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.

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.

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.

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.

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.