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Abstract

Technological advances in solid organ tissue engineering that rely on the assembly of small tissue-building parts require a novel transport method suited for soft, deformable, living objects of submillimeter- to centimeter-length scale. We describe a technology that utilizes membrane flow through a gripper to generate optimized pressure differentials across the top and bottom surfaces of microtissue so that the part may be gripped and lifted. The flow and geometry parameters are developed for automation by analyzing the fluid mechanics framework by which a gripper can lift tissue parts off solid and porous surfaces. For the axisymmetric part and gripper geometries, we examine the lift force on the part as a function of various parameters related to the gripper design, its operation, and the tissue parts and environments with which it operates. We believe our bio-gripping model can be used in various applications in high-throughput tissue engineering.

Keywords

tissue engineering spheroids assembly manipulation hydrodynamics

Introduction

Recent advances in high-throughput three-dimensional (3D) cell culture have greatly facilitated the production of cell spheroids for use in cancer research, drug discovery, and tissue engineering.^1–4 In addition to possessing cell density comparable to that of native solid organs, these self-assembled tissues are characterized by high viability, enhanced differentiation, and metabolic activity, making them well suited as building blocks for scaffold-free tissue constructs.⁵ The next step toward engineering these constructs for the ultimate goal of solid organ orthotopic transplantation lies in developing a robust method for manipulating and assembling the individual tissue-building parts.

A potentially transformative approach for the engineering of solid organs is the generation of more complex microtissue geometries beyond the spheroid. Depending on the design of the mold, cells will coalesce to form geometries such as toroids, rods, or more complex lattice-like structures.⁶ The resulting tissue “parts” are typically thin and planar and contain one or more lumens. Lumens are useful due to the diffusive transport, which limits the thickness of tissues to prevent hypoxia.⁷ Second, following the completion of self-assembly, the mold is transported to a culture media bath environment and inverted to release the part that then lies submerged under culture medium. The part is then lifted by a fluid-driven gripper, which acts as a “claw machine” to pick up the tissue part, transport it to the build head, rotate the part about the z coordinate for proper alignment, and finally place it atop a perfused tissue stack composed of previously placed parts ( Fig. 1a ).^8,9 Lastly, the aligned lumens of the placed parts form cylindrical conduits through which culture medium is drawn, thus perfusing the large tissue construct and ensuring the maintenance and viability of the tissue as the individual parts fuse into one contiguous tissue.

Figure 1.

Gripper and part schematic overview. (a) Diagram of a microtissue gripping and stacking environment. (b) Cutaway representation of the gripper acting on a toroid tissue part. Membrane is not shown. (c) Axisymmetric simulation domain.

The micromanipulation of single cells in suspension has been demonstrated extensively in many techniques, such as electrostatic deflection,¹⁰ optically actuated dielectrophoresis,¹¹ and a multitude of hydrodynamically driven microfluidic focusing schemes.¹² Strategies for the transport and manipulation of spheroids are mostly limited to pipetting, which can generate large shear stresses and is not suited for individual spheroid handling.¹³ The transport of thin sheet-like tissues composed of 10⁵ to 10⁷ compacted self-aggregated cells, however, remains an unexplored problem. Their size, deformability, and need for precise handling require a novel transport technology that could be applied to not only millimeter-scale tissues, but also any suspended soft material. In this article, we develop a basic framework for a fluid-driven claw machine for soft microtissue translocation, investigating the hydrodynamics of part gripping and obtaining the optimal operational parameters necessary for automatic tissue handling. From a fluid mechanics perspective, we describe the flow and stress fields imposed by the gripper, as well as determine the effects of various parameters related to the gripper design, operation, and part characteristics.

Theory and Methods

Problem Description

We considered a tissue part within a nutrient-rich fluid culture medium and investigated the effect of an imposed suction flow. Typical parts may range in diameter from 10⁻⁴ to 10⁻² m and are lifted by a gripper head whose width is comparable to but not exceedingly greater than that of the part. The gripper is essentially a hollow tube connected to a pump supplying negative pressure ( Fig. 1b ). The end of the gripper is sealed by a thin, low-porosity polymer membrane that can hold the part when sufficient suction flow is applied. Because the part is very thin and lies on a flat surface, an impinged radial flow is produced when the gripper is moved into place above the part and activated.

We idealized our part-gripper system as an axisymmetric two-dimensional flow as depicted in Figure 1c . We chose a cylindrical coordinate system (r, z) for simplicity, though generalization to any system is straightforward. All gripped parts were of the same density of 1050 kg/m³. Individual parts of outer radius w, inner radius $w ϕ$ , and height h were simulated to lie atop a flat, uniform surface at z = 0 while submerged in a culture medium bath. These parts were lifted by a suction flow rate Q through the gripper membrane of radius $χ^{w}$ located at $z = ζ h$ above the top surface of the part. The fluid was drawn through the membrane at a steady flow rate per unit area $υ U = Q / (π χ^{2} w^{2})$ , where U = 10⁻⁴ m/s is the characteristic velocity and $υ$ is a nondimensional scaling parameter. If the membrane was placed closer to the tissue part, it is expected to impose a higher net lift force for a set fluid flow rate. It was therefore necessary to construct a fluid mechanics model to study how part geometry, gripper geometry, and flow conditions affect the lift of the tissue part.

Governing Equations and Nondimensionalization

At steady state, we constructed the continuity and momentum conservation equations for Newtonian incompressible Stokes flow to solve for the velocity $u = u e_{r} + v e_{z}$ and pressure p:

\begin{matrix} \frac{1}{r} \frac{\partial (r u)}{\partial r} + \frac{\partial v}{\partial z} = 0, \end{matrix}

\begin{matrix} - \frac{\partial p}{\partial r} + μ [\frac{1}{r} \frac{\partial}{\partial r} (r \frac{\partial u}{\partial r}) - \frac{u}{r^{2}} + \frac{\partial^{2} u}{\partial z^{2}}] = 0, \end{matrix}

\begin{matrix} - \frac{\partial p}{\partial z} + μ [\frac{1}{r} \frac{\partial}{\partial r} (r \frac{\partial v}{\partial r}) + \frac{\partial^{2} v}{\partial z^{2}}] = 0, \end{matrix}

where $μ$ is the dynamic viscosity. When simulating, we employed a viscosity of μ = 1 cP and neglected body forces and hydrostatic pressure, thus solving only for the dynamic pressure. Equation 1 assumes viscous-dominated flows (i.e., Reynolds number Re << 1) and thus omits inertial effects.

We nondimensionalized the continuity and momentum equations to understand relevant parameters. The length scales in the r (radial coordinate) and z (axial coordinate) directions were normalized as $r = \hat{r} w$ and $z = \hat{z} w$ , respectively. The r and z components of the velocity field were normalized as $u = \hat{u} U$ and $v = \hat{v} U$ , respectively. Equation 1 can then be rewritten as

\begin{matrix} α \frac{1}{\hat{r}} \frac{\partial (\hat{r} \hat{u})}{\partial \hat{r}} + \frac{\partial \hat{v}}{\partial \hat{z}} = 0, \end{matrix}

\begin{matrix} - α \frac{\partial \hat{p}}{\partial \hat{r}} + α^{2} [\frac{1}{\hat{r}} \frac{\partial}{\partial \hat{r}} (\hat{r} \frac{\partial \hat{u}}{\partial \hat{r}}) - \frac{\hat{u}}{{\hat{r}}^{2}}] + \frac{\partial^{2} \hat{u}}{\partial {\hat{z}}^{2}} = 0, \end{matrix}

\begin{matrix} - \frac{\partial \hat{p}}{\partial \hat{z}} + α^{2} \frac{1}{\hat{r}} \frac{\partial}{\partial \hat{r}} (\hat{r} \frac{\partial \hat{v}}{\partial \hat{r}}) + \frac{\partial^{2} \hat{v}}{\partial {\hat{z}}^{2}} = 0, \end{matrix}

where a α ≡ h/w is the height-to-width aspect ratio of the tissue part and the dynamic pressure is made dimensionless by $p = \hat{p} μ U / h$ . We constrained the outer radius of the cylindrical tissue parts to w = 1 mm, while the height was held at h = 200 μm. This latter length constraint is due to the diffusion-dependent transport that occurs during part formation,⁷ thus limiting the thickness of tissue parts that can be produced.

Simulation Domain and Boundary Conditions

The problem at hand was simplified and represented by a cylindrical, axisymmetric domain Ω that extends a distance χ in the $\hat{r}$ direction ( Fig. 1c ). The $\hat{z}$ dimension of the domain is scheme dependent; for the study of part gripping off a solid, impermeable substrate, the subdomain Ω_b was omitted and Ω was composed of the gripper membrane, the top surface of the solid substrate, and the fluid between the two. The membrane has thickness L_m / h and is at a distance ζ from the top surface of the part. In contrast, when simulating part gripping off an axially permeable substrate, Ω_b was included and extends a distance L_b / h = 5 below the $\overset{⌣}{r}$ axis. The membrane and bed were of permeability K_m and K_b, respectively. The $\hat{r}$ and $\hat{z}$ dimensions of the tissue part were both equal to 1. The part lies a small distance $ε$ above $\hat{z} = 0$ and may or may not contain a central cylindrical void of radius $ϕ$ . The surface of the part imposes a no-slip boundary condition on the surrounding flow. At the outlet boundary, $Γ_{1}$ , a laminar outflow condition was set that prescribes the entrance flow profile for a Poiseuille flow that becomes fully developed after an entrance length of $χ w / 2 h$ .

Two gripping schemes, solid (SBG) and porous (PBG) bed gripping, were examined. For SBG, the $\hat{r}$ axis represents an impermeable boundary where $\hat{u} = 0$ . The inlet boundary was represented solely by $Γ_{2}$ , where $\hat{p} = 0$ and $\hat{v} = 0$ . In contrast, PBG was performed on a small raised platform with uniformly spaced cylindrical perforations aligned along the $\hat{z}$ coordinate, giving the gripping bed a permeability K_b = 1.4E-8 m². Ω_b is thus included along with a second inlet boundary at $\hat{z} = - L_{b} / h$ where $\hat{p} = 0$ and $\hat{u} = 0$ . For both gripping schemes, we treated the part as impermeable and applied the no-slip velocity condition at its surfaces.

The surface of the part was discretized into n surface elements. For each element i, the force F _i was calculated by solving for the normal and tangential stress, such that $F_{i} = (ν_{i} \cdot σ_{i}) A_{i}$ , where n _i is the normal vector, $σ_{i} = - p_{i} I + \frac{1}{2} μ (\nabla u_{i} + \nabla u_{i}^{T})$ is the Cauchy stress tensor, and A_i is the element area. We note here that the normal stress tensor p_iI does not contain the contribution from hydrostatic pressure because we stipulated instead that the sum of all surface forces must be greater than the buoyant weight $g V Δ ρ$ , where g is the gravitational acceleration, $V = π w^{2} (1 - ϕ^{2}) h$ is the part volume, and $Δ ρ$ is the difference in density between the part and the surrounding fluid (taken to be 50 kg/m³). We defined the nondimensional specific force

\hat{f} = \frac{1}{g V Δ ρ} \sum_{i}^{n} e_{z} \cdot (n_{i} \cdot σ_{i}) A_{i};

thus, in order for the hydrodynamic force to overcome the part weight and lift the part, $\hat{f} \geq 1$ .

Numerical Methods

We conducted part gripping simulations using the finite-element analysis suite COMSOL Multiphysics—specifically the Free and Porous media flow module for domain-specific solutions to the Navier–Stokes and Stokes–Brinkman equations for free and porous regions, respectively. Steady-state solutions to the nonlinear Galerkin-form differential equations were obtained with the damped Newton method. Solution of the linear system was performed with the PARDISO direct solver. Computer-aided design (CAD) geometries were constructed in the COMSOL environment, meshed, and discretized using linear elements for both the velocity and pressure fields, resulting in O(10⁴) degrees of freedom. Transport was stabilized by streamline and crosswind diffusion.

Results and Discussion

Bottom Gap Width ε

We began first with a justification for the simulation methods—particularly the choice to include a small gap of width ε between the part and substrate, which we called the ε gap. This feature is necessary to evaluate eq 3, the closed surface integral that calculates $\hat{f}$ on the tissue part. Without the presence of the ε gap, the part would lie flush against the solid bed at $\hat{z} = 0$ and no boundary would exist between the two, consequently allowing for only the evaluation of pressure on the top surface of the part. This configuration, however, is physically implausible as the part remains wetted by the surrounding fluid as it settles and rests atop the solid bed surface. The absence of an ε gap would suggest a complete evacuation of fluid between part and bed. Furthermore, self-aggregated tissue parts have surface roughness—that is, small protrusions and indentations that would likely form small conduits through which fluid could pass between the part and bed.

While the inclusion of the ε gap can be intuitively explained, the precise value of ε is difficult to determine. It is likely that part roughness, the specific gravity of the tissue to the surrounding fluid $Δ ρ$ , and the elastic modulus are factors in this value. For the purposes of this investigation, however, we were primarily concerned with the conditions affecting lift force and thus chose a somewhat arbitrary value for ε. When solving the flow and stress fields of part gripping for a range of values for ε, we found that ε has a very small (compared with other parameters discussed later in the text) effect on the normal and tangential stresses on the part ( Suppl. Fig. S1 ). Decreasing ε by three orders of magnitude registered a mere 14% reduction in $\hat{f}$ Approaching the limits of the ε range tested, $\hat{f} (ε)$ begins to flatten and approach maximum and minimum values (observed for a wide range of gripper height ζ tested). Assuming the lower gap width was smaller than the part height (ε < 1), we chose to use a value from the lower limit, ε = 0.025, for the remaining gripping simulations.

Gripper Design Parameters

We next examined the elements of gripper design and their effect on tissue part lifting. The flow field generated by the gripper can be thought of as an axisymmetric cross-flow filtration, in which the gripper membrane acts as the filter. The cross-flow, which occurs below the membrane, flows radially in the $- \hat{r}$ direction. The large pressure drop occurring across the membrane strains the flow, causing it to curve upward and pass axially through the membrane. Of the gripper design parameters, it is clear that the gripper radius, χ, and membrane permeability will have the largest impact on part gripping as these directly affect the fluid domain aspect ratio, α, and the outlet flow, respectively.

Full encapsulation of the fluid flow within the numerical model requires accounting for flow through the track-etched polycarbonate membrane of thickness L_m ≈ 20 μm. The perforated membrane consisted of small cylindrical pores of radius R_p = 4 μm. We assumed these pores to be aligned with the $\hat{z}$ coordinate and have a porosity ψ = 0.07. Utilizing a combination of the Hagen–Poiseuille equation and Darcy’s law, we calculated the permeability to be $K_{m} = κ ψ R_{p}^{2} / 8 \approx$ 1E-13 m², where κ is a permeability scaling factor.

The transport of an incompressible fluid through porous media is given by the Stokes–Brinkman equations, represented in nondimensional cylindrical form by eq 2a and the following momentum conservation equation:

- \frac{β^{2}}{{\hat{K}}_{m}} \hat{v} + α^{2} \frac{\partial^{2} \hat{v}}{\partial {\hat{r}}^{2}} + α^{2} \frac{1}{\hat{r}} \frac{\partial \hat{v}}{\partial \hat{r}} + \frac{\partial^{2} \hat{v}}{\partial {\hat{z}}^{2}} - \frac{d \hat{p}}{d \hat{z}} = 0,

where ${\hat{K}}_{m} = k_{m} / R_{p}^{2}$ is the nondimensional membrane permeability and $β \equiv h / R_{p}$ is the aspect ratio of the part height to the membrane pore radius. Assuming no radial flow within the porous membrane and thus a $\hat{z}$ -invariant velocity field, these simplify to

- \frac{β^{2}}{{\hat{K}}_{m}} \hat{v} + α^{2} \frac{\partial^{2} \hat{v}}{\partial {\hat{r}}^{2}} + α^{2} \frac{1}{\hat{r}} \frac{\partial \hat{v}}{\partial \hat{r}} - \frac{d \hat{p}}{d \hat{z}} = 0 .

The solution to this differential equation is

\hat{v} (\hat{r}) = - \frac{{\hat{K}}_{m}}{β^{2}} \frac{d \hat{p}}{d \hat{z}} + C_{1} J_{0} (\frac{i β \hat{r}}{α \sqrt{{\hat{K}}_{m}}}) + C_{2} Y_{0} (\frac{- i β \hat{r}}{α \sqrt{{\hat{K}}_{m}}}),

where J₀ and Y₀ are Bessel functions of the first and second kind, respectively, and C₁ and C₂ are constant coefficients. Y₀ is singular at $\hat{r} = 0$ , and therefore C₂ = 0. As others have previously shown,¹⁴ imposition of the no-slip boundary condition $\hat{v} (χ) = 0$ results in

\hat{v} (\hat{r}) = - \frac{{\hat{K}}_{m}}{β^{2}} \frac{d \hat{p}}{d \hat{z}} [1 - I_{0} (\frac{β \hat{r}}{α \sqrt{{\hat{K}}_{m}}}) {(I_{0} (\frac{β χ}{α \sqrt{{\hat{K}}_{m}}}))}^{- 1}],

where I₀ is the modified Bessel function of the first kind. The bracketed expression approaches 1 for small permeability ${\hat{K}}_{m}$ , producing a constant velocity profile $\hat{v} = - \frac{{\hat{K}}_{m}}{β^{2}} \frac{d \hat{p}}{d \hat{z}}$ across the membrane.

We found the ratio of the gripper to part width, χ, to have a small effect on $\hat{f}$ under certain conditions. Examining a range of χ from 1.75 to 4.75 with the bulk outlet velocity factor υ and membrane permeability factor κ both held at 1, we observed that decreasing the gripper size led to a small increase in $\hat{f}$ ( Suppl. Fig. 2a ). This effect is negligible for modest to large gripper height (ζ > 0.1) and more pronounced as the gripper is lowered closer to the part. In addition, this effect is noticeable when $\hat{f}$ is already a factor of 100 greater than the requisite lift force ( $\hat{f}$ = 1). Larger χ increases the membrane area through which fluid may enter, thus lowering the hydraulic resistance and allowing more flow to circumvent the part. A smaller c will direct more fluid over the part, thus effecting greater pressure loss and more lift. This is especially significant when ζ approaches zero, as a small χ can then partially overcome the large resistance of the narrow gap between the part and the membrane.

A critical component that conducts fluid uniformly through its surface, the membrane must be of low permeability for sufficient constriction flow to occur at small ζ. Because χ can also influence the amount of constriction flow, its effect is greatly enhanced when the gripper membrane permeability is increased. Here we examined its direct effect on lift force by simulating gripping at various heights ζ with a membrane with permeability factor κ = 100. With this larger permeability, $\hat{f}$ was significantly reduced for small ζ by roughly a factor of 100 for χ = 1.75 and 400 for χ = 4.75 ( Suppl. Fig. 2b ). The decreased hydraulic resistance across the membrane substantially weakened its ability to draw fluid through the constriction between the membrane and the top surface of the part (we will now call this the ζ gap as its width is determined by ζ), while a larger proportion of fluid takes the path of less resistance and flows through the membrane at $\hat{r}$ > 1; this in turn lowers the lift force. This allows χ to have greater influence; for high permeability, the effects of reduced ζ gap flow are such that $χ$ may determine if $\hat{f} > 1$ for a given height ζ.

Gripper Operation Parameters

We next considered how the operating parameters—namely, flow rate and gripper height—affect gripping. Due to the low Reynolds number that is characteristic of this system, nonlinear terms are comparatively small, leading to the linear eq 2. Thus from eq 3, lift force is proportional to the bulk velocity, $υ U$ . This relation was evident when evaluating $\hat{f} (υ)$ at different values of ζ ( Fig. 2a ). Though $\hat{f}$ correlated linearly with υ, the slope of that correlation was highly sensitive to the width of the ζ gap. The remainder of the results presented will be for υ = 1.

Figure 2.

Effects of flow speed υ (a) and gripper height ζ (b) on lift force $\hat{f} \cdot ϕ = 0$ . Dotted line represents theoretical calculation of the lift force assuming Poiseuille flow through the ζ gap, while black squares represent simulation output.

The vertical distance between the gripper membrane and the tissue part, ζ, exerts the greatest influence over the lift force $\hat{f}$ out of all parameters considered. As shown in Figure 3a , flow enters radially through the $\hat{r}$ = χ inlet and squeezes through the ζ gap, generating high shear along the top surface of the part. This constriction generates pressure loss that increases with smaller ζ, creating a pressure differential across the top and bottom surfaces of the part and thus creating lift ( Fig. 3c ). For parts with a central void ( $ϕ \neq 0$ ), such as that in Figure 5a , flow passes around the part rather than simply over it. The volume enclosed by the bed, the membrane, and $\hat{r}$ = φ then becomes a low-pressure zone, creating a pressure differential between the part boundaries at $\hat{z} = ε$ and $\hat{z} = ε + 1$ , as well as those at $\hat{r} = φ$ and $\hat{r} = 1$ . This in turn generates both lift in the $\hat{z}$ direction and pressure drag in the $- \hat{r}$ direction. Consequently, for parts of nonzero ϕ, an unwanted radial compressive stress is generated along with the shear stress along the top surface.

Figure 3.

Gripping off solid versus porous substrate. φ = 0.4, ζ = 0.39. Fluid streamlines and velocity magnitude are plotted for SBG (a) and PBG (b). Color bar is in units of υ. Nondimensional pressure along the top (upward-pointing triangles) and bottom (downward-pointing triangles) surfaces of the part are plotted for SBG (filled triangles) and PBG (empty triangles) (c).

The pressure along the top surface of the part can be approximately modeled if several simplifying assumptions are made, such as a vanishing radial velocity at the membrane, $\hat{z}$ -invariant dynamic pressure, and uniform flow through the membrane, such that $\hat{v} (\hat{r}, ε + 1 + ζ) = υ$ . Solving eq 2b following those assumptions (see the Appendix) gives the top pressure

{\hat{p}}_{t} = - \frac{6 {\hat{r}}^{2} \ln \hat{r}}{α^{2} ζ^{3}}

along the part surface at $\hat{z} = ε + 1$ . For the $ϕ$ = 0 part geometry, we assumed that at the bottom surface, the dynamic pressure was equal to that of the fluid entering the part-membrane gap, such that $\hat{p}$ ( $\hat{r}$ , 0) = $\hat{p}$ (1,1). The calculation of the force $\hat{f}$ can then follow, as shown in Figure 2b . This simplified model is applicable for a finite range of ζ. When the part-membrane gap was very large, the constriction flow became z dependent; as ζ shrinks to zero, the hydrodynamic resistance grows larger and impedes flow such that more flow per unit area occurs through the membrane for $\hat{r}$ > 1. Consequently, the forced derived from the full solution begins to abate below a certain membrane height; yet above that height, the force can be reasonably approximated without the full solution.

Substrate and Part Properties

Finally, we discuss how properties related to the gripping surface and part geometry can influence normal and shear stresses on the part. The purpose of the gripper is to precisely and gently handle sub-millimeter- to sub-centimeter-scale tissue parts that can be easily deformed when subjected to high-straining flows. The maximum shear rate experienced by the parts during gripping must also be considered and, if possible, minimized. The PBG method aims to address this by allowing both radial and axial entrance flows. Previously, we have limited the discussion to gripping parts on a solid bed. While those results are also applicable to PBG, several key differences arise.

We prescribed the same numerical methods and analysis for PBG as for SBG. Here the part rests on the additional subdomain Ω_b, which represents a porous platform of height L_b whose void volume is composed of identical, equally spaced cylindrical pores, giving the platform a permeability of K_b = 1.4E-8 m² along the $\hat{z}$ component. As a result, flow may enter both through the boundary $\hat{r}$ = χ and through $\hat{z} = - L_{b} / h$ , resulting in generally lower shear rates throughout the bulk of the fluid domain, as well as more isotropic flow around parts of nonzero ϕ ( Fig. 3b ). The effect of this additional $\hat{z}$ -facing inlet is that, in contrast to SBG, all transmembrane flow for $\hat{r}$ < 1 need not pass solely through the ζ gap at $\hat{r}$ = 1; instead, this flow is composed of streamlines originating from below and passing along either side ( $\hat{r}$ = ϕ and $\hat{r}$ = 1) of the part, which in turn produces a more isotropic distribution of radial normal and shear stresses. In addition, this increased axial flow realigns the dynamic pressure gradient within the submembrane domain such that $\hat{p}$ becomes mostly $\hat{z}$ dependent ( Fig. 3c ).

Because of the more even distribution of flow, PBG substantially reduces shear stress on the part. Excessive shear forces could adversely affect tissue structure and cell function. When comparing the maximum nondimensional shear rate $\hat{E} = E_{h} / U$ (where $E = \partial u_{i} / \partial x_{j}$ is the dimensional shear rate) experienced by solid and toroid parts, a clear diminishment occurs when utilizing a porous bed. In particular, at ζ = ζ_c, the critical gripper height at which $\hat{f}$ = 1, toroid parts (e.g., ϕ = 0.8) saw roughly an order of magnitude reduction in shear stress ( Fig. 4 ). This outcome was more prominent for parts with nonzero ϕ than solid parts, however. The latter lack the central void that may allow for the distribution of flow, thus directing fluid—in a similar manner to SBG—inward through the narrow opening of the ζ gap at $\hat{r}$ = 1. The maximum shear stress experienced by cells aligning small capillaries and postcapillary venules is 95.5 and 2.8 dynes/cm², respectively.¹⁵ The latter value corresponds to a nondimensional shear rate of $\hat{E}$ = 810. HepG2 liver cells maintain normal cell morphology and function when exposed to shear stress as large as 5.4 dynes/cm².¹⁶ The operating conditions of the gripper are well below these values, and previously we have shown that the function and structure of gripped tissues were not adversely affected (data not shown).⁹

Figure 4.

Porous substrate effects lower shear on parts. Maximum shear rate on the part surface is calculated with respect to gripper height for SBG (a) and PBG (b). Blue, red, and black vertical dotted lines represent $ζ_{c}$ for φ = 0.8, 0.4, and 0, respectively.

It is important to note that the reduction of the maximum shear rate at ζ_c is not only a result of less straining flows but also due to an upward shift in ζ_c for PBG, indicating that larger lift forces are generated at a given gripper height. These shifts are presented in Figure 5 , which details the critical ζ gap width, ζ_c, with respect to void radius ϕ for both SBG and PBG schemes. From these data, it is evident that PBG elicits greater force per gripper height for toroid parts (and marginally so for ϕ = 0 disks) and thus can lift parts from a greater distance. Furthermore, both gripping methods show similar responses to ϕ; as the void radius increases, $\hat{f}$ for fixed ζ decreases and so ζ must be reduced to attain the critical force $\hat{f}$ = 1. This trend is reflected in the analytical solution provided by our simplified model (eq A10). Because the sum of normal stresses in the $\hat{z}$ coordinate ultimately determines the lift force (with axial shear stresses making a much smaller contribution), the reduction of surface area facing the $\hat{z}$ direction (i.e., with normal component n such that n.e _z ≠ 0) negatively impacts the lift force.

Figure 5.

Critical gripper height ζ_c for void radius φ and spheroids. Filled squares and empty squares represent SBG and PBG data, respectively. Solid line indicates approximated analytical solution for SBG.

We include here a brief description of results for the gripping of spherical tissue parts, or spheroids. Typically 100–200 μm in size, these are by far the most prevalent form of self-aggregated 3D cell cultures currently being studied.¹⁷ Here we consider spheres of 100 μm radius, so that h = 200 μm and w_s = 100 μm. Because of the minimized surface-to-volume ratio, spheroids require a low gripper height ζ to be lifted. The velocity and pressure fields generated during both SBG and PBG are similar in nature to those of solid disks. Because they are solid, spheroids do not allow flow to pass through them as with toroids during PBG; nonetheless, a significant increase in ζ_c was observed for the spheroid using this gripping method ( Fig. 5 ). This is due to the ε gap having a short length along the $\hat{r}$ coordinate (due to the curvature of the sphere geometry) compared with that of the $ϕ$ = 0 disk. Consequently, the hydraulic resistance of this constriction is lower, which allows for a significant amount of flow to enter up through $\hat{z}$ = 0, into the ε gap and around the surface of the spheroid, thus raising the pressure along the bottom hemisphere while also contributing greater vertical viscous drag.

Because cell spheroids are often nonspherical, we considered spheroids of nonzero oblateness and its effect on lift force $\hat{f}$ during SBG. Maintaining the volume at $\frac{4}{3} ᴨ w_{s}^{3}$ , the spheroid was “flattened” to varying degrees by the oblateness factor $γ \equiv w^{'} / w_{s}$ , where $w'$ is the equatorial radius of the spheroid. From this we set the spheroid height to be $h^{'} = w / γ^{2}$ . We then determined the critical flow speed $υ_{c}$ (for which $\hat{f}$ = 1) as a function of ζ (still scaled by a factor h) and γ ( Fig. 6 ). As shown above, υ ∝ $\hat{f}$ ; thus, the solution of $υ_{c} (γ, ζ)$ correlates inversely with $\hat{f}$ (γ, ζ). More oblate spheroids were easier to lift, in that they required lower $υ_{c}$ at greater distances ζ from the gripper membrane. A larger oblateness factor γ creates a greater amount of surface area whose normal component points along the $\hat{z}$ coordinate. The top surface of the spheroid adjacent to the gripper membrane is stretched in the $\hat{r}$ direction, thus effecting increased pressure drop in the inward radial flow occurring between the two surfaces.

Figure 6.

Critical flow speed υ_c for gripper height _ζ and spheroid oblateness _γ. The y axis is represented in logarithmic scale. The color scale denotes log (υ_c). Contour lines denote curves of constant υ_c.

Lastly, we compared the results of our model to gripping experiments on toroids of various dimensions⁸ and on large hepatocyte microtissues. In the former, polystyrene parts of height 0.254 mm and varying width w and inner radius $ϕ w$ were submerged in room temperature water and gripped. The bulk flow speed U was incrementally increased until a critical flow speed U_c was reached and the part could be successfully lifted ( Fig. 7 ). Our model was able to closely predict U_c across all axisymmetric geometries. We furthermore found good agreement in U_c when comparing the results of our gripping model to those generated by gripping experiments performed on actual self-assembled microtissues composed of HepG2 cells. These complex, nonaxisymmetric 3D geometries contained roughly 5 million cells and were gripped with both SBG and PBG methods to obtain the experimental values U_c = 0.70 and 0.28 mm/s, respectively.¹⁸ Our model predicted critical flow speeds of 0.58 and 0.19 mm/s for SBG and PBG, respectively; these discrepancies are likely due to the additional force required in practice to robustly accelerate the tissue upward. In addition, the deformability of the tissue may lead to drooping during lift, thus necessitating greater flow rate.

Figure 7.

Comparison of experimental (gray) and simulated (black) critical flow speeds for torus geometries labeled by outer radius and φ.

Conclusions

We have investigated the mechanics of gripping using an impinged membrane flow for thin, axisymmetric geometries both with and without a central void. Successful gripping requires a membrane of low permeability, so that sufficient flow is drawn over the surface of the part as the gap between the membrane and part is narrowed. For such low permeabilities, flow is nearly uniform across the surface of the membrane. When parts are gripped off a solid surface, high-straining flow is generated and large shear forces appear across the surface of the parts. For part geometries with voids, PBG is both gentler and more effective, allowing for a more uniform distribution of streamlines along the surface of the part, less radial shear, and more viscous and pressure drag along the axial direction, increasing the hydrodynamic lift force.

Supplemental Material

Supplemental_figures_for_Bio-Gripper_Hydrodynamics_by_Cui,_et_al. – Supplemental material for Hydrodynamics of the Bio-Gripper: A Fluid-Driven “Claw Machine” for Soft Microtissue Translocation

Supplemental material, Supplemental_figures_for_Bio-Gripper_Hydrodynamics_by_Cui,_et_al. for Hydrodynamics of the Bio-Gripper: A Fluid-Driven “Claw Machine” for Soft Microtissue Translocation by Francis R. Cui, Blanche C. Ip, Jeffrey R. Morgan and Anubhav Tripathi in SLAS Technology

Footnotes

Appendix

We evaluate eq 3 by assuming the following: (1) the sum of stresses on the part is primarily composed of contributions from the dynamic pressure, with shear stress being negligible; (2) the pressure is axially invariant unless comparing the top and bottom surfaces of the part, such that $\hat{p} (\hat{r}, 1 + ε) = \hat{p} (\hat{r}, 1 + ε + ζ) = {\hat{p}}_{t} (\hat{r})$ and $\hat{p} (\hat{r}, 0) = \hat{p} (\hat{r}, ε) = {\hat{p}}_{b} (\hat{r})$ , but ${\hat{p}}_{t} (\hat{r}) \neq {\hat{p}}_{b} (\hat{r})$ ; (3) the axial velocity component is zero; and (4) the Reynolds number is small, and thus nonlinear terms in eq 2b can be neglected.

To calculate ${\hat{p}}_{t} (\hat{r})$ following these assumptions, we begin with eq 2a and find that the solution must take the form¹⁹

\begin{matrix} \hat{u} = \frac{γ (\hat{z})}{\hat{r}} . \end{matrix}

where γ is a nondimensional function of $\hat{z}$ . Combining this result with eq 2b yields

\begin{matrix} \frac{d^{2} γ}{d {\hat{z}}^{2}} = α \hat{r} \frac{d {\hat{p}}_{t}}{d \hat{r}} . \end{matrix}

We note that because γ is $\hat{z}$ dependent and ${\hat{p}}_{t}$ is $\hat{r}$ dependent, both sides of this equality must be constant. Integrating the left-hand side and solving for γ while imposing the no-slip boundary conditions gives

\begin{matrix} γ = \frac{1}{2} α \hat{r} \frac{d \hat{p}}{d \hat{r}} \hat{z} (\hat{z} - ζ) . \end{matrix}

For simplicity, we have taken the boundaries of the top flow to be $\hat{z} = 0$ and $\hat{z} = ζ$ . From here we may derive the average velocity

\begin{matrix} \hat{u} = - \frac{1}{12} α ζ^{2} \frac{d {\hat{p}}_{t}}{d \hat{r}} \end{matrix}

by substituting eq A1 into eq A3 and integrating from $\hat{z} = 0$ to $\hat{z} = ζ$ .

Returning to eq A2, integrating the right-hand side, and taking ${\hat{p}}_{t} (1) = 0$ , we find the top pressure to be

\begin{matrix} {\hat{p}}_{t} (\hat{r}) = {\hat{p}}_{t} (ϕ) \frac{\ln \hat{r}}{\ln ϕ} . \end{matrix}

Upon differentiating with respect to $\hat{r}$ and substituting back into eq A4, we then approximate the average velocity to be $\hat{u} U = - \frac{π {\hat{r}}^{2} w^{2} U}{2 π \hat{r} w h ζ}$ ; thus, $\hat{u} = - \frac{\hat{r}}{2 α ζ}$ and

{\hat{p}}_{t} (ϕ) = \frac{6 {\hat{r}}^{2} \ln ϕ}{α^{2} ζ^{3}} .

After obtaining an expression for ${\hat{p}}_{t}$ , we then must do the same for ${\hat{p}}_{b}$ to calculate the force,

\begin{matrix} \hat{F} = 2 π \int_{ϕ}^{1} ({\hat{p}}_{b} - {\hat{p}}_{t}) \hat{r} d \hat{r} . \end{matrix}

For parts of disk-shaped geometries ( $ϕ =$ 0), we assume that ${\hat{p}}_{b} (\hat{r}) = {\hat{p}}_{t} (1) = 0$ . Evaluating the above integral then yields

\hat{F} = \frac{3 π}{4 α^{2} ζ^{3}} (4 ϕ^{4} \ln ϕ - ϕ^{4} + 1) .

Subsequent evaluation of the indeterminate terms shows that they approach zero as $ϕ \to 0$ . For toroid geometries ( $ϕ \neq 0$ ), we require that ${\hat{p}}_{b} (1) = {\hat{p}}_{t} (1) = 0$ and ${\hat{p}}_{b} (ϕ) = {\hat{p}}_{t} (ϕ)$ . We then use a simple first-order approximation for ${\hat{p}}_{b} (\hat{r})$ , such that

\begin{matrix} \frac{d {\hat{p}}_{b}}{d \hat{r}} = - \frac{{\hat{p}}_{t} (ϕ)}{1 - ϕ} . \end{matrix}

Integrating eq A9 and substituting back into eq A7, we are left with the following expression for force generated on a toroid:

\begin{matrix} \hat{F} = \frac{3 π \ln ϕ}{α^{2} ζ^{3}} [\frac{4 ϕ^{4} \ln ϕ - ϕ^{4} + 1}{4 \ln ϕ} + \frac{4 ϕ^{5} - 5 ϕ^{4} + 1}{5 (1 - ϕ)}] . \end{matrix}

This nondimensional force can be converted to the normalized nondimensional force $\hat{f}$ as follows:

\begin{matrix} \hat{f} = \frac{μ U \hat{F}}{π Δ ρ g h^{2} (1 - ϕ^{2})} . \end{matrix}

Acknowledgements

The authors thank J. Murphy III for the fabrication of the Bio-P3 assembly.

Supplementary material is available online with this article.

Declaration of Conflicting Interests

The authors declared the following potential conflicts of interest with respect to the research, authorship, and/or publication of this article: J. Morgan has an equity interest in Microtissues, Inc. This relationship has been reviewed and managed by Brown University in accordance with its conflict of interest policies.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was funded by NSF grant CBET-1428092.

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