Dimensionless Evaluation of Cell Deformability with High Resolution Positioning in a Microchannel

Abstract

This chapter covers dimensionless evaluation for the stiffness-based deformability of a cell using a high-resolution vision system and a microchannel. In conventional approaches, the transit time of a cell through a microchannel is often utilized for the evaluation of cell deformability. However, such time includes both the information of cell stiffness and viscosity. In this work, we eliminate the effect from cell viscosity, and focus on the cell stiffness only. We find that the velocity of a cell varies when enters a channel, and eventually reaches to equilibrium where the velocity becomes constant. The constant velocity is defined as the equilibrium velocity of the cell, and it is utilized to define the observability of stiffness-based deformability. The necessary and sufficient numbers of sensing points for evaluating stiffness-based deformability are discussed. Through the dimensional analysis on the microchannel system, three dimensionless parameters determining stiffness-based deformability are derived, and a new index is introduced based on these parameters. The experimental study is conducted on the red blood cells from a healthy subject and a diabetic patient. With the proposed index, we showed that the experimental data can be nicely arranged.

Part of the materials in this chapter is from C. Tsai, S. Sakuma, F. Arai and M. Kaneko, IEEE Transactions on Biomedical Engineering, vol.61, no.4, pp1187-1195, 2014. The permission of reuse is granted by IEEE.

Appendix A Physical Quantities in Microchannel system

The model in Thin-Film Lubrication Theory [32] is adopted, and it is claimed that a very thin layer of fluid always exists between two objects. In other words, there is no direct contact between a cell and channel wall, and the interaction always through a thin layer of fluid between them. While a cell through a channel, the resistance, ${{F}_{R}}$, would be the shear force given by

$${{F}_{R}}=\mu \frac{{{u}_{eq}}}{{{d}_{g}}}{{A}_{s}}$$

(2.24)

where μ, ${{u}_{eq}}$, d_g and ${{A}_{s}}$ are fluid viscosity, equilibrium velocity, gap size, and the area where the force is exerted. By assuming the shape of deformed cell inside the channel is a cylinder, we have

$${{A}_{s}}=\pi w\lambda $$

(2.25)

where λ is the cell length inside the channel (a.k.a. in-channel length). ${{d}_{g}}$ can be regarded as a function of the compression force, ${{F}_{c}}$, acting on the cell

$${{F}_{c}}=k\left( {{D}_{c}}-w \right)$$

(2.26)

where k, ${{D}_{c}}$ and w are cell stiffness, undeformed diameter and channel width, respectively. Thus, we have

$${{F}_{R}}=\mathcal{R}\left( k,{{D}_{c}},\lambda ,{{u}_{eq}},w,\mu \right)$$

(2.27)

On the other hand, the force pushing a cell forward (the pushing force, ${{F}_{P}}$) is

$${{F}_{P}}=\Delta P{{A}_{c}}$$

(2.28)

where ΔP and ${{A}_{C}}$ are the pressure difference between two sides of the channel and cross-sectional area, respectively. ${{A}_{C}}$ is

$${{A}_{C}}=\pi {{\left( \frac{w}{2} \right)}^{2}}$$

(2.29)

Because it is difficult to directly measure the ΔP in a microchannel experimentally, fluid velocity is employed for the information of flow. We know that ΔP is a function of fluid velocity, ${{u}_{f}}$ according to Hagen-Poiseuille equation [34], thus we have

$$\Delta P=H\left( {{u}_{f}} \right)$$

(2.30)

From Eqs. (2.28)–(2.30) we have

$${{F}_{P}}=P\left( {{u}_{f}},w \right)$$

(2.31)

While a cell reaches equilibrium, the force pushing cell forward, ${{\text{F}}_{P}}$ and backward, ${{\text{F}}_{\text{R}}}$, are balanced, and can be represented by

$$P\left( {{u}_{f}},w \right)=R\left( k,{{D}_{C}},\lambda ,{{u}_{eq}},w,\mu \right)$$

(2.32)

In summary, we have

$$\mathcal{F}\left( k,{{D}_{C}},\lambda ,{{u}_{eq}},w,\mu \right)=0$$

(2.33)

which shows that 7 physical quantities, k, ${{D}_{C}}$, λ, u_eq, u_f, w and μ, determine how a cell moving inside a microchannel.