Modelling Radiation Cancer Treatment with a Death-Rate Term in Ordinary and Fractional Differential Equations.

Fractional calculus has recently been applied to the mathematical modelling of tumour growth, but its use introduces complexities that may not be warranted. Mathematical modelling with differential equations is a standard approach to study and predict treatment outcomes for population-level and patient-specific responses. Here, we use patient data of radiation-treated tumours to discuss the benefits and limitations of introducing fractional derivatives into three standard models of tumour growth. The fractional derivative introduces a history-dependence into the growth function, which requires a continuous death-rate term for radiation treatment. This newly proposed radiation-induced death-rate term improves computational efficiency in both ordinary and fractional derivative models. This computational speed-up will benefit common simulation tasks such as model parameterization and the construction and running of virtual clinical trials.

A Poroelasticity Theory Approach to Study the Mechanisms Leading to Elevated Interstitial Fluid Pressure in Solid Tumours.

Although the mechanisms responsible for elevated interstitial fluid pressure (IFP) in tumours remain obscure, it seems clear that high IFP represents a barrier to drug delivery (since the resulting adverse pressure gradient implies a reduction in the driving force for transvascular exchange of both fluid and macromolecules). R. Jain and co-workers studied this problem, and although the conclusions drawn from their idealized mathematical models offered useful insights into the causes of elevated IFP, they by no means gave a definitive explanation for this phenomenon. In this paper, we use poroelasticity theory to also develop a macroscopic mathematical model to describe the time evolution of a solid tumour, but focus our attention on the mechanisms responsible for the rise of the IFP, from that for a healthy interstitium to that measured in malignant tumours. In particular, we discuss a number of possible time scales suggested by our mathematical model and propose a tumour-dependent time scale that leads to results in agreement with experimental observations. We apply our mathematical model to simulate the effect of "vascular normalization" (as proposed by Jain in Nat Med 7:987-989, 2001) on the IFP profile and discuss and contrast our conclusions with those of previous work in the literature.

Anistropically varying conductivity in irreversible electroporation simulations.

BACKGROUND: One recent area of cancer research is irreversible electroporation (IRE). Irreversible electroporation is a minimally invasive procedure where needle electrodes are inserted into the body to ablate tumor cells with electricity. The aim of this paper is to propose a mathematical model that incorporates a tissue's conductivity increasing more in the direction of the electrical field as this has been shown to occur in experiments. METHOD: It was necessary to mathematically derive a valid form of the conductivity tensor such that it is dependent on the electrical field direction and can be easily implemented into numerical software. The derivation of a conductivity tensor that can take arbitrary functions for the conductivity in the directions tangent and normal to the electrical field is the main contribution of this paper. Numerical simulations were performed for isotropic-varying and anisotropic-varying conductivities to evaluate the importance of including the electrical field's direction in the formulation for conductivity. RESULTS: By starting from previously published experimental results, this paper derived a general formulation for an anistropic-varying tensor for implementation into irreversible electroporation modeling software. The anistropic-varying tensor formulation allows the conductivity to take into consideration both electrical field direction and magnitude, as opposed to previous published works that only took into account electrical field magnitude. The anisotropic formulation predicts roughly a five percent decrease in ablation size for the monopolar simulation and approximately a ten percent decrease in ablation size for the bipolar simulations. This is a positive result as previously reported results found the isotropic formulation to overpredict ablation size for both monopolar and bipolar simulations. Furthermore, it was also reported that the isotropic formulation overpredicts the ablation size more for the bipolar case than the monopolar case. Thus, our results are following the experimental trend by having a larger percentage change in volume for the bipolar case than the monopolar case. CONCLUSIONS: The predicted volume of ablated cells decreased, and could be a possible explanation for the slight over-prediction seen by isotropic-varying formulations.

Leakage Evaluation by Virtual Entropy Generation (VEG) Method.

Leakage through microscale or nanoscale cracks is usually hard to observe, difficult to control, and causes significant economic loss. In the present research, the leakage in a pipe was evaluated by the virtual entropy generation (VEG) method. In virtual entropy generation method, the "measured entropy generation" is forced to follow the "experimental second law of thermodynamics". Taking the leakage as the source virtual entropy generation, a new pipe leakage evaluation criterion was analytically derived, which indicates that the mass leakage rate should be smaller than the pressure drop rate inside a pipe. A numerical study based on computational fluid dynamics showed the existence of an unrealistic virtual entropy generation at a high mass leakage rate. Finally, the new criterion was used in the evaluation of leakage available in the literature. These results could be useful for leakage control or industry criteria design in the future.

Modeling NO Biotransport in Brain Using a Space-Fractional Reaction-Diffusion Equation.

Nitric oxide (NO) is a small gaseous molecule that is involved in some critical biochemical processes in the body such as the regulation of cerebral blood flow and pressure. Infection and inflammatory processes such as those caused by COVID-19 produce a disequilibrium in the NO bioavailability and/or a delay in the interactions of NO with other molecules contributing to the onset and evolution of cardiocerebrovascular diseases. A link between the SARS-CoV-2 virus and NO is introduced. Recent experimental observations of intracellular transport of metabolites in the brain and the NO trapping inside endothelial microparticles (EMPs) suggest the possibility of anomalous diffusion of NO, which may be enhanced by disease processes. A novel space-fractional reaction-diffusion equation to model NO biotransport in the brain is further proposed. The model incorporates the production of NO by synthesis in neurons and by mechanotransduction in the endothelial cells, and the loss of NO due to its reaction with superoxide and interaction with hemoglobin. The anomalous diffusion is modeled using a generalized Fick's law that involves spatial fractional order derivatives. The predictive ability of the proposed model is investigated through numerical simulations. The implications of the methodology for COVID-19 outlined in the section "Discussion" are purely exploratory.

Aspiration-assisted bioprinting for precise positioning of biologics.

Three-dimensional (3D) bioprinting is an appealing approach for building tissues; however, bioprinting of mini-tissue blocks (i.e., spheroids) with precise control on their positioning in 3D space has been a major obstacle. Here, we unveil "aspiration-assisted bioprinting (AAB)," which enables picking and bioprinting biologics in 3D through harnessing the power of aspiration forces, and when coupled with microvalve bioprinting, it facilitated different biofabrication schemes including scaffold-based or scaffold-free bioprinting at an unprecedented placement precision, ~11% with respect to the spheroid size. We studied the underlying physical mechanism of AAB to understand interactions between aspirated viscoelastic spheroids and physical governing forces during aspiration and bioprinting. We bioprinted a wide range of biologics with dimensions in an order-of-magnitude range including tissue spheroids (80 to 600 µm), tissue strands (~800 µm), or single cells (electrocytes, ~400 µm), and as applications, we illustrated the patterning of angiogenic sprouting spheroids and self-assembly of osteogenic spheroids.

Deformation-based mapping of volume change from serial brain MRI in the presence of local tissue contrast change.

This paper is motivated by the analysis of serial structural magnetic resonance imaging (MRI) data of the brain to map patterns of local tissue volume loss or gain over time, using registration-based deformation tensor morphometry. Specifically, we address the important confound of local tissue contrast changes which can be induced by neurodegenerative or neurodevelopmental processes. These not only modify apparent tissue volume, but also modify tissue integrity and its resulting MRI contrast parameters. In order to address this confound we derive an approach to the voxel-wise optimization of regional mutual information (RMI) and use this to drive a viscous fluid deformation model between images in a symmetric registration process. A quantitative evaluation of the method when compared to earlier approaches is included using both synthetic data and clinical imaging data. Results show a significant reduction in errors when tissue contrast changes locally between acquisitions. Finally, examples of applying the technique to map different patterns of atrophy rate in different neurodegenerative conditions is included.

Digital Signal Processing Methods for Ultrasonic Echoes.

Digital signal processing has become an important component of data analysis needed in industrial applications. In particular, for ultrasonic thickness measurements the signal to noise ratio plays a major role in the accurate calculation of the arrival time. For this application a band pass filter is not sufficient since the noise level cannot be significantly decreased such that a reliable thickness measurement can be performed. This paper demonstrates the abilities of two regularization methods - total variation and Tikhonov - to filter acoustic and ultrasonic signals. Both of these methods are compared to a frequency based filtering for digitally produced signals as well as signals produced by ultrasonic transducers. This paper demonstrates the ability of the total variation and Tikhonov filters to accurately recover signals from noisy acoustic signals faster than a band pass filter. Furthermore, the total variation filter has been shown to reduce the noise of a signal significantly for signals with clear ultrasonic echoes. Signal to noise ratios have been increased over 400% by using a simple parameter optimization. While frequency based filtering is efficient for specific applications, this paper shows that the reduction of noise in ultrasonic systems can be much more efficient with regularization methods.

An electromechanical model of neuronal dynamics using Hamilton's principle.

Damage of the brain may be caused by mechanical loads such as penetration, blunt force, shock loading from blast, and by chemical imbalances due to neurological diseases and aging that trigger not only neuronal degeneration but also changes in the mechanical properties of brain tissue. An understanding of the interconnected nature of the electro-chemo-mechanical processes that result in brain damage and ultimately loss of functionality is currently lacking. While modern mathematical models that focus on how to link brain mechanics to its biochemistry are essential in enhancing our understanding of brain science, the lack of experimental data required by these models as well as the complexity of the corresponding computations render these models hard to use in clinical applications. In this paper we propose a unified variational framework for the modeling of neuronal electromechanics. We introduce a constrained Lagrangian formulation that takes into account Newton's law of motion of a linear viscoelastic Kelvin-Voigt solid-state neuron as well as the classic Hodgkin-Huxley equations of the electronic neuron. The system of differential equations describing neuronal electromechanics is obtained by applying Hamilton's principle. Numerical simulations of possible damage dynamics in neurons will be presented.

A MATHEMATICAL INVESTIGATION OF THE ROLE OF INTRACRANIAL PRESSURE PULSATIONS AND SMALL GRADIENTS IN THE PATHOGENESIS OF HYDROCEPHALUS.

Cerebrospinal fluid (CSF) pulsations have been proposed as a possible causative mechanism for the ventricular enlargement that characterizes the neurological condition known as hydrocephalus. This paper summarizes recent work by the authors to anaylze the effect of CSF pulsations on brain tissue to determine if they are mechanically capable of enlarging the cerebral ventricles. First a poroelastic model is presented to analyze the interactions that occur between the fluid and porous solid constituents of brain tissue due to CSF pulsations. A viscoelastic model is then presented to analyze the effects of the fluid pulsations on the solid brain tissue. The combined results indicate that CSF pulsations in a healthy brain are incapable of causing tissue damage and thus the ventricular enlargement observed in hydrocephalus. Therefore they cannot be the primary cause of this condition. Finally, a hyper-viscoelastic model is presented and used to demonstrate that small long-term transmantle pressure gradients may be a possible cause of communicating hydrocephalus in infants.

The dynamics of brain and cerebrospinal fluid growth in normal versus hydrocephalic mice.

OBJECT: Hydrocephalus has traditionally been quantified by linear measures of ventricular size, with adjunct use of cortical mantle thickness. However, clinical outcome depends on cognitive function, which is more directly related to brain volume than these previous measures. The authors sought to quantify the dynamics of brain and ventricular volume growth in normal compared with hydrocephalic mice. METHODS: Hydrocephalus was induced in 14-day-old C57BL/6 mice by percutaneous injection of kaolin into the cisterna magna. Nine hydrocephalic and 6 normal mice were serially imaged from age 2-12 weeks with a 14.1-T MR imaging unit. Total brain and ventricle volumes were calculated, and linear discriminant analysis was applied. RESULTS: Two very different patterns of response were seen in hydrocephalic mice compared with mice with normative growth. In one pattern (3 mice) brain growth was normal despite accumulation of CSF, and in the second pattern (6 mice) abnormal brain enlargement was accompanied by increased CSF volume along with parenchymal edema. In this latter pattern, spontaneous ventricular rupture led to normalization of brain volume, implying edema from transmantle pressure gradients. These 2 patterns of hydrocephalus were significantly discriminable using linear discriminant analysis (p < 0.01). In contrast, clinically relevant measurements of head circumference or frontal and occipital horn ratios were unable to discriminate between these patterns. CONCLUSIONS: This study is, to the authors' knowledge, the first serial quantification of the growth of brain and ventricle volumes in normal versus hydrocephalic development. The authors' findings demonstrate the feasibility of constructing normative curves of brain and fluid growth as complements to normative head circumference curves. By measuring brain volumes, distinct patterns of brain growth and enlargement can be observed, which are more likely linked to cognitive development and clinical outcome than fluid volumes alone.

A nonlinear total variation-based denoising method with two regularization parameters.

The aim of the present paper is to study the effect of the regularization parameter used in the numerical implementation of the Rudin-Osher-Fatemi denoising model. By using two different regularization parameters in the numerical scheme of the Rudin-Osher-Fatemi model, we will show experimentally that when a particular relationship between the sizes of these parameters holds, the quality of the denoised image and the speed of convergence of the numerical scheme are both much improved in comparison with the classic numerical scheme of the Rudin-Osher-Fatemi model where only one regularization parameter is used.

Mechanisms determining safety and performance of brain stimulating electrodes.

Electrical current is widely used to interact with or stimulate neural systems. Current transduction from device to tissue is mediated at the electrode-tissue interface by capacitive charge and electrochemistry. This charge-passing-capacity is frequency dependent. While safety parameters have been established for high-frequencies, safety has not been fully determined for novel materials and pulse frequencies significantly lower than 100 Hz. We are explicitly interested in safety parameters and performance of charge passing at low frequencies (<<100 Hz) for neural systems. We present a visual study of pH during charge passing for electrodeposited iridium oxide electrodes. Clear reaction-diffusion waves are observed that extend many hundreds of micrometers from the electrode surface.

Intensity robust viscous fluid deformation based morphometry using regionally adapted mutual information.

This paper describes an approach to fine scale non-rigid registration for mapping patterns of tissue volume loss in serial MRI studies of the brain. Specifically it addresses the important confound of diffuse tissue contrast changes which can influence local sub-voxel estimates of volume change. Such changes can be induced by neurodegenerative or neurodevelopmental processes, which not only modify apparent tissue volume, but also modify tissue integrity and its resulting MRI contrast parameters. We derive an approach to the voxel-wise maximization of regional mutual information (RMI) and use this to drive a viscous fluid deformation model between images. This provides a topology preserving map of local changes in volume between time points that is robust to regional changes in tissue contrast. Comparisons with current methodology are included showing that the approach provides a significant reduction in errors when tissue contrast varies locally between MRI acquisitions.