Visualization and modelling of the thermal inactivation of bacteria in a model food.

A large number of incidents of food poisoning have been linked to undercooked meat products. The use of mathematical modelling to describe heat transfer within foods, combined with data describing bacterial thermal inactivation, may prove useful in developing safer food products while minimizing thermal overprocessing. To examine this approach, cylindrical agar blocks containing immobilized bacteria (Salmonella typhimurium and Brochothrix thermosphacta) were used as a model system in this study. The agar cylinders were subjected to external conduction heating by immersion in a water bath. They were then incubated, sliced open, and examined by image analysis techniques for regions of no bacterial growth. A finite-difference scheme was used to model thermal conduction and the consequent bacterial inactivation. Bacterial inactivation rates were modelled with values for the time required to reduce bacterial number by 90% (D) and the temperature increase required to reduce D by 90% taken from the literature. Model simulation results agreed well with experimental results for both bacteria, demonstrating the utility of the technique.     

Visualization and Modelling of the Thermal Inactivation of Bacteria in a Model Food

A large number of incidents of food poisoning have been linked to undercooked meat products. The use of mathematical modelling to describe heat transfer within foods, combined with data describing bacterial thermal inactivation, may prove useful in developing safer food products while minimizing thermal overprocessing. To examine this approach, cylindrical agar blocks containing immobilized bacteria (Salmonella typhimurium and Brochothrix thermosphacta) were used as a model system in this study. The agar cylinders were subjected to external conduction heating by immersion in a water bath. They were then incubated, sliced open, and examined by image analysis techniques for regions of no bacterial growth. A finite-difference scheme was used to model thermal conduction and the consequent bacterial inactivation. Bacterial inactivation rates were modelled with values for the time required to reduce bacterial number by 90% (D) and the temperature increase required to reduce D by 90% taken from the literature. Model simulation results agreed well with experimental results for both bacteria, demonstrating the utility of the technique.     

Transient behavior of a chemotaxis system modelling certain types of tissue inflammation

A spatially-distributed mathematical model for the inflammatory response to bacterial invasion of tissue is proposed which includes leukocyte motility and chemotaxis behavior and chemical mediator properties explicitly. This system involves three coupled nonlinear partial differential equations and so is not amenable to analysis. Using scaling arguments and singular perturbation techniques, an approximating system of two coupled nonlinear ordinary differential equations is developed. This system now permits analysis by phase plane methods. Using the approximating model, the dependence of the dynamic behavior of the inflammatory response upon key process parameters, including leukocyte chemotaxis, is studied.This work has been supported by the Deutsche Forschungsgemeinschaft     

Modeling capsule tissue growth around disk-shaped implants: a numerical and in vivo study

We propose a new mathematical model that describes the growth of fibrous tissue around rigid, disk-shaped implants. A solution methodology based on an efficient regularized iterative method is presented to calibrate the model from some measurements of the capsule tissue concentration. Numerical results obtained with synthetic data are presented to demonstrate the ability of the proposed solution methodology to determine the model parameters corresponding to a given implant. In addition, numerical results obtained with experimental data are presented to illustrate the validity of the proposed model.     

Theoretical study of the fibrous capsule tissue growth around a disk-shaped implant

We analyze the mathematical properties of the fibrous capsule tissue concentration around a disk-shaped implant. We establish stability estimates as well as monotonicity results that illustrate the sensitivity of this growth to the biocompatibility index parameters of the implant. In addition, we prove that the growth of the tissue increases exponentially in time toward an asymptotic regime. We also study the mathematical properties of the solution of the inverse problem consisting in the determination of the values of the biocompatibility index parameters from the knowledge of some fibrous capsule tissue measurements. We prove that this model calibration problem admits a unique solution, and establish a characterization of the index parameters. Furthermore, we demonstrate analytically that such a solution is not continuous with respect to the data, and therefore the considered inverse problem is ill-posed due to the lack of the stability requirement.     

Remarks on the Evolutionary Effect of Natural Selection

The so-called "Fundamental Theorem of Natural Selection", that the mean fitness of a population increases with time under natural selection, is known not to be true, as a mathematical theorem, when fitnesses depend on more than one locus. Although this observation may not have particular biological relevance, (so that mean fitness may well increase in the great majority of interesting situations), it does suggest that it is of interest to find an evolutionary result which is correct as a mathematical theorem, no matter how many loci are involved. The aim of the present note is to prove an evolutionary theorem relating to the variance in fitness, rather than the mean: this theorem is true for an arbitrary number of loci, as well as for arbitrary (fixed) fitness parameters and arbitrary linkage between loci. Connections are briefly discussed between this theorem and the principle of quasi-linkage equilibrium.     

Weak ergodicity of population evolution processes

The weak ergodic theorems of mathematical demography state that the age distribution of a closed population is asymptotically independent of the initial distribution. In this paper, we provide a new proof of the weak ergodic theorem of the multistate population model with continuous time. The main tool to attain this purpose is a theory of multiplicative processes, which was mainly developed by Garrett Birkhoff, who showed that ergodic properties generally hold for an appropriate class of multiplicative processes. First, we construct a general theory of multiplicative processes on a Banach lattice. Next, we formulate a dynamical model of a multistate population and show that its evolution operator forms a multiplicative process on the state space of the population. Subsequently, we investigate a sufficient condition that guarantees the weak ergodicity of the multiplicative process. Finally, we prove the weak and strong ergodic theorems for the multistate population and resolve the consistency problem.     

Modeling of laser coagulation of tissue with MRI temperature monitoring

Light energy from a laser source that is delivered into body tissue via a fiber-optic probe with minimal invasiveness has been used to ablate solid tumors. This thermal coagulation process can be guided and monitored accurately by continuous magnetic resonance imaging (MRI) since the laser energy delivery system does not interfere with MRI. This report deals with mathematical modeling and analysis of laser coagulation of tissue. This model is intended for "real-time" analysis of magnetic resonance images obtained during the coagulation process to guide clinical treatment. A mathematical model is developed to simulate the thermal response of tissue to a laser light heating source. For fast simulation, an approximate solution of the thermal model is used to predict the dynamics of temperature distribution and tissue damage induced by a laser energy line source. The validity of these simulations is tested by comparison with MRI-based temperature data acquired from in vivo experiments in rabbits. The model-simulated temperature distribution and predicted lesion dynamics correspond closely with MRI-based data. These results demonstrate the potential for using this combination of fast modeling and MRI technologies during laser heating of tissue for online prediction of tumor lesion size during laser heating.     

Analysis of a lumped model for tissue inflammation dynamics

The inflammatory response to bacterial invasion of tissue is a complex combination of chemical and physical events, and the outcome of a particular challenge is dependent upon the interaction of these. Thus a wide variety of behavior can be observed for the operation of this response. In this paper we develop and analyze a mathematical model for a generalized inflammatory response to bacterial invasion of a tissue region assumed to be homogeneous on a macroscopic scale, in order to study the dynamical behavior of the system. Our analysis allows interpretation of the outcome of a challenge in terms of key parameters representing the rate processes involved. It demonstrates how abnormalities in these processes can lead to pathological behavior. Numerical values for the parameters are estimated from experimental literature sources, and used in example computations to illustrate the predictions of the model under physiological conditions.     

The evolution of virulence in vector-borne and directly transmitted parasites

Ewald (1994) has suggested that vector-borne parasites are expected to evolve a higher level of host exploitation than directly transmitted parasites, and this should thereby result in them being more virulent. Indeed, some data do conform to this general pattern. Nevertheless, his hypothesis has generated some debate about the extent to which it is valid. I explore this issue quantitatively within the framework of mathematical epidemiology. In particular, I present a dynamic optimization model for the evolution of parasite replication strategies that explicitly explores the validity of this hypothesis. A few different model assumptions are explored and it is found that Ewald's hypothesis has only qualified support as a general explanation for why vector-borne parasites are more virulent than those that are directly transmitted. I conclude by suggesting that an alternative explanation might lie in differences in inoculum size between these two types of transmission.     

Interpretation of the Ussing flux ratio from the fluctuation theorem

The fluctuation theorem gives a mathematical expression to quantify the probability of observing events violating the second law of thermodynamics in a small system over a short period of time. The theorem predicts the ratio of forward (entropy-producing) runs to the backward (entropy-consuming) runs for a nanometer-sized molecular machine in a nonequilibrium system. However, few experimental verifications of the theorem have been carried out. In this paper, I show that the Ussing flux ratio, the ratio of outward to inward unidirectional ion fluxes across a membrane channel, can be derived from the fluctuation theorem if we consider the ion channel and the contacting solutions as a small nonequilibrium system. The entropy change due to ion electrodiffusion is expressed from the fundamental equation for the entropy change. Thus, the empirical flux ratio equation can be interpreted from the more general fluctuation theorem, and serves as a verification of the theorem.     

A stochastic model and a functional central limit theorem for information processing in large systems of neurons

The paper deals with information transmission in large systems of neurons. We model the membrane potential in a single neuron belonging to a cell tissue by a non time-homogeneous Cox-Ingersoll-Ross type diffusion; in terms of its time-varying expectation, this stochastic process can convey deterministic signals. We model the spike train emitted by this neuron as a Poisson point process compensated by the occupation time of the membrane potential process beyond the excitation threshold. In a large system of neurons 1≤i≤N processing independently the same deterministic signal, we prove a functional central limit theorem for the pooled spike train collected from the N neurons. This pooled spike train allows to recover the deterministic signal, up to some shape transformation which is explicit.     

Evaluation of Analytical Techniques to Predict Viability after Cryopreservation

Preservation of plant germplasm is important to safeguard biodiversity and to store elite plants. Cryopreservation is one of the possible preservation techniques. Research for a cryopreservation protocol is often inefficient because of slow or poor regrowth of plant material. Therefore, at least one technique, that allows a quick and accurate prognosis of viability after cryopreservation, is required. We evaluated five techniques: electrolyte leakage, triphe-nyltetrazoliumchloride (TTC) staining (visual and spectrophotometrical analysis), malondialdehyde concentrations in plant tissue and a mathematical model that relates ‘water content’ to the weight of encapsulated plant material. Electrolyte leakage and TTC-staining (if visually analysed) are efficient to predict viability. Our mathematical model allows us to save time and plant material in order to develop an efficient encapsulation—dehydration protocol. All other techniques were rejected because of the high variability of the results. This is due to the variability of biochemical activity in plant tissue and the small amount of tissue used in the experiments.     

Modelling the fracture-healing process as a moving-interface problem using an interface-capturing approach

We present a novel numerical model of the fracture-healing process using interface-capturing techniques, a well-known approach from fields like fluid dynamics, to describe tissue growth. One advantage of this method is its direct connection to experimentally observable parameters, including tissue-growth velocities. In our model, osteogenesis, chondrogenesis and revascularisation are triggered by mechanical stimuli via mechano-transduction based on previously established hypothesis of Claes and Heigele. After experimentally verifying the convergence of the numerical method, we compare the predictions of our model with those of the already established Ulm bone-healing model, which serves as a benchmark, and corroborate our results with existing animal experiments. We demonstrate that the new model can predict the history of the interfragmentary movement and forecast a tissue evolution that appears similar to the experimental results. Furthermore, we compare the relative tissue concentration in the healing domain with outcomes of animal experiments. Finally, we discuss the possible application of the model to new fields, where numerical simulations could also prove beneficial.     

Assembly systems in molecular biology: A graph theory of genetic complementation between two related species

In this report I discuss the interpretation of the data of genetic complementation in vivo of assembly systems between two related species, such as the complementation of structural proteins between related bacteriophages. It is suggested that such experiments reveal interactions between gene products that are overlooked in many other experiments. A mathematical model based on the graph theory is presented, assuming that the assembly system consists of gene products and interactions between them. The model shows that information from the complementation experiment is limited to those interactions which are mismatching if the interacting gene products are produced by different species. Moreover, it shows that the number and positions of mismatching interactions cannot always be determined uniquely by the data of complementation. However, there is a mathematical method by which one can calculate all the possible solutions for the number and positions of mismatching interactions from experimental data. Actual calculation is performed for a simple example. Thus, the model clarifies validity and limitation of complementation experimiments between related species.     

Rheological properties of myometrium: experimental quantification and mathematical modeling

This work answers some questions related to detection of rheological properties of soft tissues exemplified in myometrium, stressed by external tensile force. In the first stage of the experiment the tissue samples were ciclically stressed and response loops were recorded. This test proved severe plastical deformation of samples, which is not usually being stated for living tissues. In addition to course, growth and stabilizing this deformation also energetical losses of individual hysteresis loops of the response were evaluated. In the second stage of the experiment the tissue samples were exposed to a loading force changed in step-wise manner in four steps. The sample response to each force step was processed and evaluated separately to obtain basic properties of used model. In next step, the changes in model characteristics were obtained and evaluated for each element in subsequent force steps. By reason of following easier interpretation, the quite simple visco-elastic model, defined by differential equation with analytic solution, is used. The results prove necessary to introduce in model both spring and damper constants dependent on the magnitude of the loading force and one damper with even time dependent constant. The interindividual variability of characteristic values of the model elements is surprisingly low. On the other side, they are strongly dependent on load magnitude. Complete mathematical model of uterine wall tissue is obtained by amending the principal equation by formulas describing changes in individual components of the model.     

Propagation of soft tissue artifacts to the center of rotation: A model for the correction of functional calibration techniques

This paper presents a mathematical model for the propagation of errors in body segment kinematics to the location of the center of rotation. Three functional calibration techniques, usually employed for the gleno-humeral joint, are studied: the methods based on the pivot of the instantaneous helical axis (PIHA) or the finite helical axis (PFHA), and the “symmetrical center of rotation estimation” (SCoRE). A procedure for correcting the effect of soft tissue artifacts is also proposed, based on the equations of those techniques and a model of the artifact, like the one that can be obtained by double calibration. An experiment with a mechanical analog was performed to validate the procedure and compare the performance of each technique. The raw error (between 57 and 68 mm) was reduced by a proportion of between 1:6 and less than 1:15, depending on the artifact model and the mathematical method. The best corrections were obtained by the SCoRE method. Some recommendations about the experimental setup for functional calibration techniques and the choice of a mathematical method are derived from theoretical considerations about the formulas and the results of the experiment.     

Mathematical approaches to differentiation and gene regulation

We consider some mathematical issues raised by the modelling of gene networks. The expression of genes is governed by a complex set of regulations, which is often described symbolically by interaction graphs. These are finite oriented graphs where vertices are the genes involved in the biological system of interest and arrows describe their interactions: a positive (resp. negative) arrow from a gene to another represents an activation (resp. inhibition) of the expression of the latter gene by some product of the former. Once such an interaction graph has been established, there remains the difficult task to decide which dynamical properties of the gene network can be inferred from it, in the absence of precise quantitative data about their regulation. There mathematical tools, among others, can be of some help. In this paper we discuss a rule proposed by Thomas according to which the possibility for the network to have several stationary states implies the existence of a positive circuit in the corresponding interaction graph. We prove that, when properly formulated in rigorous terms, this rule becomes a theorem valid for several different types of formal models of gene networks. This result is already known for models of differential [C. Soulé, Graphic requirements for multistationarity, ComPlexUs 1 (2003) 123-133] or Boolean [E. Rémy, P. Ruet, D. Thieffry, Graphic requirements for multistability and attractive cycles in a boolean dynamical framework, 2005, Preprint] type. We show here that a stronger version of it holds in the differential setup when the decay of protein concentrations is taken into account. This allows us to verify also the validity of Thomas' rule in the context of piecewise-linear models. We then discuss open problems.