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\title{PATHOLOGICAL CONDITIONS RESULTING
FROM INSTABILITIES IN
PHYSIOLOGICAL CONTROL SYSTEMS}
\author{Nasir Mirzayi}


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\begin{document}


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Department of Mathematics Science\\
Tarbiat Modares University\\
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%	\tableofcontents
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\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.
In this paper, the changes in qualitative dynamics associated with the onset of
the disease are identified with bifurcations in the dynamics of mathematical models
of the physiological control systems. We shall consider in some detail dynamical
diseases in the respiratory and haematopoietic systems.
\end{description}
\end{samframe}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{samframe}{Mathematical Models of Dynamic Hematological Diseases }
\begin{center}
\nast \fontsize{70}{80} \selectfont \color{red}
In this section we consider two problems: the first is related to peripheral control
over circulating cell numbers (e.g., as exercised via erythropoietin), and the
second is related to control within the PPSC compartment. In both cases, our
models are similar to models proposed by others for coupled stem cell-peripheral
control systems in granulopoiesis, erythropiesis and thrombopoiesis
\end{center}
\end{samframe}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{samframe}{Peripheral Control in Haematopoiesis}
We first consider a simple model for the control of peripheral blood cell numbers
via a humoral feedback mechanism. Let $ x ( t ) $  be the concentration of circulating
cells (cells/kg) and assume that cells are randomly lost from the circulation
at a rate $ \gamma $ (day^{-1}) proportional to their concentration. To reproduce the effects
of poietin feedback control from the circulating population of cells, we assume
that the flux ($ \lambda $ in cells/kg/day) into the circulation from the stem cell compartment
depends on $ x $  at time$ t- \tau $, and thus the dynamics of $ x ( t ) $ is governed by
\begin{align}\label{1}
\dfrac{dx}{dt}=\lambda(x_{\tau})-\gamma x
\end{align}
\end{samframe}

%%%%%%%%%%%%%%%%%%5
\begin{samframe}
We have examined two forms for $ \lambda (x)$:
\begin{align}\label{2}
\lambda (x)=\begin{cases}\frac{ \lambda_{0}\theta^{n}}{\theta^{n}+x^{n}}\\
\\
\frac{ \lambda_{1}\theta^{n}x}{\theta^{n}+x^{n}}
\end{align}
\mathbf{Parameters}
\begin{itemize}
\item $ n, \theta $ (cells/kg)
\item \lambda_{0} (day^{-1})
\itam \lambda_{1} (kg/day-cell)
\end{itemize}
\end{samframe}

%%%%%%%%%%%%%%%%%%5
\begin{samframe}
Combining Equation (\ref{1}) with the equations for $  \lambda $ thus gives two possible equations
governing the evolution of$ x(t) $,
\begin{align}\label{3}
\dfrac{dx}{dt}=\frac{ \lambda_{0}\theta^{n}}{\theta^{n}+x_{\tau}^{n}}-\gamma x
\end{align}
and
\begin{align}\label{4}
\dfrac{dx}{dt}=\frac{ \lambda_{1}\theta^{n}x_{\tau}}{\theta^{n}+x_{\tau}^{n}}-\gamma x
\end{align}
An equation similar to  (\ref{3}) has been proposed for the control of erythropoiesis.
\end{samframe} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{samframe}{Bifurcating solutions to Equation (\ref{4})}
Here we show the numerically detetminedsolutions to this equation in the form of phase plots of$ x_{\tau} $ =$ x(2) $ versus $ x = x(1) $. The integrations were carried out with a step size of$ 0.05 $  using a predictor-corrector method, assuming $ \gamma =1 $, $ \lambda _{1}=2 $, $ \theta=1 $, $ \tau=2 $, and an initial condition on $ x $ and $ x_{\tau}$ of 0.50.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\label{a}
\centerline{ \includegraphics[scale=0.2]{fig1.jpg}}
\end{frame}\\
\mathbf{$ n = 7 $ \quad 100 \leq t \leq 150}
\end{samframe}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{samframe}
\begin{frame}
\label{b}
\centerline{ \includegraphics[scale=0.2]{fig2.jpg}}
\end{frame}\\
\mathbf{$ n = 7.75 $ \quad 150 \leq t \leq 200}
\end{samframe} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{samframe}
\begin{frame}
\label{c}
\centerline{ \includegraphics[scale=0.2]{fig3.jpg}}
\end{frame}\\
\mathbf{$ n = 8.50 $ \quad 200 \leq t \leq 250}
\end{samframe} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{samframe}
\begin{frame}
\label{d}
\centerline{ \includegraphics[scale=0.2]{fig4.jpg}}
\end{frame}\\
\mathbf{$ n = 8.79 $ \quad 300 \leq t \leq 400}
\end{samframe} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{samframe}
\begin{frame}
\label{e}
\centerline{ \includegraphics[scale=0.2]{fig5.jpg}}
\end{frame}\\
\mathbf{$ n = 9.65 $ \quad 300 \leq t \leq 600}
\end{samframe} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{samframe}
\begin{frame}
\label{f}
\centerline{ \includegraphics[scale=0.2]{fig6.jpg}}
\end{frame}\\
\mathbf{$ n =9.69715 $ \quad 300 \leq t \leq 400}
\end{samframe}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{samframe}
\begin{frame}
\label{g}
\centerline{ \includegraphics[scale=0.2]{fig7.jpg}}
\end{frame}\\
\mathbf{$ n = 9.6975 $ \quad 300 \leq t \leq400}
\end{samframe}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{samframe}
\begin{frame}
\label{h}
\centerline{ \includegraphics[scale=0.2]{fig8.jpg}}
\end{frame}\\
\mathbf{$ n = 9.76 $ \quad 300 \leq t \leq 400}
\end{samframe}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{samframe}
\begin{frame}
\label{i}
\centerline{ \includegraphics[scale=0.2]{fig10.jpg}}
\end{frame}\\
\mathbf{$ n = 10.0 $ \quad 300 \leq t \leq 600}
\end{samframe}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{samframe}
\begin{frame}
\label{j}
\centerline{ \includegraphics[scale=0.2]{fig10.jpg}}
\end{frame}\\
\mathbf{$ n =20.0 $ \quad 300 \leq t \leq 400}
\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}