A mathematical model of biological evolution

In order to understand generally how the biological evolution rate depends on relevant parameters such as mutation rate, intensity of selection pressure and its persistence time, the following mathematical model is proposed: dN _n (t)/dt=(m _n (t-)N _n (t)+N _n-1(t) (n=0,1,2,3...), where N _n (t) and m _n (t) are respectively the number and Malthusian parameter of replicons with step number n in a population at time t and is the mutation rate, assumed to be a positive constant. The step number of each replicon is defined as either equal to or larger by one than that of its parent, the latter case occurring when and only when mutation has taken place. The average evolution rate defined by is rigorously obtained for the case (i) m _n (t)=m _n is independent of t (constant fitness model), where m _n is essentially periodic with respect to n, and for the case (ii) (periodic fitness model), together with the long time average m of the average Malthusian parameter . The biological meaning of the results is discussed, comparing them with the features of actual molecular evolution and with some results of computer simulation of the model for finite populations.An early version of this study was read at the International Symposium on Mathematical Topics in Biological held in kyoto, Japan, on September 11–12, 1978, and was published in its Procedings.     

A generalized mathematical model of biological oscillators

Biological rhythms such as circadian rhythms, biochemical rhythms and neural oscillators are based on the mathematical model of the theory of harmonic oscillators. These are solutions of certain second-order differential equations. They can also be viewed as spherical harmonics on the circle in the two-dimensional Euclidean space. The spherical harmonics on (n-1)-spheres and, more generally, the Stiefel harmonics can represent oscillatory phenomena, and we expect that they can serve as models for more complex biological rhythms.     

Effect of oxygen tension and rate of pressure reduction during decompression on central gas bubbles

Reinertsen, R. E., V. Flook, S. Koteng, and A. O. Brubakk.Effect of oxygen tension and rate of pressure reduction duringdecompression on central gas bubbles. J. Appl.Physiol. 84(1): 351-356, 1998.Reduction inascent speed and an increase in theO₂ tension in the inspired airhave been used to reduce the risk for decompression sickness. It haspreviously been reported that decompression speed andO₂ partial pressure are linearly related for human decompressions from saturation hyperbaric exposures. The constant of proportionality K(K = rate/partial pressure of inspiredO₂) indicates the incidence ofdecompression sickness. The present study investigated the relationshipamong decompression rate, partial pressure of inspiredO₂, and the number of central gasbubbles after a 3-h dive to 500 kPa while breathing nitrox with an O₂ content of 35 kPa. Weused transesophageal ultrasonic scanning to determine the number ofbubbles in the pulmonary artery of pigs. The results show that, for agiven level of decompression stress, decompression rate andO₂ tension in the inspired air canbe traded off against each other by using pulmonary artery bubbles asan end point. The results also seem to confirm that decompressions thathave a high K value are morestressful.

     

A phase-field model for fracture in biological tissues

This work presents a recently developed phase-field model of fracture equipped with anisotropic crack driving force to model failure phenomena in soft biological tissues at finite deformations. The phase-field models present a promising and innovative approach to thermodynamically consistent modeling of fracture, applicable to both rate-dependent or rate-independent brittle and ductile failure modes. One key advantage of diffusive crack modeling lies in predicting the complex crack topologies where methods with sharp crack discontinuities are known to suffer. The starting point is the derivation of a regularized crack surface functional that \({\varGamma }\)-converges to a sharp crack topology for vanishing length-scale parameter. A constitutive balance equation of this functional governs the crack phase-field evolution in a modular format in terms of a crack driving state function. This allows flexibility to introduce alternative stress-based failure criteria, e.g., isotropic or anisotropic, whose maximum value from the deformation history drives the irreversible crack phase field. The resulting multi-field problem is solved by a robust operator split scheme that successively updates a history field, the crack phase field and finally the displacement field in a typical time step. For the representative numerical simulations, a hyperelastic anisotropic free energy, typical to incompressible soft biological tissues, is used which degrades with evolving phase field as a result of coupled constitutive setup. A quantitative comparison with experimental data is provided for verification of the proposed methodology.     

A mathematical model for the freezing process in biological tissue

A mathematical model has been developed to study the process of freezing in biological organs. The model consists of a repetitive unit structure comprising a cylinder of tissue with an axial blood vessel (Krogh cylinder) and it is analysed by the methods of irreversible thermodynamics. The mathematical simulation of the freezing process in liver tissue compares remarkably well with experimental data on the structure of tissue frozen under controlled thermal conditions and the response of liver cells to changes in cooling rate. The study also supports the proposal that the damage mechanism responsible for the lack of success in attempts to preserve tissue in a frozen state, under conditions in which cells in suspension survive freezing, is direct mechanical damage caused by the formation of ice in the vascular system.     

Towards an analytical model of soft biological tissues

In the past years, soft-tissue modelling research has seen substantial developments, a significant part of which can be ascribed to the refinement of numerical techniques, such as Finite Element analysis. A large class of physico-mechanical properties can be effectively simulated and predictions can be made for a variety of phenomena. However, there is still much that can be conceptually explored by means of fundamental theoretical analysis. In the past few years, driven by our interest in articular cartilage mechanics, we have developed theoretical microstructural models for linear elasticity and permeability that accounted for the presence and arrangement of collagen fibres in cartilage. In this paper, we investigate analytically the non-linear elasticity of soft tissues with collagen fibres arranged according to a given distribution of orientation, a problem that, aside from the case of fibres aligned in a finite number of distinct directions, has been treated exclusively numerically in the literature. We show that, for the case of a tissue with complex fibre arrangement, such as articular cartilage, the theoretical framework commonly used leads to an integral expression of the elastic strain energy potential. The present model is a first attempt in the development of a unified analytical microstructural model for non-linear elasticity and permeability of hydrated, fibre-reinforced soft tissues.     

A mathematical model of alveolar gas exchange in partial liquid ventilation

In partial liquid ventilation (PLV), perfluorocarbon (PFC) acts as a diffusion barrier to gas transport in the alveolar space since the diffusivities of oxygen and carbon dioxide in this medium are four orders of magnitude lower than in air. Therefore convection in the PFC layer resulting from the oscillatory motions of the alveolar sac during ventilation can significantly affect gas transport. For example, a typical value of the Péclet number in air ventilation is Pe approximately 0.01, whereas in PLV it is Pe approximately 20. To study the importance of convection, a single terminal alveolar sac is modeled as an oscillating spherical shell with gas, PFC, tissue and capillary blood compartments. Differential equations describing mass conservation within each compartment are derived and solved to obtain time periodic partial pressures. Significant partial pressure gradients in the PFC layer and partial pressure differences between the capillary and gas compartments (P(C)-Pg) are found to exist. Because Pe> 1, temporal phase differences are found to exist between P(C)-Pg and the ventilatory cycle that cannot be adequately described by existing non-convective models of gas exchange in PLV The mass transfer rate is nearly constant throughout the breath when Pe>1, but when Pe<1 nearly 100% of the transport occurs during inspiration. A range of respiratory rates (RR), including those relevant to high frequency oscillation (HFO) +PLV, tidal volumes (V(T)) and perfusion rates are studied to determine the effect of heterogeneous distributions of ventilation and perfusion on gas exchange. The largest changes in P(C)O2 and P(C)CO2 occur at normal and low perfusion rates respectively as RR and V(T) are varied. At a given ventilation rate, a low RR-high V(T) combination results in higher P(C)O2, lower P(C)CO2 and lower (P(C)-Pg) than a high RR-low V(T) one.     

A model for influence of exercise on formation and growth of tissue bubbles during altitude decompression

In response to exercise performed before or after altitude decompression, physiological changes are suspected to affect the formation and growth of decompression bubbles. We hypothesized that the work to change the size of a bubble is done by gas pressure gradients in a macro- and microsystem of thermodynamic forces and that the number of bubbles formed through time follows a Poisson process. We modeled the influence of tissue O(2) consumption on bubble dynamics in the O(2) transport system in series against resistances, from the alveolus to the microsystem containing the bubble and its surrounding tissue shell. Realistic simulations of experimental decompression procedures typical of actual extravehicular activities were obtained. Results suggest that exercise-induced elevation of O(2) consumption at altitude leads to bubble persistence in tissues. At the same time, exercise-enhanced perfusion leads to an overall suppression of bubble growth. The total volume of bubbles would be reduced unless increased tissue motion simultaneously raises the rate of bubble formation through cavitation processes, thus maintaining or increasing total bubble volume, despite the exercise.     

Quantitative analysis of volatile halothane metabolites in biological tissues by gas chromatography

A simple and sensitive gas chromatographic method for the determination of 2-chloro-1, 1-difluoroethylene (CDE) and 2-chloro-1,1,1-trifluoroethane (CTE), two highly volatile metabolites of halothane, in blood, liver and isolated hepatic microscomes is described. The entire head-space in equilibrium with a known volume or weight of the sample is injected into the gas chromatograph equipped with a flame ionization detector. Quantification is accomplished with standards prepared by fortifying blank samples with known concentrations of CDE and CTE which are treated under the same conditions as the samples. Detection limits for CDE and CTE were 2 pmole/ml in blood and 10 pmole/g in liver and the mean relative standard deviations are no greater than ± 6% except for CTE in hepatic microsomes (± 9%). A preliminary study of blood CDE and CTE levels in humans anesthetized with halothane is reported.     

A variational constitutive model for soft biological tissues

In this paper, a fully variational constitutive model of soft biological tissues is formulated in the finite strain regime. The model includes Ogden-type hyperelasticity, finite viscosity, deviatoric and volumetric plasticity, rate and microinertia effects. Variational updates are obtained via time discretization and pre-minimization of a suitable objective function with respect to internal variables. Genetic algorithms are used for model parameter identification due to their suitability for non-convex, high dimensional optimization problems. The material behavior predicted by the model is compared to available tests on swine and human brain tissue. The ability of the model to predict a wide range of experimentally observed behavior, including hysteresis, cyclic softening, rate effects, and plastic deformation is demonstrated.