Analysis of a mathematical model for the growth of tumors

 In this paper we study a recently proposed model for the growth of a nonnecrotic, vascularized tumor. The model is in the form of a free-boundary problem whereby the tumor grows (or shrinks) due to cell proliferation or death according to the level of a diffusing nutrient concentration. The tumor is assumed to be spherically symmetric, and its boundary is an unknown function r=s(t). We concentrate on the case where at the boundary of the tumor the birth rate of cells exceeds their death rate, a necessary condition for the existence of a unique stationary solution with radius r=R ₀ (which depends on the various parameters of the problem). Denoting by c the quotient of the diffusion time scale to the tumor doubling time scale, so that c is small, we rigorously prove that (i) lim inf _t→∞ s(t)>0, i.e. once engendered, tumors persist in time. Indeed, we further show that (ii) If c is sufficiently small then s(t)→R ₀ exponentially fast as t→∞, i.e. the steady state solution is globally asymptotically stable. Further, (iii) If c is not “sufficiently small” but is smaller than some constant γ determined explicitly by the parameters of the problem, then lim sup _t→∞ s(t)<∞; if however c is “somewhat” larger than γ then generally s(t) does not remain bounded and, in fact, s(t)→∞ exponentially fast as t→∞. Received: 25 February 1998 / Revised version: 30 April 1998     

Interaction of maturation delay and nonlinear birth in population and epidemic models

 A population with birth rate function B(N) N and linear death rate for the adult stage is assumed to have a maturation delay T>0. Thus the growth equation N′(t)=B(N(t−T)) N(t−T) e⁻ d ₁ T−dN(t) governs the adult population, with the death rate in previous life stages d ₁≧0. Standard assumptions are made on B(N) so that a unique equilibrium N _e exists. When B(N) N is not monotone, the delay T can qualitatively change the dynamics. For some fixed values of the parameters with d ₁>0, as T increases the equilibrium N _e can switch from being stable to unstable (with numerically observed periodic solutions) and then back to stable. When disease that does not cause death is introduced into the population, a threshold parameter R ₀ is identified. When R ₀<1, the disease dies out; when R ₀>1, the disease remains endemic, either tending to an equilibrium value or oscillating about this value. Numerical simulations indicate that oscillations can also be induced by disease related death in a model with maturation delay. Received: 2 November 1998 / Revised version: 26 February 1999     

The root–soil system of Norway spruce subjected to turning moment: resistance as a function of rotation

The reactions of trees to wind, rockfall, and snow and debris flow depend largely on how strong and deformable their anchorage in the soil is. Here, the resistive turning moment M of the root–soil system as a function of the rotation ϕ at the stem base plays the major role. M(ϕ) describes the behavior of the root–soil system when subject to rotational moment, with the maximum M(ϕ) indicating the anchorage strength M _a of the tree. We assessed M(ϕ) of 66 Norway spruce (Picea abies L. Karst) by pulling them over with a winch. These 45- to 170-year-old trees grew at sites of low and high elevation, with a diameter at breast height DBH = 14–69 cm and a height H = 9–42 m. M(ϕ) displayed a strong nonlinear behavior. M _a was reached at a lower ϕ for large trees than for small trees. Thus overhanging tree weight contributed less to M _a for the large trees. Overturning also occurred at a lower ϕ for the large trees. These observations show that the rotational ductility of the root–soil system is higher for small trees. M _a could be described by four monovariate linear regression equations of tree weight, stem weight, stem volume and DBH ² ·H (0.80 < R ² < 0.95), and ϕ at M _a, ϕ _a, by a power law of DBH²·H (R ² = 0.85). We found significantly higher M _a for the low-elevation spruces than for the high-elevation spruces, which were more shallowly anchored, but no significant difference in ϕ _a. The 66 curves of M(ϕ), normalized (_n) by M _a in M-direction and by ϕ _a in ϕ-direction, yielded one characteristic average curve: . Using and the predictions of M _a and ϕ _a, it is shown that M(ϕ) and the curves associated with M(ϕ) can be predicted with a relative standard error ≤25%. The parameterization of M(ϕ) by tree size and weight is novel and provides useful information for predicting with finite-element computer models how trees will react to natural hazards.     

Chemotactic collapse for the Keller-Segel model

 This work is concerned with the system (S) {u _t =Δu − χ∇ (u∇v) for x∈Ω, t>0Γ v _t =Δv+(u−1) for x∈Ω, t>0 where Γ, χ are positive constants and Ω is a bounded and smooth open set in ℝ². On the boundary ∂Ω, we impose no-flux conditions: (N) ∂u∂n =∂v∂n =0 for x∈∂ Ω, t>0 Problem (S), (N) is a classical model to describe chemotaxis corresponding to a species of concentration u(x, t) which tends to aggregate towards high concentrations of a chemical that the species releases. When completed with suitable initial values at t=0 for u(x, t), v(x, t), the problem under consideration is known to be well posed, locally in time. By means of matched asymptotic expansions techniques, we show here that there exist radial solutions exhibiting chemotactic collapse. By this we mean that u(r, t) →Aδ(y) as t→T for some T<∞, where A is the total concentration of the species. Received 9 March 1995; received in revised form 25 December 1995     

Extinction probabilities in branching processes: A note on holgate and Lakhani's paper

Within the class of offspring distributions with given meanm>1 and probability of no offspringp _o, the probabilityq of ultimate extinction in a Galton-Watson branching process starting from one individual satisfiesp ₀<q(m,p₀)≤q<1. A short table illustrates the lower boundq(m,p ₀). Work done at the Mathematics Department, University of Washington, Seattle, U.S.A.     

Development of optical criteria to discriminate various types of highly turbid lake waters

Our aim was to refine the optical classification of turbid waters in order to improve the performance of water color algorithms. Bio-optical measurements and sampling of optically active substances were performed in highly turbid lakes Taihu, Chaohu, and Dianchi, and in Three Gorges reservoir in China. Based on strong correlations between trough depths of remote sensing reflectance (R _rs(λ)) near 680 nm (denoted as TD680) and the ratios of inorganic suspended matter (ISM) to total suspended matter (TSM) concentrations, an empirical model was developed for water classification. In the 400–900 nm spectral range, different correlations between R _rs(λ), TSM and chlorophyll a (Chla) concentrations indicate discrepancies among the following ISM/TSM ranges: ISM/TSM ≤ 0.5, 0.5 < ISM/TSM < 0.8, and ISM/TSM ≥ 0.8. Corresponding findings support an important conclusion that only high ISM/TSM ratios, usually above 0.5, and using the more sensitive and stable near infrared spectral range (730–820 nm), can assure good performances of the TSM remote sensing algorithms. Meanwhile, the particulate absorption a _p(λ) and scattering b _p(λ) were strongly influenced by different ranges of ISM/TSM ratios. Typically the a _p(675) peaks became more and more vague as ISM/TSM increased, and the b _p(λ) values first decreased and then increased with a marked inflexion at ISM/TSM ≈ 0.5. The TD680 threshold values were derived to discriminate three types of turbid waters, i.e., Type 1 (TD680 ≥ 0.0082 sr⁻¹), Type 2 (0.0082 sr⁻¹ > TD680 > 0 sr⁻¹), and Type 3 (TD680 ≤ 0 sr⁻¹). This study provides a promising tool for identifying various types of highly turbid waters, and is significant for the development of class-based algorithms of water color remote sensing.     

The minimum effort required to eradicate infections in models with backward bifurcation

We study an epidemiological model which assumes that the susceptibility after a primary infection is r times the susceptibility before a primary infection. For r = 0 (r = 1) this is the SIR (SIS) model. For r > 1 + (μ/α) this model shows backward bifurcations, where μ is the death rate and α is the recovery rate. We show for the first time that for such models we can give an expression for the minimum effort required to eradicate the infection if we concentrate on control measures affecting the transmission rate constant β. This eradication effort is explicitly expressed in terms of α,r, and μ As in models without backward bifurcation it can be interpreted as a reproduction number, but not necessarily as the basic reproduction number. We define the relevant reproduction numbers for this purpose. The eradication effort can be estimated from the endemic steady state. The classical basic reproduction number R ₀ is smaller than the eradication effort for r > 1 + (μ/α) and equal to the effort for other values of r. The method we present is relevant to the whole class of compartmental models with backward bifurcation.Dedicated to Karl Peter Hadeler on the occasion of his 70th birthday.     

Approximation of the Basic Reproduction Number <Emphasis Type="Italic">R</Emphasis><Subscript>0</Subscript> for Vector-Borne Diseases with a Periodic Vector Population

The main purpose of this paper is to give an approximate formula involving two terms for the basic reproduction number R ₀ of a vector-borne disease when the vector population has small seasonal fluctuations of the form p(t) = p ₀ (1+ε cos (ωt − φ)) with ε ≪ 1. The first term is similar to the case of a constant vector population p but with p replaced by the average vector population p ₀. The maximum correction due to the second term is (ε²/8)% and always tends to decrease R ₀. The basic reproduction number R ₀ is defined through the spectral radius of a linear integral operator. Four numerical methods for the computation of R ₀ are compared using as example a model for the 2005/2006 chikungunya epidemic in La Réunion. The approximate formula and the numerical methods can be used for many other epidemic models with seasonality. MSC 92D30 ⋅ 45C05 ⋅ 47A55     

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.     

Anaerobic contribution to the time to exhaustion at the minimal exercise intensity at which maximal oxygen uptake occurs in elite cyclists, kayakists and swimmers

Using 23 elite male athletes (8 cyclists, 7 kayakists, and 8 swimmers), the contribution of the anaerobic energy system to the time to exhaustion (t _lim) at the minimal exercise intensity (speed or power) at which maximal oxygen uptake (V˙O_{2 max}) occurs (I _{V˙O2 max}) was assessed by analysing the relationship between the t _lim and the accumulated oxygen deficit (AOD). After 10-min warming up at 60% of V˙O_{2 max}, the exercise intensity was increased so that each subject reached his I _V˙O2max in 30 s and then continued at that level until he was exhausted. Pre-tests included a continuous incremental test with 2 min steps for determining the I _V˙O2max and a series of 5-min submaximal intensities to collect the data that would allow the estimation of the energy expenditure at I _V˙O2max . The AOD for the t _lim exercise was calculated as the difference between the above estimation and the accumulated oxygen uptake. The mean percentage value of energy expenditure covered by anaerobic metabolism was 15.2 [(SD 6)%, range 8.9–24.1] with significant differences between swimmers and kayakists (16.8% vs 11.5%, P≤0.05) and cyclists and kayakists (16.4% vs 11.5%, P≤0.05). Absolute AOD values ranged from 26.4 ml · kg⁻¹ to 83.6 ml · kg⁻¹ with a mean value of 45.9 (SD 18) ml · kg⁻¹. Considering all the subjects, the t _lim was found to have a positive and significant correlation with AOD (r = 0.62, P≤0.05), and a negative and significant correlation with V˙O_{2 max} (r = −0.46, P≤0.05). The data would suggest that the contribution of anaerobic processes during exercise performed at I _V˙O2max should not be ignored when t _lim is used as a supplementary parameter to evaluate specific adaptation of athletes. Accepted: 17 December 1996     

Mathematical analysis of cancer chemotherapy: The effects of chemotherapeutic agents on the cell cycle traverse

For the tumor model of Skipper and Zubrod, which has been analyzed previously for the theoretical FLM function and the effect of chemotherapy against tumors of known or assumed kinetic characteristics, the theoretical continuous labeling (CL) function is derived by considering an equivalent tumor (in terms of unlabeled cell populations) in which the density function of phase duration of cells inS-phasef ₂(a ₂)=δ(a ₂−∞) and the loss functionL ₂(t)=0. This mathematical concept of blocking is applied to the analysis of synchronization in tumor growth and blocking effects in cancer chemotherapy. These effects of chemical agents on the cell cycle progression are being incorporated into a previously written computer simulation program for cancer chemotherapy. Whereas, a program is written and used to simulate the CL functions for L1210 leukemia, and primary and metastatic Lewis lung carcinoma.     

Proportion of mated females, female mating experience, and sex ratio of the osmund sawfly, Strongylogaster osmundae (Hymenoptera, Tenthredinidae)

The proportion of mated females (M _f) of the osmund sawfly, Strongylogaster osmundae, and the sex ratio of the eggs they deposited (r, proportion of males) were estimated in the wild by collecting egg masses. The proportion of mated females at oviposition varied from 0 to 1.0. M _f was high (often 1.0) among the females that emerged after hibernation, and lower in the subsequent generations. Mated females of the hibernated generation deposited equal numbers of eggs of both sexes. Mated females of the first and subsequent generations produced more female than male eggs. These results qualitatively agreed with the prediction provided by an evolutionarily stable strategy (ESS) model (if M _f < 1 then r < 0.5). However, the quantitative prediction provided by the model [M _f (1 − r) = 0.5] was not always observed in the wild, especially where the population density and M _f were high. The value of r was often lower than the predicted one. The following simple hypothesis was tested by experimentation: “Females that encounter males frequently estimate the proportion of mated females to be high and deposit eggs with a 1:1 sex ratio.” However, results did not support this hypothesis. Females that copulated soon after emergence and were courted by males two or more times did not show a higher offspring sex ratio than those which mated 1 or 2 days after emergence and experienced no other sexual encounter. Another mechanism for determination of r is suggested, and the reason why the population sex ratio of sawflies is often female-biased (r < 0.5) is discussed.     

Diffusion with attrition

This article treats the problem of the sharp front observed when a diffusing substance interacts irreversibly with binding sites within the medium. The model consists of two simultaneous partial differential equations that are nonlinear and cannot be solved in closed form. The parameters are the diffusion coefficient D in the direction under consideration (x), the interaction constant k, the binding-site concentration μ and the boundary concentration of the diffusing ion c ₀. Our aim is to develop methods to enable the estimation of these parameters from the experimental data. An analytical solution for the case k → ∞, as found by others, is given first and then a finite element analysis package is used to obtain numerical solutions for the general case. Graphs are presented to illustrate the effects of the various parameters. Simple graphical procedures are described to compute μ and c ₀. The position of the advancing front ξ then provides, together with μ, a way to estimate D. A mathematical identity relating D and x and a second one involving D, k and t help to reduce the complexity of the problem. A new, measurable quantity S(t) is defined as where f is the total concentration (free + bound) of the diffusing ion at time t, and detailed plots are furnished that permit the computation of k directly from S(t), μ and D. The accuracy with which such methods can be expected to determine the various parameters of the model is considered at some length. Finally, in a concluding section, we simulate typical experimental data, examine the validity of our methods, and see how their accuracy is affected by controlled amounts of various kinds of noise.     

Theory of transport in linear biological systems: I. Fundamental integral equation

The conditions under which the output,γ _b (t), of a biological system is related to the input,γ _a (t), by an integral equation of the typeγ _b (t) = ∫ ₀ ^t γ _a (ω)w(t−ω)dω, where ω(t) is a transport functioncharacteristic of the system, are analyzed in detail. Methods of solving this type of integral equation are briefly discussed. The theory is then applied to problems in tracer kinetics in which input and output are sums of exponentials, and explicit formulae, which are applicable whether or not the pool is uniformly mixed, are derived for “turnover time” and “pool” size.     

A model of photosystem II for the analysis of fast fluorescence rise in plant leaves

The polyphasic patterns of fluorescence induction rise in pea leaves in vivo and after the treatment with ionophores have been studied using a Plant Efficiency Analyzer. To analyze in detail photosystem II (PS II) electron transfer processes, an extended PS II model was applied, which included the sums of exponential functions to specify explicitly the light-driven formation of the transmembrane electric potential (ΔΨ(t)) as well as pH in the lumen (pH_L(t)) and stroma (pH_S(t)). PS II model parameters and numerical coefficients in ΔΨ(t), pH_L(t), and pH_S(t) were evaluated to fit fluorescence induction data for different experimental conditions: leaf in vivo or after ionophore treatment at low or high light intensity. The model imitated changes in the pattern of fluorescence induction rise due to the elimination of transmembrane potential in the presence of ionophores, when ΔΨ = 0 and pH_L(t), pH_S(t) changed to small extent relative to control values in vivo, with maximum ΔΨ(t) ∼ 90 mV and ΔΨ(t) ∼ 40 mV for the stationary state at ΔpH ≅ 1.8. As the light intensity was increased from 300 to 1200 μmol m⁻² s⁻¹, the heat dissipation rate constants increased threefold for nonradiative recombination of P680⁺Phe⁻ and by ∼30% for P680⁺Q_A⁻. The parameters ΔΨ, pH_S and pH_L were analyzed as factors of PS II redox state populations and fluorescence yield. The kinetic mechanism of fluorescence quenching is discussed, which is related with light-induced lumen acidification, when ⁺Q_A⁻ and P680⁺ recombination probability increases to regulate the Q_A reduction.     

Electroporative deformation of salt filled lipid vesicles

Membrane electroporation, vesicle shape deformation and aggregation of small, NaCl-filled lipid vesicles (of radius a = 50 nm) in DC electric fields was characterized using conductometric and turbidimetrical data. At pulse durations t_E≤ 55 ± 5 ms the increase in the conductivity of the vesicle suspension is due to the field-induced efflux of electrolyte through membrane electropores. Membrane electroporation and Maxwell stress on the vesicle membrane lead to vesicle elongation concomitant with small volume reduction (up to 0.6% in an electric field of E = 1 MV m^–1). At t_E > 55 ± 5 ms, further increases in the conductivity and the optical density suggest electroaggregation and electrofusion of vesicles. The conductivity changes after the electric pulse termination reflect salt ion efflux through slowly resealing electropores. The analysis of the volume reduction kinetics yields the bending rigidity κ = (4.1 ± 0.3) ⋅ 10^–20 J of the vesicle membrane. If the flow of Na⁺ and Cl^– ions from the vesicle interior is treated in terms of Hagen-Poiseuille's equation, the number of permeable electropores is N = 39 per vesicle with mean pore radius r_p = 0.85 ± 0.05 nm at E = 1 MVm^–1 and t_E≤ 55 ± 5 ms. The turbidimetric and conductometric data suggest that small lipid vesicles (a ≤ 50 nm) are not associated with extensive membrane thermal undulations or superstructures. In particular with respect to membrane curvature, the vesicle results are suggestive for the design and optimization of electroporative delivery of drugs and genes to cell tissue at small field strengths (≤1 MVm^–1) and large pulse durations (≤100 ms). Received: 8 July 1997 / Accepted: 15 September 1997     

Field measurement of soil water repellency and its impact on water flow under different vegetation

Numerous recent laboratory studies have shown that vegetation can influence soil water flow by inducing very low levels of water repellency. In this study we extended on this previous research by developing a field-based test using a miniature infiltrometer to assess low levels of water repellency from physically based measurements of liquid flow in soil. The field-based test was verified through a simple laboratory experiment and then applied to determine the impact of vegetation and antecedent soil water content. The soil hydraulic properties determined were hydraulic conductivity, sorptivity, as well as the persistence and index of water repellency. Tests were conducted following a dry spell and wet spell on (1) forest soil (0 cm depth), (2) glade soil (0 cm depth) and (3) glade soil (50 cm depth). It was found that both the persistence and index of water repellency, R, decreased in the order as follows: forest soil > glade soil (0 cm) > glade soil (50 cm) for both dry and wet spell. The range of values of R was 0.28 (wettable) to 360 (highly water repellent), which affected hydraulic conductivity k _r(−2 cm). R increased and hence k _r(−2 cm) decreased in the order: forest soil < glade soil (0 cm) < glade soil (50 cm) for both the dry and wet spell. There were clear interactions between vegetation and changes to water flow caused by presence of repellency. Presented at the International Conference on Biohydrology, Prague, Czech Republic, 20–22 September 2006.     

Cross-Correlation Functions for a Neuronal Model

Cross-correlation functions, R_XY(t,τ), are obtained for a neuron model which is characterized by constant threshold θ, by resetting to resting level after an output, and by membrane potential U(t) which results from linear summation of excitatory postsynaptic potentials h(t). The results show that: (1) Near time lag τ = 0, R_XY(t,τ) = f_U [θ-h(τ), t + τ] {h′(τ) + E_U [u′(t + τ)]} for positive values of this quantity, where f_U(u,t) is the probability density function of U(t) and E_U [u′(t + τ)] is the mean value function of U′(t + τ). (2) Minima may appear in R_XY(t,τ) for a neuron subjected only to excitation. (3) For large τ, R_XY(t,τ) is given approximately by the convolution of the input autocorrelation function with the functional of point (1). (4) R_XY(t,τ) is a biased estimator of the shape of h(t), generally over-estimating both its time to peak and its rise time.     

A Generalised Age- and Phase-Structured Model of Human Tumour Cell Populations Both Unperturbed and Exposed to a Range of Cancer Therapies

We develop a general mathematical model for a population of cells differentiated by their position within the cell division cycle. A system of partial differential equations governs the kinetics of cell densities in certain phases of the cell division cycle dependent on time t (hours) and an age-like variable τ (hours) describing the time since arrival in a particular phase of the cell division cycle. Transition rate functions control the transfer of cells between phases. We first obtain a theoretical solution on the infinite domain −∞ < t < ∞. We then assume that age distributions at time t=0 are known and write our solution in terms of these age distributions on t=0. In practice, of course, these age distributions are unknown. All is not lost, however, because a cell line before treatment usually lies in a state of asynchronous balanced growth where the proportion of cells in each phase of the cell cycle remain constant. We assume that an unperturbed cell line has four distinct phases and that the rate of transition between phases is constant within a short period of observation (‘short’ relative to the whole history of the tumour growth) and we show that under certain conditions, this is equivalent to exponential growth or decline. We can then gain expressions for the age distributions. So, in short, our approach is to assume that we have an unperturbed cell line on t ≤ 0, and then, at t=0 the cell line is exposed to cancer therapy. This corresponds to a change in the transition rate functions and perhaps incorporation of additional phases of the cell cycle. We discuss a number of these cancer therapies and applications of the model.     

An urn model study of variability within a compartment

Formulas are derived for the mean and variance of the number of radioactive atoms present in a compartment (or urn). Initally,n ₁ radioactive atoms andb stable atoms are placed in the urn; and subsequently,r stable atoms are added and an equal number,r, of a random mixture of stable and radioactive atoms is removed per unit time. The expected number of radioactive atoms,E(t), present at timet is, as expected,n ₁ e^−λt where λ=(r/Δt)/(b+r+n ₁). The relative variance, σ²(t)/n ₁ ² , vanishes to zero forr=1, atoms per unit time and for a large number ofn ₁ radioactive atoms; but for a large number of bothr andn ₁ atoms the relative variance is ∼e ^−λt , equal to the fractional retention, fort>1/λ. Thus in studies where radionuclides are injected into animals and a single compartment represents the data, if a large variance is observed it might be due to the fact that large numbers of atoms are transferred out in unit time. When a small variance is observed, this is probably due to the fact that few atoms are transferred in smaller units of time (such that λ is the same in both cases). Research sponsored by the Energy Research and Development Administration under contract with Union Carbide Corporation.