Modeling of bone adaptative behavior based on cells activities

Bone remodeling occurs in an adult’s skeleton to adapt its architecture to external loadings. This involves bone resorption by osteoclasts cells followed by formation of new bone by osteoblasts cells. During bone remodeling, osteoclasts and osteoblasts interact with each other by expressing autocrine and paracrine factors that regulate cells’ population. Therefore, changes in bone density depend on the amount of each acting cell population. The aim of this paper is to propose a model for the bone remodeling process, which takes into account the opposite activity of both types of cells. For this purpose, a system of differential equations, proposed by Komarova et al. (Bone 33:206–215, 2003), is introduced to describe bone cell interactions using parameters which characterize the autocrine and paracrine factors. Such equations allow us to determine how the autocrine and paracrine factors vary in response to an external stimulus. It is assumed that an equilibrium state can be obtained for values of stimulus near to some reference quantity. Far from this value, unbalanced activity of osteoblasts and osteoclasts is observed, which leads to bone apposition or resorption. The proposed model has been implemented into the finite element software ABAQUS to analyze the qualitative response of a bone structure when subjected to certain mechanical loadings. Obtained results are satisfactory and in accordance with the expected bone remodeling behavior.     

A mathematical model of cell-to-cell spread of HIV-1 that includes a time delay

 We consider a two-dimensional model of cell-to-cell spread of HIV-1 in tissue cultures, assuming that infection is spread directly from infected cells to healthy cells and neglecting the effects of free virus. The intracellular incubation period is modeled by a gamma distribution and the model is a system of two differential equations with distributed delay, which includes the differential equations model with a discrete delay and the ordinary differential equations model as special cases. We study the stability in all three types of models. It is shown that the ODE model is globally stable while both delay models exhibit Hopf bifurcations by using the (average) delay as a bifurcation parameter. The results indicate that, differing from the cell-to-free virus spread models, the cell-to-cell spread models can produce infective oscillations in typical tissue culture parameter regimes and the latently infected cells are instrumental in sustaining the infection. Our delayed cell-to-cell models may be applicable to study other types of viral infections such as human T-cell leukaemia virus type 1 (HTLV-1). Received: 18 November 2000 / Published online: 28 February 2003 RID="*" ID="*" Research was partially supported by the NSERC and MITACS of Canada and a start-up fund from the College of Arts and Sciences at the University of Miami. On leave from Dalhousie University, Halifax, Nova Scotia, Canada. Current address: Department of Mathematics, Clarke College, Dubuque, Iowa 52001, USA Key words or phrases: HIV-1 – Cell-to-cell spread – Time delay – Stability – Hopf bifurcation – Periodicity     

Simple testing procedures for the Holling type II model

In this paper, we consider predator–prey data that can be viewed as solutions to a planar system of ordinary differential equations (ODE) observed with random error. The ODE system admits a limit cycle, while the random error is supposed to act additively in the log-scale. One of the oldest such systems is Holling’s type II model. In spite of its simplicity, it is still very popular in data analyses, although more sophisticated models have been introduced in the literature. We propose a simple way of deciding whether a set of predator–prey pairs is indicative or not of a departure from this basic model by exploiting the geometric properties of the solution in the phase plane. To illustrate our method, we use simulated and real data.     

An Efficient Algorithm for Some Highly Nonlinear Fractional PDEs in Mathematical Physics

In this paper, a fractional complex transform (FCT) is used to convert the given fractional partial differential equations (FPDEs) into corresponding partial differential equations (PDEs) and subsequently Reduced Differential Transform Method (RDTM) is applied on the transformed system of linear and nonlinear time-fractional PDEs. The results so obtained are re-stated by making use of inverse transformation which yields it in terms of original variables. It is observed that the proposed algorithm is highly efficient and appropriate for fractional PDEs and hence can be extended to other complex problems of diversified nonlinear nature.     

Lyapunov functions for Lotka–Volterra predator–prey models with optimal foraging behavior

 The theory of optimal foraging predicts abrupt changes in consumer behavior which lead to discontinuities in the functional response. Therefore population dynamical models with optimal foraging behavior can be appropriately described by differential equations with discontinuous right-hand sides. In this paper we analyze the behavior of three different Lotka–Volterra predator–prey systems with optimal foraging behavior. We examine a predator–prey model with alternative food, a two-patch model with mobile predators and resident prey, and a two-patch model with both predators and prey mobile. We show that in the studied examples, optimal foraging behavior changes the neutral stability intrinsic to Lotka–Volterra systems to the existence of a bounded global attractor. The analysis is based on the construction and use of appropriate Lyapunov functions for models described by discontinuous differential equations. Received: 23 March 1999     

Vigorous allograft rejection in the absence of danger

Tolerance to self is a necessary attribute of the immune system. It is thought that most autoreactive T cells are deleted in the thymus during the process of negative selection. However, peripheral tolerance mechanisms also exist to prevent development of autoimmune diseases against peripheral self-Ags. It has been proposed that T cells develop tolerance to peripheral self-Ags encountered in the absence of inflammation or "danger" signals. We have used immunodeficient Rag 1-/- mice to study the response of T cells to neo-self peripheral Ags in the form of well-healed skin and vascularized cardiac allografts. In this paper we report that skin and cardiac allografts without evidence of inflammation are vigorously rejected by transferred T cells or when recipients are reconstituted with T cells at a physiologic rate by nude bone graft transplantation. These results provide new insights into the role of inflammation or "danger" in the initiation of T cell-dependent immune responses. These findings also have profound implications in organ transplantation and suggest that in the absence of central deletional tolerance, peripheral tolerance mechanisms are not sufficient to inhibit alloimmune responses even in the absence of inflammation or danger.     

Coexistence of different serotypes of dengue virus

 We formulate a non–linear system of differential equations that models the dynamics of dengue fever. This disease is produced by any of the four serotypes of dengue arbovirus. Each serotype produces permanent immunity to it, but only a certain degree of cross–immunity to heterologous serotypes. In our model we consider the relation between two serotypes. Our interest is to analyze the factors that allow the invasion and persistence of different serotypes in the human population. Analysis of the model reveals the existence of four equilibrium points, which belong to the region of biological interest. One of the equilibrium points corresponds to the disease–free state, the other three equilibria correspond to the two states where just one serotype is present, and the state where both serotypes coexist, respectively. We discuss conditions for the asymptotic stability of equilibria, supported by analytical and numerical methods. We find that coexistence of both serotypes is possible for a large range of parameters. Received: 7 July 1998 / Revised version: 12 July 2002 / Published online: 26 September 2002 Keywords or phrases: Dengue fever – Primary and secondary infections – Serotype – Coexistence – Threshold – Basic reproduction number – Persistence     

Complex dynamic behaviour of autonomous microbial food chains

 The dynamic behaviour of food chains under chemostat conditions is studied. The microbial food chain consists of substrate (non-growing resources), bacteria (prey), ciliates (predator) and carnivore (top predator). The governing equations are formulated at the population level. Yet these equations are derived from a dynamic energy budget model formulated at the individual level. The resulting model is an autonomous system of four first-order ordinary differential equations. These food chains resemble those occuring in ecosystems. Then the prey is generally assumed to grow logistically. Therefore the model of these systems is formed by three first-order ordinary differential equations. As with these ecosystems, there is chaotic behaviour of the autonomous microbial food chain under chemostat conditions with biologically relevant parameter values. It appears that the trajectories on the attractors consists of two superimposed oscillatory behaviours, a slow one for predator–top predator and a fast one for the prey–predator on one branch at which the top predator increases slowly. In some regions of the parameter space there are multiple attractors. Received 8 November 1995; received in revised form 7 January 1997     

Function and dysfunction of CD4+ T cells in the immune response to melanoma

A major difficulty for tumor immunotherapy derives from the phenomenon that the encounter of the immune system with an antigen does not necessarily result in activation, but may also be followed by the induction of tolerance either by anergy or physical deletion. It is well established that the immune system becomes alerted only in the face of danger, i.e. upon ligand recognition in the context of increased expression of costimulatory molecules, adhesion molecules, and MHC molecules on antigen-presenting cells (APC). The pivotal role of CD4⁺ T lymphocytes in this process has been established. However, encounter of CD4⁺ T cells with either MHC class II-expressing melanoma cells or certain tumor antigen-presenting APC has been reported to induce antigen-specific tolerance. Thus, as more is learned about the molecular regulation of immune responses and the role of CD4⁺ T cells in particular, additional strategies to block inhibitory pathways of T-cell activation will be developed. Such strategies are likely to be based on a modulation of the context in which antigen is encountered by the immune system, e.g. in situ cytokine therapy, induction of costimulatory molecules or the simulation of `danger' signals. Received: 20 March 1999 / Accepted: 3 May 1999     

On fractional order models for Hepatitis C

In this paper we present a fractional order generalization of Perelson et al. basic hepatitis C virus (HCV) model including an immune response term. We argue that fractional order equations are more suitable than integer order ones in modeling complex systems which include biological systems. The model is presented and discussed. Also we argue that the added immune response term represents some basic properties of the immune system and that it should be included to study longer term behavior of the disease.     

Development of a biologically-based controlled growth and differentiation model for developmental toxicology

 A mathematical model is developed with a highly controlled birth and death process for precursor cells. This model is both biologically- and statistically-based. The controlled growth and differentiation (CGD) model limits the number of replications allowed in the development of a tissue or organ and thus, more closely reflects the presence of a true stem cell population. Leroux et al. (1996) presented a biologically-based dose-response model for developmental toxicology that was derived from a partial differential equation for the generating function. This formulation limits further expansion into more realistic models of mammalian development. The same formulae for the probability of a defect (a system of ordinary differential equations) can be derived through the Kolmogorov forward equations due to the nature of this Markov process. This modified approach is easily amenable to the expansion of more complicated models of the developmental process such as the one presented here. Comparisons between the Leroux et al. (1996) model and the controlled growth and differentiation (CGD) model as developed in this paper are also discussed. Received: 8 June 2001 / Revised version: 15 June 2002 / Published online: 26 September 2002 Keywords or phrases: Teratology – Multistate process – Cellular kinetics – Numerical simulation     

Parametric and non-parametric modeling of short-term synaptic plasticity. Part I: Computational study

Parametric and non-parametric modeling methods are combined to study the short-term plasticity (STP) of synapses in the central nervous system (CNS). The nonlinear dynamics of STP are modeled by means: (1) previously proposed parametric models based on mechanistic hypotheses and/or specific dynamical processes, and (2) non-parametric models (in the form of Volterra kernels) that transforms the presynaptic signals into postsynaptic signals. In order to synergistically use the two approaches, we estimate the Volterra kernels of the parametric models of STP for four types of synapses using synthetic broadband input–output data. Results show that the non-parametric models accurately and efficiently replicate the input–output transformations of the parametric models. Volterra kernels provide a general and quantitative representation of the STP.     

Relevance of nonlinear lumped-parameter models in the analysis of depth-EEG epileptic signals

In the field of epilepsy, the analysis of stereoelectroencephalographic (SEEG, intra-cerebral recording) signals with signal processing methods can help to better identify the epileptogenic zone, the area of the brain responsible for triggering seizures, and to better understand its organization. In order to evaluate these methods and to physiologically interpret the results they provide, we developed a model able to produce EEG signals from “organized” networks of neural populations. Starting from a neurophysiologically relevant model initially proposed by Lopes Da Silva et al. [Lopes da Silva FH, Hoek A, Smith H, Zetterberg LH (1974) Kybernetic 15: 27–37] and recently re-designed by Jansen et al. [Jansen BH, Zouridakis G, Brandt ME (1993) Biol Cybern 68: 275–283] the present study demonstrates that this model can be extended to generate spontaneous EEG signals from multiple coupled neural populations. Model parameters related to excitation, inhibition and coupling are then altered to produce epileptiform EEG signals. Results show that the qualitative behavior of the model is realistic; simulated signals resemble those recorded from different brain structures for both interictal and ictal activities. Possible exploitation of simulations in signal processing is illustrated through one example; statistical couplings between both simulated signals and real SEEG signals are estimated using nonlinear regression. Results are compared and show that, through the model, real SEEG signals can be interpreted with the aid of signal processing methods. Received: 3 January 2000 / Accepted: 24 March 2000     

Tumors and the danger model

This article reviews the evidence for the danger model in the context of immune response to tumors and the insufficiency of the immune system to eliminate tumor growth. Despite their potential immunogenicity tumors do not induce significant immune responses which could destroy malignant cells. According to the danger model, the immune surveillance system fails to detect tumor antigens because transformed cells do not send any danger signals which could activate dendritic cells and initiate an immune response. Instead, tumor cells or antigen presenting cells turn off the responding T cells and induce tolerance. The studies reviewed herein based on model tumor antigens, recombinant viral vectors and detection of tumor specific T cells by MHC/peptide tetramers underscore the critical role of tumor antigen presentation and the context in which it occurs. They indicate that antigen presentation only by activated but not by cancer or resting dendritic cells is necessary for the induction of immune responses to tumor antigens. It becomes apparent that the inability of dendritic cells to become activated provides a biological niche for tumor escape from immune destruction and seems to be a principal mechanism for the failure of tumor immune surveillance.     

P2 Receptors of microglia: sensors for danger signals in the CNS

Following tissue damage or invasion by pathogens a number of soluble signals are generated to alert the immune system of the impending danger and initiate inflammation. Some danger signals are released from injured or dying cells. Once released, danger signals activate a autocrine/paracrine network that recruits inflammatory cells, stimulates cytokine production, promotes dendritic cell maturations and increases the antigen (Ag) presenting efficiency. These events also occurs in the central nervous system (CNS) where cytokines and cytokine-releasing cells have a central role in spreading inflammation. P2 receptors of microglia are the focus of increasing interest, especially after they were shown to mediate chemotaxis, cytokine release and cell death in microglia. We propose that P2 receptors may function in microglia as sensors of the ATP/UTP concentration in the pericellular space, and therefore as sensors of danger signals in the CNS. Furthermore, microglia itself can release ATP when stimulated by inflammatory stimuli. Thus extracellular nucleotides may be included in the family of the early inflammatory mediators acting via P2 receptors to spread inflammation in the CNS.
References 1. Ferrari D., Villalba M., Chiozzi P., Falzoni S., Ricciardi-Castagnoli P. and Di Virgilio F. (1996) Mouse microglia cells express a plasma membrane pore gated by extracellular ATP. J. Immunol. 156 , 1531–1539.
2. Ferrari D., Chiozzi P., Falzoni S., Hanau S. and Di Virgilio F. (1997) Purinergic modulation of interleukin-1B release from microglia cells stimulated with bacterial endotoxin. J. Exp. Med. 185 , 579–582.     

A sense of danger from radiation

Tissue damage caused by exposure to pathogens, chemicals and physical agents such as ionizing radiation triggers production of generic "danger" signals that mobilize the innate and acquired immune system to deal with the intrusion and effect tissue repair with the goal of maintaining the integrity of the tissue and the body. Ionizing radiation appears to do the same, but less is known about the role of "danger" signals in tissue responses to this agent. This review deals with the nature of putative "danger" signals that may be generated by exposure to ionizing radiation and their significance. There are a number of potential consequences of "danger" signaling in response to radiation exposure. "Danger" signals could mediate the pathogenesis of, or recovery from, radiation damage. They could alter intrinsic cellular radiosensitivity or initiate radioadaptive responses to subsequent exposure. They may spread outside the locally damaged site and mediate bystander or "out-of-field" radiation effects. Finally, an important aspect of classical "danger" signals is that they link initial nonspecific immune responses in a pathological site to the development of specific adaptive immunity. Interestingly, in the case of radiation, there is little evidence that "danger" signals efficiently translate radiation-induced tumor cell death into the generation of tumor-specific immunity or normal tissue damage into autoimmunity. The suggestion is that radiation-induced "danger" signals may be inadequate in this respect or that radiation interferes with the generation of specific immunity. There are many issues that need to be resolved regarding "danger" signaling after exposure to ionizing radiation. Evidence of their importance is, in some areas, scant, but the issues are worthy of consideration, if for no other reason than that manipulation of these pathways has the potential to improve the therapeutic benefit of radiation therapy. This article focuses on how normal tissues and tumors sense and respond to danger from ionizing radiation, on the nature of the signals that are sent, and on the impact on the eventual consequences of exposure.     

Inflammasomes in infection and inflammation

Two of the main challenges that multicellular organisms faced during evolution were to cope with invading microorganisms and eliminate and replace dying cells. Our innate immune system evolved to handle both tasks. Key aspects of innate immunity are the detection of invaders or tissue injury and the activation of inflammation that alarms the system through the action of cytokine and chemokine cascades. While inflammation is essential for host resistance to infections, it is detrimental when produced chronically or in excess and is linked to various diseases, most notably auto-immune diseases, auto-inflammatory disorders, cancer and septic shock. Essential regulators of inflammation are enzymes termed “the inflammatory caspases”. They are activated by cellular sensors of danger signals, the inflammasomes, and subsequently convert pro-inflammatory cytokines into their mature active forms. In addition, they regulate non-conventional protein secretion of alarmins and cytokines, glycolysis and lipid biogenesis, and the execution of an inflammatory form of cell death termed “pyroptosis”. By acting as key regulators of inflammation, energy metabolism and cell death, inflammatory caspases and inflammasomes exert profound influences on innate immunity and infectious and non-infectious inflammatory diseases. Christian R. McIntire and Garabet Yeretssian have contributed equally to this review.     

A Simple Mathematical Model Inspired by the Purkinje Cells: From Delayed Travelling Waves to Fractional Diffusion

Recently, several experiments have demonstrated the existence of fractional diffusion in the neuronal transmission occurring in the Purkinje cells, whose malfunctioning is known to be related to the lack of voluntary coordination and the appearance of tremors. Also, a classical mathematical feature is that (fractional) parabolic equations possess smoothing effects, in contrast with the case of hyperbolic equations, which typically exhibit shocks and discontinuities. In this paper, we show how a simple toy-model of a highly ramified structure, somehow inspired by that of the Purkinje cells, may produce a fractional diffusion via the superposition of travelling waves that solve a hyperbolic equation. This could suggest that the high ramification of the Purkinje cells might have provided an evolutionary advantage of “smoothing” the transmission of signals and avoiding shock propagations (at the price of slowing a bit such transmission). Although an experimental confirmation of the possibility of such evolutionary advantage goes well beyond the goals of this paper, we think that it is intriguing, as a mathematical counterpart, to consider the time fractional diffusion as arising from the superposition of delayed travelling waves in highly ramified transmission media. The case of a travelling concave parabola with sufficiently small curvature is explicitly computed. The new link that we propose between time fractional diffusion and hyperbolic equation also provides a novelty with respect to the usual paradigm relating time fractional diffusion with parabolic equations in the limit. This paper is written in such a way as to be of interest to both biologists and mathematician alike. In order to accomplish this aim, both complete explanations of the objects considered and detailed lists of references are provided.