Three-Dimensional Spatiotemporal Modeling of Colon Cancer Organoids Reveals that Multimodal Control of Stem Cell Self-Renewal is a Critical Determinant of Size and Shape in Early Stages of Tumor Growth

Abstract

We develop a three-dimensional multispecies mathematical model to simulate the growth of colon cancer organoids containing stem, progenitor and terminally differentiated cells, as a model of early (prevascular) tumor growth. Stem cells (SCs) secrete short-range self-renewal promoters (e.g., Wnt) and their long-range inhibitors (e.g., Dkk) and proliferate slowly. Committed progenitor (CP) cells proliferate more rapidly and differentiate to produce post-mitotic terminally differentiated cells that release differentiation promoters, forming negative feedback loops on SC and CP self-renewal. We demonstrate that SCs play a central role in normal and cancer colon organoids. Spatial patterning of the SC self-renewal promoter gives rise to SC clusters, which mimic stem cell niches, around the organoid surface, and drive the development of invasive fingers. We also study the effects of externally applied signaling factors. Applying bone morphogenic proteins, which inhibit SC and CP self-renewal, reduces invasiveness and organoid size. Applying hepatocyte growth factor, which enhances SC self-renewal, produces larger sizes and enhances finger development at low concentrations but suppresses fingers at high concentrations. These results are consistent with recent experiments on colon organoids. Because many cancers are hierarchically organized and are subject to feedback regulation similar to that in normal tissues, our results suggest that in cancer, control of cancer stem cell self-renewal should influence the size and shape in similar ways, thereby opening the door to novel therapies.

Appendix

1.1 Model Non-Dimensionalization

Let L be the length scale and $\nabla '=\nabla /L$ be the dimensionless gradient. Following Wise et al. (2011), we choose L to be the nutrient diffusion length $L=\sqrt{D_n/u_n^\mathrm{SC}}$, typically on the order of 200 $\upmu$m. Since the nutrient concentration is measured against that in the microenvironment, we non-dimensionalize n as $n'=n/\bar{n}$. We rewrite Eq. (14) for the dimensionless nutrient concentration $n'$:

$$\begin{aligned} 0=\varDelta 'n'-\left( \phi _\mathrm{SC}+u_n'^\mathrm{CP}\phi _\mathrm{CP}+u_n'^\mathrm{TD}\phi _\mathrm{TD}\right) n'+p_n'Q(\phi _T)(1-n'), \end{aligned}$$

(20)

where $u_n'^\mathrm{CP}=u_n^\mathrm{CP}/u_n^\mathrm{SC}$, $u_n'^\mathrm{TD}=u_n^\mathrm{TD}/u_n^\mathrm{SC}$ and $p_n'=p_n/u_n^\mathrm{SC}$.

Next, we non-dimensionalize Eq. (1). Let T be the timescale and $t'=t/T$ be the dimensionless time. Denote $\mathbf u '=\mathbf u /(L/T)$, $M'=M/\bar{M}$ and $\mu '=\mu /\bar{\mu }$ as the non-dimensionalized cell velocity, mobility and chemical potential, respectively. We write Eq. (1) for $\phi _T$ as

$$\begin{aligned} \displaystyle {\frac{1}{T\lambda _m^\mathrm{SC}\bar{n}}}\left( \displaystyle {\frac{\partial \phi _T}{\partial t'}}+\nabla '\cdot \left( \mathbf u '\phi _T\right) \right) =\displaystyle {\frac{\bar{M}\bar{\mu }}{L^2\lambda _m^\mathrm{SC}\bar{n}}}\nabla '\cdot \left( M'\phi _T\nabla '\mu '\right) +n'\phi _\mathrm{SC}+\lambda _m'^\mathrm{CP}n'\phi _\mathrm{CP}-\lambda '_L\phi _D. \end{aligned}$$

(21)

We choose timescale $T=\left( \lambda _m^\mathrm{SC}\bar{n}\right) ^{-1}$ and $\displaystyle {\frac{\bar{M}\bar{\mu }}{L^2\lambda _m^\mathrm{SC}\bar{n}}}=1$. $\lambda _m'^\mathrm{CP}=\lambda _m^\mathrm{CP}/\lambda _m^\mathrm{SC}$, $\lambda _L=\lambda _L/\lambda _m^\mathrm{SC}$. Analogously, the dimensionless equations for other cell species are

$$\begin{aligned} \displaystyle {\frac{\partial \phi _i}{\partial t'}}+\nabla '\cdot \left( \mathbf u '\phi _i\right) =\nabla '\cdot \left( M'\phi _i\nabla '\mu '\right) +\hbox {Src}'_i, \end{aligned}$$

(22)

where $i=\hbox {SC},\hbox {CP},\hbox {TD}$ or D, and the dimensionless source terms are

$$\begin{aligned} \begin{aligned} \hbox {Src}'_\mathrm{SC}&=n'\phi _\mathrm{SC}\cdot (2p_0-1)-\lambda _n'^\mathrm{SC}\mathscr {H}({\tilde{n}}_\mathrm{SC}-n)\phi _\mathrm{SC}\\ \hbox {Src}'_\mathrm{CP}&=n'\phi _\mathrm{SC}\cdot 2(1-p_0)+\lambda _m'^\mathrm{CP}n'\phi _\mathrm{CP}\cdot (2p_1-1)-\lambda _n'^\mathrm{CP}\mathscr {H}({\tilde{n}}_\mathrm{CP}-n)\phi _\mathrm{CP}\\ \hbox {Src}'_\mathrm{TD}&=\lambda _m'^\mathrm{CP}n'\phi _\mathrm{CP}\cdot 2(1-p_1)-\lambda _n'^\mathrm{TD}\mathscr {H}({\tilde{n}}_\mathrm{TD}-n)\phi _\mathrm{TD}-\lambda _a'^\mathrm{TD}\phi _\mathrm{TD}\\ \hbox {Src}'_D&=\lambda _n'^\mathrm{SC}\mathscr {H}({\tilde{n}}_\mathrm{SC}-n)\phi _\mathrm{SC}+\lambda _n'^\mathrm{CP}\mathscr {H}({\tilde{n}}_\mathrm{CP}-n)\phi _\mathrm{CP}+\lambda _n'^\mathrm{TD}\mathscr {H}({\tilde{n}}_\mathrm{TD}-n)\phi _\mathrm{TD}\\&\quad +\lambda _a'^\mathrm{TD}\phi _\mathrm{TD}-\lambda _L'\phi _D, \end{aligned} \end{aligned}$$

(23)

where $\lambda _n'^\mathrm{SC}=\lambda _n^\mathrm{SC}/\lambda _m^\mathrm{SC}$, $\lambda _n'^\mathrm{CP}=\lambda _n^\mathrm{CP}/\lambda _m^\mathrm{SC}$, $\lambda _n'^\mathrm{TD}=\lambda _n^\mathrm{TD}/\lambda _m^\mathrm{SC}$ and $\lambda _L'=\lambda _L/\lambda _m^\mathrm{SC}$.

The dimensionless velocity $\mathbf u '$ satisfies $\displaystyle {\frac{L}{T}}{} \mathbf u '=-\displaystyle {\frac{\bar{p}}{L}}\nabla 'p'+\displaystyle {\frac{\bar{\mu }}{L}}\displaystyle {\frac{\lambda }{\varepsilon }}\mu '\nabla '\phi _T$, where $p'=p/\bar{p}$ is the dimensionless pressure. We choose $\bar{p}=\bar{\mu }=L^2/T$, then

$$\begin{aligned} \mathbf u '=-\nabla 'p'+\displaystyle {\frac{\lambda }{\varepsilon }}\mu '\nabla '\phi _T. \end{aligned}$$

(24)

The dimensionless equation for $T_1$ is

$$\begin{aligned} 0=\varDelta ' C_{T_1})-\left( u_{T_1}'^\mathrm{SC}\phi _\mathrm{SC}+d'_{T_1}\right) C_{T_1}+p'_{T_1}\phi _\mathrm{TD}, \end{aligned}$$

(25)

where $u_{T_1}'^\mathrm{SC}=u_{T_1}^\mathrm{SC}/D_{T_1}$, $d'_{T_1}=d_{T_1}/D_{T_1}$ and $p'_{T_1}=p_{T_1}/D_{T_1}$. The equation of $T_2$ can be non-dimensionalized similarly.

We now non-dimensionalize Eq. (12):

$$\begin{aligned} \begin{aligned} \displaystyle {\frac{\partial C_W}{\partial t'}}+\nabla '\cdot (\mathbf u '_w C_W)&=\nabla '(D_W'\nabla ' C_W)+{\bar{\gamma }}' F'(C_W,C_{WI}),\\ \displaystyle {\frac{\partial C_{WI}}{\partial t'}}+\nabla '\cdot (\mathbf u '_w C_{WI})&=\nabla '(D_{WI}'\nabla C_{WI})+{\bar{\gamma }}' G'(C_W,C_{WI}), \end{aligned} \end{aligned}$$

(26)

where $D_W'=\displaystyle {\frac{T}{L^2}}D_W$ and $D_{WI}'=\displaystyle {\frac{T}{L^2}}D_{WI}$. We take ${\bar{\gamma }}'=\displaystyle {\frac{T}{L^2p_W\bar{n}}}\bar{\gamma }$ and reaction terms

$$\begin{aligned} \begin{aligned} F'(C_W,C_{WI})&=\displaystyle {\frac{C_W^2}{C_{WI}}}n'\phi _\mathrm{SC}-d_W'C_W+u_0'n'(\phi _T-\phi _D),\\ G'(C_W,C_{WI})&=p_{WI}'C_W^2n'\phi _\mathrm{SC}-d_{WI}'C_{WI}, \end{aligned} \end{aligned}$$

(27)

where $d_W'=\displaystyle {\frac{d_W}{p_W\bar{n}}}$, $u_0'=\displaystyle {\frac{u_0}{p_W\bar{n}}}$, $p_{WI}'=\displaystyle {\frac{p_{WI}}{p_W\bar{n}}}$ and $d_{WI}'=\displaystyle {\frac{d_{WI}}{p_W\bar{n}}}$.

The non-dimensionalized equations can be obtained by dropping the prime notation in Eqs. (20)–(27). Model parameters for Figs. 2 and 8 are listed in Tables 1 and 2 respectively.