A structured population model of clonal selection in acute leukemias with multiple maturation stages

Abstract

Recent progress in genetic techniques has shed light on the complex co-evolution of malignant cell clones in leukemias. However, several aspects of clonal selection still remain unclear. In this paper, we present a multi-compartmental continuously structured population model of selection dynamics in acute leukemias, which consists of a system of coupled integro-differential equations. Our model can be analysed in a more efficient way than classical models formulated in terms of ordinary differential equations. Exploiting the analytical tractability of this model, we investigate how clonal selection is shaped by the self-renewal fraction and the proliferation rate of leukemic cells at different maturation stages. We integrate analytical results with numerical solutions of a calibrated version of the model based on real patient data. In summary, our mathematical results formalise the biological notion that clonal selection is driven by the self-renewal fraction of leukemic stem cells and the clones that possess the highest value of this parameter are ultimately selected. Moreover, we demonstrate that the self-renewal fraction and the proliferation rate of non-stem cells do not have a substantial impact on clonal selection. Taken together, our results indicate that interclonal variability in the self-renewal fraction of leukemic stem cells provides the necessary substrate for clonal selection to act upon.

Appendices

Multi-compartmental ODE model

In a number of previous papers (Stiehl et al. 2014a, 2015, 2016, 2018; Stiehl and Marciniak-Czochra 2012), it was shown that mathematical models defined in the framework of the following ODE system can effectively recapitulate clinical data from leukemia patients

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle {\frac{\mathrm{d}}{\mathrm{d} t} N^j_1(t) = \left( \frac{2 \, a^j_{1}}{1 + K \, Z_M(t)} - 1 \right) \, p^j_1 \, N^j_1(t),} \\ \\ \displaystyle {\frac{\mathrm{d}}{\mathrm{d} t} N^j_i(t) = 2 \, \left( 1 - \frac{a^j_{i-1}}{1 + K \, Z_M(t)} \right) \, p^j_{i-1} \, N^j_{i-1}(t)} \\ \displaystyle {\phantom {\frac{\mathrm{d}}{\mathrm{d} t} N^j_i(t) =} + \left( \frac{2 \, a^j_{i}}{1 + K \, Z_M(t)} - 1 \right) \, p^j_i \, N^j_i(t),} \\ \\ \displaystyle {\frac{\mathrm{d}}{\mathrm{d} t} N^j_M(t) = 2 \, \left( 1 - \frac{a^j_{M-1}}{1 + K \, Z_M(t)} \right) \, p^j_{M-1} \, N^j_{M-1}(t) - d \, N^j_M(t),} \end{array} \right. \end{aligned}$$

(68)

with $i=2, \ldots , M-1$, $j=0, \ldots , J$ and ${Z_M(t) = {\sum \nolimits _{j=0}^J} N^j_M(t)}$.

The index i denotes the cell maturation stage while the index j indicates to which leukemic clone the cells belong. In particular, the stem-cell compartment is labelled by the index $i=1$, the indices $i=2, \ldots , M-1$ correspond to increasingly mature progenitor-cell compartments and the mature-cell/blast compartment is labelled by the index $i=M$. Moreover, the index $j=0$ refers to healthy cells, whereas the different leukemic clones are labelled by the indices $j=1, \ldots , J$.

In the system of ODEs (68), the function $N^j_i(t)$ models the density of cells of clone j at the maturation stage i at the time instant $t \ge 0$. Cells in the compartment $i=M$ do not divide and are cleared from the system at rate $d>0$, which is assumed to be the same for healthy and leukemic cells (Busse et al. 2016; Malinowska et al. 2002; Savitskiy et al. 2003; Stiehl et al. 2014a, 2018). The parameters $p^j_i > 0$ and $a^j_i > 0$ model, respectively, the proliferation rate and the self-renewal fraction of cells of clone j at the maturation stage i. Coherently with biological findings (Kondo et al. 1991; Layton et al. 1989; Shinjo et al. 1997), the terms $a^j_i$ are multiplied by the factor $\frac{1}{1 + K Z_M(t)}$ to model the fact that the signal which promotes the self-renewal of dividing cells is absorbed by mature cells and leukemic blasts at a rate that depends on the total density of these cells $Z_M(t)$. The parameter $K>0$ is related to the degradation rate of the feedback signal by mature cells and leukemic blasts. This has proved to be a biologically consistent way of modelling the effects of feedback signals that control cell self-renewal (Marciniak-Czochra et al. 2009; Stiehl et al. 2014c, 2018; Stiehl and Marciniak-Czochra 2011, 2012). In principle, the effects of feedback signals that control cell proliferation could also be included. However, it has been demonstrated that such signals have little impact on the dynamics of the blood system (Marciniak-Czochra et al. 2009; Stiehl et al. 2014b, c).

A version of the ODE model (68) with only one leukemic clone and three maturation stages (i.e. $M=3$ and $J=1$) has been fully analysed by Stiehl and Marciniak-Czochra (2012), while a two compartmental version of the model for healthy hematopoiesis (i.e. $M=2$ and $J=0$) has been analysed by Getto et al. (2013); Nakata et al. (2012); Stiehl and Marciniak-Czochra (2011). Possible applications of this model to clinical data can be found in the works by Stiehl et al. (2014a, 2015), whereas applications to healthy hematopoiesis are provided in the publications by Stiehl et al. (2014b, c). Finally, a version of this model with a continuous differentiation structure has been proposed and studied by Doumic-Jauffret et al. (2011).

Proof of the uniform upper bounds (15)

In this appendix we prove the upper bounds (15) through a suitable development of the method of proof presented in (Busse et al. 2016). Using the system of IDEs (7), straightforward calculations give the following ODEs

$$\begin{aligned} \varepsilon \, \frac{\mathrm{d}}{\mathrm{d}t}\frac{\rho _{1 \, \varepsilon }}{\rho _{2 \, \varepsilon }}= & {} \frac{1}{\rho _{2 \, \varepsilon }} \int _0^1 P_{1}(\rho _{M \varepsilon }(t),x)\, n_{1 \varepsilon } \, \mathrm{d}x \nonumber \\&- \frac{\rho _{1 \, \varepsilon }}{\rho ^2_{2 \, \varepsilon }}\! \int _0^1 Q_{1}(\rho _{M \varepsilon }(t),x)\, n_{1 \varepsilon } \, \mathrm{d}x \!-\! \frac{\rho _{1 \, \varepsilon }}{\rho ^2_{2 \, \varepsilon }} \int _0^1 P_{2}(\rho _{M \varepsilon }(t),x)\, n_{2 \varepsilon } \, \mathrm{d}x, \qquad \end{aligned}$$

(69)

$$\begin{aligned} \varepsilon \, \frac{\mathrm{d}}{\mathrm{d}t}\frac{\rho _{i \, \varepsilon }}{\rho _{i+1 \, \varepsilon }}\!= & {} \! \frac{1}{\rho _{i+1 \, \varepsilon }} \int _0^1 Q_{i-1}(\rho _{M \varepsilon }(t),x)\, n_{i-1 \varepsilon } \, \mathrm{d}x \!+\! \frac{1}{\rho _{i+1 \, \varepsilon }} \int _0^1 P_{i}(\rho _{M \varepsilon }(t),x)\, n_{i \varepsilon } \, \mathrm{d}x \nonumber \\&- \frac{\rho _{i \, \varepsilon }}{\rho ^2_{i+1 \, \varepsilon }} \int _0^1 Q_{i}(\rho _{M \varepsilon }(t),x)\, n_{i \varepsilon } \, \mathrm{d}x \nonumber \\&- \frac{\rho _{i \, \varepsilon }}{\rho ^2_{i+1 \, \varepsilon }} \int _0^1 P_{i+1}(\rho _{M \varepsilon }(t),x)\, n_{i+1 \varepsilon } \, \mathrm{d}x, \quad \text {for } \; i = 2, \ldots , M-2 \end{aligned}$$

(70)

and

$$\begin{aligned} \varepsilon \, \frac{\mathrm{d}}{\mathrm{d}t}\frac{\rho _{M-1 \, \varepsilon }}{\rho _{M \, \varepsilon }}= & {} \frac{1}{\rho _{M \, \varepsilon }} \int _0^1 Q_{M-2}(\rho _{M \varepsilon }(t),x)\, n_{M-2 \varepsilon } \, \mathrm{d}x \nonumber \\&+ \frac{1}{\rho _{M \, \varepsilon }} \int _0^1 P_{M-1}(\rho _{M \varepsilon }(t),x)\, n_{M-1 \varepsilon } \, \mathrm{d}x \nonumber \\&- \frac{\rho _{M-1 \, \varepsilon }}{\rho ^2_{M \, \varepsilon }} \int _0^1 Q_{M-1}(\rho _{M \varepsilon }(t),x)\, n_{M-1 \varepsilon } \, \mathrm{d}x + d \, \frac{\rho _{M-1 \, \varepsilon }}{\rho _{M \, \varepsilon }}. \end{aligned}$$

(71)

Under assumptions (2)–(4), for all $i = 1, \ldots , M-1$ we have

$$\begin{aligned}&P_{i}(\rho _{M \, \varepsilon }(t),x) > - \Vert p_{i}\Vert _{\mathrm{L}^\infty ([0,1])},\\&\quad P_{i}(\rho _{M \, \varepsilon }(t),x) \le 2 \, \Vert a_{i}\Vert _{\mathrm{L}^\infty ([0,1])} \Vert p_{i}\Vert _{\mathrm{L}^\infty ([0,1])} =: {\overline{P}}_i, \end{aligned}$$

and

$$\begin{aligned}&Q_{i}(\rho _{M \, \varepsilon }(t),x) \ge 2 \, \left( 1-\Vert a_{i}\Vert _{\mathrm{L}^\infty ([0,1])}\right) \inf _{x \in [0,1]} p_{i} =: {\underline{Q}}_i > 0,\\&Q_{i}(\rho _{M \, \varepsilon }(t),x) \le 2 \, \Vert p_{i}\Vert _{\mathrm{L}^\infty ([0,1])}, \end{aligned}$$

for all $t \ge 0$ and all $x \in [0,1]$. Hence, estimating from above the right-hand sides of the ODEs (69)–(71) using the above estimates on $P_i$ and $Q_i$ along with the non-negativity of $n_{i \, \varepsilon }$ and $\rho _{i \, \varepsilon }$ for all $i = 1, \ldots , M$ gives the following differential inequalities

$$\begin{aligned} \varepsilon \, \frac{\mathrm{d}}{\mathrm{d}t}\frac{\rho _{1 \, \varepsilon }}{\rho _{2 \, \varepsilon }}\le & {} {\overline{P}}_1 \, \frac{\rho _{1 \, \varepsilon }}{\rho _{2 \, \varepsilon }} - {\underline{Q}}_1 \, \frac{\rho ^2_{1 \, \varepsilon }}{\rho ^2_{2 \, \varepsilon }} + \Vert p_{2}\Vert _{\mathrm{L}^\infty ([0,1])} \, \frac{\rho _{1 \, \varepsilon }}{\rho _{2 \, \varepsilon }}\nonumber \\\le & {} \left[ \left( {\overline{P}}_1 + \Vert p_{2}\Vert _{\mathrm{L}^\infty ([0,1])}\right) - {\underline{Q}}_1 \frac{\rho _{1 \, \varepsilon }}{\rho _{2 \, \varepsilon }}\right] \frac{\rho _{1 \, \varepsilon }}{\rho _{2 \, \varepsilon }}, \end{aligned}$$

(72)

$$\begin{aligned} \varepsilon \, \frac{\mathrm{d}}{\mathrm{d}t}\frac{\rho _{i \, \varepsilon }}{\rho _{i+1 \, \varepsilon }}\le & {} 2 \, \Vert p_{i-1}\Vert _{\mathrm{L}^\infty ([0,1])} \, \frac{\rho _{i-1 \, \varepsilon }}{\rho _{i+1 \, \varepsilon }} + {\overline{P}}_i \, \frac{\rho _{i \, \varepsilon }}{\rho _{i+1 \, \varepsilon }} - {\underline{Q}}_i \, \frac{\rho ^2_{i \, \varepsilon }}{\rho ^2_{i+1 \, \varepsilon }} \nonumber \\&+ \Vert p_{i+1}\Vert _{\mathrm{L}^\infty ([0,1])} \frac{\rho _{i \, \varepsilon }}{\rho _{i+1 \, \varepsilon }} \nonumber \\\le & {} \left[ \left( 2 \, \Vert p_{i-1}\Vert _{\mathrm{L}^\infty ([0,1])} \frac{\rho _{i-1 \, \varepsilon }}{\rho _{i \, \varepsilon }} + {\overline{P}}_i + \Vert p_{i+1}\Vert _{\mathrm{L}^\infty ([0,1])}\right) \right. \nonumber \\&\left. - {\underline{Q}}_i \frac{\rho _{i \, \varepsilon }}{\rho _{i+1 \, \varepsilon }}\right] \frac{\rho _{i \, \varepsilon }}{\rho _{i+1 \, \varepsilon }} \qquad \end{aligned}$$

(73)

for $i = 2, \ldots , M-2$,

$$\begin{aligned} \varepsilon \, \frac{\mathrm{d}}{\mathrm{d}t}\frac{\rho _{M-1 \, \varepsilon }}{\rho _{M \, \varepsilon }}\le & {} 2 \, \Vert p_{M-2}\Vert _{\mathrm{L}^\infty ([0,1])} \, \frac{\rho _{M-2 \, \varepsilon }}{\rho _{M \, \varepsilon }} + {\overline{P}}_{M-1} \, \frac{\rho _{M-1 \, \varepsilon }}{\rho _{M \, \varepsilon }} \nonumber \\&- {\underline{Q}}_{M-1} \, \frac{\rho ^2_{M-1 \, \varepsilon }}{\rho ^2_{M \, \varepsilon }} + d \, \frac{\rho _{M-1 \, \varepsilon }}{\rho _{M \, \varepsilon }} \nonumber \\\le & {} \left[ \left( 2 \, \Vert p_{M-2}\Vert _{\mathrm{L}^\infty ([0,1])} \frac{\rho _{M-2 \, \varepsilon }}{\rho _{M-1 \, \varepsilon }} + {\overline{P}}_{M-1} + d\right) \right. \nonumber \\&\left. - {\underline{Q}}_{M-1} \frac{\rho _{M-1 \, \varepsilon }}{\rho _{M \, \varepsilon }}\right] \frac{\rho _{M-1 \, \varepsilon }}{\rho _{M \, \varepsilon }}. \qquad \end{aligned}$$

(74)

The differential inequality (72) yields

$$\begin{aligned} \frac{\rho _{1 \, \varepsilon }(t)}{\rho _{2 \, \varepsilon }(t)} \le \max \left( \frac{\Vert n^0_{1}\Vert _{\mathrm{L}^\infty ([0,1])}}{\Vert n^0_{2}\Vert _{\mathrm{L}^\infty ([0,1])}}, \frac{{\overline{P}}_1 + \Vert p_{2}\Vert _{\mathrm{L}^\infty ([0,1])}}{{\underline{Q}}_1} \right) =: B_1, \end{aligned}$$

(75)

for all $t \ge 0$. Substituting the estimate (75) into the differential inequality (73) for $i=2$ we find

$$\begin{aligned}&\frac{\rho _{2 \, \varepsilon }(t)}{\rho _{3 \, \varepsilon }(t)} \nonumber \\&\quad \le \max \left( \frac{\Vert n^0_{2}\Vert _{\mathrm{L}^\infty ([0,1])}}{\Vert n^0_{3}\Vert _{\mathrm{L}^\infty ([0,1])}}, \frac{2 \, \Vert p_{1}\Vert _{\mathrm{L}^\infty ([0,1])} B_1 + {\overline{P}}_2 + \Vert p_{3}\Vert _{\mathrm{L}^\infty ([0,1])}}{{\underline{Q}}_2} \right) =: B_2,\nonumber \\ \end{aligned}$$

(76)

for all $t \ge 0$. In a similar way, substituting the estimate (76) into the differential inequality (73) for $i=3$ gives

$$\begin{aligned}&\frac{\rho _{3 \, \varepsilon }(t)}{\rho _{4 \, \varepsilon }(t)} \nonumber \\&\quad \le \max \left( \frac{\Vert n^0_{3}\Vert _{\mathrm{L}^\infty ([0,1])}}{\Vert n^0_{4}\Vert _{\mathrm{L}^\infty ([0,1])}}, \frac{2 \, \Vert p_{2}\Vert _{\mathrm{L}^\infty ([0,1])} B_2 + {\overline{P}}_3 + \Vert p_{4}\Vert _{\mathrm{L}^\infty ([0,1])}}{{\underline{Q}}_3} \right) =: B_3,\nonumber \\ \end{aligned}$$

(77)

for all $t \ge 0$. Using a bootstrap argument based on the method of proof that we have used for the case $i=3$, one can prove that

$$\begin{aligned}&\frac{\rho _{i \, \varepsilon }(t)}{\rho _{i+1 \, \varepsilon }(t)} \nonumber \\&\quad \le \max \left( \frac{\Vert n^0_{i}\Vert _{\mathrm{L}^\infty ([0,1])}}{\Vert n^0_{i+1}\Vert _{\mathrm{L}^\infty ([0,1])}}, \frac{2 \, \Vert p_{i-1}\Vert _{\mathrm{L}^\infty ([0,1])} B_{i-1} + {\overline{P}}_i + \Vert p_{i+1}\Vert _{\mathrm{L}^\infty ([0,1])}}{{\underline{Q}}_i} \right) =: B_i,\nonumber \\ \end{aligned}$$

(78)

for all $t \ge 0$ and for all $i=4, \ldots , M-2$. Finally, substituting the estimate (78) with $i=M-2$ into the differential inequality (74) we obtain

$$\begin{aligned}&\frac{\rho _{M-1 \, \varepsilon }(t)}{\rho _{M \, \varepsilon }(t)}\nonumber \\&\quad \le \max \left( \frac{\Vert n^0_{M-1}\Vert _{\mathrm{L}^\infty ([0,1])}}{\Vert n^0_{M}\Vert _{\mathrm{L}^\infty ([0,1])}}, \frac{2 \, \Vert p_{M-2}\Vert _{\mathrm{L}^\infty ([0,1])} B_{M-2} + {\overline{P}}_{M-1} + d}{{\underline{Q}}_{M-1}} \right) =: B_{M-1}\nonumber \\ \end{aligned}$$

(79)

for all $t \ge 0$. Combining the estimates (75)-(79) yields

$$\begin{aligned} \rho _{i \, \varepsilon }(t) \le \rho _{M \, \varepsilon }(t) A_i \quad \text {with} \quad A_i := \prod _{k=i}^{M-1} B_k > 0 \quad \text {for } \; i=1,\ldots ,M-1 \end{aligned}$$

for all $t \ge 0$, which allows us to conclude that

$$\begin{aligned} P_{i}(\rho _{M \varepsilon }(t),x) \le \left( \frac{2 \, a_i(x)}{1 + \frac{K}{A_i} \rho _{i \varepsilon }(t)} - 1 \right) p_i(x) \; \text{ for } \; i=1, \ldots , M-1 \end{aligned}$$

for all $t \ge 0$ and all $x \in [0,1]$. The latter inequality ensures that $P_{i}(\rho _{M \varepsilon }(t),x)$ satisfies the following relations for all $t \ge 0$ and each $i=1, \ldots , M-1$

$$\begin{aligned}&\text {if } \; \displaystyle {\rho _{i \varepsilon }(t) \le \frac{A_i}{K} \, \left( 2 \, \Vert a_{i}\Vert _{\mathrm{L}^\infty ([0,1])} -1 \right) } \; \text { then } \nonumber \\&\quad 0 \le \Vert P_{i}(\rho _{M \varepsilon }(t),\cdot )\Vert _{\mathrm{L}^\infty ([0,1])} \le \left( \frac{2 \, \Vert a_{i}\Vert _{\mathrm{L}^\infty ([0,1])}}{1 + \frac{K}{A_i} \rho _{i \varepsilon }(t)} - 1 \right) \, \Vert p_{i}\Vert _{\mathrm{L}^\infty ([0,1])} \quad \quad \end{aligned}$$

(80)

while

$$\begin{aligned}&\text {if } \; \displaystyle {\rho _{i \varepsilon }(t) > \frac{A_i}{K} \, \left( 2 \, \Vert a_{i}\Vert _{\mathrm{L}^\infty ([0,1])} -1 \right) } \; \text { then } \nonumber \\&\quad \Vert P_{i}(\rho _{M \varepsilon }(t),\cdot )\Vert _{\mathrm{L}^\infty ([0,1])} \le \left( \frac{2 \, \Vert a_{i}\Vert _{\mathrm{L}^\infty ([0,1])}}{1 + \frac{K}{A_i} \rho _{i \varepsilon }(t)} - 1 \right) \, \inf _{x \in [0,1]} p_{i} < 0. \end{aligned}$$

(81)

Integrating over [0, 1] both sides of the IDEs (7) for $n_{1 \, \varepsilon }$ and estimating from above gives the following differential inequality

$$\begin{aligned} \varepsilon \, \frac{\mathrm{d}}{\mathrm{d}t}\rho _{1 \, \varepsilon }(t) \le \Vert P_{1}(\rho _{M \varepsilon }(t),\cdot )\Vert _{\mathrm{L}^\infty ([0,1])} \, \rho _{1 \, \varepsilon }(t) \end{aligned}$$

(82)

from which, using (80) and (81) with $i=1$, we find that for any $\varepsilon >0$

$$\begin{aligned} \rho _{1 \, \varepsilon }(t) \le \max \left( \Vert n^0_{1}\Vert _{\mathrm{L}^\infty ([0,1])}, \frac{A_1}{K} \, \left( 2 \, \Vert a_{1}\Vert _{\mathrm{L}^\infty ([0,1])} -1 \right) \right) =: {\overline{\rho }}_1 \quad \text {for all } \; t \ge 0, \end{aligned}$$

that is, the upper bound (15) on $\rho _{1 \, \varepsilon }$ is verified.

Furthermore, integrating over [0, 1] both sides of the IDEs (7) for $n_{2 \, \varepsilon }$ and estimating from above using the fact that

$$\begin{aligned} Q_{1}(\rho _{M \, \varepsilon }(t),x) \le 2 \, \Vert p_{1}\Vert _{\mathrm{L}^\infty ([0,1])} \quad \text {for all } (t,x) \in [0,\infty ) \times [0,1] \end{aligned}$$

along with the upper bound (15) on $\rho _{1 \, \varepsilon }$ gives

$$\begin{aligned} \varepsilon \, \frac{\mathrm{d}}{\mathrm{d}t}\rho _{2 \, \varepsilon }(t) \le 2 \, \Vert p_{1}\Vert _{\mathrm{L}^\infty ([0,1])} {\overline{\rho }}_1 + \Vert P_{2}(\rho _{M \varepsilon }(t),\cdot )\Vert _{\mathrm{L}^\infty ([0,1])} \, \rho _{2 \, \varepsilon }(t). \end{aligned}$$

(83)

The above differential inequality along with the estimates (80) and (81) with $i=2$ ensures that there exists $C>0$ such that for any $\varepsilon >0$

$$\begin{aligned} \rho _{2 \, \varepsilon }(t) \le \max \left( \Vert n^0_{2}\Vert _{\mathrm{L}^\infty ([0,1])}, \frac{A_2}{K} \, \left( 2 \, \Vert a_{2}\Vert _{\mathrm{L}^\infty ([0,1])} -1 \right) , \frac{2}{C} \, \Vert p_{1}\Vert _{\mathrm{L}^\infty ([0,1])} \, {\overline{\rho }}_1 \right) =: {\overline{\rho }}_2 \end{aligned}$$

for all $t \ge 0$. Hence, the upper bound (15) on $\rho _{2 \, \varepsilon }$ is verified. The upper bounds (15) on $\rho _{i \, \varepsilon }$ for $i=3,\ldots ,M-1$ can be proved in a similar way using a bootstrap argument.

Finally, integrating over [0, 1] both sides of the IDEs (7) for $n_{M \, \varepsilon }$ and estimating from above using the fact that

$$\begin{aligned} Q_{M-1}(\rho _{M \, \varepsilon }(t),x) \le 2 \, \Vert p_{M-1}\Vert _{\mathrm{L}^\infty ([0,1])} \quad \text {for all } (t,x) \in [0,\infty ) \times [0,1] \end{aligned}$$

along with the upper bound (15) on $\rho _{M-1 \, \varepsilon }$ gives the following differential inequality

$$\begin{aligned} \varepsilon \, \frac{\mathrm{d}}{\mathrm{d}t}\rho _{M \, \varepsilon }(t) \le 2 \, \Vert p_{M-1}\Vert _{\mathrm{L}^\infty ([0,1])} {\overline{\rho }}_{M-1} - d \, \rho _{M \varepsilon }(t), \end{aligned}$$

which ensures that for any $\varepsilon >0$ we have

$$\begin{aligned} \rho _{M \, \varepsilon }(t) \le \max \left( \Vert n^0_{M}\Vert _{\mathrm{L}^\infty ([0,1])}, \frac{2}{d} \, \Vert p_{M-1}\Vert _{\mathrm{L}^\infty ([0,1])} \, {\overline{\rho }}_{M-1} \right) =: {\overline{\rho }}_M \quad \text {for all } \; t \ge 0, \end{aligned}$$

that is, the upper bound (15) on $\rho _{M \, \varepsilon }$ is verified.