3D printer (Stratasys 3D printing company, Fortus 380 mc 3D printer, Minneapolis, MN, USA). Using the 3D-printed mold, we created 14 gelatin-based liver phantoms with 14 different recipes, using water, cast sugar and dehydrated gelatin, 32% fat bovine milk cream with intravenous lipid solution and technical alcohol in different amounts. We tested all these phantoms as well as ex vivo pig liver and human normal, fatty and cirrhotic liver by measuring the elasticity, shear wave speed, ultrasound attenuation, CT-scan density, MRI signal intensity and fracture force. We assessed the results of the testing performed, as well as the optical appearance on ultrasound, CT and MRI, in order to find the best recipe for gelatin-based phantoms for image-guided procedure training. Results: After the assessment of all phantom recipes, we selected as the best recipe for transparent phantoms one with 14 g of gelatin/100 mL water and for opaque phantom, the recipes with 25% cream. Conclusions: These liver gelatin-based phantom recipes are an inexpensive, reproducible and accessible alternative for training in image-guided and diagnostic procedures and will meet most requirements for valuable training.

Mueller-Jones matrices as representing conformal Lorentz transformations.

It is shown that Mueller-Jones matrices represent conformal Lorentz transformations. Thus the necessary and sufficient condition of a polarization device to be deterministic is to be describable by a conformal Lorentz transformation.

On the enpolarization/depolarization effects of deterministic devices.

A Poincaré sphere geometric approach to the enpolarization/depolarization effects of dichroic devices/media is presented. The degree of polarization (DoP) of the emergent light is expressed as a function of two parameters: the DoP of the incident light and the dichroic strength of the device. The polar plots of this dependence give an intuitive grasp on how the alternation of the two effects depends on the competition between these parameters. The same approach can be transposed in an adapted terminology in various fields of physics, in all the problems whose underpinning mathematics is that of Lorentz transformation.

Nonnormal operators in physics, a singular-vectors approach: illustration in polarization optics.

The singular-vectors analysis of a general nonnormal operator defined on a finite-dimensional complex vector space is given in the frame of a pure operatorial ("nonmatrix," "coordinate-free") approach, performed in a Dirac language. The general results are applied in the field of polarization optics, where the nonnormal operators are widespread as operators of various polarization devices. Two nonnormal polarization devices representative for the class of nonnormal and even pathological operators-the standard two-layer elliptical ideal polarizer (singular operator) and the three-layer ambidextrous ideal polarizer (singular and defective operator)-are analyzed in detail. It is pointed out that the unitary polar component of the operator exists and preserves, in such pathological case too, its role of converting the input singular basis of the operator in its output singular basis. It is shown that for any nonnormal ideal polarizer a complementary one exists, so that the tandem of their operators uniquely determines their (common) unitary polar component.

On a quasi-relativistic formula in polarization theory.

In a pure operatorial (nonmatrix) Pauli algebraic approach, this Letter shows that the Poincaré vector of the light transmitted by a dichroic device can be expressed as function of the Poincaré vectors of the incoming light and of the device by a composition law of the same kind as the composition law of the noncolinear relativistic velocities. This is, in fact, a general law of composition for three-dimensional (3D) vectors remaining in the Poincaré ball (where they have a group-like structure). The differences between this problem and that of the composition law of two dichroic devices are pointed out and justified.

Partial isometries, unitary operators, and complementary operators in polarization optics.

We show that for nonnormal singular operators corresponding to the nonorthogonal polarizers, the unitary polar component is constituted by two partial isometries, one of them "active" and the other "hidden" or "mute." For each such operator there exists a complementary one, corresponding also to a nonorthogonal polarizer, which has the same unitary polar component and whose partial isometries reverse their roles.

Skew projectors and generalized observables in polarization optics: a biorthogonal analyses [Invited].

This paper constitutes an application of the polarization optics in the problem of quantum measurement. The non-Hermitian operators of the nonorthogonal multilayer optical polarizers represent observables in the sense of the generalized quantum theory of measurement. The intimate spectral structure of these polarizers can be disclosed in the frame of skew-angular vector bases and biorthonormal vector systems. We show that these polarizers correspond to skew projectors; their operators are "generated" by skew projectors in the sense of the spectral theorem of linear operators theory. Thus the common feature of all the polarizers (Hermitian and non-Hermitian) is that their "nuclei" are (orthogonal or skew) projectors--the generating projectors.

Nonorthogonal polarizers: a polar analysis.

An analysis of the operators of some widespread nonorthogonal polarizers is performed on the basis of the polar factorization theorem, in pure operatorial (nonmatrix) Dirac algebraic language. The role of the unitary polar component as a converter of the two sets of singular eigenvectors of the operator, one in the other, is emphasized in each case; this role is maintained for the singular operators corresponding to these special nonorthogonal polarizers.

Vectorial pure operatorial Pauli algebraic approach in polarization optics: a theoretical survey and some applications.

In the last decade we have elaborated a mathematical tool for the description of the interaction of polarized light with polarization devices, alternative to the standard matrix (Jones and Mueller) formalisms, namely a vectorial pure operatorial Pauli algebraic one. After a brief, coherent survey of this formalism, we present some applicative results obtained in this frame, referring to the gain and the modification of the state of polarization at the interaction of the polarized light with deterministic devices. Due to an adequate parameterization of the problem, specific to this method, symmetric expressions of the gain and of the generalized Malus' law are obtained. On the other hand, the equation of the ellipsoid in which a Poincaré sphere of a given degree of polarization is mapped by such a device can be established.

Disorientated optical polarization devices.

By using a pure operatorial (nonmatrix) approach to some of the variable composite polarization devices, we show that for some values of the variable internal parameters, these devices lose their anisotropy. Their principal axes, or more generally their eigenvectors, become undetermined, i.e., in this sense, the anisotropic devices get "disorientated."

Pauli algebraic analysis of polarized light modulation.

The modification of the polarization and spectral structure of light by electro-optic modulation with longitudinal effect in crystals of class 42 m is analyzed in the frame of a Pauli algebraic and Poincaré geometric approach. The results are generalized, in a vectorial Pauli algebraic form, for any birefringent time-varying device.

Pauli algebraic forms of normal and nonnormal operators.

A unified treatment of the Pauli algebraic forms of the linear operators defined on a unitary linear space of two dimensions over the field of complex numbers C(1) is given. The Pauli expansions of the normal and nonnormal operators, unitary and Hermitian operators, orthogonal projectors, and symmetries are deduced in this frame. A geometrical interpretation of these Pauli algebraical results is given. With each operator, one can associate a generally complex vector, its Pauli axis. This is a natural generalization of the well-known Poincaré axis of some normal operators. A geometric criterion of distinction between the normal and nonnormal operators by means of this vector is established. The results are exemplified by the Pauli representations of the normal and nonnormal operators corresponding to some widespread composite polarization devices.

Non-Hermitian polarizers: a biorthogonal analysis.

The non-Hermitian operators of the ideal nonorthogonal multilayer optical polarizers are spectrally analyzed in the framework of skew-angular biorthonormal vector bases. It is shown that these polarizers correspond to skew projectors and their operators are generated by skew projectors, exactly as the canonical ideal polarizers correspond to Hermitian projectors. Thus the common feature of all the polarizers (Hermitian and non-Hermitian) is that their "nuclei" are (orthogonal or skew) projectors--the generating projectors. It is shown that if these nonorthogonal polarizers are looked upon as variable devices, two kinds of degeneracy may occur for suitable values of the inner parameter of the device: The corresponding operators may become normal (more precisely, Hermitian) or, on the contrary, very pathological--defective and singular. In the first case their eigenvectors and biorthogonal conjugate eigenvectors collapse into a unique pair of eigenvectors; in the second case their eigenvectors (as well as their biorthogonal conjugates) collapse into a single vector.

Dirac-algebraic approach to the theory of device operators in polarization optics.

An alternative approach to the theory of polarization optical devices, based on the Dirac-algebraic formalism, is presented. The advantages of this treatment over the widespread Jones-matrix formalism are discussed. The operators of some basic homogeneous optical devices are expanded in their eigenbases and in the Cartesian basis. The global and spectral properties of some cascades of polarization devices are analyzed.