\documentclass[12pt,oneside]{bidipresentation} %By Sayyed Ahmad Mousavi (s.a.mousavi@hotmail.com) \usepackage{SBUKPresentation} %%رنگ​ها \sidebartc{rgb}{0,0,0}%{cmyk}{0,0,0,1} رنگ متن سایدبار \linktc{rgb}{0,0,0}%{cmyk}{0,0,0,0} رنگ لینك​ها \rtopbarc{rgb}{0.9,1,0.6}%{cmyk}{0.94,0.54,0,0} رنگ نوار بالا راست \ltopbarc{RGB}{0,61,100}%{cmyk}{0.15,0.15,0,0} رنگ نوار بالا چپ \ltopbartc{rgb}{1,1,1}%{cmyk}{0,0,0,1} رنگ متن نوار بالا \rbotbarc{RGB}{0,21,30}%{cmyk}{0.15,0.15,0,0} رنگ نوار پایین راست \lbotbarc{rgb}{0.28,0.76,0.99}%{cmyk}{0.94,0.54,0,0} رنگ نوار پایین چپ \lbotbartc{rgb}{1,1,1}%{cmyk}{0,0,0,0} رنگ متن نوار پایین %\settextfont[Scale=1]{XB Zar} %\setlatintextfont[Scale=1]{Times New Roman} %\setdigitfont{XB Zar} %\defpersianfont\titr{XB Titre} %\defpersianfont\nast{IranNastaliq} \title{PATHOLOGICAL CONDITIONS RESULTING FROM INSTABILITIES IN PHYSIOLOGICAL CONTROL SYSTEMS} \author{Nasir Mirzayi} \renewcommand*{\baselinestretch}{1.5} \begin{document} \begin{staticcontents*}{botbar1} \begin{center} \begin{footnotesize} \makeatletter\@title\makeatother \end{footnotesize} \end{center} \end{staticcontents*} \begin{staticcontents*}{botbar2} \begin{center} \begin{footnotesize} \rm\makeatletter\@author\makeatother \end{footnotesize} \end{center} \end{staticcontents*} \begin{plainslide} \AddToShipoutPicture*{\put(-50,50){ \includegraphics[height=\paperheight]{besm} }} \end{plainslide} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{plainslide} %\AddToShipoutPicture*{\put(-683,0){\includegraphics[height=\paperheight,width=\paperwidth]{tex.jpg}}} \distance{1} \centering% \LARGE \color[rgb]{0,0,0.6}{\Huge\titr{\makeatletter\@title\makeatother}} \distance{2} \color{black}\rm \begin{LARGE}\bfseries \makeatletter\@author\makeatother\\[2ex] \end{LARGE} \begin{large}\bfseries \end{large} \vfill \begin{center}\color[rgb]{0,0,0.6} Department of Mathematics Science\\ Tarbiat Modares University\\ \end{center} \distance{2} % \tableofcontents \end{plainslide} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \fancyfoot{\thepage} \begin{samframe}{INTRODUCTION} \begin{description}\Large A large number of human diseases are characterized by changes in the qualitative dynamics of physiological control systems: Systems that normally oscillate, stop oscillating, or begin to oscillate in a new and unexpected fashion, and systems that normally do not oscillate, begin oscillating. These changes in qualitative dynamics often have a sudden onset, and in many instances it has not been possible to identify the factors that lead to the disease. By dynarnical disease we mean a disease that occurs in an intact physiological control system operating in a range of control parameters that leads to abnormal dynamics and human pathology. \end{description} \end{samframe} %%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{samframe}{INTRODUCTION } consider the ordinary differential equation \begin{align}\label{1} \dfrac{dx}{dt}=\lambda-\gamma x \end{align} \begin{itemize} \item $ x $ variable of interest \\ \item $ \lambda $ production rate for x \\ \item $ \gamma $ destruction rate of x \\ \item $ t $ is the time. \end{itemize} \end{samframe} %%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{samframe}{INTRODUCTION } For $ \lambda $ and $ \gamma $ constant, $ x \rightarrow \frac{\lambda}{\gamma} $ in the limit$ t \rightarrow \infty $. However, in many physiological systems $ \lambda $ and $ \gamma $ at $ t $ may depend on $ x $ and/or $ x $, (the value of $ x $ at a time$ t- \tau $, where $ \tau $ is the time lag). We show that instabilities analogous to those found in pathological conditions in humans can be reproduced by assuming that $ \lambda $ and $ \gamma $ in Equation (\ref{1}) are appropriate nonlinear functions of $ x $ and/or $ x_{\tau} $. \end{samframe} %%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{samframe}{RESPIRATION} Respiratory oscillations in mammals are generated in the brainstem. Several groups have shown that this region is essential for respiration, and that cells located in the brainstem fire phasically during the respiratory cycle. Several different classes of cells have been identified (e.g., inspiratory cells and expiratory cells which fire during inspiration and expiration, respectively), but the number of different classes of cells, their anatomical location, and interconnections are not agreed upon by workers in the field. A number of mathematical models of the respiratory oscillator have been suggested. \end{samframe} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{samframe}{RESPIRATION} Experimental studies have shown that both the frequency and amplitude of the respiratory oscillations can be modulated by a variety of factors including activity in the cerebral cortex, $ pH $ and concentrations of $ CO_{2} $, and $ O_{2} $ in arterial blood and cerebrospinal fluid, and the amount of stretching in the intercostal muscle in the chest. In healthy humans, these inputs act to maintain arterial concentrations of $ O_{2} $ and $ CO_{2} $ at constant levels. \end{samframe} %%%%%%%%%%%%%%%%%% \begin{samframe}{Respiratory Disorders} Rapid shallow breathing (panting, tachypnea, or polypnea) occurs in a variety of pathological conditions, for example, as a result of pain in structures moved by breathing, during fever, and under severe hypoxia (low oxygen tension) of long duration. In dogs, panting is a normal response to heat stress, and brief periods of panting (frequency $ 300-400 min^{-1} $) alternate with periods of normal breathing (frequency $ 20-40 min^{-1} $ ). Superimposed on the panting rhythm may be an occasional deep breath to give a sighing pattern. \end{samframe} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{samframe}{variety of patterns} There are a variety of patterns in which periods of apnea alternate with periods of breathing. We call these apneic patterns. Apneic patterns are referred to by clinicians generically as “periodic breathing.” \mathbf{variety of apneic patterns} \begin{enumerate} \item Cheyne-Stokes respiration \\ \item Biot breathing \\ \item infant apnea \\ \end{enumerate} \end{samframe} %%%%%%%%%%%%%%%%%% \begin{samframe}{Cheyne-Stokes breathing} \begin{itemize} \item Cheyne-Stokes breathing is characterized by a regular waxing and waning of breathing amplitude separated by periods of distinct apnea \item This is the most common apneic pattern encountered clinically, and is often found in obese patients, patients with congestive heart failure, and patients with certain neurological deficits \item It is interesting that a regular waxing and waning of breathing amplitude wilhout apnea (a “wavy” pattern), is more commonly observed than Cheyne- Stokes respiration and is not necessarily associated with pathological conditions. \end{itemize} \end{samframe} %%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{samframe}{Cheyne-Stokes breathing diagram} \begin{frame} \centerline{ \includegraphics[scale=0.8]{fig11.jpg}} \caption{A wavelike respiration pattern} \end{frame} \end{samframe} %%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{samframe}{Cheyne-Stokes breathing diagram} \begin{frame} \centerline{ \includegraphics[scale=0.8]{fig12.jpg}}\\ \caption{Cheyne-Stokes respiration in a 29-yearold man} \end{frame}\\ \end{samframe} %%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{samframe}{Cheyne-Stokes breathing diagram} \begin{frame} \centerline{ \includegraphics[scale=0.6]{fig13.jpg}} \end{frame}\\ \end{samframe} %%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{samframe}{Biot breathing} \begin{itemize} \item Biot breathing refers to alternating periods of breathing with apnea. The regular alternations of Cheyne-Stokes respiration are absent, and marked irregularity is observed \item Biot breathing is often observed just prior to death \item Biot's breathing is characterized by periods, or "clusters", of fairly rapid respirations of close to equal depth followed by reular periods of apnea that can last between 15 seconds to 120 seconds. Biot's breathing is very imilar to Cheyen-Stokes except the spontaneous tidal volume is equal throughout the period of respiration \end{itemize} \end{samframe} %%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{samframe}{Biot breathing diagram} \begin{frame} \centerline{ \includegraphics[scale=0.6]{fig14.jpg}} \end{frame}\\ \end{samframe} %%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{samframe}{Biot breathing diagram} \begin{frame} \centerline{ \includegraphics[scale=0.6]{fig15.jpg}} \end{frame}\\ \end{samframe} %%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{samframe}{Infant apnea} Infant apnea refers to the pronounced periods of apnea found in most premature and many full term infants. The apnea generally occurs during rapid eye movement sleep. It has been speculated that there is a causal relation between the sudden infant death syndrome and infant apnea. \end{samframe} %%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{samframe}{Infant apnea diagram} \begin{frame} \centerline{ \includegraphics[scale=0.7]{fig16.jpg}} \caption{(A) breathing spontaneously returned, (B) physical stimulation was applied. \end{frame}\\ \end{samframe} %%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{samframe}{Infant apnea diagram} \begin{frame} \centerline{ \includegraphics[scale=0.6]{fig17.png}} \end{frame}\\ \end{samframe} %%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{samframe}{Mathematical Models of Respiratory Disorders} Theoretical studies of the mechanism of Cheyne-Stokes respiration ascribe the slow oscillations observed to instabilities in the respiratory control system. It is known that the total ventilation increases monotonically as the $ CO_{2} $ concentration in arterial blood increases. However, since the blood is oxygenated in the lungs but the receptors, which are sensitive to the $ CO_{2} $ concentration, are believed to be present in the brainstem, there is an inherent time lag $ \tau $ in the respiratory control system. Several investigators have developed complex systems of differential- delay equations to describe the production, transport, and elimination of $ CO_{2} $ in humans. Since the mathematical properties of these complex systems of equations are not easily deduced, we have proposed a simplified schematic model which displays similar qualitative features to the more complex models. \end{samframe} %%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{samframe}{Mathematical Models of Respiratory Disorders} The ventilation $ V $ at time f is assumed to depend on $ x(t- \tau) $ , the $ CO_{2} $ concentration at time $ t- \tau $. We also assume that $ CO_{2} $ elimination is proportional to the product of $ CO_{2} $ concentration $ (x) $ and ventilation. Experimental studies indicate that ventilation is an increasing monotonic function of $ CO_{2} $ concentration. Assuming that the dependence of the ventilation on $ CO_{2} $ concentration is described by the Hill function $ V=\frac{V_{max}x_{\tau}^{n}}{\theta^{n}+x_{\tau}^{n}}$, we obtain: \begin{align}\label{2} \dfrac{dx}{dt}= \lambda - {\dfrac{\alpha V_{max}x_{\tau}^{n}x}{\theta^{n}+x_{\tau}^{n}}} \end{align} \begin{itemize} \item $ V_{max} $ is the maximum ventilation. \item $ n $, $ \theta $ and $ \alpha $ are parameters chosen to agree with experimental data. \end{itemize} \end{samframe} %%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{samframe}{Mathematical Models of Respiratory Disorders} The stability of (\ref{2}) in the neighborhood of the steady state can be analyzed (at the steady state $ dx/dt = 0 $). Denoting the values of $ x $ and $ V $ at the steady state by $ x_{0} $ and $ V_{0} $, and setting $ S_{0} = (dV/dx)_{x_{0}} $ and $\alpha =\lambda /x_{0}V_{0} $, the first-order equation in $ x $ and $ x_{ \tau} $ is, \begin{align}\label{3} \frac{dx}{dt}=-\frac{\lambda}{x_{0}}(x-x_{0})-\frac{\lambda S _{0}}{V_{0}}(x_{\tau}-x_{0}) \end{align} The stability criteria for (\ref{3}) are well known. \end{samframe} %%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{samframe}{Mathematical Models of Respiratory Disorders} In general, for the first-order linear differential-delay equation, \begin{align}\label{4} \frac{dz}{dt}=Az+Bz_{\tau} \end{align} the eigenvalues of the steady state $ z = 0 $ have negative real parts if and only if, $$ A \tau < 1 $$\\ \begin{align}\label{5} A \tau < -B \tau < \sqrt{(A \tau)^{2}+a_{1}^{2}} \end{align} where $ a_{1} \in (0, \pi) $ is the root of the equation, \begin{align}\label{6} a \cot a = A \tau \end{align} and $ a_{1} = \dfrac{\pi}{2} $ if $ A \tau =0 $ \end{samframe} %%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{samframe}{Mathematical Models of Respiratory Disorders} Applying (\ref{5}) to determine the stability of the steady state of (\ref{2}), we find that the steady state will be stable provided: \begin{align}\label{7} \frac{{\lambda {S_0}\tau }}{{{V_0}}} < \sqrt {{{\left( {\frac{{\lambda \tau }}{{{x_0}}}} \right)}^2} + a_1^2} \end{align} where $ a_{1} $ is found by solving : \begin{align}\label{8} a \cot a = -\frac{\lambda \tau}{x_{0}} \end{align} Further analysis requires numerical values for the parameters in (\ref{7}) and (\ref{8}). \end{samframe} %%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{samframe} The nonzero steady states $ x_{0} $ of these equations may be calculated and the local behavior of the solutions examined near $ x_{0} $ as in the section on respiratory models. The results of these computations indicate that increases in $ n $, $ \tau $, or $ \lambda_{0} $, or decreases in $ \gamma $ or $ \lambda_{1} $ may lead to a loss of stability at $ x_{0} $ and the appearance of oscillatory solutions about $ x_{0} $. To investigate the behavior of (\ref{3}) and (\ref{4}) away from the region of applicability of this linear analysis, we have numerically integrated the equations using either a predictor-corrector or a \mathbf{Runge-Kutta} integration scheme. For Equation (\ref{3}), we have found only two qualitatively different behaviors: 1) either a stable steady state or 2) a stable limit cycle oscillation-for any set of parameters only one or the other behavior is found. \end{samframe} %%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{samframe} However, the qualitative behavior of (\ref{4}) in response to parameter changes are quite different. To illustrate this behavior we assume that y = I , XI = 2, 0 = 1 (so xo = I), and T = 2. Equation 3.5 was integrated starting from an initial condition x(r) = 0.50, -7 < f < 0, using a predictor-corrector integration routine with a step size of A = 0.05 for various values of n. The linear analysis of(3.5) indicates that with these parameters, xo = I should be stable for n < 5.0404, and, as stability is lost for n - 5.0404, periodic solutions of period T - 5.49 should appear. Numerical solutions of (3.5) in the neighborhood of n = 5.04 bear out the accuracy of this analysis, and indicate that the periodic solutions are stable. \end{samframe} %%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{samframe} I n FIGURE6 we show the dynamics of (3.5) in the x, - x phase plane for several values ofn. Notice that as n is increased, the oscillation undergoes a sequence of bifurcations. Further, this sequence is analogous to the sequence of bifurcations observed in a class of finite-difference equations in Notice that the oscillatory patterns in FIGUR6Ea and b are analogous to the “period 2” oscillation; FIGURE6b and 6c are analogous to the “period 4” oscillation: FIGU R E 6d is analogous to the “period 8” oscillation; FIGURE6g is analogous to the “period 3” oscillation; FIGURE6 h is analogous to the “period 6” oscillation: FIGUR6Ee and 6i are analogous to the ”chaotic regimes.” In FIGUR6Ef w e observe that the “period 3” oscillation has almost “condensed” out of the chaotic regime. FIGUR6Ej shows a stable oscillation, which appears for large n. \end{samframe} %%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{samframe} To examine the possible origins of PH within the PPSC, we assume I ) cells in the PPSC are either proliferating or in the resting phase (Go) cells, 2 ) cells travel through proliferation to undergo mitosis at a fixed time 7 (days) from their time of entry into the proliferative phase, 3) all cells enter Go upon the completion of mitosis, and 4) cells in Go exit randomly to differentiate either irreversibly into one of the haematopoietic lines (myeloid, erythroid, or thromboid) at a rate a (day-’) or reenter proliferation at a rate fi (day-‘) proportional to their concentration (in certain pathological states, proliferating phase cells die at a rate k (day-') proportional to their concentration). \end{samframe} %%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{samframe} We further assume that control in the PPSC is exercised over the rate p of cell reentry into proliferation and that @ is a monotonic decreasing function of Go cells. Calling the population density of the Go cells x (cells/kg), we obtain, d-s- 2@008nx, pOenx dt 8" + x," en + Xn exp(-kr) - ax - ~ -- (3.6) where 0 (cells/kg), Po (day-'), and n are parameters characterizing the PPSC. PH and A A . Based on several lines of evidence, there is good reason to believe that the strange dynamics of PH is intimately connected with the death of cells from the proliferating phase of the PPSC. Further, at least some cases of AA may involve the death of proliferating PPSC cells.' \end{samframe} %%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{samframe} It is possible to estimate the values of the parameters characterizing a PPSC population in a normal (k = 0) state. Depending on the values of these parameters, the linear analysis of the stability of the nonzero steady state xo of (3.6) predicts two possible responses in the PPSC to increases in k (FIGURE7) . For humans, taking n = 3, T = 2.22 days, a = 0.05 day-', and Po = 1.77 days, an increase in k leads to a depression of the steady state xo. At about k = 0.235 day-', the steady state is no longer stable and periodic solutions of period 19.00 days are predicted. At k = 0.287 day-' a stable steady state reappears. For 0.235 < k < 0.287, numerical studies indicate stable limit cycle oscillation. These behaviors are illustrated in FIGUR8E. Since both depression of cell densities and periodic dynamics can be accounted for in this single model and the numerical values obtained are reasonable, it has been suggested that both AA and PH can be accounted for by varying rates of cell destruction during proliferation of stem \end{samframe} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{samframe}{مراجع} \begin{itemize} \item دیوان حافظ، انتشارات سروش. \item دیوان حافظ، انتشارات سروش. \item دیوان حافظ، انتشارات سروش. \item دیوان حافظ، انتشارات سروش. \begin{LTRitems} \item Johann Wolfgang von Goethe, Faust. \item Johann Wolfgang von Goethe, Faust. \item Johann Wolfgang von Goethe, Faust. \item Johann Wolfgang von Goethe, Faust. \item Johann Wolfgang von Goethe, Faust. \end{LTRitems} \end{itemize} \end{samframe} %%%%%%%%%%%%%%%%%%%%% \begin{samframe}{مراجع} \begin{itemize} \item دیوان حافظ، انتشارات سروش. \item دیوان حافظ، انتشارات سروش. \item دیوان حافظ، انتشارات سروش. \item دیوان حافظ، انتشارات سروش. \begin{LTRitems} \item Johann Wolfgang von Goethe, Faust. \item Johann Wolfgang von Goethe, Faust. \item Johann Wolfgang von Goethe, Faust. \item Johann Wolfgang von Goethe, Faust. \item Johann Wolfgang von Goethe, Faust. \end{LTRitems} \end{itemize} \end{samframe} %%%%%%%%%%%%%%%%%%%%% \begin{samframe}{ تشکر و قدردانی } \begin{center} \nast \fontsize{70}{80} \selectfont با تشکر از توجه شما \end{center} \end{samframe} %%%%%%%%%%%%%%%%%%%555 \end{document}