Aldo Rescigno

A neuronal model for the discharge patterns produced by cyclic inputs

To understand the patterns of nerve impulses produced by sinusoidal stimuli, a simple model is considered which integrates input currents with a finite time constant until a threshold voltage is reached, whereupon an output pulse is produced and the process is restarted. We show here that (a) a general analytic solution exists for this model driven by sinusoidal stimuli, determining the interval between every member of the pulse train, (b) for all values of the parameters of the model a pattern exists which repeats periodically after a finite number of pulses in the absence of noise, (c) the system will approach a stable pattern which, if perturbed, will be recovered once the perturbation is removed, (d) the linear integrator or relaxation oscillator and the curren multiplier are limiting cases of this model.

Immune surveillance and neoplasia—1 a minimal mathematical model

A deterministic predator-prey model is presented for describing the dynamics of a solid tumor in the presence of a specifically reactive lymphocyte population which is stimulated by, and antagonistic to, the tumor. The qualitative behavior of the solutions is developed and briefly compared to the results of transplantation experiments. Although the model is primitive, it leads to predictions that are in general agreement with observation and intuitive expectations. In particular, it is found that: (1) very low levels of transplanted tumor will not survive in the recipient. (2) At somewhat higher levels, tumor growth will be uncontrolled in the syngeneic recipient. However, immune intervention if early enough, can lead to control and elimination of the tumor. (3) At still higher levels of transplanted tumor, no amount of immune intervention will be effective in controlling the tumor. (4) If the recipients immune system is suppressed prior to transplantation, or is debilitated for any reason, the chance that the tumor will grow increases. (5) If the recipients immune system is stimulated prior to transplantation, the chance of tumor survival decreases. (6) The survival of the tumor is much more sensitive to changes in tumor parameters (for example, antigenicity) than in lymphocyte parameters. In addition it makes the unintuitive prediction that (7) There areisolated instances under which anincrease in the number of lymphocytes canincrease the chance of tumor survival.

On transfer times in tracer experiments

Abstract A compartment is defined as a pool of material whose behavior can be described by a deterministic or by a stochastic equation; these two equations are used to define the transit time through the compartment, the total residence time , the time of entrance and the time of exit . If in a complex system one or more compartments are accessible, the transport of material through it can be studied using a tracer. Then the transfer time between any two compartments, or through the cycle around a compartment, can be analyzed under certain hypotheses, even if the transport along the route considered cannot be described by compartment equations.

On the stochastic theory of compartments: II. Multi-compartment systems

A stochastic model is developed for a system of interconnected compartments. The generating function of the random variable of any compartment can be constructed from a flow graph involving the expectations of the random variables of all compartments of the system.

On the stochastic theory of compartments: I. A single-compartment system

A stochastic model is developed for a compartment with a single time-dependent input, and generalized to include inputs from several sources. With the number of particles of a given molecular species in the compartment as the random variable, the mean, variance and third central moment of this variable are calculated from its generating function, and compared with previous results. The behavior of the calculated moments is discussed, and the possibility of applying the model to chemical and biological systems is considered.

Immune surveillance and neoplasia—II A two-stage mathematical model

In a previous paper (DeLisi and Rescigno, 1977) a model for the interaction of tumor cells and killer lymphocytes was presented. Although that model was highly simplified, the qualitative behavior was in accord with intuitive expectations and a wide range of data. It could not however account forde novo tumor development. In this paper a slightly more realistic model is presented by introducing a delay in the formation of killer lymphocytes. This is done by requiring two stages in the production of a killer. We show that introduction of this second stage allows tumor development from even a single cell, thus removing an important limitation of two variable systems.

The struggle for life: III. A predator-prey chain

Ann species predator-prey chain is analyzed to determine what oscillations occur in population sizes. It is found that only the populations of the first and second species in the chain must necessarily oscillate around the point of equilibrium if they do not come to equilibrium. The other species may or may not oscillate.

On the use of pharmacokinetic models

Extensive use of models in pharmacology, in physiology and in radiotherapy raises some questions on the nature and utility of models in general and of compartmental models in particular. In this paper I will define in a simple and logical way a set of useful pharmacokinetic parameters and show how their estimation depends on the assumed model. A special problem arises when some parameters are not identifiable; in that case I will show how it is possible to determine a range for them. Two examples are used to illustrate how to compute the value of the identifiable parameters and the range of the non-identifiable ones, when the available experimental data are not sufficient to identify a model.

Kinetic modeling in support of radionuclide dose assessment.

In this review, we trace the origins of mathematical modeling methods and pay particular attention to radiotracer applications. Nuclear medicine has been advanced greatly by the efforts of the Society of Nuclear Medicines Medical Internal Radiation Dose Committee. Well-developed mathematical methods and tools have been created in support of a wide range of applications. Applications of mathematical modeling extend well beyond biology and medicine and are essential to analysis is a wide range of fields that rely on numerical predictions, eg, weather, economic, and various gaming applications. We start with the discovery of radioactivity and radioactive transformations and illustrate selected applications in biology, physiology, and pharmacology. We discuss compartment models as tools used to frame the context of specific problems. A definition of terms, methods, and examples of particular problems follows. We present models of different applications with varying complexity depending on the features of the particular system and function being analyzed. Commonly used analysis tools and methods are described, followed by established models which describe dosimetry along gastrointestinal and urinary excretory pathways, ending finally with a brief discussion of bone marrow dose. We conclude pointing to more recent, promising methods, not yet widely used in dosimetry applications, which aim at coupling pharmacokinetic data with other patient data to correlate patient outcome (benefits and risk) with the type, amount, kind and timing of the therapy the patient received.

The struggle for life—V. One species living in a limited environment

The general properties of the equations describing a single species living in a limited environment in the presence of its own pollutant are discussed.