Dimension Estimate of Uniform Attractor for a Model of High Intensity Focussed Ultrasound-Induced Thermotherapy.

High intensity focussed ultrasound (HIFU) has emerged as a novel therapeutic modality, for the treatment of various cancers, that is gaining significant traction in clinical oncology. It is a cancer therapy that avoids many of the associated negative side effects of other more well-established therapies (such as surgery, chemotherapy and radiotherapy) and does not lead to the longer recuperation times necessary in these cases. The increasing interest in HIFU from biomedical researchers and clinicians has led to the development of a number of mathematical models to capture the effects of HIFU energy deposition in biological tissue. In this paper, we study the simplest such model that has been utilized by researchers to study temperature evolution under HIFU therapy. Although the model poses significant theoretical challenges, in earlier work, we were able to establish existence and uniqueness of solutions to this system of PDEs (see Efendiev et al. Adv Appl Math Sci 29(1):231-246, 2020). In the current work, we take the next natural step of studying the long-time dynamics of solutions to this model, in the case where the external forcing is quasi-periodic. In this case, we are able to prove the existence of uniform attractors to the corresponding evolutionary processes generated by our model and to estimate the Hausdorff dimension of the attractors, in terms of the physical parameters of the system.

Modeling Invasion Dynamics with Spatial Random-Fitness Due to Micro-Environment.

Numerous experimental studies have demonstrated that the microenvironment is a key regulator influencing the proliferative and migrative potentials of species. Spatial and temporal disturbances lead to adverse and hazardous microenvironments for cellular systems that is reflected in the phenotypic heterogeneity within the system. In this paper, we study the effect of microenvironment on the invasive capability of species, or mutants, on structured grids (in particular, square lattices) under the influence of site-dependent random proliferation in addition to a migration potential. We discuss both continuous and discrete fitness distributions. Our results suggest that the invasion probability is negatively correlated with the variance of fitness distribution of mutants (for both advantageous and neutral mutants) in the absence of migration of both types of cells. A similar behaviour is observed even in the presence of a random fitness distribution of host cells in the system with neutral fitness rate. In the case of a bimodal distribution, we observe zero invasion probability until the system reaches a (specific) proportion of advantageous phenotypes. Also, we find that the migrative potential amplifies the invasion probability as the variance of fitness of mutants increases in the system, which is the exact opposite in the absence of migration. Our computational framework captures the harsh microenvironmental conditions through quenched random fitness distributions and migration of cells, and our analysis shows that they play an important role in the invasion dynamics of several biological systems such as bacterial micro-habitats, epithelial dysplasia, and metastasis. We believe that our results may lead to more experimental studies, which can in turn provide further insights into the role and impact of heterogeneous environments on invasion dynamics.

Efficacy of dose escalation on TCP, recurrence and second cancer risks: a mathematical study.

OBJECTIVE: We investigated the effects of conventional and hypofractionation protocols by modelling tumour control probability (TCP) and tumour recurrence time, and examined their impact on second cancer risks. The main objectives of this study include the following: (a) incorporate tumour recurrence time and second cancer risks into the TCP framework and analyse the effects of variable doses and (b) investigate an efficient protocol to reduce the risk of a secondary malignancy while maximizing disease-free survival and tumour control. METHODS: A generalized mathematical formalism was developed that incorporated recurrence and second cancer risk models into the TCP dynamics. RESULTS: Our results suggest that TCP and relapse time are almost identical for conventional and hypofractionated regimens; however, second cancer risks resulting from hypofractionation were reduced by 22% when compared with the second cancer risk associated with a conventional protocol. The hypofractionated regimen appears to be sensitive to dose escalation and the corresponding impact on tumour recurrence time and reduction in second cancer risks. The reduction in second cancer risks is approximately 20% when the dose is increased from 60 to 72 Gy in a hypofractionated protocol. CONCLUSION: Our results suggest that hypofractionation may be a more efficient regimen in the context of TCP, relapse time and second cancer risks. Overall, our study demonstrates the importance of including a second cancer risk model in designing an efficient radiation regimen. ADVANCES IN KNOWLEDGE: The impact of various fractionation protocols on TCP and relapse in conjunction with second cancer risks is an important clinical question that is as yet unexplored.

Spatial invasion dynamics on random and unstructured meshes: implications for heterogeneous tumor populations.

In this work we discuss a spatial evolutionary model for a heterogeneous cancer cell population. We consider the gain-of-function mutations that not only change the fitness potential of the mutant phenotypes against normal background cells but may also increase the relative motility of the mutant cells. The spatial modeling is implemented as a stochastic evolutionary system on a structured grid (a lattice, with random neighborhoods, which is not necessarily bi-directional) or on a two-dimensional unstructured mesh, i.e. a bi-directional graph with random numbers of neighbors. We present a computational approach to investigate the fixation probability of mutants in these spatial models. Additionally, we examine the effect of the migration potential on the spatial dynamics of mutants on unstructured meshes. Our results suggest that the probability of fixation is negatively correlated with the width of the distribution of the neighborhood size. Also, the fixation probability increases given a migration potential for mutants. We find that the fixation probability (of advantaged, disadvantaged and neutral mutants) on unstructured meshes is relatively smaller than the corresponding results on regular grids. More importantly, in the case of neutral mutants the introduction of a migration potential has a critical effect on the fixation probability and increases this by orders of magnitude. Further, we examine the effect of boundaries and as intuitively expected, the fixation probability is smaller on the boundary of regular grids when compared to its value in the bulk. Based on these computational results, we speculate on possible better therapeutic strategies that may delay tumor progression to some extent.

Treatment of distal radial fractures with the DVR-A plate--the early Bristol experience.

AIM: The aim of the study was to assess how adequately distal radial fracture reduction was reproduced and maintained with the distal volar radius anatomic DVR-A (Biomet, Inc) locking plate. METHODS: We looked at a consecutive series of 111 patients treated with the DVR-A plate at our institution from 2007-2010. The preoperative, intra-operative, and postoperative films were reviewed. The AO and Frykman classification was recorded. The sagittal tilt, radial inclination, and radial length were measured on intra-operative X-rays and compared with final follow-up X-rays. The last recorded range of motion at follow-up and a functional assessment using the Disabilities of the Arm, Shoulder, and Hand (DASH) score was recorded. RESULTS: At final follow-up a mean radial inclination of 22.22 degrees (11 to 38 degrees), radial height of 11.85 mm (6 to 18 mm), and sagittal tilt of 6.71 degrees volar (-9 to 19 degrees) was achieved. From initial post-operative films, to final follow-up X-rays there was a mean increase of 0.17 degrees in radial inclination, a mean loss of 0.36 mm radial height, and a mean loss of 2.17 degrees volar tilt. The mean extension was 46.8 degrees, flexion 48.3 degrees, pronation 77.4 degrees, supination 74.8 degrees, radial deviation 15.3 degrees and the ulnar deviation 19 degrees. The mean DASH score was 12.8 (0-68). CONCLUSION: The DVR-A plate achieved a highly satisfactory reduction of radial length and radial inclination, with a small loss of volar sagittal tilt at final follow-up. A good functional outcome was reported, with a satisfactory range of motion achieved. The DVR-A plate is a safe and effective treatment for unstable and intra-articular displaced distal radius fractures, particularly in younger patients, in the short term.

Aging impact on brain biomechanics with applications to hydrocephalus.

Hydrocephalus is a neurological disorder whose clinical symptoms and treatment outcome are correlated with patient age. In Wilkie et al. (2010, A theoretical study of the effect of intraventricular pulsations on the pathogenesis of hydrocephalus. Appl. Math. Comput., 215, 3181-3191), the fractional Zener model was used to investigate the role of cerebrospinal fluid pressure pulsations in the development of hydrocephalus in infants and adults. In this paper, we determine the mechanical parameters of the fractional Zener model for the infant and adult brains using age-dependent shear complex modulus data (Thibault, K. L. & Margulies, S. S. (1998) Age-dependent material properties of the porcine cerebrum: effect on pediatric inertial head injury criteria. J. Biomech., 31, 1119-1126). The displacement of brain tissue under conditions representing the onset of hydrocephalus are then calculated. The infant brain was found to produce tissue displacements that are unphysical for our model geometry and a new boundary condition is proposed to replace the stress-free outer boundary condition used in Wilkie et al. (2010). The steadystate elastic modulus is identified as the parameter of interest in the development of hydrocephalus: it is found to increase from the infant value of 621 Pa to the young adult value of 955 Pa and we hypothesize that it then decreases with age. The low steady-state elastic modulus of the infant brain (and possibly the aged brain) increases the tissue's susceptibility to large deformations and thus to the ventricular expansion characteristic of hydrocephalus.

Quantitative model for efficient temporal targeting of tumor cells and neovasculature.

The combination of cytotoxic therapies and antiangiogenic agents is emerging as a most promising strategy in the treatment of malignant tumors. However, the timing and sequencing of these treatments seem to play essential roles in achieving a synergic outcome. Using a mathematical modeling approach that is grounded on available experimental data, we investigate the spatial and temporal targeting of tumor cells and neovasculature with a nanoscale delivery system. Our model suggests that the experimental success of the nanoscale delivery system depends crucially on the trapping of chemotherapeutic agents within the tumor tissue. The numerical results also indicate that substantial further improvements in the efficiency of the nanoscale delivery system can be achieved through an adjustment of the temporal targeting mechanism.

Poroelastic theory of transcapillary flow: effects of endothelial glycocalyx deterioration.

The luminal surface of endothelial cells is lined with a carbohydrate-rich layer known as the endothelial glycocalyx. Identification of the structural properties of the glycocalyx has led investigators to examine its various functions and it has since been recognized as playing a role in many physiological processes, one of which is the regulation of fluid and protein exchange across the capillary wall. Experimental observations in which the glycocalyx was degraded in rat myocardial capillaries showed fluid accumulation in the tissue, suggesting that the glycocalyx acts as a protective barrier against edema. In this work we seek to quantify the observed edema formation by using our earlier poroelastic model to examine the consequences of glycocalyx deterioration on transcapillary filtration. Upon enzymatic treatment the properties of the glycocalyx, such as its thickness and permeability, will be modified, and our purpose here is to investigate quantitatively how changes in these parameters affect the magnitude of the fluid filtration through the capillary wall. We compare our results with both experimental data as well as other theoretical models where applicable, discussing the implications of the models as well as the limitations of comparison. This work provides the basis for further experiments that may better characterise many of the parameters involved in this process.

Characterization of brain cancer stem cells: a mathematical approach.

OBJECTIVE: In recent years, support has increased for the notion that a subpopulation of brain tumour cells in possession of properties typically characteristic of stem cells is responsible for initiating and maintaining the tumour. Unravelling details of the brain tumour stem cell (BTSC) hierarchy, as well as interactions of these cells with various therapies, will be essential in the design of optimal treatment strategies. MATERIALS AND METHODS: Motivated by this, we have developed a mathematical model of the BTSC hypothesis that may aid in characterization of brain tumours, as well as in prediction of effective therapeutic strategies, which can be further validated in experimental and clinical studies. At the level of a small number of cells, the model developed herein is stochastic. For larger populations of cancer cells, the model is handled from a deterministic approach. RESULTS AND CONCLUSIONS: In the stochastic regime, importance of a relationship between the likelihoods of two distinct types of symmetric BTSC divisions in determining BTSC survival rates becomes apparent, consequently emphasizing the need for a set of biomarkers that are able to better characterize the BTSC hierarchy. At the large scale, we predict the importance of the aforementioned symmetric division rates in dictating brain tumour composition. Furthermore, we demonstrate possible therapeutic benefits of considering combination treatments of radiotherapy and putative BTSC inhibitors, such as bone morphogenetic proteins, while reinforcing the importance of developing novel treatment strategies that specifically target the BTSC subpopulation.

A poroelastic model of transcapillary flow in normal tissue.

Starling's seminal work on the absorption of fluids from connective tissue spaces (and Starling's hypothesis that the energy for transcapillary flow lies in the difference between hydrostatic and osmotic pressures across the capillary wall) has long formed the basis of much of experimental physiology. Related recent experimental evidence points to a more active role of the interstitium in controlling interstitial fluid pressure (IFP) which has significant implications for clinical oncology. In light of these considerations, it is clearly of importance to reconsider the relationship between IFP and transcapillary transport, in addition to the regulation of IFP in normal tissue. In this paper, we adopt the Michel-Weinbaum viewpoint on the locality of Starling forces and model the capillary wall as a poroelastic solid using Biot's consolidation theory. However, the incorporation of the Michel-Weinbaum hypothesis requires an extension of Darcy's law to include the effects of oncotic pressure in the mechanism of filtration through the capillary wall. A unique feature of the model of transcapillary flow developed here is its ability to predict the stress and strain distribution across the capillary wall, which to our knowledge has not been attempted previously. We are optimistic that, with rapidly advancing technological capabilities, experimentalists will soon be able to test many of the model predictions.

Dynamics of tumor growth and combination of anti-angiogenic and cytotoxic therapies.

Tumors cannot grow beyond a certain size (about 1-2 mm in diameter) through simple diffusion of oxygen and other essential nutrients into the tumor. Angiogenesis, the formation of blood vessels from pre-existing vessels, is a crucial and observed step, through which a tumor obtains its own blood supply. Thus, strategies that interfere with the development of this tumor vasculature, known as anti-angiogenic therapy, represent a novel approach to controlling tumor growth. Several pre-clinical studies have suggested that currently available angiogenesis inhibitors are unlikely to yield significant sustained improvements in tumor control on their own, but rather will need to be used in combination with conventional treatments to achieve maximal benefit. Optimal sequencing of anti-angiogenic treatment and radiotherapy or chemotherapy is essential to the success of these combined treatment strategies. Hence, a major challenge to mathematical modeling and computer simulations is to find appropriate dosages, schedules and sequencing of combination therapies to control or eliminate tumor growth. Here, we present a mathematical model that incorporates tumor cells and the vascular network, as well as their interplay. We can then include the effects of two different treatments, conventional cytotoxic therapy and anti-angiogenic therapy. The results are compared with available experimental and clinical data.

Mathematical modeling of brain tumors: effects of radiotherapy and chemotherapy.

Gliomas, the most common primary brain tumors, are diffusive and highly invasive. The standard treatment for brain tumors consists of a combination of surgery, radiation therapy and chemotherapy. Over the past few years, mathematical models have been applied to study untreated and treated brain tumors. In an effort to improve treatment strategies, we consider a simple spatio-temporal mathematical model, based on proliferation and diffusion, that incorporates the effects of radiotherapeutic and chemotherapeutic treatments. We study the effects of different schedules of radiation therapy, including fractionated and hyperfractionated external beam radiotherapy, using a generalized linear quadratic (LQ) model. The results are compared with published clinical data. We also discuss the results for combination therapy (radiotherapy plus temozolomide, a new chemotherapy agent), as proposed in recent clinical trials. We use the model to predict optimal sequencing of the postoperative (combination of radiotherapy and adjuvant, neo-adjuvant or concurrent chemotherapy) treatments for brain tumors.

Mathematical modeling of ovarian cancer treatments: sequencing of surgery and chemotherapy.

Ovarian cancer has long been one of the most common forms of cancer in women. The main treatment for ovarian cancer comprises a combination of surgery and chemotherapy. In an effort to improve treatment strategies, a variety of mathematical models have been developed in the literature. In this paper, we consider a simple mathematical model that incorporates tumor growth as well as the effects of chemotherapeutic and surgical treatments in ovarian cancer. We consider several growth models and combine them with different cell-kill hypotheses. Surgery is assumed to eliminate a fixed fraction of tumor cells instantaneously. We discuss how different models predict the optimal sequencing of chemotherapeutic and surgical treatments. This work has been carried out in the context of ovarian cancer; however, the results may also be useful for other kind of cancers.

The constitutive properties of the brain paraenchyma Part 2. Fractional derivative approach.

Fractional models have proven to be very useful for studying viscoelastic materials. We consider the fractional Zener model (also called four-parameter model) to study both the relaxation function and creep compliance. The analytical results are compared with the known experimental results of the human brain tissue to obtain the best fit and brain mechanical parameters. The results are also compared to the non-fractional Zener model and four-parameter Burgers model, indicating that the four-parameter fractional model gives a substantially better fit for the all experimental data.

The constitutive properties of the brain parenchyma Part 1. Strain energy approach.

In this paper we study several constitutive equations for the brain based on the strain energy density function. We use the polynomial function and hyper-elastic Ogden model for the strain energy and include the energy dissipation by a Prony series expansion. The models are compared with known unconfined compression experimental results of the human brain tissue to obtain the best fitted model and brain mechanical parameters. Finite element simulations are also performed using the given constitutive equations, and numerical solutions match the analytical results very closely. The results are compared with other analytical and numerical calculations.

Frequency dependence of complex moduli of brain tissue using a fractional Zener model.

Brain tissue exhibits viscoelastic behaviour. If loading times are substantially short, static tests are not sufficient to determine the complete viscoelastic behaviour of the material, and dynamic test methods are more appropriate. The concept of complex modulus of elasticity is a powerful tool for characterizing the frequency domain behaviour of viscoelastic materials. On the other hand, it is well known that classical viscoelastic models can be generalized by means of fractional calculus to describe more complex viscoelastic behaviour of materials. In this paper, the fractional Zener model is investigated in order to describe the dynamic behaviour of brain tissue. The model is fitted to experimental data of oscillatory shear tests of bovine brain tissue to verify its behaviour and to obtain the material parameters.

Brain biomechanics: mathematical modeling of hydrocephalus.

The considerable amount of literature on mathematical models of hydrocephalus and other brain abnormalities is critically reviewed. These models have various degrees of mathematical sophistication, and have influenced not only the diagnosis of hydrocephalus, but also its treatment with CSF shunts. The mathematical models are classified into two classes, pressure-volume models, and consolidation models. Advantages and disadvantages of both types are pointed out with a view to removing the confusion frequently generated by the technical aspects of the subject. The conclusion is reached that, while none of the current models are good enough to be of immediate use to the neurosurgeon, mathematical models are likely in the future to be a powerful tool for the understanding and the treatment of hydrocephalus, as well as other conditions related to brain biomechanics. The amount of mathematics has been kept to the absolute minimum, but it is cited and appended for those who would like to dig further into this fascinating area of research.

Correlation of survival time with size of axonal swellings in diffuse axonal injury.

Widespread damage to axons in the white matter of the brain is a well-recognised consequence of non-missile head injury. This diffuse axonal injury is characterised by a gradual swelling of the axon associated with an accumulation of cellular organelles and proteins. We have investigated the relationship between the size of the swellings of the damaged axon with survival time in post-mortem brain tissue. Sixty-six cases of head injury with known length of post-traumatic survival were selected for study, and immunohistochemistry for beta-amyloid precursor protein (betaAPP) was carried out. The minimum diameter of the betaAPP-immunolabelled damaged axons was measured in micrometers using the IBAS image analysis system. There was a strong, positive and significant relationship between the mean size of axonal swelling and survival time which plateaued at around 85 h post injury. With longer survival times the situation becomes more complex. betaAPP immunolabelling of damaged axons can contribute evidence about trauma and post-injury survival time in the forensic setting but should always be assessed with other evidence.

Death from diesel fumes.

A rare fatality from inhalation of diesel motor exhaust fumes is reported and the toxicity of diesel fumes is discussed briefly.