Miljenko Marušić

Tumor growth in vivo and as multicellular spheroids compared by mathematical models

In vivo volume growth of two murine tumor cell lines was compared by mathematical modeling to their volume growth as multicellular spheroids. Fourteen deterministic mathematical models were studied. For one cell line, spheroid growth could be described by a model simpler than needed for description of growthin vivo. A model that explicitly included the stimulatory role for cell-cell interactions in regulation of growth was always superior to a model that did not include such a role. The von Bertalanffy model and the logistic model could not fit the data; this result contradicted some previous literature and was found to depend on the applied least squares fitting method. By the use of a particularly designed mathematical method, qualitative differences were discriminated from quantitative differences in growth dynamics of the same cells cultivated in two different three-dimensional systems.

Sharp error bounds for interpolating splines in tension

Abstract Sharp error bounds for interpolating splines in tension with variable tension parameters are considered. Error bounds given are the best possible ones in the limit case of the tension parameter p, that is for linear (p → ∞) and cubic (p → 0) spline. Error bounds sharper than those previously published are also developed for derivatives.

PREDICTION POWER OF MATHEMATICAL MODELS FOR TUMOR GROWTH

We compared the Gompertz model, the generalized Gompertz model, the Piantadosi model, the autostimulation model and the polynomials for the power to predict growth of multicellular tumor spheroids as paradigms of the prevascular phase of tumor growth. For the comparison of models we developed a criterion that established the Gompertz model as the model with the best prediction power. The prediction power of the remaining models was ranked in declining order: the generalized Gompertz model; the mutually indistinguishable Piantadosi model and the autostimulation model; and the polynomials. The ranking of models was not affected by the applied minimization criteria of weighted least squares, unweighted least squares and fitting to logarithmically transformed data, but the prediction power was affected by these criteria. The best predictions were obtained by weighted least squares, closely followed by fitting to logarithmically transformed data. The unweighted least-squares minimization was much less applicable for prediction (and description) of growth.

A collocation method for singularly perturbed two-point boundary value problems with splines in tension

An error bound for the collocation method by spline in tension is developed for a nonlinear boundary value problemay″+by′+cy=f(·,y),y(0)=y0,y(1)=y1. Sharp error bounds for the interpolating splines in tension are used in conjunction with recently obtained formulae for B-splines, to develop an error bound depending on the tension parameters and net spacing. For singularly perturbed boundary value problems (|a|=ε≪1), the representation of the error motivates a choice of tension parameters which makes the convergence of the collocation method problem at least linear. The B-representation of the spline in tension is also used in the actual computations. Some numerical experiments are given to illustrate the theory.

The effects of the trematode Bucephalus polymorphus on the reproductive cycle of the zebra mussel Dreissena polymorpha in the Drava River

The effects of the trematode Bucephalus polymorphus on the reproductive cycle of the zebra mussel Dreissena polymorpha were examined in mussel populations from the Drava River. The reproductive cycle was studied by histological examination of the gonads and quantified by an image analysing system to determine changes in volume of the entire visceral mass, gonads, digestive glands and in particular the volume of trematodes. Results confirmed that (1) gonads of D. polymorpha were affected by B. polymorphus infection more than any other organ and (2) development of cercariae in sporocysts of B. polymorphus coincides with host gonad maturation. This is the first study in which the image analysing system was used to determine the effect of trematodes on the reproductive cycle of D. polymorpha. Also, this is the first record of sporocysts of B. polymorphus in D. polymorpha in this part of Europe.

Solving Parabolic Singularly Perturbed Problems by Collocation Using Tension Splines

Tension spline is a function that, for given partition x0 < x1 < … < xn, on each interval [xi, xi+1] satisfies differential equation (D4 − ρ i 2 D2)u = 0, where ρis are prescribed nonnegative real numbers. In the literature, tension splines are used in collocation methods applied to two-points singularly perturbed boundary value problems with Dirichlet boundary conditions.

Asymptotic Behaviour of Tension Spline Collocation Matrix

Tension spline of order k is a function that, for given partition x 0 < x 1 < ... < x n , on each interval [x i , x i+1] satisfies differential equation (D k − p i 2 /h i 2 D k−2)u = 0, where p i ‘s are prescribed nonnegative real numbers.