text:
\section{Numerical simulations}
\label{Sec:Simulation}

We present in this section two applications of the theory presented earlier. The first is the simple scenario in which two straight and idealized blood vessels are considered, one representing an artery and the other a vein. This means that the tissue block containing the two vessels is supplied nutrients by a single artery and drained of nutrients by a single vein. In between the two vessels, a tumor core is present which accepts nutrients injected through an inlet of the artery. The second scenario deals with a small blood vessel network described by data  in \cite{secomb2000theoretical} and on the following web page: \url{https://physiology.arizona.edu/sites/default/files/brain99.txt}. At four inlets of this network, nutrients are injected and transported through the network. As in the first scenario, the impact of the nutrients on a small tumor core surrounded by the network is investigated. We note that in all cases we consider a stationary vessel structure; for evolving and bifurcating vessels we refer to the further work \cite{fritz2020part2}.

To solve the one-dimensional partial differential eqautions \cref{Eq:Model1D} numerically, the vascular graph model (VGM) is employed \cite{reichold2009vascular,vidotto2019hybrid}. This method corresponds in principle to a vertex centered finite volume method with a two-point flux approximation. For the three-dimensional partial differential equations presented in \cref{sec:3Dmodel} a mixed finite volume-finite element discretization method is employed. The equations governing the pressure and nutrients in tissue, which are directly coupled with the one-dimensional system, are solved by a standard cell-centered finite volume scheme. Since the permeability of the cancerous and healthy tissue is given by a scalar field, a two-point flux approximation of the fluxes is used.

For the remaining species, we consider a continuous and piecewise linear finite element approximation over a uniform cubic mesh. The coupled nonlinear partial differential equations are discretized in time using the semi-implicit Euler method. To solve the nonlinear system of equations arising in each time step, a fixed point iteration method is applied. We consider following double-well potential in free energy functional in \cref{Eq:GinzburgLandau}
\begin{align}
	\Psi(\phi_P,\phi_H,\phi_N)&=C_{\Psi_T} \phi_T^2(1-\phi_T)^2.
\end{align}

\subsection{Tumor between two straight vessels}
\label{Sec:Simulation2Straight}
We consider a tissue domain $\Omega = (0,2)^3$ containing two blood vessels aligned along the $z$-axis. The center lines of the vessels are located diagonally opposite to each other.

The center line of the Vessel $1$ and $2$ pass through $(0.2, 0.2, 1)$ and $(1.8,1.8,1)$, respectively.
We choose a radius of $R=0.08$ and $R = 0.1$ for Vessel $1$ and $2$. At the inlets of Vessel $1$, located at $(0.2, 0.2, 0)$ and  $(0.2, 0.2, 2)$, pressure values of $10000$ and $5000$ are prescribed. The inlets of Vessel $2$ are located at $(1.8, 1.8, 0)$ and $(1.8, 1.8, 2)$. Here, we consider the pressure values $1000$ and $2000$, respectively. Thus, Vessel $2$ will act as a vein taking up nutrients and blood plasma from the tissue domain. On the other hand Vessel $1$ has the function of an artery transporting nutrients into the tissue block $\Omega$.  \blue{We note that we choose the boundary values such that the velocities are sufficiently large in order to ensure that the transport processes are visible in the simulations.} Based on the pressure boundary conditions, we choose the boundary conditions for the nutrients as follows:
\ \\
\begin{itemize} \itemsep0em
	\item $\phi_v = \phi_{v,inlet} = 1$ at $(0.2, 0.2, 0)$.
	\ \\
	\item $\phi_v = 0$ at $(1.8, 1.8, 2)$.
	\ \\
	\item At all the remaining boundaries, we consider free flow boundary conditions.
\end{itemize}
\ \\
The initial tumor core is given by a ball of radius $0.3$ and centered at $(1,1,1)$. Within the tumor core, the total tumor volume fraction, $\phi_T$, decays smoothly  from $1$ in the center to $0$ on the boundary of the ball. Thereby, the necrotic and hypoxic volume fractions, $\phi_N$ and $\phi_H$, are set to zero. In the rest of the domain all the volume fractions for the tumor species are set to $0$ at $t=0$. The nutrient volume fraction, in the tissue domain $\Omega$, is initially fixed to a constant initial value of $0.6$, which is below the threshold values for prolific-to-hypoxic transition and above the threshold value for hypoxic-to-prolific transition. We also set $\phi_{\ECM} = 1$ at $t=0$. According to \cref{Eq:Initial3D} homogeneous Neumann boundary conditions are prescribed on $\partial \Omega$.

As a simulation time period, the interval $\left(0,T \right)$ with $T=5$ is considered and the size of the time step is given by $\Delta t = 0.025$. The spatial domain $\Omega$ is discretized by cubic elements with an edge length of $h=0.025$.  We choose the parameters as listed in \cref{tab:params}.
\begin{table}[H]
	\begin{center}
		\caption{List of parameters and their values for the numerical experiments described in \cref{Sec:Simulation2Straight} and \cref{Sec:SimulationNetwork}. Parameters not mentioned below are set to zero.}\label{tab:params} \vspace{-.2cm}
				\begin{tabular}{|l | l || l | l || l | l |}
			\hline
			\textbf{Parameter} &
			\textbf{Value} & \textbf{Parameter} &
			\textbf{Value} & \textbf{Parameter} &
			\textbf{Value}  \\ \hline
			$\lambda_P$ & 5 & $\lambda_{P_h}$ & 0.5 & $\lambda_A$ & 0.005   \\
			$\lambda_{A_h}$ & 0.005  & $\lambda_{P\!H}$ & 1 & $\lambda_{H\!P}$ & 1 \\
			$\lambda_{H\!N}$ & 1 & $\sigma_{P\!H}$ & 0.55 & $\sigma_{H\!P}$ & 0.65  \\
			$\sigma_{H\!N}$ & 0.44 & $\varepsilon_P$ & 0.005 & $\varepsilon_H$ & 0.005\\
			$M_P$ & 50 & $M_H$ & 25 &  $\lambda_{\TAF_{\!P}}$ & 10 \\
			$D_{\TAF}$ & 0.5 & $m_{\TAF}$ & 1 & $L_p$  & $10^{-7}$ \\
			$D_\sigma$ & 1 & $m_\sigma$ & 1 & $K$ & $10^{-9}$ \\
			$D_{\MDE}$ & 0.5 & $m_{\MDE}$ & 1 &  $D_v$  & $0.1$ \\
			$\mu_{\text{bl}}$ & 1 & $L_\sigma $ & 10 & $\lambda_{\ECM_{\!D}}$ & 5 \\
			$\lambda_{\ECM_{\!P}}$ & 0.01 & $\lambda_{\MDE_{\!D}}$ & 1 & $\lambda_{\MDE_{\!D}}$ & 1 \\
			$\phi_{\ECM_{\!P}}$ & 0.5 & $C_{\Psi_T}$ & 0.045 & - & - \\
			\hline
		\end{tabular}

	\end{center}
\end{table}
\vspace{-.3cm}

Plots of the tumor species $\phi_T, \phi_P, \phi_H$ at the $z=1$ plane in the domain $\Omega$ at time points $t\in \{3,4,5\}$ are shown in \cref{fig:two-vessels-vars}. In \cref{fig:two-vessels-line-plot}, the tumor phases are shown along a one-dimensional line. It can be observed that the tumor separates into its three phases and moves towards the nutrient-rich regions of the domain. Moreover, the contour lines of the total tumor and its phases is presented at different times in \cref{fig:two-vessels-multi-contour}. We see that tumor is growing towards the artery. As expected, the proliferation is higher near the artery. In \cref{fig:two-vessels-taf}, plots of TAF, MDE, ECM at the time point $t=5$ in the $z=1$ plane of $\Omega$ are shown.  \vspace{-.3cm}

\begin{figure}[H]
	\begin{center}
		\includegraphics[width=.8\textwidth]{simulation1.pdf}
	\end{center}
	\vspace{-0.4cm}
	\caption{\label{fig:two-vessels-vars} Evolution of the total tumor $\phi_T$ (top), prolific $\phi_P$ (middle), and hypoxic $\phi_H$ (bottom) volume fractions at the times $t \in \{3,4,5\}$ (left, middle,right) in the $z=1$ plane of the domain $\Omega$. On the two vessels the nutrients are described by the 1D constituent $\phi_v$. }
\end{figure}

\begin{figure}[H]
	\begin{center}
	\hfill \includegraphics[width=.28\textwidth]{color1.pdf} \hspace{1cm} ~  \\
	\includegraphics[page=1,height=.22\textheight,valign=t]{simulation2.pdf} \hspace{-0.38cm}
	\includegraphics[page=2,height=.22\textheight,valign=t]{simulation2.pdf}
	\hspace{-0.38cm}
	\includegraphics[page=3,height=.22\textheight,valign=t]{simulation2.pdf}
		\includegraphics[height=.17\textheight,valign=t]{color2.pdf}  \hfill
		\\
\includegraphics[width=\textwidth]{simulation3.pdf}
	\end{center}
	\vspace{-0.4cm}
	\caption{\label{fig:two-vessels-line-plot} Left: Evolution of the tumor volume fraction ($\phi_T$) in the $z=1$ plane of the domain $\Omega$. Right: Plots of tumor ($\phi_T$), prolific ($\phi_P$), hypoxic ($\phi_H$) and necrotic ($\phi_N$) volume fractions along the line passing through the points $(0,0,1)$ and $(2,2,1)$ in the domain $\Omega$. From top row to bottom row, the plots correspond to time $t\in \{3,4,5\}$. }
\end{figure}
\vspace{-.5cm}
\begin{figure}[H]
	\begin{center}
		\includegraphics[width=.28\textwidth]{color1.pdf} \\
				\includegraphics[page=1,width=.3\textwidth]{simulation4.pdf}
				\includegraphics[page=2,width=.3\textwidth]{simulation4.pdf}
				\includegraphics[page=3,width=.3\textwidth]{simulation4.pdf} \\[-.5cm]

				\includegraphics[page=1,width=.3\textwidth]{simulation5.pdf}
\includegraphics[page=2,width=.3\textwidth]{simulation5.pdf}
\includegraphics[page=3,width=.3\textwidth]{simulation5.pdf}

	\end{center}
	\vspace{-0.4cm}
	\caption{\label{fig:two-vessels-multi-contour} Evolution of contour plots of the tumor volume fraction $\phi_T$ with vessels (top) at $0.8$ (light red) and $0.95$ (red) at the time $t \in \{4.25,4.75,5\}$ (left to right); the necrotic core is plotted at the contour line $\phi_N=0.42$ (black). Contour plots of the tumor phases without vessels (bottom) at $\phi_P=0.5$ (green), $\phi_H=0.45$ (orange), $\phi_N=0.4$ (dark red).}
\end{figure}

\begin{figure}[H]
	\begin{center}
		\includegraphics[width=.9\textwidth]{simulation6.pdf}
	\end{center}
	\vspace{-0.4cm}
	\caption{\label{fig:two-vessels-taf} Plots of $\taf$ (left), $\mde$ (middle), and $\ecm$ (right) at time $t=5$ in the $z=1$ plane of the domain $\Omega$. The colors (horizontal color bar) on the vessels show the transport of the 1D nutrient ($\phi_v$). The production of TAF and MDE is maximal where the hypoxic tumor phase is located. The decay of ECM happens in the regions where MDE is produced.}
\end{figure}

\cref{fig:two-vessels-Dsigma}  contains simulation results for the different values of parameters at $t=T=5$. Among the numerous model parameters, we focused on the chemotactic constant $\chi_c$, mobility $M_P$, proliferation rate $\lambda_P$, nutrient diffusion coefficient $D_\sigma$, and permeability constant $L_\sigma$. We vary one of these parameters while keeping other parameters fixed to their respective values listed in \cref{tab:params}.

It can be observed that all selected parameters strongly affect the tumor growth. Except for the parameter $D_\sigma$, the larger the remaining parameters, the faster the tumor cells move away from the vein and towards the nutrient rich artery. This means that for the chosen parameter values, the fluxes $J_\alpha,\;\alpha \in \CH$
given by \cref{Eq:Flux}, dominate the corresponding convective terms in \cref{Eq:Model3D} so that the tumor cells can move against the velocity field. It can be stated that for these parameter choices, the model simulates the migration of  tumor cells towards the nutrient sources in the vicinity.

\begin{figure}[H]
	\begin{center}
		\includegraphics[width=.7\textwidth]{simulation7.pdf}
	\end{center}
	\vspace{-0.4cm}
	\caption{\label{fig:two-vessels-mob}\label{fig:two-vessels-lamp} Effect of the mobility $M_P$ (top) for $M_P \in \{10,25,50\}$ (left to right) and proliferation rate $\lambda_P$ (bottom) for $\lambda_P \in \left\{2,5,10 \right\}$ (left to right) on the growth of the tumor volume.

		The color bar for 1D nutrients (left) and total tumor volume fraction (right) in included at the bottom.}

\end{figure}

\begin{figure}[H]
	\begin{center}

		\includegraphics[width=0.7\textwidth]{simulation8.pdf}
	\end{center}
	\vspace{-0.4cm}
	\caption{\label{fig:two-vessels-chi} \label{fig:two-vessels-Ls} \label{fig:two-vessels-Dsigma} Effect of the chemotactic constant $\chi_c$ (top) for $\chi_c \in \{0,0.01,0.05\}$ (left to right), permeability of the vessel wall $L_\sigma$ (middle) for $L_\sigma \in \{0.5,5,10\}$ (left to right), and diffusivity constant $D_\sigma$ (bottom) for $D_\sigma \in \{0.2,1,5\}$ (left to right) on the growth of the tumor volume.

		The color bar for 1D nutrients (left) and total tumor volume fraction (right) in included at the bottom.}

\end{figure}

\begin{figure}[H]
	\begin{center}
		\includegraphics[width=0.66\textwidth]{simulation9.pdf}
	\end{center}
	\vspace{-1cm}
	\caption{\label{fig:two-vessels-pres-vel} Pressure (left) and velocity field (right) in a plane perpendicular to the $z$-axis $\left(\text{at } z = 1 \right)$ of the domain $\Omega$. The artery is located in the right corner with a pressure decay from $10000$ to $5000$. The vein is located in the left corner with a pressure decay from $2000$ to $1000$. The velocity field induced by the pressure distribution is directed from the artery toward the vein.}
\end{figure}
\cref{fig:two-vessels-pres-vel} shows the pressure distribution in the vessels as well as the tissue pressure and velocity field within a plane that is perpendicular to the $z$-axis and located at $z=1$. The tissue pressure ranges between $1500$ and $7500$, which means that it is bounded by the extreme pressures in the vascular system. Furthermore, a gradient in the tissue pressure can be detected pointing from the artery to the vein. As a result, the velocity field is orientated from the artery to the vein.

\subsection{Tumor surrounded by a network}
\label{Sec:SimulationNetwork}
In the second subsection, we consider a small capillary network given by the data in \cite{secomb2000theoretical}. To keep the same computational domain as in the previous subsection, the network is scaled such that it fits into $\Omega = \left(0,2\right)^3$. After scaling, the resulting network has maximum, minimum, and mean vessel radius $0.0613$, $0.0307$, $0.0418$ respectively. At the inlets that are marked by an arrow, see \cref{fig:BoundariesNetwork}, we prescribe the pressure $p_{in}=25000$, while for all the other inlets, we use $p_{out}=10000$ as a boundary value. \blue{Again, we choose the boundary values synthetically in order to ensure that the transport processes are visible in the simulations.} Further, the boundary condition for $\phi_v$ is given by $\phi_v = \phi_{v,inlet} = 1$ if it holds $p_v = p_{in}$, and free outflow boundary if $p_v = p_{out}$.

\begin{figure}[H]
	\begin{center}
		\includegraphics[width=0.45\textwidth]{network1.pdf}
	\end{center}
	\caption{\label{fig:BoundariesNetwork} Outline of the scaled blood vessel network with an initial tumor core. The tumor core is represented by a contour surface with respect to $\phi_T$ (at $\phi_T = 0.1$). At the four inlets, indicated by an arrow, nutrients are injected i.e. at these boundaries, we set $\phi_v = \phi_{v,inlet} =1$ and $p_v = p_{in}$.}
\end{figure}

Contrary to the previous subsection, the spherical tumor core has a radius of $0.25$ and the center $(1.3, 0.9, 0.7)$. The same model parameters are employed. The domain $\Omega$ is discretized using cubic elements of mesh size $0.025$ and final time and time step of the simulation are $T = 5$ and $\Delta t = 0.025$.

 In \cref{fig:complex-network-t2-times}, the tumor cell volume fraction $\phi_T$, prolific cell volume fraction $\phi_P$, and hypoxic cell volume fraction $\phi_H$ are shown at $z=0.8$ plane and at time points $t \in \left\{3,4,5\right\}$. Finally in \cref{fig:complex-network-t2-times-contour-multi}, the contour plots for $\phi_T = 0.8$ and $\phi_T=0.95$ are presented. Further, the hypoxic phase is shown inside the tumor. In \cref{fig:complex-network-taf}, plots of TAF, MDE, ECM at $t=5$ in $z=0.8$ plane are shown.

The behavior of the tumor cells is similar to the two-vessel scenario. It seems that for the given parameter set the tumor cells are attracted by the nutrient rich blood vessels of the network. As can be observed in \cref{fig:complex-network-t2-times} (last row), the chemical potential of the tumor exhibits high gradients at the interface between tumor and healthy tissue. Therefore, the corresponding flux of the chemical potential given by \cref{Eq:Flux} is potentially high at this location. As a result the tumor cells are pulled towards the interface between cancerous and healthy tissue. Apparently, the flux is particularly high near the nutrient rich vessels such that the tumor cells move preferably towards the nutrient rich vessels.

\begin{figure}[H]
\begin{center}
\includegraphics[width=.8\textwidth]{network2.pdf}
\end{center}
\vspace{-0cm}
\caption{\label{fig:complex-network-t2-times} Distribution of the tumor cell volume fraction $\phi_T$ (top), prolific cell volume fraction $\phi_P$ (middle), and hypoxic cell volume fraction $\phi_H$ (bottom)  for $t \in \left\{3,4,5\right\}$. The tumor cells migrate towards to nutrient rich vessels.}
\end{figure}

\begin{figure}[H]
	\begin{center}

		\includegraphics[width=.28\textwidth]{color1.pdf} \\
		\includegraphics[page=1,width=.31\textwidth]{network3.pdf}
		\includegraphics[page=2,width=.31\textwidth]{network3.pdf}
		\includegraphics[page=3,width=.31\textwidth]{network3.pdf}

	\end{center}

	\caption{\label{fig:complex-network-t2-times-contour-multi}
		Evolution of contour plots of the tumor volume fraction $\phi_T$  at the values $0.8$ (light red) and $0.95$ (red), and of the hypoxic phase $\phi_H$ at $0.35$ at times $t \in \{3,3.5,4\}$ (left to right). The tumor growth is directed towards the nutrient rich vessels. }
\end{figure}

\begin{figure}[H]
	\begin{center}
		\includegraphics[width=.87\textwidth]{network4.pdf}
	\end{center}
	\vspace{-0.4cm}
	\caption{\label{fig:complex-network-taf} Plots of $\taf$ (left), $\mde$ (middle), and $\ecm$ (right) at time $t=5$ in the $z=0.8$ plane of the domain $\Omega$. The colors (horizontal color bar) on the vessels show the transport of the 1D nutrient ($\phi_v$). As in the case of two-vessels setting, the production of TAF and MDE is maximal where the hypoxic region is located.}
\end{figure}

\cref{fig:complex-network-t2-pres-vel} shows the tissue pressure and the corresponding velocity fields in $z=0.8$ plane. Just as in the two-vessel scenario, the pressure distribution induces a velocity field that goes from the high pressure region to the low pressure region.
\begin{figure}[H]
	\begin{center}
		\includegraphics[width=.75\textwidth,clip,trim={0 0 0 95}]{network5.pdf}
	\end{center}
	\vspace{-1cm}
	\caption{\label{fig:complex-network-t2-pres-vel} Plots of pressure (left) and velocity field (right) in the $z=0.8$ (top) and $z=1.2$ (bottom) planes in the domain $\Omega$. The plots correspond to the simulation time $t=5$. As in the two-vessel case, the velocity field is pointing from the high pressure region to the low pressure region.}
\end{figure}


TITLE: Calculating the interval and radius of convergence
QUESTION [0 upvotes]: I have $$s_n=\frac{-6^n(x+8)^n}{\sqrt{n}}$$  and I have to find the interval and radius of convergence.
Using ratio test:
$$\frac{-6^{n+1}(x+8)^{n+1}}{\sqrt{n+1}}\cdot \frac{\sqrt{n}}{-6^n(x+8)^n}$$
$$\lim_{n\rightarrow \infty}-6(x+8)\frac{\sqrt{n}}{\sqrt{n+1}}$$
The last term goes to 1.
Since the interval must be smaller than 1, we get:
$$-6(x+8)<1$$
This gives $$x<-\frac{43}{6}$$, which should be the interval of convergence. Half of that again is the radius of converge, so $$x<-\frac{43}{12}$$
But this is wrong,
any ideas where the error lies and what the right solution is?
Thanks

REPLY [2 votes]: The correct typing for $s_n$ seems to be
$$s_n=\frac{(-6)^n(x+8)^n}{\sqrt n}.$$
Compute the same ratio test but for $|s_n|$ (not $s_n$ which takes negative values).
Eventually, you get that your series is absolutely convergent if
$$6|x+8|<1,$$
i.e.
$$x\in\left(-8-\frac16,-8+\frac16\right),$$
and trivially divergent if $x\notin\left[-8-\frac16,-8+\frac16\right].$
The radius of convergence $R$ is half the length of the interval of convergence $\left(-8-\frac16,-8+\frac16\right)$:$$R=\frac16.$$
If $x=-8-\frac16$, $\sum_{n=1}^\infty s_n=\sum_{n=1}^\infty\frac1{\sqrt n}$ is divergent.
If $x=-8+\frac16$, $\sum_{n=1}^\infty s_n=\sum_{n=1}^\infty\frac{(-1)^n}{\sqrt n}$ is a conditionally convergent alternating series.


TITLE: Showing one definition of semiring implies the other (or maybe are equivalent?)
QUESTION [1 upvotes]: There are two definitions of a semiring that I have seen.
Let $X$ be a set, and $S$ be a set of subsets of $X$. Then $S$ is a semiring if and only if:

$\varnothing \in S$
$A, B \in S \implies A \cap B \in S$
If $A, B \in S$, there are sets $A_1, \dots, A_n \in S$ which are pairwise disjoint and satisfy $A \setminus B = \bigcup_{i=1}^{n}A_i$.

Another definition I've seen is identical for the first two conditions above, but changes the third condition above to:

If $A, A^{\prime} \in S$ with $A^{\prime} \subset A$, there are sets $A_1, \dots, A_n \in S$ which are pairwise disjoint and are disjoint from $A^{\prime}$, satisfying $A = A^{\prime} \cup \left(\bigcup_{i=1}^{n}A_i\right)$.

Are these definitions equivalent? If so, how? Or at the very least, does the second definition imply the first definition?
I thought there might be a trick perhaps considering $A, B \in S$, let $A^{\prime} = A \setminus B \subset A$. But we can't do anything with this unless we know $A^{\prime} \in S$. How would we know? I'm not sure.

REPLY [2 votes]: Yes, they’re equivalent. Assume the first definition, and let $A,A'\in S$ with $A'\subseteq A$. Then $A\setminus A'=\bigcup_{i=1}^nA_i$ for some pairwise disjoint sets $A_i\in S$. Clearly the sets $A_i$ are disjoint from $A'$, and $A=A'\cup\bigcup_{i=1}^nA_i$.
Now assume the second definition, and let $A,B\in S$. Let $A'=A\cap B$; $A'\in S$, so $A=A'\cup\bigcup_{i=1}^nA_i$ for some pairwise disjoint set $A_i\in S$ that are all disjoint from $A'$, and $A\setminus B=A\setminus A'=\bigcup_{i=1}^nA_i$.