R Vasudevan

The inverse problem of estimating heart parameters from cardiograms

Abstract A model of the electrical activity of the heart relating the ventricular dipoles to cardiogram measurements naturally leads to the inverse problem of estimating the heart parameters from the observation of skin potentials. The quasilinearization method for handling this multipoint boundary value problem necessitates the use of an initial guess for some of the parameters and the use of a large digital computer for the solution of a large system of linear equations. An alternate technique involving differential quadrature which obviates the use of any initial guess, if feasible. In this technique, a small on-line computer will produce the results on a real-time basis for as many as 20 myocardial segments. The details of this procedure and some numerical experiments form the contents of this article.

Splines via dynamic programming

Abstract Approximating a function with prescribed values at given points on a real interval by a cubic spline is based on the minimum curvature property of the approximation. This essential feature can be used as the criterion to determine the cubic polynomial approximation in each interval in a sequential manner by methods of dynamic programming. A stable system of recurrence relations for the coefficients of the spline in successive intervals is obtained by the methods of dynamic programming and they are shown to be identical with the usual relations of the spline approximation. Extension of this method to other types of splines is also considered.

On moment behavior of a class of stochastic difference equations

Abstract We consider a class of stochastic difference equations whose solutions are projections of vector Markov processes. It is shown that the Chapman-Kolmogorov equation leads to useful recurrence integral relations for determining the moments of the solution process; the simple moments at a given time can be generated directly and the mixed moments can be determined using the method of Kronecker products. This formulation also has the advantage that, under certain conditions, the asymptotic behavior of the moments can be predicted by means of the Jentzschs theorem.

Invariant imbedding and radiation dosimetry: IX. inverse problem of determining a plane source in a finite isotropically scattering target slab

Abstract Considering a homogeneous, isotropically scattering target slab of optical thickness x, containing an internal plane emitting source, the aim of the present paper is to determine the distribution of the internal emitting source by measuring the angular distribution of the intensity of radiation emergent from the slab. First, a system of differential recurrence relations for the intensity of finite order scattered radiation emergent from the top is deduced. In other words, a Cauchy system for determining the finite order emergent intensity is expressed in terms of a source function in the diffuse radiation field and the Fredholm resolvent that were computed in a preceding paper [cf. Bellman, Ueno, and Vasudevan, Math. Biosci. 15, 195–203 (1972)]. Then, the quasilinearization technique is used to solve the inverse problem of determining characteristics of the internal source by measuring the radiation field emergent from the top. This method will be employed in future computations relating to realistic situations in the radiation diagnosis and therapy.

On the matrix Riccati equation of transport processes

Abstract By keeping count of the number of interactions, we obtain some novel perturbation expansions for the Riccati equation. Similar results were obtained in radiative transfer theory by Mingle.

Microdiffusive and competitive modes of the decaying hemopoietic stem cell

Abstract The model of stem cell renewal according to a microdiffusing stimulus has been adapted to take into account myeloid and erythroid differentiation. These differons are produced by freely diffusible hormones (viz. erythropoietin and myelopoietin) which interact with stem cells having different concentrations of stimulus for renewal. These concentrations induce the stem cells into discrete differentiation states, which become expressed after exposure to the differentiating hormones. The decay of the stem cells into erythrons and myelons may be interpreted as two differentiating domains with a phase separation generated by the competition of the two modes of decay and conditioned by the concentration gradient along the stem cell chain. The model accounts for certain features of hemopoietic differentiation as studied in vivo and offers a mechanism for the leukemic stem cell reversion to seemingly normal growth when erythroleukemic stem cells are grown into lethally irradiated syngeneic hosts.

Invariant imbedding and radiation dosimetry: VIII. reflection function from a double layer— finite order functions☆

Abstract In computing the total reflection function for two media of given scattering properties adjoined together, the usual techniques require a knowledge of reflection and transmission functions of both the media. An alternate method developed here depends only on solving a Riccati equation with suitable initial conditions. By following this procedure, we are able to calculate the reflection function of a given finite order, for a particle that has suffered a specified number of scatterings in the two portions of the total medium. Such calculations may be useful in radiation dosimetry problems where the body to be irradiated is placed on reflecting materials.

Dynamic programming and bicubic spline interpolation

Abstract Dynamic programming techniques were used to obtain the spline approximation for a function with prescribed values on the knot points along a line. Extending this procedure to two dimensions, the bicubic spline approximation defined over a two-dimensional region is obtained in this paper employing the methods of dynamic programming. A regular rectangular region as well as a region with irregular boundaries can be handled by this method, avoiding the difficulties of large storage and high dimensionality.

Mean square spline approximation

Abstract The approximation to a specified function on the real line by fitting a cubic in a piecewise fashion is achieved by minimizing the deviations in the mean square sense. The coefficients of the cubic are determined sequentially employing the method of dynamic programming. Employing this method a known function is approximated and the results of the computation are tabulated.

Invariant imbedding and radiation dosimetry VII: finite order scattering and transmission functions of the two radiation approximations in a target slab☆

Abstract This paper treats the two component radiation approximation for the calculation of absorbed dose when γ radiation passes through irradiated material. We describe a method of solving in a successive fashion the vector transfer equation for the two component radiation. We use a two-by-two matrix to represent the differential attenuation coefficient. Employing the imbedding principle, we obtain a set of integro-differential recurrence relations for the two-by-two matrices of finite order, scattering and transmission functions. Making use of these functions, the two component radiation field within the homogeneous, anisotropically and noncoherently scattering target slab (Bellman, Ueno, and Vasudevan, to be published), can be computed. This enables us to determine the absorbed dose.