Optimal nutritional intake for fetal growth

View online at publisher's website: http://www.aimsciences.org/journals/displayArticles.jsp?paperID=6249The regular nutritional intake of an expectant mother clearly affects the weight development of the fetus. Assuming the growth of the fetus follows a deterministic growth law, like a logistic equation, albeit dependent on the nutritional intake, the ideal solution is usually determined by the birth-weight being pre-assigned, for example, as a percentage of the mother's average weight. This problem can then be specified as an optimal control problem with the daily intake as the control, which appears in a Michaelis-Menten relationship, for which there are well-developed procedures to follow. The best solution is determined by requiring minimum total intake under which the preassigned birth weight is reached. The algorithm has been generalized to the case where the fetal weight depends in a detailed way on the cumulative intake, suitably discounted according to the history. The optimality system is derived and then solved numerically using an iterative method for the specific values of parameter. The procedure is generic and can be adapted to any growth law and any parameterisation obtained by the detailed physiology

The logic behind neural control of breathing pattern

The respiratory rhythm generator is spectacular in its ability to support a wide range of activities and adapt to changing environmental conditions, yet its operating mechanisms remain elusive. We show how selective control of inspiration and expiration times can be achieved in a new representation of the neural system (called a Boolean network). The new framework enables us to predict the behavior of neural networks based on properties of neurons, not their values. Hence, it reveals the logic behind the neural mechanisms that control the breathing pattern. Our network mimics many features seen in the respiratory network such as the transition from a 3-phase to 2-phase to 1-phase rhythm, providing novel insights and new testable predictions

Robust Unidirectional Airflow through Avian Lungs: New Insights from a Piecewise Linear Mathematical Model

<div><p>Avian lungs are remarkably different from mammalian lungs in that air flows unidirectionally through rigid tubes in which gas exchange occurs. Experimental observations have been able to determine the pattern of gas flow in the respiratory system, but understanding how the flow pattern is generated and determining the factors contributing to the observed dynamics remains elusive. It has been hypothesized that the unidirectional flow is due to aerodynamic valving during inspiration and expiration, resulting from the anatomical structure and the fluid dynamics involved, however, theoretical studies to back up this hypothesis are lacking. We have constructed a novel mathematical model of the airflow in the avian respiratory system that can produce unidirectional flow which is robust to changes in model parameters, breathing frequency and breathing amplitude. The model consists of two piecewise linear ordinary differential equations with lumped parameters and discontinuous, flow-dependent resistances that mimic the experimental observations. Using dynamical systems techniques and numerical analysis, we show that unidirectional flow can be produced by either effective inspiratory or effective expiratory valving, but that both inspiratory and expiratory valving are required to produce the high efficiencies of flows observed in avian lungs. We further show that the efficacy of the inspiratory and expiratory valving depends on airsac compliances and airflow resistances that may not be located in the immediate area of the valving. Our model provides additional novel insights; for example, we show that physiologically realistic resistance values lead to efficiencies that are close to maximum, and that when the relative lumped compliances of the caudal and cranial airsacs vary, it affects the timing of the airflow across the gas exchange area. These and other insights obtained by our study significantly enhance our understanding of the operation of the avian respiratory system.</p></div

The position of zero flow points <i>q</i>_<i>T</i> = 0, <i>q</i>₁ = 0, and <i>q</i>₂ = 0 shown in the phase plane.

<p>The grey shaded region above the long dashed line <i>x</i>₂ = <i>x</i>₁ is where <i>q</i>_<i>P</i> < 0, and the unshaded region below the line <i>x</i>₂ = <i>x</i>₁ is where <i>q</i>_<i>P</i> > 0. The blue shaded region indicates where <i>q</i>_<i>T</i> > 0, and the green shaded region where <i>q</i>_<i>T</i> < 0. The value of <i>R</i>₁ changes on the line <i>q</i>₁ = 0 such that: <i>R</i>₁ = <i>R</i>_{1,<i>insp</i>} in region (1), and <i>R</i>₁ = <i>R</i>_1,<i>exp</i> in regions (2), (3), and (4). The value of <i>R</i>₂ changes on the line <i>q</i>₂ = 0 such that: <i>R</i>₂ = <i>R</i>_2,<i>exp</i> in region (4), and <i>R</i>₂ = <i>R</i>_{2,<i>insp</i>} in regions (1), (2), and (3).</p

Unidirectional flow is robust to changes in amplitude.

<p>In this figure we plot the flow through the parabronchi, <i>q</i>_<i>P</i>, against time for a range of <i>P</i>_<i>amp</i> values. The flow rates increase linearly as the amplitude of breathing, <i>P</i>_<i>amp</i> increases. Inspiration and expiration are labelled (INSP and EXP). All parameters, except <i>P</i>_<i>amp</i>, are as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004637#pcbi.1004637.t001" target="_blank">Table 1</a>.</p

The relative resistance of <i>R</i>_{1,<i>insp</i>}/<i>R</i>_2,<i>exp</i> affects the duration of the expiration and inspiration phases.

<p>Here we plot the ratio of the inspiration and expiration phase durations (I:E time ratio) against the relative resistance of <i>R</i>_{1,<i>insp</i>}/<i>R</i>_2,<i>exp</i> whilst keeping the total resistance constant (<i>R</i>_{1,<i>insp</i>} + <i>R</i>_2,<i>exp</i> = 6 cmH₂O/L⋅s). The same effect is seen for a range of <i>γ</i> values. The dashed line indicates where the period of expiration and inspiration are equal, <i>T</i>_<i>e</i> = <i>T</i>_<i>i</i>.</p

Varying the ratio of compliances <i>γ</i> = <i>C</i>₁/<i>C</i>₂ changes the timing of the flow through the parabronchi.

<p>As <i>γ</i> increases (<i>C</i>₁ increases relative to <i>C</i>₂) more air flows through the parabronchi during expiration. The dashed line at 1 indicates when the total flow during expiration and inspiration are equal. All the parameters are as given in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004637#pcbi.1004637.t001" target="_blank">Table 1</a>. The vertical dotted line shows the selected default <i>γ</i> value.</p

The shape of the flow <i>q</i>_<i>P</i> smoothes out as <i>C</i>_<i>tot</i> increases.

<p>This figure plots the flow through the parabronchi <i>q</i>_<i>P</i> against time for a range of <i>C</i>_<i>tot</i>, with all traces aligned such that the beginning of inspiration is at <i>t</i> = 0. The parameter <i>γ</i> = 1, other parameters are given in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004637#pcbi.1004637.t001" target="_blank">Table 1</a>. Note: the transition from inspiration to expiration happens at close to the same time for <i>C</i>_<i>tot</i> ≥ 90 mL/cmH₂O and is shown as a black dashed line. The time of the transition for <i>C</i>_<i>tot</i> = 9 mL/cmH₂O is shown with a dark blue dot-dashed line.</p

Changing the relative resistance of <i>R</i>_{1,<i>insp</i>} / <i>R</i>_2,<i>exp</i> affects the efficiency of the system.

<p>Plot of the overall efficiency when <i>R</i>_{1,<i>insp</i>} / <i>R</i>_2,<i>exp</i> is varied whilst keeping the total resistance of the system constant (<i>R</i>_{1,<i>insp</i>} + <i>R</i>_2,<i>exp</i> = 6 cmH₂O/L⋅s). The effect is similar for a range of <i>γ</i> values. All other parameters are given in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004637#pcbi.1004637.t001" target="_blank">Table 1</a>.</p