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\fancyhead[L]{Supplementary Information}
\fancyhead[R]{Alvaro Martínez-Rubio et. al.}
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\author{Álvaro Martínez-Rubio $^{1,2}$, Salvador Chulián $^{1,2}$, Cristina Blázquez-Goñi $^{2,4}$, \\
 Manuel Ramírez-Orellana $^{5}$,  Antonio Pérez-Martínez $^{6,7}$, Alfonso Navarro-Zapata $^{6}$, \\
 Cristina Ferreras $^{6}$, Victor M. Pérez-García $^{3}$ and María Rosa $^{1,2}$*}
 
 \date{}

\title{\Large{A Mathematical Description of Bone Marrow Dynamics of CAR-T Cell Therapy in B-cell Childhood Acute Lymphoblastic Leukemia}
\\\textbf{\LARGE{Supplementary Information}}}
 
\begin{document}
\maketitle


\footnotesize{
\noindent $^{1}$ \quad Department of Mathematics, Universidad de C\'{a}diz, Puerto Real, C\'{a}diz, Spain\\
$^{2}$ \quad Biomedical Research and Innovation Institute of C\'adiz (INiBICA), Hospital Universitario Puerta del Mar, C\'{a}diz, Spain\\
$^{3}$ \quad Department of Mathematics, Mathematical Oncology Laboratory (MOLAB), Universidad de Castilla-La Mancha, Ciudad Real, Spain\\
$^{4}$ \quad Department of Pediatric Hematology and Oncology, Hospital de Jerez C\'adiz, Spain\\
$^{5}$ \quad Department of Paediatric Haematology and Oncology, Hospital Infantil Universitario Ni\~no Jes\'us, Instituto
Investigaci\'on Sanitaria La Princesa, Madrid, Spain\\
$^{6}$ \quad Translational Research in Pediatric Oncology, Hematopoietic Transplantation and Cell Therapy, IdiPAZ, Hospital Universitario La Paz, Madrid, Spain\\
$^{7}$ \quad Pediatric Hemato-Oncology Department, Hospital Universitario La Paz, Madrid, Spain\\}
\\
$*$ maria.rosa@uca.es


\begin{figure}
\centering
\includegraphics[width=\textwidth]{lowTB.pdf}
\caption{\textbf{Influence of low CD19 burden.} Evolution in time of leukemic cells (gray) and total CAR T-cells (orange) for different initial configurations (In decreasing order: solid, dashed, dotted lines). Parameters are those of Table 1 with $\alpha=3\times10^{-9}$ day\textsuperscript{-1} $\cdot$ cell\textsuperscript{-1}.  In general, leukemia recurrence is more frequent and occurs sooner for higher values of activation threshold and lower CD19 burden.}
\label{lowTB}
\end{figure}

\clearpage

\begin{figure}
\centering
\includegraphics[width=\textwidth]{propcar2.pdf}
\caption{\textbf{Proportion of newly manufactured CAR after second infusion.} Evolution of the proportion of the second CAR infused after lack of response to first infusion, for the case of the improved product. The three cases correspond to the three examples shown in Figure 7, with new attributes (A) $h=5\times10^8$ and $\rho_C=0.9$, (B) $h=5\times10^8$ and $\alpha=6\times10^{-10}$, (C) $\alpha=5\times10^{-8}$ and $\rho_C=0.7$. Rest of parameters as in Figure 7.}
\label{propcar2}
\end{figure}


\clearpage

\begin{figure}
\centering
\includegraphics[width=\textwidth]{hm_reinfuse.pdf}
\caption{\textbf{Exploration of parameter ranges for response to second infusion.} Number of leukemic cells at day +30 after the second infusion of $5\times10^8$ CAR T-cells of the initial product, in logarithmic scale. Unchanged parameters and initial state as in Figure 6. Discontinuities correspond to the thresholds (in either cell number or time) set for the moment of reinfusion. Subfigures (A), (B) and (C) correspond to the three heatmaps in Figure 6: Top, middle and bottom respectively.}
\label{HM_reinfuse}
\end{figure}

\clearpage

\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{reB0.pdf}
\caption{\textbf{Effect of CD19+ B-cell infusion in non-responding patients.} Dynamics of tumor cells (solid gray line), CD19- and CD19+ B-cells (solid and dotted blue lines respectively) and activated and memory CAR T-cells (solid and dotted orange line, respectively), for the three non-responding patients of Figure 6. (A) Infusion of one order of magnitude of CD19+ B-cells at the time of relapse. (B) Infusion of two orders of magnitude of CD19+ B-cells at the time of relapse. }
\label{reB0}
\end{figure}


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