A Bayesian Framework to Estimate Fluid and Material Parameters in Micro-swimmer Models

Abstract

To advance our understanding of the movement of elastic microstructures in a viscous fluid, techniques that utilize available data to estimate model parameters are necessary. Here, we describe a Bayesian uncertainty quantification framework that is highly parallelizable, making parameter estimation tractable for complex fluid–structure interaction models. Using noisy in silico data for swimmers, we demonstrate the methodology’s robustness in estimating fluid and elastic swimmer parameters, along with their uncertainties. We identify correlations between model parameters and gain insight into emergent swimming trajectories of a single swimmer or a pair of swimmers. Our proposed framework can handle data with a spatiotemporal resolution representative of experiments, showing that this framework can be used to aid in the development of artificial micro-swimmers for biomedical applications, as well as gain a fundamental understanding of the range of parameters that allow for certain motility patterns.

Appendices

Appendix

Details on Micro-swimmer Simulations

As shown in (4) and (5), the forces that the swimmer exerts on the surrounding fluid are based on a variational derivative of the energy, which has several components. The bending energy of the flagellum is

$$\begin{aligned} E^{j}_{F,\mathrm{bend}}=K^j_{C}\int _{\Gamma _{F}^{j}}\left( \zeta ^j(s,t)-{\hat{\zeta }}^j(s,t)\right) ^2\mathrm{d}s, \end{aligned}$$

(13)

where $\Gamma _F^j$ is the centerline curve corresponding to the jth flagellum and $K_{C}$ is a stiffness coefficient enforcing the curvature or bending constraint. The preferred curvature ${\hat{\zeta }}^j$, based on (6), and actual curvature $\zeta ^j(s,t)$ of the flagellum are given as

$$\begin{aligned} {\hat{\zeta }}^j=\frac{\partial ^2 {\hat{y}}^j}{\partial s^2},\zeta ^j(s,t)= \frac{\frac{\partial ^2 y^j}{\partial s^2}\frac{\partial x^j}{\partial s} - \frac{\partial ^2 x^j}{\partial s^2}\frac{\partial y^j}{\partial s}}{\left( \left( \frac{\partial x^j}{\partial s}\right) ^2 + \left( \frac{\partial y^j}{\partial s}\right) ^2\right) ^{3/2}}, \end{aligned}$$

(14)

where $\varvec{X}_F^j=[x^j,y^j]$, the portion of $\varvec{X}$ corresponding to the tail.

In addition to the bending component, we will account for an additional energy component that will tend to maintain the inextensibility of the flagellum. This results in

$$\begin{aligned} E_{F,\mathrm{tens}}^j=\int _{\Gamma _F^j}K_{T}^j\left( \left| \left| \frac{\partial ^2\varvec{X}_F^j}{\partial s^2}\right| \right| -1\right) ^2\mathrm{d}s, \end{aligned}$$

(15)

which, in a discretized form, corresponds to Hookean springs between points on the flagellum with stiffness coefficient $K_{T}$.

Similar to the flagellum, we assume a preferred shape or curvature of the head. In this simple model, we will assume a head shape with radius $H_r$ and preferred curvature ${\hat{\kappa }}=1/H_r$. The corresponding energy is

$$\begin{aligned} E_{H,\mathrm{bend}}^j=\int _{\Gamma _H^j}K_{H,C}^j\left( \kappa ^j(s,t)-{\hat{\kappa }}(s,t)\right) ^2\mathrm{d}s, \end{aligned}$$

where $\Gamma _H^j$ corresponds to the circular head. Here, the actual curvature $\kappa ^j(s,t)$ is calculated using the same equation as $\zeta$ in (14), but now $\varvec{X}_H^j=[x^j,y^j]$, the portion of $\varvec{X}$ corresponding to the head. In addition, we also have an energy to maintain inextensibility in the head, the same as (15) using $\Gamma _H^j$, $\varvec{X}_H^j$, and $H_{C,\mathrm{tens}}^j$, where we envision Hookean springs between points on the membrane of the head as well as springs connecting points on the circular head that are $\pi$ apart (we choose ${\mathcal {N}}_H$, the number of points on the head, to be even to ensure points and springs exactly $\pi$ apart).

The swimmer is initialized (left to right) to have the center of the circular head be placed with a y-coordinate the same as the rightmost point on the flagellum and an x-coordinate that is shifted to the right of the rightmost point by $H_r$ and an additional small distance apart, dN. To ensure that the passive head remains attached to the actively bending flagellum, and to represent the stiff neck region of a sperm, we connect the head and flagellum with five springs. These springs connect the rightmost point (the ${\mathcal {N}}_F$th point) of the flagellum to the points on the circle with $\theta =(\pi -\mathrm{d}\theta ),\pi ,(\pi +\mathrm{d}\theta )$ where $\mathrm{d}\theta$ is the angular spacing between the ${\mathcal {N}}_H$ points on the head. Additionally, there are two springs connecting the second rightmost point on the flagellum (${\mathcal {N}}_F-1$) to the points on the circle with $\theta =\pi \pm \mathrm{d}\theta$. These springs will have a stiffness coefficient $K_{N,\mathrm{tens}}$. There is also an energy based on the desired angle between the flagellum and the head. Let $\mathbf {z}_1$ be the vector connecting the ${\mathcal {N}}_F$th point on the flagellum and the point on the head with $\theta =\pi$ and let $\mathbf {z}_2$ be the vector connecting the points on the head with $\theta =\pi \pm \mathrm{d}\theta$. In general, we wish for these vectors to be approximately orthogonal, and we can derive an energy and hence forces that penalize this deviation, tending to maintain $\varvec{z}_1\cdot \varvec{z}_2=0$ with stiffness coefficient $K_{N,\mathrm{ang}}$ (Fauci and McDonald 1995).

Table 6 Parameters for swimmer model

Given a configuration for each of the ${\mathcal {M}}_S$ swimmers at the initial time point, we determine the forces on the ${\mathcal {M}}_S{\mathcal {N}}_T$ discretized points using (4), where each of the components are calculated using (13)–(15). Second order finite difference approximations are utilized in the calculation of all derivatives in the energy components and a trapezoidal rule is used to calculate the integrals. The forces are then used to calculate the resulting velocity at points along the discretized swimmer, (3b). The location of the swimmer is updated using the no-slip condition, numerically implemented with a forward Euler method. The next time step is reached, where this calculation is repeated.

In these simulations, when there is more than one swimmer, we assume that the beat form parameters such as the amplitude and beat frequency are the same for each swimmer. In addition, we assume that all stiffness parameters are the same. All parameters are given in Table 6, previously benchmarked on experimental data and asymptotic swimming speeds (Ho et al. 2016, 2019; Leiderman and Olson 2016; Olson et al. 2011).