A Phase-field Model for Apoptotic Cell Death

TL;DR

A phase-field framework for simulating intrinsic or extrinsic apoptosis induced by an activation field is proposed, including deriving the configurational mechanics underlying such phenomena.

Abstract

The process of programmed cell death, namely apoptosis, is a natural mechanism that regulates healthy tissue, multicellular structures, and homeostasis. An improved understanding of apoptosis can significantly enhance our knowledge of biological processes and systems. For instance, pathogens can manipulate the apoptotic process to either evade immune detection or to facilitate their spread. Furthermore, of particular clinical interest is the ability of cancer cells to evade apoptosis, hence allowing them to survive and proliferate uncontrollably. Thus, in this work, we propose a phase-field framework for simulating intrinsic or extrinsic apoptosis induced by an activation field, including deriving the configurational mechanics underlying such phenomena. Along with exploring varying conditions needed to initiate or reduce apoptosis, this can serve as a starting point for computational therapeutic testing. To showcase model capabilities, we present simulations exhibiting different types of cellular dynamics produced when varying the mechanisms underlying apoptosis. The model is subsequently applied to probe different morphological transitions, such as cell shrinkage, membrane blebbing, cavity formation and fragmentation. Lastly, we compare the characteristics observed in our simulations to electron microscopy images, providing additional support for the model.

Figures (14)

• Figure 1: Schematic representation of apoptosis stages. The diagram illustrates the morphological progression from a healthy cell to apoptotic bodies. The stages include (left) normal morphology, (middle) early apoptosis characterised by cell shrinkage and membrane blebbing, and (right) advanced apoptosis showing fragmentation and formation of apoptotic bodies.
• Figure 2: Progression of the $\varphi$ and $\sigma$ phase fields at the following nondimensionalised time points at $\Bar{t}=0.00 \ \text{(A)}, \ 3.98 \ \text{(B)}, \ 7.99 \ \text{(C)}, \ 11.76 \ \text{(D)}$. Using parameter values of: $\ell_{\varphi}=1.333\times10^{-4}$, $\ell_{2}=1.5$, $\ell^{\varphi}_{r}=960$, $\ell^{\sigma}_{r}=0.28$ and $k_{2}=10.0$. The remaining parameters in the simulation are listed in Table \ref{['tab:Parameter_List']}.
• Figure 3: Evolution of the phase field $\varphi$ at the nondimensionalised time points $\bar{t}=0.00$, (A) 4.14 (B), 8.35 (C), 11.98 (D) with $\ell_{\varphi}=1.333\times10^{-4}$, $\ell_{2}=1.5$, $\ell^{\varphi}_{r}=960$, $\ell^{\sigma}_{r}=0.28$, and $k_{2}=8.0$. The remaining parameters required to recreate the simulation are as listed in Table \ref{['tab:Parameter_List']}. Distinct finger formation can be observed.
• Figure 4: Evolution of the phase field $\varphi$ at the time points at $\Bar{t}=0$ (A), 4.08 (B), 7.51 (C), 11.95 (D) of $\varphi$ with interface width increased to $\ell_{\varphi}=5.333\times10^{-4}$ in comparison to the simulation presented in Figure \ref{['fig:finger_formation']}. The following parameters $\ell_{2}=1.5$, $\ell^{\varphi}_{r}=960$, $\ell^{\sigma}_{r}=0.28$ and $k_{2}=8.0$. The remaining parameters required to recreate the simulation are as listed in Table \ref{['tab:Parameter_List']}. Finger formation in $\varphi(\boldsymbol{x},t)$ can be observed as forming during the degradation of the phase field. A fragment can be seen to form from one finger.
• Figure 5: Evolution of the phase field $\varphi$ at nondimensionalised time $\Bar{t}=0.00$ (A), 2.57 (B), 4.42 (C), 11.42 (D) of $\varphi$ with an interface width $\ell_{\varphi}=2.133\times10^{-3}$, $\ell_{2}=1.5$, $\ell^{\varphi}_{r}=960$, $\ell^{\sigma}_{r}=0.28$, and $k_{2}=8.0$. The remaining parameters required to recreate the simulation are as listed in Table \ref{['tab:Parameter_List']}. No finger formation observed around the interface and fragmentation of the phase field $\varphi(\boldsymbol{x},t)$ is present during this simulation of apoptosis.