Mathematical model of non-basal testosterone regulation in the male by pulse modulated feedback

Abstract

A parsimonious mathematical model of pulse modulated regulation of non-basal testosterone secretion in the male is developed. The model is of third differential order, reflecting the three most significant hormones in the regulation loop, but is yet shown to be capable of sustaining periodic solutions with one or two pulses of gonadotropin-releasing hormone (GnRH) on each period. Lack of stable periodic solutions is otherwise a main shortcoming of existing low-order hormone regulation models. Existence and stability of periodic solutions are studied. The periodic mode with two GnRH pulses on the least period has not been described in medical literature but is found to explain experimental data well.

A note on semi-discrete modelling in the life sciences

“…The dynamical patterns produced by the model were very close to observations made on ewes. A similar approach was proposed by Shurilov et al (2009) to model the regulation of non-basal testosterone secretion in males through pulses of gonadotropin-releasing hormone (GnRH).…”

An impulsive model of endocrine regulation with two negative feedback loops * *The work was supported in part by the European Research Council (ERC-StG-307207)

“…The model from Churilov et al (2009) inherits the cyclic structure of the classical Goodwin oscillator. When applied to describe the GnRH-LH-Te axis, it implies that Te inhibits the secretion of GnRH directly, and influences the production of LH indirectly.…”

“…So a continuous Hill-type nonlinearity should be replaced by a discontinuous map such as a Heaviside function (Cartwright and Husain, 1986). A more complicated model, based on the Goodwin oscillator, has been proposed in Churilov et al (2009). The feedback from Te to GnRH is described by a pulse-amplitudefrequency modulator (Gelig and Churilov, 2012), where the modulating amplitude function can be a Hill nonlinearity.…”

“…To use the benefits of the framework from Churilov et al (2009), we suppose that the additional feedback does not destroy this structure, that is, the additional feedback from Te to LH is linear. This structure introduces only one uncertain parameter (a scalar feedback gain is introduced), compared to the previous model from Churilov et al (2009), which makes it possible to suggest an identification procedure, similar to (Mattsson and Medvedev, 2013).…”

“…The important property of the model, developed in Churilov et al (2009), is the possibility to represent it in the Lur'e form with a single scalar nonlinearity (standing for the pulsatile feedback from Te to GnRH).…”