Characterization of red blood cell deformability induced by acoustic radiation force

Abstract

A cyclic coupling computational model is developed to investigate the large deformation of swollen red blood cells (RBCs) induced by the acoustic radiation force arising from an ultrasonic standing wave field. The RBC consists of an internal fluid enclosed by a thin elastic membrane. Based on the acoustic radiation stress tensor theory, the acoustic radiation force exerted on the cell membrane is calculated. A continuum mechanical theory is adopted to model the mechanical response of the membrane, which is capable of accounting for the in-plane and bending deformation of the cell membrane. The cyclic coupling computation of the acoustic fields and mechanical deformation is realized in a finite element model. With the developed model, the acoustic deformation of a single cell is calculated and results are compared with the semi-analytical solutions for validation purposes. Then, the multiple cell deformation is considered, showing that the multiple cell deformation is influenced by the secondary acoustic radiation force arising due to cell–cell interaction. This work provides an accurate numerical approach to predict the acoustic deformability of cells, which might help explore the application of the ultrasonic technique in disease diagnosis and in promoting stem cell differentiation.

Appendix: comparison of acoustic radiation stress distribution in 2D and 3D

Denoting the density contrast between the cell and the medium as $\tilde{\rho } = {{\rho_{i} } \mathord{\left/ {\vphantom {{\rho_{i} } {\rho_{o} }}} \right. \kern-\nulldelimiterspace} {\rho_{o} }}$, the jump of normal acoustic radiation stress acting on the cell membrane in Eq. (26) can be written as

$$ f_{n}^{a,2D} = \frac{{p_{a}^{2} }}{{\rho_{o} c_{o}^{2} }}\frac{{\left( {\tilde{\rho } - 1} \right)}}{{\left( {1 + \tilde{\rho }} \right)^{2} }}\left[ {\tilde{\rho } - \left( {\tilde{\rho } - 1} \right)sin^{2} \left( \theta \right)} \right]. $$

(34)

Here, the superscript “2D” is added to highlight that this formula is derived for 2D cells. The corresponding expression of 3D cells has been derived by Silva et al. (2019) and is given by

$$ f_{n}^{a,3D} = \frac{{9p_{a}^{2} }}{{4\rho_{o} c_{o}^{2} }}\frac{{\left( {\tilde{\rho } - 1} \right)}}{{\left( {1 + 2\tilde{\rho }} \right)^{2} }}\left[ {\tilde{\rho } - \left( {\tilde{\rho } - 1} \right)sin^{2} \left( \theta \right)} \right]. $$

(35)

Here, the angle $\theta$ for 2D and 3D cases are defined so that the direction $\theta = {{\uppi } \mathord{\left/ {\vphantom {{\uppi } 2}} \right. \kern-\nulldelimiterspace} 2}$ is consistent with the acoustic wave propagation direction, as shown in Fig. 

Fig. 9

Schematic illustration of the angle $\theta$ in 2D and 3D cases

9.

For the low-density contrast between the cell and the surrounding medium, we Taylor expand the above expressions of $f_{n}^{a,2D}$ and $f_{n}^{a,3D}$ and find

$$ f_{n}^{a,2D} = f_{n}^{a,3D} \approx \frac{{p_{a}^{2} }}{{4\rho_{o} c_{o}^{2} }}\left( {\tilde{\rho } - 1} \right)\left[ {\tilde{\rho } - \left( {\tilde{\rho } - 1} \right)sin^{2} \left( \theta \right)} \right]\;{\text{for}}\;\tilde{\rho } \to 1. $$

(36)

This consistency shows that when the acoustic contrast between the cell and the surrounding medium is low, the acoustic radiation stress distribution in 2D and 3D can be exactly the same.