Flow Physics Explains Morphological Diversity of Ciliated Organs

Feng Ling

Nawroth Mechanobiology

Helmholtz Pioneer Campus

2023 APS March Meeting - PP08.08

Motile cilia as Active porous media

Specifically, we model cilia activity as a porous media with prescribed quasi-1D density wave that transfer ciliary waves to fluid motion via drag forces
- $h/H$ - cilia-to-lumen ratio
- $H/L$ - lumen diameter / ciliary wavelength
Remaining inputs to the model are reference density of cilia $\bar\rho_c$ and the adverse pressure $\Delta P$ .

Nawroth, Ling, et al (2023)

Motile cilia as Active porous media

Cilia-to-lumen Ratio h/H

Lumen Diameter H [μm]

Nawroth, Ling, et al (2023)

We fix the total amount of input activity by constraining the total amount of active material present inside a channel by
- $(\bar\rho_cL)(h/2r_{pore})=\bar\rho_ch=\text{const.}$
This implies that carpet-like designs will need to be more densely distributed to be comparable to flame-like designs with the SAME lumen diameter

Quasi-1D metachronal wave

Prescribing the 1D wave by setting

$\bm{v}_c = \begin{cases} \bm{0} & y\in (h/2, H-h/2)\\ v_c(x,t) {\bm{e}}_x & \text{otherwise} \end{cases}$

$\zeta_c =\begin{cases} 0 & y \in (h/2, H-h/2)\\ \bar\zeta_c\left(\partial_x x_c(x,t)\right)^{-1} & \text{otherwise} \end{cases}$

with $\bar\zeta_c =\bar\rho_c\dfrac1{Lh}\dfrac{F_{drag}}{v_c}=3\pi\mu\bar\rho_c^2$ , and

$F_{drag}=(\bar\rho_cL)( h/{2r_{pore}})(6\pi\mu r_{pore}v_c)$

Since $x_c=x_c(kx-\omega t)$ is a traveling wave, only need to solve for $\bm{u}$ at $t=t_o$

$v_c(x,t)$

$x_c(x,t)$

$x_c(x,t) = x + \epsilon \cos(2\pi x/L+\omega t)$

$\langle v_c\rangle_x=\frac{1}{L}\int_0^L v_c(x,0)\,\mathrm{d} x=\pi \omega\epsilon^2/L$

$v_c(x_c,t) = \dfrac{\partial x_c}{\partial t}, \rho_c(x_c,t) = \bar\rho_c \left( \dfrac{\partial x_c}{\partial x} \right)^{-1}$

Taylor (1951) & Blake (1971)

Pak, Lauga, Proc. R. S. A (2009)

Nawroth, Ling, et al. ???

Brinkman-Stokes equation

For simplicity, we only consider one-way coupling of cilia driven velocity as drag forces on to the fluid

Resulting equations are solved numerically via mixed FEM

$p(0,y,t)=p(L,y,t)$

$\bm{u}(0,y,t)=\bm{u}(L,y,t)$

$\bm{u}(x,0,t)=\bm{u}(x,H,t)={0}$

$-\nabla^2 \bm{u} + \nabla p + \zeta_c(\bm{u}-\bm{v}_c) = \Delta p/L\cdot\bm e_x$

$\nabla \cdot \mathbf{u} = 0$

Nawroth, Ling, et al. ???

$\mu$

$\omega$

$L$

$\epsilon$

$h/H$

$H$

$\Delta P$

$\bar\rho_ch$

$U=\int_0^L\bm u\cdot\bm e_x$

$10^{-3}$ [Pa $\cdot$ s]

$15\sim30$ [Hz]

$100$ [μm]

$L/2\pi$

$0-1$

$5-200\times10^{-6}$ [m]

$0-100$ $Pa$

$1\sim1000$

$\leq \omega L /(4\pi)$

fluid viscosity

wave frequency

imposed wavelength (wave number $=1$ )

wave amplitude $\in(0,\lambda/2\pi)$

cilia-to-lumen ratio

duct lumen diameter

adverse pressure gradient

material constraint constant

mean flow speed

Model parameters

Nawroth, Ling, et al. ???

Flow Physics Explains Morphological Diversity of Ciliated Organs Feng Ling Nawroth Mechanobiology Helmholtz Pioneer Campus 2023 APS March Meeting - PP08.08

Flow Physics Explains Morphological Diversity of Ciliated Ducts

By Feng Ling

Flow Physics Explains Morphological Diversity of Ciliated Ducts

defense weeeeeeee

2 years ago
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Feng Ling

Postdoc @ Helmholtz Pioneer Campus, studying biophysics, cilia, and mucus flow. Looking for Universality from Diversity.

Flow Physics Explains Morphological Diversity of Ciliated Organs

Flow Physics Explains Morphological Diversity of Ciliated Ducts

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