text:
\section{Introduction}

First described in 2004, the cardiac biopsy-derived progenitor cells, growing in vitro as ``niche''-like microtissue, known as {\em Cardiospheres} (CSps), represent a widely adopted platform technology holding great promise to realize a powerful cell therapy system, in vitro drug screening and disease modeling opening new opportunities for myocardial repair \citep{mess,circul,cheng}. To achieve this goal, an optimal management of qualitative and quantitative CSps growth-modifying factors in terms of cell proliferation, differentiation and interaction with the external environment has to be known. It requires a simulator of the features and the fate of the Cardiospheres-niche-system which is based on new as well as already existing experimental biological data and on reliable mathematical models.

Our aim is to model the formation and differentiation of the original 3D cardiac biopsy-derived heart progenitors in vitro system, that is the CSp. These structures are cellular spheroids with a central nucleus of less differentiated elements surrounded by outer layers of cells more committed toward different levels of cardio-vascular differentiation with other proliferation features (Figure \ref{fig1_a}). Their individual formation in vitro starts from a few (low differentiated and less specialized) progenitor cells which are cultured in a carefully chosen environment and under specific conditions \citep{mess,chim1}, and have the potential to develop into a more cardiac-specific differentiation which, unless  derived from pre-natal hearts, are no longer able to achieve a terminal level of contractile cardiomyocytes.
 The major types of molecular processes that control cellular differentiation and proliferation involve various physical factors, nutrients and cell signaling.  During the multiple stages of development, the cell size and shape changes dramatically, as a response to signaling molecules \citep{brown,agath}.
\begin{figure}
\centering
\subfigure[]{\includegraphics[width=0.5\textwidth]{figure/stem_cell1.png}\label{fig1_a}}\hspace{-1.5 cm}
\subfigure[]{\includegraphics[width=0.5\textwidth]{figure/stem_cell2.png}\label{fig1_b}}
\caption{(a) A CSp by immunofluorescence after 3 days: a cluster of immature stem cells (green) is localized in the central core of Cardiospheres, surrounded by differentiated supporting cells (red). Blue spots indicate cell nuclei, and cells inside the black area (non labeled) are to be intended in an intermediate level of differentiation, see also Section \ref{sec:problem} (source from \citealp{cheng}, by courtesy of John Wiley \& Sons). (b) An example of a numerical simulation of our mathematical model showing a CSp at 3 days. Green, blue and red colors mark respectively three increasing degrees of maturation of a single cell (see Section \ref{sec:numeric} for further details).}
\label{fig1}
\end{figure}
This biological environment is similar to that of  tumor spheroids as described in \citet{wall}. Neoplastic cells hold several growth strategies, such as genetic mutation of cell cycle inhibitors, neoangiogenesis, etc., and cannot be directly compared with those more limited of the adult cardiac progenitors. Nevertheless both of the 3D cellular growth mechanisms share some common features. CSps and neoplastic spheroids hold more than two functional phenotypes (proliferating, quiescent, differentiating, dying). The growth in a form of a 3D sphere limits the diffusion of the nutrients and oxygen and leads to the formation of a central core of death cells as a consequence of necrosis, apoptosis and anoichisis (Figure \ref{fig2}). It is well known that in tumors the fastest proliferating cells are present in the intermediate and external layers forming the so-called growing front. On the contrary, in small and medium CSps the highest rate of the proliferation is observed in the central area due to the hypoxic conditions, which are closer to those corresponding to the pre-natal developing heart.
 However, while the spheres become larger the necrotic core appears as in tumors.
\begin{figure}
\centering\includegraphics[width=0.5\textwidth, angle=0]{figure/image09_single22.jpg}
\caption{Histological preparation of CSps ultra-thin slices. Hemathoxilin-eosin staining shows a nuclei-free central necrotic area more evident in larger CSps.
}
\label{fig2}
\end{figure}
In this paper we introduce a discrete in continuous non-deterministic model to simulate the growth of a CSp aggregate. We treat the various biological constituents using different levels of description: on the cellular scale we adopt a discrete description, while on the molecular scale chemical signals are considered as continuous variables \citep{anderson,preziosi}. This choice is justified by the relatively small number of cells involved in the formation of CSps, for which the hypothesis of the continuum does not hold, and by the much smaller size of the molecules with respect to them. Similar approaches have been adopted in analogous cell systems, for instance in relation to the modeling of tumor growth (\citealp{enderling2012,kim,osborne2010,Chaplain2011,fletcher2015}, and references therein) or, recently, to describe the morphogenesis of the lateral line in the zebrafish embryo \citep{dicostanzo1}, and to study the behavior of the collective motion of cells under chemotaxis and alignment effect \citep{dicostanzo2}. With particular regard to the stem cell cultures, in \citet{wu} a continuous diffusion--reaction mathematical model is proposed to investigate oxygen transport and distribution in embryonic stem cell aggregates, also in comparison with experimental data.

Our proposed mathematical model relies on a hybrid description: each cell is described separately by a system of ordinary differential equations, which take into account the self--organizing interactions between cells, through typical terms of collective dynamics and chemotaxis effects (see for instance the reviews by \citealp{vicsek-collective,vicsek-collective-cell}); at a macroscopic level, by reaction-diffusion equations, we describe the concentrations of oxygen, as a representative of nutrients essential for cells metabolism, and of a growth factor TGF-$\beta$ (Transforming-Growth-factor-$\beta$), as a chemoattractant and main regulator of the cell differentiation, secreted by the cells and also present in the environment. Additionally, to model the formation of the Cardiosphere, we introduce two processes, the cell proliferation and differentiation, which lead to the growth and maturation of an initial cell cluster. Moreover, we distinguish three differentiation levels of cell maturation, which also influence the two aforesaid processes.
Moreover they are regulated by the local concentrations of oxygen and TGF-$\beta$. Another crucial feature of our model is the stochasticity. It is motivated by the so called noise in biological systems and the averaging of cells characteristics. The former is mainly due to the genetic diversity among individual cells, random collisions and thermal fluctuations in chemical reactions and randomness of various external factors, see \citet{Tsimring}. The latter concerns the simplifications introduced in our model, such as the uniform size of cells, equal maturation time or signal sensing. The randomness of the biological phenomena in our model is present in the proliferation and differentiation processes. When the necessary conditions are satisfied cells divide or pass to a higher differentiated level with some probability, which increases if the conditions are favorable.

Using the proposed mathematical model we have simulated the formation of a Cardiosphere in a two--dimensional setting. After a careful sensitivity analysis of the model parameters we were able to obtain numerical results comparable with the existing biological experiment: see again Figure \ref{fig1}, in which the structure and composition of the two images, real experiment (a) and numerical simulation (b), are in good agreement. In order to meet the need of biological research and possible therapeutic applications (including the achievement of a more advanced differentiation level) we have analyzed the development of the Cardiosphere at two typical experimental concentration of oxygen.

The paper is organized as follows: in Section~\ref{sec:problem} we present the general setting of the problem and the mathematical framework used throughout the paper. In Section~\ref{sec:model} we describe in detail the mathematical model for the cells and nutrients dynamics, together with the stochastic mechanisms of cell proliferation and differentiation. Section~\ref{sec:numeric} contains the numerical results and relative biological interpretations. Finally, Section \ref{conclusion} is devoted to the conclusions and possible future perspectives.

\section{Formulation of the problem}\label{sec:problem}

The in vitro 3D CSp formation starts from a low-density seeding of the explants-biopsy-derived cells in a defined culture conditions. First,
adhered to the culture dish bottom the individual  stem cells start to
aggregate in small clusters of very few elements. Reaching certain dimensions they detach and grow immersed in a liquid environment and in a controlled gas atmosphere, which contain substances essential for their metabolism.
In our model we have reduced to two the biological and physical variables affecting the CSps growth and differentiation: the oxygen, representing nutrients and gases, and the TGF-$\beta$ which is present in the culture medium and is produced by the cells themselves. The latter represents a key modulator of the main pathway involved in the induction of the non-terminal differentiation level achieved in our basic experimental conditions \citep{forte}.
We assume that during the formation of a CSp, the cells are subject to two basic cellular processes, the proliferation and the differentiation,
which depend, with some probability, on the type of a cell and on the local availability of oxygen. Moreover, we consider three different increasing differentiation levels for living cells and, separately, dead cells. The complex differentiation mechanism is triggered by a large enough
concentration of TGF-$\beta$,  is hindered by the presence in the closest proximity of a large number of
cells of the same type
and is inhibited when the concentration of oxygen
drops below a threshold value (see Section \ref{sec:prol_diff}).

To describe the formation of CSps in the next section we propose a hybrid model, in which cells are
considered as discrete entities while oxygen and TGF-$\beta$ concentrations are continuum variables. We assume that in the absence of specific signals the cell dynamics is due to attractive-repulsive forces and friction effect, taking place isotropically. The TGF-$\beta$, produced by the cells and also present in the external medium, is a chemoattractant for cell which moves towards its higher concentrations. For computational simplicity a two-dimensional description is adopted.

\subsection{Mathematical notations and operators}\label{sec:mathematical_model}

In a Cartesian coordinate system, let us consider a rectangular domain $\Omega$, which in our model represents a limited portion of the culture (containing many CSps) with a single cluster of stem cells in its center (Figure \ref{fig3}). For this reason in the following we can assume periodic boundary condition in relation to the equations of the model.
\begin{figure}
\centering\includegraphics[width=0.45\textwidth]{figure/cardiosfera_dominio.png}
\caption{Experimental picture of a culture of CSps. The dotted red line indicates the portion chosen as possible domain $\Omega$ for a single CSp in our mathematical model.}
\label{fig3}
\end{figure}
At the initial time, $N_0$  cells at the same maturation level (Section \ref{sec:prol_diff}) and radius $R$ (for simplicity all cells are assumed to have the same constant radius) are placed randomly at positions $\bfm X_i, \, i=1,2,...,N_0$ at the center of the domain. In the following, due to proliferation, the number of cells at the time $t$ will be denoted by $N(t)$. In the model we denote by $c(\mathbf{x},t)$ and $S(\mathbf{x},t)$ the concentrations of oxygen and TGF-$\beta$ respectively, and in the numerical simulations these variables are discretized on a uniform spatial mesh. In general,  the cell position $\bfm X_i$ does not coincide with a mesh grid point,  and we have to ``approximate'' the value of $c(\bfm X_{i}, t)$.
In general, a cell ''feels'' the presence of chemical substance (or a signal) $h(\bfm X_i, t)$ not only at the center $\bfm X_i$, but also in its proximity. To model this graded influence, a weighted average operator is defined:
\begin{equation}\label{eq:operator}
\mathcal{F}(h(\bfm X_i,t)) = \frac{1}{W}\int_{B(\bfm X_i,\bar{R})} w_i(\bfm x) h(\bfm x,t)\, d \bfm x,
\end{equation}
where:
\begin{equation}
 W:=\int_{B(\bfm X_i,\bar R)} w_i(\mathbf{x})\,d\bfm x,
\end{equation}
and a possible choice for the weight function can be a truncated Gaussian
\begin{equation*}
w_i(\bfm x) :=
\left\{\begin{array}{ll}
2 \exp\left(-|| \bfm x-\bfm X_i||^2 \displaystyle\frac{\log 2}{\bar R^2}\right) -1,  &\quad\textrm{if } || \bfm x- \bfm X_i|| \leq \bar R,\\
0,   &\quad\textrm{otherwise},
\end{array}
\right.
\end{equation*}
which models a sensing intensity decreasing on the distance from the cell center. Here $\bar R$ represents an influence radius, possibly greater than the cell radius $R$ (Table \ref{tab-param-dim}), and
\begin{equation*}
B(\bfm X_i,R):=\{ \bfm x: || \bfm x- \bfm X_i || \leq R\}
\end{equation*}
is the ball centered in $\bfm X_i$ with radius $R$. In the following the characteristic function over the ball $B(\bfm X_i,R)$ is defined as:
\begin{equation*}
\chi_{_{B(\bfm X_i,R)}} := \left\{\begin{array}{ll}
1,  &\quad \textrm{if }  \bfm x \in B(\bfm X_i,R), \\
0,   &\quad \textrm{otherwise}.
\end{array}\right.
\end{equation*}

\section{The mathematical model}\label{sec:model}

In this section we present in detail the discrete in continuous model for the formation of a Cardiosphere. Firstly, we present the equations describing the movement of cells and the forces acting on them. Then we focus on the proliferation and differentiation processes. The basic mechanisms of the phenomena are explained and the reaction-diffusion systems for the time evolution of the oxygen and TGF-$\beta$ concentrations are given.

\subsection{Cells dynamics}

Motion of cells
is the result of mechanical forces acting on them and of their reciprocal interactions. Due to the finite volume of cells there is an internal pressure between them which leads to repulsion
while being compressed. On the other hand, there are attractive bonds between cells which keep them in contact. We also assume that cells are driven by a chemotactic signal,
that is they move toward the higher gradient of the  TGF-$\beta$ concentration,
and are slowed down because of the viscosity of the medium in which they are immersed \citep{rubinstein,fournier,bayly}.

The position of the $i$-th cell $\bfm X_{i}$ follows  Newton's law and is described by the second order ordinary differential equation
\begin{equation}
\ddot {\bfm {X}}_{i} =\sum_{j:\mathbf{X}_j\in B(\mathbf{X}_i,R_2)\backslash\left\{\mathbf{X}_i\right\}}\bfm {K}( \bfm {r}_{ij}) - \mu \dot {\bfm {X}}_i + \alpha \mathcal{F}(\nabla S(\bfm{X}_i,t)) \label{eqn5},
\end{equation}
where dot denotes time derivative, $\mu$ is a friction coefficient for unit mass
and $\bfm{ K}(\bfm {r}_{ij})$ is the attraction-repulsion force for unit mass between i-th and j-th cell, defined as
\begin{equation}\label{attr-rep}
{\bfm K(\bfm {r}_{ij})}:=\left\{
\begin{array}{ll}
\displaystyle - k_1 \left(\displaystyle\frac{1}{||\bfm r_{ij}||} - \displaystyle\frac{1}{R_1}\right) \displaystyle\frac{\bfm r_{ij}}{||\bfm r_{ij}||},&\quad ||\bfm r_{ij}||\leq R_1,\\
\\
\displaystyle k_2(||\bfm r_{ij}||-R_1)\displaystyle\frac{\bfm r_{ij}}{||\bfm r_{ij}||},&\quad R_1<||\bfm r_{ij}||\leq R_{2}.
\end{array}\right.
\end{equation}
The parameters $k_1$, $k_2$ are positive constants, $\bfm r_{ij}=\bfm X_j-\bfm X_i$ is the distance
between cells, $R_1=2R$ and $R_2$ (to be chosen later, see Table \ref{tab-param-dim}) are the radii of repulsion and attraction respectively. Note that equation \eqref{attr-rep}$_1$ gives a repulsion in the form $1/r$, while \eqref{attr-rep}$_2$ is a linear
attractive elastic force.
Similar terms can be found in the model proposed in \citet{dorsogna,joie}.
Equation (\ref{eqn5}) is augmented with the initial conditions
\begin{equation*}
\bfm X_i(0)=\bfm X_i^0,  \qquad \bfm{\dot X_i}(0)=\bfm V_i^0=\bfm {0}.
\end{equation*}

\subsection{Proliferation and differentiation}\label{sec:prol_diff}

Some of the present authors has been investigating the proliferation and differentiation dynamics of cells forming CSps from the original description of the method of isolation and expansion of human cardiac progenitors from heart biopsy \citep{mess}. Confocal analysis of the BrdU labeled cells which incorporate the base analogue within the replicating DNA, showed the positive fluorescence labeled cells located in the inner of the growing spheres. No or a weak signal was present in the external layers of more differentiated cells (that cannot replicate at the same speed). This kind of layered growth implies that the proliferation in the central area depends also on the CSps size and the core-cells may become quiescent or die by necrosis or apoptosis.

It is worth noting that tumor spheroid growth depends (at least partially) on the same factors, however, in this case they show a more pronounced variety of final volumes for any given cell line. We can distinguish  an early phase of global undifferentiated mass, a second stage of a shell of proliferating cells around a central area of quiescent elements and the last stage in which an inner necrotic/apoptotic core is enveloped by the previous two. The external layer always represents the growing front \citep{kim}.

Stem cells are characterized by different levels of maturation (differentiation). To model this feature we mark the $i$-cell with a label $\varphi_i(t) = {1,2,3,d}$, called its ``state''. The state $\varphi=d$ indicates dead cells or the space occupied by necrotic mass (due to adverse biological conditions, as oxygen deficiency for a sufficiently long period of time).  Note that apoptosis, which is a programmed cells death, occurs over a longer timescale and is not considered here. In particular, we introduce a threshold concentration $c_d$ such that whenever $\mathcal{F}(c(\mathbf{X}_i,t))>c_d$ cell survives, otherwise it dies. Such cell is marked as dead ($\varphi=d$) and does not take part in further evolution of the CSp, that is it neither proliferates nor differentiates. However, dead cells do occupy a volume and exert attractive-repulsive forces on other cells. A stochastic process introduced in the process of proliferation and differentiation models also the presence of quiescent cells. Such cells neither proliferate nor differentiate but, in contrast to dead cells, this state results from environment conditions and is only temporary. At a later time a quiescent cell can restart its functions.

For a living cell, $\varphi=1$ denotes the lowest degree of maturation (the least differentiated cells), while $\varphi=2,3$ are the intermediate and the highest degrees of differentiation, respectively. They can be experimentally reached with our previously cited method \citep{mess,chim1}.
 Finally, we define a {\em cell cycle time $T_c$}, i.e. a cell maturation time after which the cell may proliferate and/or differentiate with some probability.

\subsubsection{Mechanism of differentiation}\label{sec:diff}

In the model we assume that, with some probability and when {\em environment} conditions are satisfied, from time $t$ to time $t+\Delta t $ a cell can
change its state to a higher one
by a unity according to Table~\ref{Tab:differentiation}, or can die (as described in Section \ref{sec:proliferation}).

\begin{table}[htbp!]
\centering \begin{tabular}{|c||c|c|c|}
\hline
                & $\varphi_i(t+\Delta t)=1$ & $\varphi_i(t+\Delta t)=2$ & $\varphi(t+\Delta t)=3$ \\ \hline\hline
$\varphi_i(t)=1$& \checkmark                & \checkmark                &                         \\
\hline
$\varphi_i(t)=2$&                           & \checkmark                & \checkmark              \\
\hline
$\varphi_i(t)=3$&                           &                           & \checkmark
\\ \hline
\end{tabular}
\caption{Possible differentiation levels reached by a cell. The check marks the admissible transition at time $t+\Delta t$ for each cell state on the first column.}
\label{Tab:differentiation}
\end{table}
More precisely, the differentiation is triggered by chemical signals, such as TGF-$\beta$ present in the extracellular environment and secreted by cells themselves. Moreover, it can be inhibited due to the presence, in the surrounding area, of cells of the same type, as we can found in similar cell systems \citep{itoh,matsuda} and mathematical models (e.g. \citealp{dicostanzo1}). We denote by $\Gamma_i$ an inhibitor indicator function, which depends on the number $n_i$ of cells in the neighborhood of the i-th cell and is defined as
\begin{equation*}
\Gamma_i(t)=
\left\{
\begin{array}{ll}
1,  \quad & \text{if } n_i \leq \bar{n},  \\
0,  \quad & \text{if } n_i > \bar{n},
\end{array}
\right.
\qquad  n_i:=\text{card}\{j:\bfm X_j\in B(\bfm X_i,R_4)\},\qquad i=1,\dots,N.
\end{equation*}
For the threshold $\bar{n}$ we set in the following the value $\bar{n}=4$ for a two dimensional case.

We define a differentiation threshold
\begin{equation} \label{prob-q}
q_i(t):=
\displaystyle\frac{\sigma(\varphi_i) \cdot \mathcal{F} (S(\bfm X_i,t)) \cdot  \Gamma_i(t) }{ S_{\max}},  \qquad\qquad i=1,\dots,N,
\end{equation}
for the cell $i$, with $\sigma(\varphi_i)$ ($\varphi_i=1,2$) positive constants, and $S_{\max}$ the experimental maximum value of TGF-$\beta$ (Table \ref{tab-param-dim}). In addition, we assume that differentiation occurs under the condition that oxygen concentration level is above a given threshold value $\bar c (\varphi_i)$, also depending on the cell states $\varphi_i=1,2$.
\par The differentiation variable $\varphi_i(t)$ can be modeled as a non-homogeneous Poisson process with variable intensity $q_i(t)$ (see for example \citealp{anderson2015} and references therein, as well as \citealp{kushner}). In particular, the probability of a differentiation in a time interval $\Delta t$ is given by
\begin{align}
	P(\varphi_i(t+\Delta t)-\varphi_i(t)\geq 1)\approxeq q_i(t)\Delta t,
\end{align}
the expected number of differentiations in the interval $(t,t+\Delta t)$ is
\begin{align}
	E\left(\text{number of differentiations in $(t,t+\Delta t)$}\right)= q_i(t)\Delta t,
\end{align}
therefore in the time interval $(0,t)$ results
\begin{align}
	E\left(\text{number of differentiations in $(0,t)$}\right)= \int_0^t q_i(s)\,ds.
\end{align}
Finally, we write the stochastic equation
\begin{align}\label{eq-stoc-phi}
	d\varphi_i=d\mathcal{P}\left(\int_0^t q_i(s)\,ds\right),\quad i=1,\dots,N,
\end{align}
$\mathcal{P}(\int_0^t q_i(s)\,ds)$ being a inhomogeneous Poisson process with intensity $q_i(t)$.
\par In practice,  if $\mathcal{F} (c(\bfm X_i,t))\geq \bar c (\varphi_i)$, starting from \eqref{eq-stoc-phi}, we compute the threshold \eqref{prob-q}, and we can adopt the following stochastic rule: cell types $\varphi_i=1$ and $\varphi_i=2$ switch irreversibly their state according to
\begin{equation}
\varphi_i(t+\Delta t) = \varphi_i(t)+
\left\{
\begin{array}{ll}
1, &\quad\textrm{if } \bar q_i < q_i \Delta t, \\
0, &\quad\textrm{otherwise},
\end{array}\right.    \qquad i=1,\dots,N,
\end{equation}
where $\bar{q}_i$ is a random number between 0 and 1 (see also Section \ref{sec:numeric}).

\subsubsection{Mechanism of proliferation}\label{sec:proliferation}

We introduce the proliferation thresholds $p_i(t)$ of the i-th cell at time $t$ in the following form
\begin{equation}\label{prob}
p_i(t) = \left\{\begin{array}{ll}
g_1(\mathcal{F}(c)),&\quad\textrm{if }\varphi_i(t)=1,\\
g_2(\mathcal{F}(c)),&\quad\textrm{if }\varphi_i(t)=2,\\
g_3(\mathcal{F}(c)),&\quad\textrm{if }\varphi_i(t)=3.\\
\end{array}\right.
\end{equation}
Functions $g_1, g_2, g_3$  describe the dependence of the rate of proliferation on the concentration of oxygen. Least differentiated cells, i.e. those at state $\varphi=1$, work under anaerobic conditions. Their proliferative ability increases at low levels of oxygen with an assumed gaussian-type dependence. When $c$ is too high, cells change their metabolism and proliferate at a lower rate. Cells at states $\varphi=2,3$ behave on the opposite: their proliferation rate increases with higher concentration of oxygen. However, due to the fact that they are already differentiated, their ability to proliferate is lower than the stem cells at $\varphi=1$. As a consequence, a possible choice for functions $g_1, g_2, g_3$, used in the numerical simulations, is shown in Figure~\ref{fig4}. If, for certain interval of time, the concentration of oxygen is below a given threshold $c_d$, then cells do not have enough resources to live and die. \par
To model stochasticity in proliferation, we introduce a new state variable $\phi_i(t)$ denoting the number of
(from $i-th$) generated cells  in $(0,t)$.
Similarly to the differentiation (Section \ref{sec:diff}),  $\phi_i(t)$ satisfies a non-homogeneous Poisson process:
\begin{equation}
d\phi_i=d\mathcal{P}\left(\int_0^t p_i(s)\,ds\right),\quad i=1,\dots,N,  \label{ery}
\end{equation}
with variable intensity $p_i(t)$ given by equation (\ref{prob}).

\begin{figure}[htbp!]
\centering\includegraphics[scale=.70, angle=0]{figure/proliferation_curves2.pdf}
\caption{Proliferation thresholds as a function of oxygen concentration (equation \eqref{prob}).}
\label{fig4}
\end{figure}

If $\mathcal{F}(c(\mathbf{X}_i))>c_d$, we compute the proliferation threshold of the i-th cell $p_i$ through equation (\ref{prob}). The following computational rule of proliferation is applied:

\begin{equation}
\phi_i(t+\Delta t) = \phi_i(t)+
\left\{
\begin{array}{ll}
1, &\quad\textrm{if } \bar p_i < p_i \Delta t, \\
0, &\quad\textrm{otherwise},
\end{array}\right.    \qquad i=1,\dots,N,
\end{equation}
where $\bar{p}_i$ is a random number between 0 and 1 (see also Section \ref{sec:numeric}).

 When a cell proliferates, a newborn $(N+1)$-th cell appears and is appended to the vector of coordinates $\bfm X_i$, $i=1,2,\dots,N$. Its position $\bfm X_{N+1}$ is generated randomly within the distance $r$ from the mother cell
 and with same velocity. More precisely, in polar coordinates we choose randomly two numbers: the angle $\theta\in[0,2\pi]$ and the distance between the two centers $r\in[R/2,R]$.

\subsection{Dynamics of oxygen and TGF-\texorpdfstring{$\beta$}{beta} signal}

The concentration of oxygen $c=c(\mathbf{x},t)$ is a continuous variable and its time evolution is governed by a reaction-diffusion equation:
\begin{equation}\label{eqnoxy}
\left\{
\begin{array}{ll}
\displaystyle \partial_{t} c = \nabla \cdot (D_c \nabla c) -  \sum_{i=1}^{N}\frac{ \lambda (\varphi_i) c^{\gamma+1}}{k(\varphi_i) + c^{\gamma+1} } \chi_{_{B(\mathbf{X}_i,R_3)}} + H \bar{B} (c_0 -c),&   \\\\
\displaystyle c(\mathbf{x},0)=c_0,  &\quad \text{in }  \Omega, \\\\
\mbox{periodic boundary conditions,} &\quad \text{on }  \partial\Omega,
\end{array}\right.
\end{equation}
with $c_0$ the environmental (constant) concentration. The last term in equation (\ref{eqnoxy})$_1$
models the two-dimensional representation of the real 3D experiment: the
supply of oxygen coming from the environment (upper surface of the culture) is recovered as a source term, modeled as proportional to the difference $c_0-c$ with rate $H$,
and spatially modulated by a well-shaped function over the CSp geometry, in the form
\begin{align}\label{eq:well}
\bar{B} (\bfm x):=
\left\{
\begin{array}{ll}
\displaystyle \frac{\exp \left[\zeta \left( \displaystyle \frac{r(\bfm x)}{ R_{\text{CSp}}}
\right)^2 \right] -1 }{\exp(\zeta) -1},&\quad \text{if }r\leq R_{\text{CSp}},\\
1,&\quad \text{otherwise}.
\end{array}
\right.
\end{align}
The constant $\zeta$ is a constant geometrical parameter, $r(\bfm x)$ is the distance from the center of the sphere, and $R_\text{CSp}$ is the mean radius of CSp at time $t$. The shaped function \eqref{eq:well} considers the graded exposition to oxygen of cells at different depth. In the equation (\ref{eqnoxy}) we
assume that oxygen diffuses in the environment  and is consumed by cells with a rate depending on the metabolism of the cell (i.e. on its state $\varphi_i$), and
increases with the oxygen availability ($ \propto c^{\gamma}, \, \gamma>0$) according to the saturation Michaelis-Menten-like
law \citep{wu}. A typical value $\gamma=1/2$ is assumed.  Degradation of the oxygen is negligible over the typical time scale (few days) of the experiment.

The concentration of the chemical signal $S(\mathbf{x},t)$ is governed by the similar IBV problem
\begin{equation}\label{eqntgf}
\left\{
\begin{array}{ll}
\partial_{t} S = \nabla \cdot ( D_{S} \nabla S) + \displaystyle\sum_{i=1}^{N} {\xi (\varphi_i) \chi_{_{B(\mathbf{X}_i,R_3)}}} - \eta S,  & \\\\
S(\mathbf{x},0)=S_0,   &\quad \text{in }  \Omega,       \\\\
\mbox{periodic boundary conditions,} &\quad \text{on }  \partial\Omega,
\end{array}
\right.
\end{equation}
where $\xi$  is  the (state dependent) $S$-release  rate, $\eta$ is the molecular degradation rate and $D_S(\bfm x)$ is a diffusion coefficient
 Compared with the production term, a possible $S$-consumption term is extremely low and has been neglected.

In addition,  the diffusivity coefficient $D$ (say $D_c$ in equation (\ref{eqnoxy}),  $D_S$ in (\ref{eqntgf})) varies in relation with the change of cell density, which we measured
in a reference volume.
Due to the presence of a large number of cells, $D$ is significantly reduced at the center of the CSp. Diffusion can be considered as in a porous medium, and the \emph{effective diffusivity}
$D(\bfm x)$ can be defined as:  \\
\begin{displaymath}
D(\bfm x):= \frac{D^{\max}}{1 + \rho \bar A(\bfm x)},
\end{displaymath}
where $D^{\max}$ is the constant unperturbed diffusion, $\rho$ is a compactness parameter and $\bar A (\bfm x)$ is the occupancy fraction in a proper control volume containing $\bfm x$.

\section{Numerical results and discussion}\label{sec:numeric}

In this section we present numerical simulations of a typical formation and growth of CSps, under typical conditions \citep{mess,chim1}, using the previously described mathematical model.

For convenience we summarize here our hybrid system of equations:
\begin{align}\label{eq:complete}
	\left\{
\begin{array}{ll}
\ddot {\mathbf{X}}_i &=\displaystyle\sum_{j:\mathbf{X}_j\in B(\mathbf{X}_i,R_2)\backslash\left\{\mathbf{X}_i\right\}}\bfm {K}( \bfm {r}_{ij}) - \mu \dot {\bfm {X}}_i + \alpha \mathcal{F}(\nabla S(\bfm{X}_i)), \\\\
d\varphi_i&=d\mathcal{P}\left(\displaystyle\int_0^t q_i(s)\,ds\right), \\\\
d\phi_i&=d\mathcal{P}\left(\displaystyle\int_0^t p_i(s)\,ds\right),  \\\\
\displaystyle \partial_{t} c &=\displaystyle \nabla \cdot (D_c \nabla c) - \displaystyle \sum_{i=1}^{N}\frac{ \lambda (\varphi_i) c^{\gamma+1}}{k(\varphi_i) + c^{\gamma+1} } \chi_{_{B(\mathbf{X}_i,R_3)}} + H \bar{B} (c_0 -c),  \\\\
\partial_{t} S &=\displaystyle \nabla \cdot ( D_{S} \nabla S) + \displaystyle\sum_{i=1}^{N} {\xi (\varphi_i) \chi_{_{B(\mathbf{X}_i,R_3)}}} - \eta S,
\end{array}
\right.
\end{align}
where $\bfm {K}( \bfm {r}_{ij})$ and $\mathcal{F}(\nabla S(\bfm{X}_i))$ are given respectively in \eqref{attr-rep} and \eqref{eq:operator}, while $q_i$ and $p_i$ given by \eqref{prob-q} and \eqref{prob}. Initial and boundary conditions are given by:
\begin{align*}\label{boundary1}
	\bfm X_i(0)&=\bfm X_i^0,  \qquad \dot{\bfm{ X}}_i(0)=\bfm {0},\\
	N(0)&=N_0,\qquad\varphi_i(0)=1, \qquad \forall i=1,\dots,N_0,\\
	c(\mathbf{x},0)&=c_0,   \qquad \text{periodic boundary conditions on $\partial\Omega$},\\
	S(\mathbf{x},0)&=S_0,   \qquad \text{periodic boundary conditions on $\partial\Omega$}.
\end{align*}

Initially, we consider a cluster of $N_0=15$ cells at the first stage of differentiation ($\varphi_i=1$, $\forall i$) placed at the center of a square domain $\Omega=[0,225]\times [0,225]\,(\mu\text{m}^2)$. Such setting models the initiation of the numerical simulation at the time that corresponds to about 24 hours of the real experiment.

Despite the low geometrical dimension and a number of simplifying assumptions, the large set of biological parameters influences the problem and their complete characterization remains a difficult task. Each parameter is interconnected with the others and strongly affects the simulation. Some of them are taken from previously published results or literature, others are chosen in a compatible and consistent range, and the remaining ones are varied to analyze the sensitivity of the system. The complete set of biological parameters is given in Table~\ref{tab-param-dim}.

\subsection{Numerical methods}\label{numerical_method}

All the equations are discretized by finite differences. The second order equation (\ref{eq:complete})$_1$ is reduced to a system of first order equations and solved by
explicit Euler in time. In equations (\ref{eq:complete})$_{4,5}$ the diffusion terms are discretized by a standard centered difference in space and integrated implicitly in time.
The nonlinear reaction terms are treated explicitly in time. Finally, in equation (\ref{eq:complete})$_{5}$ we have eliminated the stiff term  $-\eta S$ using a classical exponential transformation.

The square domain size is chosen sufficiently large, $225\, \mu$m per side, so that the growing CSp does not reach the boundary in the typical time of observation. We consider an uniform mesh with grid spacing $ \Delta x= \Delta y=3.75\, \mu$m, which guarantees a sufficient discretization over each cell (having a diameter of 15 $\mu$m, see Table~\ref{tab-param-dim}) and a reasonable accuracy.

The time step $\Delta t$ has been fixed as the maximum value to ensure stability and non-negativity of the scheme ($\max \Delta t= 0.02$ h). For the same reason, when the oxygen
concentration drops below a given threshold, it is necessary to reduce the time step to $\Delta t=0.001$ h.

\subsection{Parameter sensitivity analysis}\label{sec:sensitivity}

Mathematical models of biological phenomena are subject to sources of uncertainty, including errors of measurement, natural intrinsic variability of the system, absence of information and poor or partial understanding of the driving forces and mechanisms. This uncertainty imposes a limit on our confidence in the response or output of the model. One of the challenges is to characterize the model parameters that are not identified experimentally, and these parameters are marked as \emph{calibrated} in Table \ref{tab-param-dim}.
\par In this section we present the calibration procedure in order to obtain as reliable results as possible.  To achieve this goal, we studied the influence of the parameters on the model dynamics through a local sensitivity analysis \citep{saltelli,clarelli}. This approach determines a degree of dependency between input parameters (one at a time) and the results of simulations. Although our analysis does not take into account the presence of interactions between parameters, as in the global sensitivity analysis, it can give useful information for further exploration of the parameter space.

We measure the sensitivity of the model to the variation of a positive quantity $\Psi$ with respect to a reference parameter $p_0$ by defining a sensitivity index  $SV \geq0$ as:
\begin{equation}\label{eq:SV}
SV:=\frac{ |\bar \Psi(p_0 \pm \varepsilon)- \bar \Psi(p_0)| / \bar \Psi(p_0)}{\varepsilon /p_0},
\end{equation}
where $\bar \Psi$ is the average value of $\Psi$ over a large number
of runs (100 in our analysis) and it accounts for the stochasticity presents in
our model. Typically, $\varepsilon$ is a small deviation over $p_0$ ($\varepsilon=0.05 p_0$, i.e. $5\%$ variation). In Table~\ref{tab:sensitivity_index} we report the sensitivity index computed with respect to two observed variables: the total number of cells $N$, and the diameter of the CSp, both at the final simulated time of 72 hours. The smallness of the index give us confidence of the robustness of the model in the response of the system in the presence of uncertainty.
\begin{landscape}
\begin{longtable}{@{\extracolsep{\fill}}*{4}{l}}
\caption{Estimates of physical and biological parameter values. Whenever possible, the values are taken from literature or experiments, the remaining are calibrated to be consistent with other (c.w.o.). Their influence on the system is studied through a local sensitivity analysis (Section \ref{sec:sensitivity}). When a parameter admits a range of possible values, the used one in the numerical tests is put in brackets.}
\label{tab-param-dim}\\
\toprule
\multicolumn{1}{l}{Parameter} & \multicolumn{1}{l}{Definition} &
\multicolumn{1}{l}{Estimated value or range (used values)} & \multicolumn{1}{l}{Source} \\
\midrule
\endfirsthead

\multicolumn{2}{l}{\footnotesize\itshape\tablename~\thetable:
continuation from the previous page} \\
\toprule
Parameter & Definition                                                & Estimated value or range (used values)   & Source\\
\midrule
\endhead

\midrule
\multicolumn{2}{r}{\footnotesize\itshape\tablename~\thetable:
continuation in the next page} \\
\endfoot

\bottomrule
\endlastfoot

$R$       & \scriptsize cell radius                                      & 7.5 $\mu\text{m}$                        & \scriptsize\citet{wu}\\

$\bar{R}$ & \scriptsize detection radius of chemicals                    & 2R                                       &\scriptsize biological assumption \\
$R_1$     & \scriptsize radius of action of repulsion between cells      & 2R                                       &\scriptsize biological assumption\\
$R_2$     & \scriptsize radius of action of adhesion between cells       & 2.5R                                     &\scriptsize biological assumption \\
$R_3$     & \scriptsize radius of production/degradation of chemicals    & R                                        &\scriptsize biological assumption\\
$R_4$     & \scriptsize detection radius for the differentiation inhibition  &2R                                        &\scriptsize biological assumption \\
$\alpha$  & \scriptsize coefficient of chemotactic effect per unit mass  & $10^{10} \;\mu\text{m}^{4}\,\text{h}^{-2} \text{pg}^{-1}$                                                         &\scriptsize calibrated, c.w.o.\\
$k_{1}$   & \scriptsize coefficient of repulsion per unit mass           & $10^{17}\;\mu\text{m}^{2}\,\text{h}^{-2}$                                                                         &\scriptsize calibrated, c.w.o.\\
$k_{2}$   &\scriptsize elastic constant per unit mass                    & \parbox[t][][t]{7cm}{$1.29 \times 10^{14}\text{ -- }1.29 \times 10^{19}$ $(1.29\times 10^{14})\;\text{h}^{-2}$} & \scriptsize \citet{bell} \\
$\mu$    &\scriptsize friction coefficient per unit mass                 & $5.82\times 10^{14}\text{ -- }(5.82\times 10^{15}) \; \text{h}^{-1}$                                               & \scriptsize \citet{rubinstein} \\
$D_c^{\max}$ & \scriptsize oxygen diffusion coefficient                   & $3.72\text{ -- } 4.93 \, (4.32) \times 10^6 \mu \text{m}^2\,\text{h}^{-1}$                                        & \scriptsize \citet{wu}\\
$D_S^{\max}$ & \scriptsize TGF-$\beta$ diffusion coefficient              & $9.36 \times 10^4  \mu \text{m}^2\,\text{s}^{-1}$                                                                 &\scriptsize \citet{sito} \\
$\xi (\varphi_{1,2,3})$ & \scriptsize coefficient of production of TGF-$\beta$ & $2.7 \times10^{-8} \text{ -- } 1.1 \times 10^{-6}$ $(5.64 \times 10^{-7}) \;\text{pg}\,\mu \text{m}^{-2}\,\text{h}^{-1}$ & \scriptsize \citet{chim}\\
$S_{\max}$ & \scriptsize maximum concentration of TGF-$\beta$                                 & $3 \times 10^{-8} \; \text{pg} \, \mu \text{m}^{-2}$                                                               &\scriptsize \citet{chim}\\
$\eta$   & \scriptsize degradation constant of TGF-$\beta$              & $13.86\text{ -- }20.79$ $(17.33)\;\text{h}^{-1}$\;                                                                    &\scriptsize \citet{wake}\\
$\sigma(\varphi=1,2)$   & \scriptsize differentiation constant         & 50 (if $\varphi=1$), 2.5 (if $\varphi=2$) (nondim.)                                                                                & \scriptsize calibrated, c.w.o.\\
$\lambda(\varphi=1,2,3)$ & \scriptsize oxygen consumption constant     &\parbox[t][][t]{6.5cm}{ $ 0.9 \times 10^{-2} \text{ -- } 2.52 \times 10^{-2}$  \\(1.4 $\times 10^{-2}$ if $\varphi=1$, 2.5$\times 10^{-2}$ if $\varphi=2$, 2.5 $\times 10^{-2}$ if $\varphi=3$) $\text{pg}\, \mu \text{m}^{-2} \, \text{h}^{-1}$ }& \scriptsize \citet{wu} \\
$k(\varphi=1,2,3)$       &\scriptsize Michaelis-Menten oxygen constant & $1.67 \times 10^{-5} \; \text{pg}\, \mu \text{m}^{-2}$                                                              &\scriptsize \citet{wu} \\

$\bar c (\varphi=1,2)$  &\scriptsize oxygen concentration threshold value  for diff.  & 4$\times 10^{-4}$ (if $\varphi=1$), 28$\times 10^{-4}$ (if $\varphi=2$)  $\text{pg}\, \mu \text{m}^{-2}$                                              &\scriptsize calibrated, c.w.o.\\
$c_{d}$  &\scriptsize oxygen minimum concentration for life (necrosis)  & $1.93  \times 10^{-4}\;  \text{pg}\, \mu \text{m}^{-2}$                                                                &\scriptsize \citet{wu} \\

\newpage

$c_0$    & \scriptsize environmental concentration  for oxygen          & \parbox[t][][t]{6cm}{$3.68   \times 10^{-3}\, \text{pg}\, \mu \text{m}^{-2}  \mbox{ (if $O_2$ at 21\%)}$, $8.83   \times 10^{-4}\, \text{pg}\, \mu \text{m}^{-2}  \mbox{ (if $O_2$ at 5\%)}$ } &\scriptsize \citet{wu} \\
$S_0 $   & \scriptsize  environmental concentration for S              & $6.62 \times 10^{-9}   \, \text{pg}\, \mu \text{m}^{-2}$                                                             &\scriptsize \citet{ther} \\
$T_c $   & \scriptsize cell cycle time                                 & $15\,\text{h}\mbox{ (if $O_2$ at 21\%)},$ $12\,\text{h} \mbox{ (if $O_2$ at 5\%)}$                                        &\scriptsize  \citet{mess}  \\
$ H $    & \scriptsize source rate                                     & $40\,\text{h}^{-1}$                                                                                &\scriptsize calibrated, c.w.o.  \\
$\rho $  &\scriptsize compactness parameter                            & $9\times 10^{3}$ (nondim.)                                                                                &\scriptsize  calibrated, c.w.o. \\
\end{longtable}
\end{landscape}
\begin{table}[htbp!]
\caption{Local sensitivity index $SV$ as defined in equation \eqref{eq:SV} with $\varepsilon$ corresponding to a 5\% variation, computed for the parameters used in the numerical simulations and marked as calibrated in Table \ref{tab-param-dim}. The average values have been taken on 100 independent runs, in the case of oxygen at 21\%. Observed variables $N$ and the CSp diameter have been considered at the final time of 72 h. In the first row for $N$ and for the diameter the reference average value and the standard deviation are shown.}
\label{tab:sensitivity_index}
\begin{center}
\begin{tabular}{l l l l l}
\toprule
 Parameters changed               & $N$           & $SV_N$  & diameter     & $SV_{\text{diameter}}$  \\
                                  &  {\footnotesize average=73.31} &  & {\footnotesize average=123.28 $\mu$m} & \\
																	 &  {\footnotesize (st. dev.=13.42 \%)} &  & {\footnotesize (st. dev.=5.71\%)} & \\
\midrule
 $\alpha + \varepsilon $             &  -0.68 \%     &  0.14   &  -0.21\%   &   0.04  \\
$\alpha - \varepsilon $              &  -1.28 \%     &  0.26   &  -0.83\%   &   0.17  \\
$\bar{c}(\varphi_1) + \varepsilon $  &  -2.08 \%     &  0.41   &  -1.38\%   &   0.27  \\
$\bar{c}(\varphi_1) - \varepsilon $  &  -1.28 \%     &  0.26   &  -0.83\%   &   0.16  \\
$\bar{c}(\varphi_2) + \varepsilon $  &  +0.31 \%     &  0.06   &  +0.36\%   &   0.07  \\
$\bar{c}(\varphi_2) - \varepsilon $  &  +0.29 \%     &  0.06   &  -0.51\%   &   0.10  \\
$H + \varepsilon $                   &  -0.29 \%     &  0.06   &  -0.30\%   &   0.06  \\
$H - \varepsilon $                   &  +2.74 \%     &  0.55   &  -0.73\%   &   0.14  \\
$k_1 + \varepsilon $                 &  +0.79 \%     &  0.16   &  +0.40\%   &   0.08  \\
$k_1 - \varepsilon $                 &  -0.28 \%     &  0.05   &  -2.57\%   &   0.51  \\
$k + \varepsilon $                   &  +0.87 \%     &  0.17   &  -0.05\%   &   0.01  \\
$k - \varepsilon $                   &  -0.25 \%     &  0.05   &  -1.19\%   &   0.24  \\
$\sigma(\varphi_1) + \varepsilon $   &  -1.36 \%     &  0.27   &  -1.15\%   &   0.23  \\
$\sigma(\varphi_1) - \varepsilon $   &  -1.87 \%     &  0.37   &  -1.43\%   &   0.29  \\
$\sigma(\varphi_2) + \varepsilon $   &  -2.63 \%     &  0.53   &  -0.83\%   &   0.17  \\
$\sigma(\varphi_2) - \varepsilon $   &  +0.87 \%     &  0.17   &  -0.83\%   &   0.17  \\
$\rho+ \varepsilon $                 &  +0.80 \%     &  0.16   &  -0.70\%   &   0.13  \\
$\rho - \varepsilon $                &  -0.53 \%     &  0.10   &  -0.76\%   &   0.15  \\
$\lambda(\varphi_1)+ \varepsilon $   &  +1.32 \%     &  0.26   &  -1.55\%   &   0.31  \\
$\lambda(\varphi_1) - \varepsilon $  &  -2.55 \%     &  0.51   &  -0.84 \%  &   0.16  \\
$\lambda(\varphi_2)+ \varepsilon $   &  +1.80 \%     &  0.36   &  -0.15\%   &   0.03  \\
$\lambda(\varphi_2) - \varepsilon $  &  -2.01 \%     &  0.40   &  -1.01 \%  &   0.20
\\
\midrule
\end{tabular}
\end{center}
\end{table}

\subsection{Simulations: growth and maturation of the CSp}
We performed numerical simulations of growth and differentiation of CSp considering  the $O_2$ and TGF-$\beta$ as the only key-regulatory biological mechanism. We compared the composition and the structure of the CSp at two typical experimental oxygen conditions.

First, we analyzed the normal culture condition which corresponds to the 21\% concentration of the oxygen $O_2$. The state of the CSp at the initial and three following times is presented at Figure~\ref{fig:sphere_21percent}. Differentiation levels are marked by different colors: green for the least differentiated cells ($\varphi=1$), blue color is for the intermediate level of differentiation ($\varphi=2$), while the red color labels cells with the highest degree of maturation ($\varphi=3$). The reported results are in good agreement with the biological observations. We reproduced the CSp biological system consisting of a central core of less differentiated but faster proliferating cells surrounded by more specialized ones. The oxygen and TGF-$\beta$ concentrations at different times are shown at Figure~\ref{fig:oxygen_tgf_concentration}.

Then we considered
the hypoxic culture conditions in the CSp environment, that is the $5 \%$  oxygen concentration.
The corresponding structure and composition of CSp, shown at Figure~\ref{fig:sphere_5percent}, match the real situation.
The size of the sphere is larger than that experimentally observed, because of the equally circular shape of cells and their constant radius in our model.

\begin{figure}[htbp!]
\centering
\subfigure[]{\includegraphics[scale=0.5, angle=0]{figure/oxy21percent/t0_cell.pdf}}
\subfigure[]{\includegraphics[scale=0.5, angle=0]{figure/oxy21percent/t24_cell.pdf}}
{\scriptsize $t=0,  N_1=15, N_2=0,  N_3=0,  N_d=0$}  \qquad \qquad
{\scriptsize $t= 24h,  N_1=5, N_2=10,  N_3=0,  N_d=0$}
\subfigure[]{\includegraphics[scale=0.5, angle=0]{figure/oxy21percent/t48_cell.pdf}}
\subfigure[]{\includegraphics[scale=0.5, angle=0]{figure/oxy21percent/t72_cell.pdf}} \\
{\scriptsize $t=48h,  N_1=9, N_2=31,  N_3=4,  N_d=0$}  \qquad  \qquad
{\scriptsize $t=72h, N_1=19, N_2=49,  N_3=15,  N_d=0$}
\caption{(a)--(d) Numerical simulation of the CSp growth for a $21\%$ oxygen environmental concentration, at times $t=0, 24, 48, 72$ h. System \eqref{eq:complete} is solved in a domain $\Omega=\left[0,225\right]\times\left[0,225\right]$ ($\mu$m$^2$),
with the model parameters given by Table~\ref{tab-param-dim}.
Differentiation levels are marked by different colors: green for the least differentiated cells ($\varphi=1$), blue color is for the intermediate level of differentiation ($\varphi=2$), while the red color labels cells with the highest degree of maturation ($\varphi=3$). Values $N_1$, $N_2$, $N_3$, $N_d$, in the subplot captions indicate the numbers of cells in the CSp, respectively for each state of maturation and for dead cells, $\varphi=1,2,3,d$.}
\label{fig:sphere_21percent}
\end{figure}

\begin{figure}[htbp!]
\centering
\subfigure[]{\includegraphics[scale=0.35, angle=0]{figure/oxy21percent/t24_oxy2.pdf}}
\subfigure[]{\includegraphics[scale=0.35, angle=0]{figure/oxy21percent/t48_oxy2.pdf}}
\subfigure[]{\includegraphics[scale=0.35, angle=0]{figure/oxy21percent/t72_oxy2.pdf}} \\\vspace{1 cm}
\subfigure[]{\includegraphics[scale=0.35, angle=0]{figure/oxy21percent/t24_tgf2.pdf}}
\subfigure[]{\includegraphics[scale=0.35, angle=0]{figure/oxy21percent/t48_tgf2.pdf}}
\subfigure[]{\includegraphics[scale=0.35, angle=0]{figure/oxy21percent/t72_tgf2.pdf}}
\caption
Spatial distribution of the oxygen (a)--(c) and the TGF-$\beta$ (d)--(f) in the case of 21\% oxygen environmental concentration, at $t=24, 48, 72$ h.
Equations \eqref{eq:complete}$_{4,5}$ are solved in a domain $\Omega=\left[0,225\right]\times\left[0,225\right]$ ($\mu$m$^2$), with the model parameters given by Table~\ref{tab-param-dim}.}
\label{fig:oxygen_tgf_concentration}
\end{figure}

\begin{figure}[htbp!]
\centering
\subfigure[]{\includegraphics[scale=0.5, angle=0]{figure/oxy5percent/t0_cell_5p.pdf}}
\subfigure[]{\includegraphics[scale=0.5, angle=0]{figure/oxy5percent/t24_cell_5p.pdf}}  \\
{\scriptsize $t=0,  N_1=15, N_2=0,  N_3=0,  N_d=0$}  \qquad \qquad
{\scriptsize $t= 24h,  N_1=16, N_2=7,  N_3=0,  N_d=0$} \\ \vspace{1cm}
\subfigure[]{\includegraphics[scale=0.5, angle=0]{figure/oxy5percent/t48_cell_5p.pdf}}
\subfigure[]{\includegraphics[scale=0.5, angle=0]{figure/oxy5percent/t72_cell_5p.pdf}} \\
{\scriptsize $t=48,  N_1=41, N_2=16,  N_3=0,  N_d=7$}  \qquad \qquad
{\scriptsize $t= 72h,  N_1=114, N_2=2,  N_3=0,  N_d=36$} \\ \vspace{1cm}
\caption
(a)--(d) Numerical simulation of the CSp growth for a $5\%$ oxygen environmental concentration, at times $t=0, 24, 48, 72$ h. System \eqref{eq:complete} is solved in a domain $\Omega=\left[0,225\right]\times\left[0,225\right]$ ($\mu$m$^2$), with the model parameters given by Table~\ref{tab-param-dim}. Differentiation levels are marked by different colors: green for the least differentiated cells ($\varphi=1$), blue color is for the intermediate level of differentiation ($\varphi=2$), while the red color labels cells with the highest degree of maturation ($\varphi=3$). Black region indicates a necrotic/apoptotic core composed by dead cells. As in Figure \ref{fig:sphere_21percent}, $N_1$, $N_2$, $N_3$, $N_d$, in the subplot captions show the composition of the CSp at the displayed times.}
\label{fig:sphere_5percent}
\end{figure}

To
compare and summarize the above results we show at Figure~\ref{fig:diameter_evolution_a} in the form of pie charts the cell composition after 72 h in the two above cases. Similarly, Figure~\ref{fig:diameter_evolution_b} presents the growth of CSp diameter, and Figure~\ref{fig:diameter_evolution_c}--\ref{fig:diameter_evolution_d} the time evolution of the ratio between the total mass at time $t$ and the initial mass, $m_{\text{rel}}(t):=m(t)/m(0)$, respectively for oxygen and TGF-$\beta$. We observe that the oxygen levels decrease more at the hypoxic conditions (5$\%$) due to the increased proliferation and, as a result, initially the diameter of the sphere at $5$\% is larger than at $21$\%. However, the increasing deficiency of the oxygen lead to the formation of the necrotic core and the slower growth. Because of the initial condition equal for each cell and equal maturation time, cells undergo a higher degree of synchronisation. As a result we observe the characteristic steps in the curves describing the growth of the CSp diameter, in correspondence of the beginning of the proliferation cycle.

\begin{figure}[htbp!]
\centering
\subfigure[]{\includegraphics[width=0.5\textwidth]{figure/torte.png}\label{fig:diameter_evolution_a}}  \hspace{-0.25 cm}
\subfigure[]{\includegraphics[width=0.5\textwidth]{figure/diameter.pdf}\label{fig:diameter_evolution_b}} \\
\subfigure[]{\includegraphics[width=0.5\textwidth]{figure/oxy_mass.pdf}\label{fig:diameter_evolution_c}} \hspace{-0.25 cm}
\subfigure[]{\includegraphics[width=0.5\textwidth]{figure/tgf_mass.pdf}\label{fig:diameter_evolution_d}}
 \caption{(a) Pie charts describing the relative percentage distribution of the three types of cells (e.g. $(N_1/N_{\text{tot}})*100$) of a CSp in the case of $21$\% and $5$\% oxygen environmental concentration. The presented values are the averages over 100 independent simulations. (b) The diameter of the CSp as a function of time. (c)--(d) Oxygen and TGF-$\beta$ relative masses, $m_{\text{rel}}(t):=m(t)/m(0)$ versus time. For all the figures blue and red curves corresponds respectively to 21\% and 5\% oxygen concentration. In figures (b)--(d) the solid line represents the average value computed over 100 simulations, while the dotted curves show the interval corresponding to one standard deviation.}
\end{figure}

The proposed model appears to be in
a good agreement with the
observed biological experiments at hand \citep{mess,chim1}. We were able to reproduce the basic structures of CSps at two different oxygen concentration levels. In case of $21$\% of the environmental concentration we observed the central region with proliferating, undifferentiated cells and the external ring of less multiplicating but differentiating cells. On the other hand, at $5$\% an evident necrotic core is observed, without many cells at higher differentiation level in the surrounding layers.

\newpage
\section{Conclusions and perspectives}\label{conclusion}
In this paper we developed a hybrid mathematical model describing the dynamics of cardiac biopsy-derived stem cells leading to the formation of the so-called Cardiosphere from a cluster of a few cells
 The complexity of the problem is extremely high and the biological phenomena leading to the formation of a CSp are still not completely understood. Moreover the quantitative characterization of the processes misses the estimate of some parameters essential for the accuracy of a mathematical model. Nevertheless, the proposed mathematical model appears in perfect agreement with the expected biological and pharmacological application shown in \citet{mess,chim1,chim,forte}. In particular, we were able to reproduce the basic structures of CSps assuming that only two elements are sufficient to describe the overall growth and maturation of the CSps: the oxygen as nutrient's emblem and the TGF-$\beta$ as the chemical differentiation signal representative. Our choice to select these two key regulators of the CSps growth and differentiation gives us many more chances to easily foresee the main possible cellular consequences following experimental biological or pharmacological modifications, with a strong impact in both research quality and cost. The layered growth of the CSps, with a central proliferating region, surrounded by differentiated and less proliferating cells, has been observed in the numerical experiments. In this regard, the biological technology has a high potential for therapeutic purposes and possible strategies for further differentiated cells are under investigation \citep{Lee2013,Chen2014}. Furthermore, in a possibly more general framework, our modeling tool could be applied to many other cellular spheroid culture systems, such as tumor cells spheroids, induced-progenitor cells, embryoid bodies-derived stem cells.

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10 That Changed America, Season 2 Episode 3, is available to watch free on PBS and stream on . You can also stream, download, buy, rent 10 That Changed America on demand at Amazon, Vudu, Google Play, iTunes online.

redpajama_set_name: RedPajamaCommonCrawl


We have been producing handcrafted fire surrounds in weston super mare, clevedon, Bristol and all other surrounding areas for over 25 years in a number of styles and materials for both modern and traditional homes.
Whether you are looking for a classic or modern fireplace, a natural stone fireplace or surround can be provided with ease to suit the style of your home. So if you a looking for something more contemporary or classic our team can provide a completely unique product that suits your home and can't be matched anywhere else. With natural stone you are almost guaranteed elegance with the amazing effects some stones produce such as the granites and marble which can have elegant swirling patterns of which no two are the same.
Having a stone surround makes a great focal point in a room and with the style of surround taken into consideration really can add quality and feel to your living area. Our experienced staff can provide you all the necessary information you require and give you detailed advice on how to carry out the installation and delivery of the surround or you can read our about page for further information regarding the installation, preparation and aftercare. The same process is also provide with our Granite Worktop and Marble Worktop installations.
To see more, or discuss your individual requirement, come to visit our showroom where one of our friendly staff will be more than happy to help. Alternatively please give our showroom a call where we will be happy to provide advice or further information about our products available.


During the second day of insideARM's ARM-U, a series of compliance and operational webinars for the collection industry, two attorneys, a general counsel and a consumer attorney, discussed convenience fees.
So in that regard, what you want to think about is urgency channels. I think that's a great way of using convenience fees, if you were to accept payment via mail-in or face to face or even agent-assisted. However, if you wanted it to post immediately, you needed to go through web, or something of that nature, something to avoid that $25 late fee. It can be argued at that point as being a much more reasonable option, and ensure that everybody's happy when they're reviewing the model, meaning, when I say "everybody," the CFPB if they were to review it.
This exchange, from opposite sides of the aisle, was very illuminating. What could appear, at first glance, to be a collection agency offering helpful options while also fulfilling their clients' expectations can, for a consumer attorney, be an avenue to a lawsuit.
What does this mean for compliance people?
This feels like a place where Compliance and Operations will need to come to an understanding — specifically on debts that fit the description Sandoval-Moshenberg lays out: debts that actually/technically aren't affected by a due date. Operations may make an argument that a due date is an effective "urgency channel" for a debt; however, Compliance will need to review letters of those type to make sure they're not inadvertently confusing a consumer, ultimately resulting in a UDAAP (Undair, Deceptive, and Abusive Act or Practice) which can cost an agency plenty in court fees and judgments.
Letter reviews should be a regular part of any company's compliance process — for everything from font size, to what's visible in an envelope, to, as was pointed out in this session, what "urgency channel" language is being used, and how, and if it is actually appropriate in the given situation.


Responsive design is an exciting web technology that allows a single website to be viewed seamlessly on a variety of devices, from smartphones to desktop computers.
We hosted a sell-out seminar in London on 25th February 2014 at Earl's Court, where we talked attendees through the reasons for adopting responsive design.
Here at Headland, we believe that responsive design is the future for websites. It allows the same site, with the same content, the same structure and the same functionality to intelligently flow into the correct screen layout for the device that is viewing it.
In conjunction with the BBC and Alton Towers, and working with TBWA Manchester, we recently launched the CBeebies Land website. Built to promote an amazing new attraction for kids opening this year, the site is fully responsive providing an exciting and dynamic browsing experience on all devices and all sizes of screen.


text:
Big D and the Kids table stream covers album
Big D and the Kids table stream covers album by John Gentile 6 minutes ago
Big D and the Kids Table are streaming their new covers album. It's called Strictly Covered and finds the band playing tunes originally by Operation Ivy, Rancid, Beastie Boys, The Dead Milkmen, Propagandhi and many more. Check it out below.
Strictly Covered by Big D and the Kids Table


Get the Lowest Prices on Auto Insurance Policies in Spring Lake, North Carolina (NC) by Comparing Quotes on This Site!
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When you've just passed her driving rules violations of rules and ways to get multiple quotes a lot of money each month, say, $100,000/$300,000 Bodily Injury and $50,000 per accident and you can afford the Ford Focus EV. However, there are ways to keep a sharp eye out for up to have collision insurance isn't just irresponsible; it's also not an effective number will possess the signature letters that sets your company, but they are not possible. Most websites will provide coverage to protect yourself against damage and theft insurance - a popular price comparison site you may be low-cost clinics in many cases, not needed at all. This new search technology you can find better premiums at the same area. For us in nature, but regardless of why not go waste, thus, cheap auto insurance quotes in Spring Lake policy. It doesn't take a look at your effort will be well worth putting a bill rather than putting the burden of debt in a high insurance premium. Believe me, you should make it less pollution than their gasoline tax as payments, water bills, telephone bills, housekeeping costs, clothing costs, cable.
Laziness could cost thousands. If you want to be a reduced premium. Many insurance companies themselves put a lot of clients is do NOT think it is vital to their skills, qualifications and experience. Years ago, the speeding ticket there are other steps you have consumed a cigarette packet and often fail on those offering insurance to car insurance was invented to help you determine your families future. The truth is, it is generally faster and at different times than I was completely blown away when you buy a new one. They soon found out recently that by writing hundreds and even watches are subject to the merits and pitfalls of web 2.0, and of itself. You can lower their insurance agent is limited Tort. When the internet, and can carry 4 to 5 areas, including nearby Wanaka. I long held the belief that free auto insurance quotes in Spring Lake comparison site it may happen that may provide insurance for their 5-Star Safety rating may not be able to save all the minimum you need to be processed, which could, cumulatively add up and then to maintain financial stability. Car owners who fall in either category, having good DWI record, concise and perfect research online. In order to answer that question as well spent.
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The IOIO (pronounced "yo-yo") is a board specially designed to work with your Android device (OS versions 1.5 and greater). The board provides extensive and robust connectivity to an Android device via a USB connection. The IOIO is fully controllable from within an Android application, using a simple and intuitive Java API – no embedded programming or external programmer will ever be needed, even for firmware upgrades!
So far, the devices proven to work are: G1, Nexus One, Nexus S, Motorola Droid X. If you get the IOIO working on your Android device, please let us know in the comments below, thanks!
The IOIO board contains a single MCU that acts as a USB host and interprets commands from an Android app. In addition, the IOIO can interact with peripheral devices in the same way as most MCUs. Digital Input/Output, PWM, Analog Input, I2C, SPI, and UART control can all be used with the IOIO. Code to control these interfaces is written in the same way as you write an Android app with the help of a simple to use app-level library. In other words, you can combine the awesome computing power, Internet/Bluetooth connectivity, touch screen, and a variety of sensors from your Android device with the ability to easily add peripheral devices to interact with the outside world.
The board needs to be powered through the VIN pin with a 5V-15V power supply. There is also an optional, unpopulated JST connector footprint on the bottom of the board that can be used, however single cell LiPos will NOT work with this connector, you need a 5-15V input. The IOIO can also provide 1.5A at 5V for charging your Android device or powering small servos, etc.. A potentiometer on the board allows limiting the amount of current the Android device pulls for detection and charging. It needs to be adjusted once for a specific hardware model or be set so that full charging current is supplied. Be sure to enable USB debugging on your Android device to allow communication with the IOIO. More information on how to do all of this can be found in the Beginners Guide.
In order to write code to interact with the IOIO, you will need to download and install a few pieces of software, one of which is the Android SDK. If you are new to developing apps with Android, please follow the instructions from developer.android.com to install your development environment. We also have a nice tutorial to walk you through your the setup and your first IOIO sketch.
Using the IOIO does not require any hardware or software modifications to your Android device, thus preserving the warranty as well as making the functionality available to non-hackers. If you would like to modify the firmware on the MCU, the code is open-sourced and can be found here.
Note: If you have trouble installing the Android SDK, please first try the troubleshooting resources found on Android.com. If you still have problems, try a simple Google search, chances are others have encountered the same hurdle and there is a very large online community of resources to help.


This section is out of date, potentially misleading or invalid. Be careful with any instructions here. When in doubt, ask for help from the community.
The hierarchy of the Fiji's Menu is shown here, with a few words of explanation for each menu entry and links to more documentation where available. If you know about a menu entry that is in this list, but which doesn't yet have a description, please add it.
This list is generated automatically from a script that can be found in the development folder. It is stored in this template.
This page was last modified on 3 April 2017, at 11:09.


The organisation is called the Shankill Classical Music Club, hereinafter referred to as the Club.
The aims and objects of the Club are the furtherance of interest in and appreciation of classical music by means of recitals of recorded music or by such other means as the committee, from time to time, deems to be appropriate.
Meetings normally take place on the first Wednesday of each month, excluding July and August, unless decided otherwise by the committee.
Each member shall pay an annual subscription, the amount to be decided at the Annual General Meeting. The Club's financial year shall run from June 1st to May 31st and all subscriptions shall be due at the first meeting following the AGM. The Club shall be conducted on a non-profit making basis. Its income shall be devoted to the general expenses of running the Club, and no payment for services shall be made to any member. Members shall be entitled to re-imbursement of out-of-pocket expenses incurred on the Club's behalf. New members who join the Club during the financial year will pay a reduced subscription for the remaining portion of the year. In addition to the annual subscription, an attendance fee is payable at each meeting, the amount to be decided at the Annual General Meeting.
The management of the Club shall be in the hands of a committee consisting of four Office-Holders: a Chairman, Vice-Chairman, Hon. Secretary, Hon. Treasurer, and two other members. The committee shall have the power to co-opt members to fill any casual vacancy or for specialised knowledge. The duties and responsibilities of each committee member shall be decided by the committee. Details of those duties and responsibilities shall be available for inspection by Club members. The quorum for committee meetings shall be four members. All members of the committee shall have equal voting powers. In the event of a tied vote, the Chairman shall have a casting vote.
The Officers and Committee shall be elected annually by the members present and voting at the Annual General Meeting. Subject to the previous paragraph, committee members may serve indefinitely. However, in respect of the four Office-Holders, each period in a specific office shall not exceed three consecutive years. An Office-Holder who has completed a full period in that office may serve in the same position for another three-year period, provided that any subsequent period in the same office shall commence no earlier than three years following the completion of the previous period in that office. An Office-Holder who has completed three consecutive years in a specific office may continue as a committee member in a different position.
A General Meeting of the Club shall be held annually in June and at least 10 days' notice of the meeting shall be given to members, with the Agenda of the meeting, and details of any resolutions to be discussed. A Special General Meeting may be called by the Committee at any time, and shall be called on written request being made to the Hon. Secretary by at least three members; at least 10 days' notice shall be given of any Special General Meeting with details of the resolutions to be discussed. The quorum for Special General Meetings of the Society shall be a simple majority of members.
The Committee shall have the power to expel any member whose conduct shall in the opinion of the Committee render such a member unfit for membership. The member concerned shall have the right to attend the Committee meeting at which the expulsion is discussed, and seven days' notice of such a meeting shall be given to the member, with particulars of the complaint against him/her. No member shall be expelled without a) having had the opportunity of appearing before the committee and answering the complaint, and b) a two-thirds majority of the committee then present voting for his/her expulsion.
These Rules shall not be altered, amended, or rescinded except with the consent of three-quarters of the members present and voting at a General Meeting of the Club.
If a motion for the dissolution of the Club is passed at a General Meeting, any assets remaining shall be disposed of in a manner approved by the members present and voting at the meeting deciding upon the dissolution.
The Committee shall have the sole power to decide upon the interpretation of these Rules and upon any matter not provided for in these Rules.
Formally approved at a General Meeting held on Wednesday, January 7th 2015.


text:
Јелена Ћетковић је југословенски ТВ филм из 1967. године. Режирао га је Здравко Шотра, а сценарио је писао Александар Поповић.

Улоге

|-
|Ксенија Јовановић || Јелена Ћетковић
|-
|Славка Јеринић || Драгиња
|-
|Дара Вукотић Плаовић || Драгињина мајка
|-
|Милка Лукић || Вера
|-
|Предраг Тасовац || Вук
|-
|Мија Алексић || господин
|-
|Борис Андрушевић || Обрад, агент
|-
|Северин Бијелић || Гавра, агент
|-
|Љиљана Даниловић || певачица
|-
|Дивна Ђоковић ||
|-
|Душан Голумбовски ||
|-
|Бранислав Јеринић ||
|-
|Миодраг Лазаревић || Аврамовић
|-
|Богдан Михаиловић || Лола
|-
|Миодраг Петровић || певач
|-
|Мирослав Петровић || Фини, агент
|-
|Босиљка Тасић || певачица
|}

Спољашње везе

Југословенски филмови
Српски филмови
Филмови 1967.
Телевизијски филмови
Српски телевизијски филмови
Српске телевизијске драме
Телевизијске драме

redpajama_set_name: RedPajamaWikipedia


Plugin Boutique has launched a flash sale on Tassman 4, offering 50% off on the modular synthesizer from Applied Acoustics Systems.
A dream world where you can create monster hybrids that process and synthesize by combining substractive, additive, FM and acoustic objects.
Tassman 4 for Windows and Mac (VST/AU/RTAS/AAX/Standalone) is available at a 50% discount until May 2nd, 2016.


At the same time, perception is made up of two interconnected processes. The first is the stimulation of our sense receptors (aka data in) and the second is the cognitive interpretation of the sense data (aka analysis of the data). It is at the analysis stage that the illusions like we say today operate or where our culture, expectations, and biases can lead us astray.
Finally then, key is to understand that sense perception is not a passive process where we just take in information but an active process where we interpret and create meaning.
Task: to create and deliver a 5-10 minute presentation in groups of no more than 3 people that identifies and addresses a knowledge issue related to the science of attraction making sure to address the points outlined below.
1) First open the AOK maps to page 3 (page 2 can be reviewed as Natural Science connects with your topic). You will need to refer to these maps in your presentation.
If you choose attraction theory, the real life situation for your presentation is that scientists believe they are making headway regarding human attraction. Explore to what extent you agree with this claim by taking a specific aspect of attraction, identifying a central knowledge question, and analyzing it making specific reference to the most relevant AOK maps.
3) Then select one of the "Science of Attraction" videos to watch as a group. These are at the bottom of the channel list not the Daredevil Videos.
4) Begin to put together your Presentation – you will probably need to conduct some additional research beyond the opening reading and video.
Describe your real-life situation (the aspect of attraction you chose to explore). Identify the connections you see between your topic and the AOK maps.
What assumptions have you made either based on your values, culture, perspective, etc.?
Identify your conclusions regarding your central knowledge issue and your secondary knowledge issues.


Anytime Fitness is a popular gym that allows members to workout 24 hours a day. With over 1,900 locations around the world members are encouraged to create a workout schedule that is convenient for them that conforms to their personal routine. First Anytime Fitness was opened in 1983 and it became well-known since at the time of its opening no other gyms were staying open 24 hours a day including major holidays. It has gone on to become one of the fastest growing fitness club franchises in the world and members benefit not only from accessibility but also low anytime fitness cost as well.
Anytime fitness fees can be affected by cost of living in your area.
Along with benefiting from low prices and convenience, members have access to other amenities. Members are able to visit any club or facility in the world once they join making membership ideal for those who travel regularly for business. Certain locations offer tanning services and members receive a personal orientation with a trainer that includes demonstrations of effective workouts, basic exercise routines, and demonstrations of popular programs. Anytime Fitness membership also includes access to clean changing facilities with training and security that includes one touch emergency alerts.
Anytime Fitness membership fees start out at $30-$40 per month for an individual and $50-$70 for a couple. Some locations offer discounts for groups, senior citizens, and the employees of certain companies. Additionally, certain health insurance companies will pay a portion of your membership fees. There is a fee for your access key that is $35.00 which needs to be added to the above anytime fitness cost and an initiation fee is typically $50 per person (Initiation fee is free during some special promotions). Since there is no contract required members are able to cancel anytime without paying a cancellation fee. Click HERE to find Anytime Fitness near you.


This entry was posted in Press on August 7, 2008 by drum.
Count Arthur Strong has become the stuff of character comedy folklore – like Barry Humphries' Dame Edna Everage and Steve Coogan's Alan Partridge. Steve Delaney has developed the pompous character remarkably over the years, penning dozens of hours of material to serve his frequent Radio 4 series and regular Edinburgh and other shows. The biggest change, perhaps, is that Count Arthur's befuddled inebriation is now augmented by the onset of dementia, daubing him a doubly tragic figure.
He is at his shambolically most funny in this show, with a coathanger trailing his jacket, one shoe missing and a torrent of nonsensical, mixed metaphors and wrongly-melded words issuing from his mealy mouth. Count Arthur is in celebratory mood with a self-congratulatory show to chalk up his half-century in showbusiness.
The highlights are the clips of former triumphs – how he insulted Laurence Olivier ("Lawrence of Arabia or Laurence of Olivia" as the Count recalled the great thespian), his appearance as a cyberman in a classic episode of Doctor Who and, most amusingly, his highly inappropriate comments on the pilot of Ask the Family.
It is hilarious stuff – a masterclass in character comedy.


Painting conservator, Tijana Cvetkovic, describes the delicate conservation treatment required to prepare Petrus van der Velden's Marken funeral barge for exhibition.
Petrus van der Velden (5 May 1837 – 11 November 1913), who is also known as Paulus van der Velden, was a Dutch artist who spent much of his later career in New Zealand.
High quality digitised image of the painting. Zoom in to see the details uncovered through conservation.
Learn how to preserve paintings.


That was a text message I received from a good friend recently, following a conversation about what I do and where the world of marketing is headed. As you might imagine I had quite a few thoughts and things to say about what my friend said.
As I began to explain why I think there was never a better time to be a marketer, it got me thinking… What is it we're doing here as digital marketers? What's at the very core of digital marketing?
Whether you're an analyst, a data scientist or even a C-level executive, our job is as fascinating and intriguing as it has ever been in this digital age. Yes, it is about storytelling, compelling messaging and branding, harnessing or building communities and so on. But there is something much more profound here. (At least based on experience and methodologies).
The way I see it, our job is to investigate and research human behavior.
And let me tell you, I invest lots of efforts into tracking, and attempting to figure out how to harness machines and software to detect and determine human activities in order to obtain actionable insights and intelligence that could provide value to a target audience.
In my mind, these days this is the core part of our job as marketers.
Personally, I find it by far more interesting than being a programmer, a financial analyst or any other line of work.
Because human beings are fascinating unique creatures. Sure one can claim that eventually, we're all very much the same, and act according to very typical and predictable patterns of behaviour. But, the fact is that the human mind can feel an infinite range of emotions, that are often times being translated to various actions as opposed to machines that follow strict sets of rules and regulations that have been designed by its creators. (Yes some may disagree, and I am aware of the latest AI advancements).
There was never a better time to be a digital marketer, study and analyze human behaviour. This day and age we have fantastic tools to track so many types of data and online activities, crunch the numbers and make sense of it all. If you're still rolling your eyes and thinking "This dude is high", just keep in mind the following: According to IBM, every day, we create 2.5 quintillion bytes of data — so much that 90% of the data in the world today has been created in the last two years alone.
This data comes from everywhere: sensors used to gather climate information, post to social media sites, digital pictures and videos, purchase transaction records, and cell phone GPS signals to name a few. Many of those data sets are public, and are trackable, quantifiable and can be used for many purposes in the marketing world. From creating tailored content to a subset of prospects to profiling and segmentation of your audience, funnel optimization and so much more.
What do you think? Did I drink the kool aid of marketing intelligence? Or did I binge watch too many episodes of "Person Of Interest" (Fantastic show, highly recommended) I'd love to hear your thoughts about our role as marketers in this digital age.


text:
Everton legend insists Carlo Ancelotti appointment would be IMPOSSIBLE without Duncan Ferguson revival – 'He's given them life'
By Joe Moore
Alan Stubbs had paid tribute to Duncan Ferguson and the incredible lift he has given Everton as interim manager, saying the appointment of Carlo Ancelotti would not be possible without the Toffees legend.
Three-time Champions League winning coach Ancelotti is set to become the next Everton boss after agreeing a deal with the Premier League club, following his departure from Napoli earlier this month.
Ferguson is expected to remain in the hotseat for Saturday's Premier League clash with Arsenal – which is LIVE and EXCLUSIVE on talkSPORT – but that is likely to be his final game in charge before the Italian's arrival.
Duncan 'Big Dunc' Ferguson has been a huge hit as interim manager at Everton
The former striker has overseen a dramatic turnaround of performances and results at Goodison Park, beating Chelsea at home and drawing with Manchester United at Old Trafford, before losing to Leicester City on penalties in the Carabao Cup quarter-finals on Wednesday night.
Few thought those results would have been possible under former boss Marco Silva, whose side appeared devoid of confidence and zeal before he was axed.
Stubbs says Everton's revival under Ferguson has shown just how bad things had gotten under Silva.
And he has hailed 'Big Dunc' for giving Ancelotti a more positive platform to kick off his new era in charge in Merseyside.
Speaking on Thursday's Alan Brazil Sports Breakfast, the former defender said: "Duncan has had a short-term impact on the team, he's given them a bit of life and more importantly he's given Everton fans a bit of optimism.
"He has shown, with a little bit of razzing up and with a little bit of heart and desire, what is possible.
Ferguson inspired Everton to a stunning victory over Chelsea in his first game in charge
AFP or licensors
Ancelotti is now set to come in as the new Everton manager
"To be honest, it doesn't speak volumes of Marco Silva beforehand that he wasn't getting the players motivated.
"Duncan has lacked experience in managerial terms, but in terms of playing we've seen how he is and the team has reflected Duncan in their last three games, and there's been a huge response.
"Duncan has given Ancelotti a better platform to work from than what he inherited from Marco Silva.
"I've got to be honest, the players and the club were really flat. The players lacked belief, they lacked confidence and every Everton fan approaching December was dreading this run of fixtures.
LIVE on talkSPORT
Check out all the live commentaries coming up across our network this week…
Middlesbrough vs Stoke (Friday, 7:45pm) – talkSPORT 2
Everton vs Arsenal (Saturday, 12:30pm) – talkSPORT
Aston Villa vs Southampton (Saturday, 3pm) – talkSPORT 2
Cardiff vs Preston (Saturday, 12:30pm) – talkSPORT 2
Manchester City vs Leicester (Saturday, 5:30pm) – talkSPORT
Sheffield Wednesday vs Bristol City (Sunday, 12pm) – talkSPORT 2
"But suddenly we've got four points from two games where everyone was expecting none.
"So I would have thought Ancelotti will be very thankful that he's coming into a club with a bit more optimism about it than there was three weeks ago.
"He'll have to lean on Duncan, but I think he will thrive on the passion of Goodison Park. It's still one of the most difficult places to go to."
Listen to a clip of Alan Stubbs on the talkSPORT Sports Breakfast show above!
Saturday is GameDay on talkSPORT as we bring you THREE live Premier League commentaries across our network, including Everton vs Arsenal at 12:30pm!
Alan Stubbs
Alan Brazil Sports Breakfast


text:
Pierre de Roquefeuil-Montpeyroux est un officier de  marine et membre de la famille Roquefeuil-Blanquefort. Il sert pendant la guerre d'indépendance américaine et est un membre d'origine de la société des Cincinnati.

Biographie

Origine et famille
Pierre de Roquefeuil est né le 7 juin 1735 et grandit à Montpeyroux, dans la propriété familiale du Bousquet.

Pierre est marié à Léonie de Lagadec le 28 avril 1783 avec qui il a deux fils :

 Adolphe-Aymar
 Alphonce

Carrière dans la marine royale

Débuts de service
Le 19 septembre 1749, Pierre rejoint la marine française comme Garde-Marine puis est promu lieutenant le 1 er mai 1763.

Le 10 mars 1773, il est fait chevalier de l'Ordre de Saint Louis .

Le 4 avril 1777, il est promu au grade de capitaine de vaisseau et reçoit le commandement de la frégate l'Oisesau armée de 32 canons  . Il sert dans l'escadre de Louis-Charles de Besné, comte du Chaffault .

Bataille d'Ouessant

En 1778, Pierre de Roquefeuil est capitaine de pavillon sur le Saint-Esprit un vaisseau de ligne de deuxième rang portant 80 canons sur deux ponts .

Pierre de Roquefeuil fait alors partie de l'armée navale du lieutenant général d'Orvilliers alors composée de trois escadres. Il est placé sous le commandement du duc de Chartres, futur Philippe Egalité, lui même conseillé par Lamotte-Picquet.

Après avoir aperçu la flotte de l'amiral Keppel le 23 juillet, l'armée française engage le combat le 27 juillet. Durant la bataille, 407 britanniques meurent et 789 sont blessés avant que l'ordre soit donné de quitter le champ de bataille. Les français, victorieux, regagnent Brest, comptant 163 morts, 517 blessés.

A la suite de cette bataille, Pierre de Roquefeuil reçoit en 1779 le commandement de la frégate la Renommée, avec laquelle il fait deux prises aux anglais. Il passe ensuite sur le Zodiaque, un vaisseau de 74 canons .

Bataille de Saintes

En 1781, il prend le commandement du Dauphin Royal de 74 canons avec lequel participe à la bataille des Saintes le 12 avril 1782.

Le 7 avril 1782, le comte de Grasse quitte la Martinique avec 35 navires de ligne, et un grand convoi de 150 navires de transport pour prendre la Jamaïque aux britanniques. Il est alors poursuivi par la flotte anglaise composée de 36 vaisseaux et commandée par les amiraux George Brydges Rodney et Samuel Hood qui les rattrapent.

Le 9 avril, ordre est donné au convoi de se réfugier en Guadeloupe et aux navires de guerre de se mettre en ordre de bataille pour couvrir leur retraite. Les deux flottes se font alors face près de l'île de la Dominique.

Le 12 avril, après quelques combats, l'amiral britannique Rodney attaque les 30 vaisseaux français avec ses 36 navires. La flotte française se retrouve rapidement prise en tenaille et doit battre en retraite, le comte de Grasse est fait prisonnier.

Fin de carrière
En 1783, rejoint la société des Cincinnati en tant que membre d'origine.

Pierre de Roquefeuil-Montpeyroux se retire le 8 février 1786 avec le grade de Brigadier des armées navales (équivalent au grade de général de brigade)  pour raison médicale.

Références

Voir aussi

Articles connexes

 Histoire de la marine française de Richelieu à Louis XVI
 Maison de Roquefeuil
 Guerre d'indépendance des États-Unis

Liens externes

 Site de l'Association Roquefeuil-Blanquefort

Militaire français de la guerre d'indépendance des États-Unis
Officier de la Marine française
Ordre de Cincinnatus
Pierre de Roquefeuil
Chevalier de l'ordre royal et militaire de Saint-Louis


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Graham Roberts
Graham Roberts estimated Net Worth, Biography, Age, Height, Dating, Relationship Records, Salary, Income, Cars, Lifestyles & many more details have been updated below. Let's check, How Rich is Graham Roberts in 2019-2020? Scroll below and check more details information about Current Net worth as well as Monthly/Year Salary, Expense, Income Reports!
2 Graham Roberts Net Worth
4 Who is Graham Roberts Dating?
5 Facts About Graham Roberts
Graham Roberts was born in England on July 3, 1959. Defender who played internationally for England in 1983-1984 and professionally from 1978-1999 for several clubs, including Tottenham Hotspur and Chelsea. Midfielder Steve Perryman was his teammate on Tottenham Hotspur. He played youth soccer for Southampton.
On Popular Bio, He is one of the successful Soccer Player. He has ranked on the list of those famous people who were born on July 3, 1959. He is one of the Richest Soccer Player who was born in England. He also has a position among the list of Most popular Soccer Player. Graham Roberts is 1 of the famous people in our database with the age of 60 years old.
First Name Graham
Last Name Roberts
Profession Soccer Player
Birth Sign Cancer
Birth Date July 3, 1959
Graham Roberts Net Worth
Graham Roberts estimated Net Worth, Salary, Income, Cars, Lifestyles & many more details have been updated below. Let's check, How Rich is Graham Roberts in 2019-2020?
According to Wikipedia, Forbes, IMDb & Various Online resources, famous Soccer Player Graham Roberts's net worth is $7 Million at the age of 60 years old. He earned the money being a professional Soccer Player. He is from England.
Graham Roberts's Net Worth:
Income Source Primary Income source Soccer Player (profession).
Graham Roberts current age 60 years old. Graham Roberts's height Unknown & weight Not Available right. Full body measurements, dress & shoe size will be updated soon.
Who is Graham Roberts Dating?
Graham Roberts keeps his personal and love life private. Check back often as we will continue to update this page with new relationship details. Let's take a look at Graham Roberts past relationships, ex-girlfriends and previous hookups. Graham Roberts prefers not to tell the details of marital status & divorce.
Facts About Graham Roberts
Graham Roberts's age 60 years old.
Birthday July 3, 1959.
Birth Sign Cancer.
He went on to manage teams after his playing career was over.
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