fluid dynamic design principles for flow … at ynus.pdfelements of fluid dynamics the micro / nano...

LABORATORY OF BIOLOGICAL STRUCTURE MECHANICS

Fluid dynamic design principles for flow microchambers Gabriele Dubini

Department of Chemistry, Materials and Chemical Engineering ‘Giulio Natta’ Politecnico di Milano, Milan, Italy

自己紹介

皆さん、こんにちは。はじめまして、ガブリエーレ・ドゥビニと申します。どうぞよろしくお願いします。

バイオエンジニアリング学部の教師です。ご招待ありがとうございます。日本では初めて、楽しみにしています。

Outline Introduction to the Laboratory of Biological Structure Mechanics – LaBS, Politecnico di Milano, Milan, Italy Elements of fluid dynamics The micro / nano scale environment Design of a parallel-plate microfluidic chamber

Laboratory of Prosthesis

Biomechanics

Laboratory of Tissue Mechanics

Laboratory of Medical

Microdevices

Mechanobiology Lab

Laboratory of Life Support

Systems

Computer center

EXPERIMENTAL

COMPUTATIONAL

Validation of computational predictions Physical measurements of quantities

Evaluation of quantities hard to assess experimentally Design and study of experimental activity

10-6 mm 103 mm

1m 1nm 10-3 mm 100 mm

1mm 1µm

10ton 0.01mg 1kg 0.1g 10-7 N 105 N 10-3 N 101 N

10-4 mm 10-5 mm 10-1 mm 10-2 mm 102 mm 101 mm

10-5 N 10-1 N 103 N

Displacements

Forces

www.labsmech.polimi.it

Laminar and turbulent flow: the Reynolds number

water

ink

µρ

= cLwRe

x

y

z

wx

τy+dy

τy

dxdydzx

wwdxdydzDtDw x

x ∂∂

ρ=ρ= forces (inertial) convective

( ) dxdydzywdxdzdy

yw

ydxdzdy

ydxdz xx

ydyy 2

2

forces viscous∂

∂µ=

∂

∂µ

∂∂

=

∂

τ∂=τ−τ= +

carat

x

Lw

xw

∝∂

∂

22

2

caratt

x

Lw

yw

∝∂

∂

conduitsin diameter hydraulic 4length sticcharacteri ==== ht

c DPAL

wwx ∝

r

z

h

h

DmDw

πµ=

µρ

=4Re

−=

2

max 1)(Rrwrw 7

1

max 1)(

−=

Rrwrw

In steady-state conditions:

Laminar and turbulent flow: the velocity profiles

µρπ

⋅=µ

ρω⋅=

FLL cc 222

Wo

( ) dxdydzywdxdzdy

yw

ydxdzdy

ydxdz xx

ydyy 2

2

forces viscous∂

∂µ=

∂

∂µ

∂∂

=

∂

τ∂=τ−τ= +

Fwwt

wx π=ω∝∂

∂ 2

22

2

c

x

Lw

yw

∝∂

∂

dxdydzt

wx

∂∂

ρ= forcestransient

x

y

z

wx

τy+dy

τy

Pulsatile flow: the Womersley number

flowslaminar in Re056.0 ⋅≈hD

x

flowsnt in turbule 10≈hD

x

r

x

The entry length

0=⋅∇ w

( ) gwwwwρ+∇µ+−∇=

∇⋅+

∂∂

ρ 2pt

(mass conservation)

(momentum conservation)

Hypotheses: incompressible, homogeneous, Newtonian fluid

Incompressible, Newtonian fluids: the Navier-Stokes equations

A particular case: 2-D Navier-Stokes equations for steady-state flow

𝜌𝜕𝑤𝑥𝜕𝜕

+ 𝑤𝑥𝜕𝑤𝑥𝜕𝑥

+ 𝑤𝑧𝜕𝑤𝑥𝜕𝑧

= −𝜕𝜕𝜕𝑥

+ 𝜇𝜕2𝑤𝑥𝜕𝑥2

+𝜕2𝑤𝑥𝜕𝑧2

𝜌𝜕𝑤𝑧𝜕𝜕

+ 𝑤𝑥𝜕𝑤𝑧𝜕𝑥

+ 𝑤𝑧𝜕𝑤𝑧𝜕𝑧

= −𝜕𝜕𝜕𝑧

+ 𝜇𝜕2𝑤𝑧𝜕𝑥2

+𝜕2𝑤𝑧𝜕𝑧2

𝜕𝑤𝑥𝜕𝑥

+𝜕𝑤𝑧𝜕𝑧

= 0

𝜕𝜕𝜕𝑥

= 𝜇𝜕2𝑤𝑥𝜕𝑧2

and, if the pressure gradient ∂P/∂x is constant and equal to ∆P/L:

𝑤𝑥 𝑧 =∆𝜕ℎ2

8𝜇𝜇1 −

4𝑧2

ℎ2= 𝑣𝑚𝑚𝑥 1 −

4𝑧2

ℎ2 −

ℎ2≤ 𝑧 ≤ +

ℎ2

for

x y

z

L

w h

h « L h « w

𝑤𝑥 =1𝐴𝑡� 𝑤𝑥 𝑧 𝑑𝑧+ℎ2

−ℎ2

=23∆𝜕ℎ2

8𝜇𝜇

Why miniaturization?

• It reduces reagent, energy consumption, and waste handling • It enables faster, cheaper, and better processes (same as in

microelectronics) • It yields better performances (speed and output) • It allows integration of multiple processes (including

parallelization) • It automates processes • It enables new functionalities, often impossible at the

macroscopic level

The lab-on-a-chip concept

An early integrated device with two liquid samples and electrophoresis gel present

Burns et al., Science, 1998

Blue, liquid sample (ready for metering) Green, hydrophobic surfaces Purple, polyacrylamide gel

Comparison between volume densities of culture conditions in traditional, macroscale culture in 6-well plates and in microscale, microchannel culture (750 µm wide, 5 mm long, and 250 µm tall).

Paguirigan and Beebe, BioEssays, 2008

1 mm 1 mm3 = 1 µl

10 µm 103 µm3 = 1 pl

100 µm 106 µm3 = 1 nl

Capillary pressure

Pcap > Patm Pcap < Patm

Hydrophilic microchannel 100 µm (water-air): Pcap = 0,015 bar Nanochannel 100 nm (water-air): Pcap = 15 bar

Definition

Range of channel dimension

Conventional channels Dh > 3 mm

Minichannels 3 mm ≥ Dh > 200 µm

Microchannels 200 µm ≥ Dh > 10 µm

Transitional microchannels 10 µm ≥ Dh > 1 µm

Transitional nanochannels 1 µm ≥ Dh > 0,1 µm

Nanochannels Dh ≤ 0,1 µm = 100 nm

𝐻𝐻𝑑𝐻𝐻𝐻𝐻𝐻𝐻 𝑑𝐻𝐻𝑑𝑑𝜕𝑑𝐻 𝐷ℎ =4𝐴𝑡𝑝

Design considerations for microflows: driving force for fluid motion and the channel characteristics can be chosen independently

A flow driven by either a pressure gradient, an electric field, or a surface tension gradient.

A surface modified chemically in stripes. A surface modified with topography.

Stone et al. Annu. Rev. Fluid Mech., 2004

Microfluidic devices for manipulating fluids: a vast experience! What about fluids with suspended cells?

Cell size vs channel size

Ex. #1 - In vivo: Lymphocyte homing

Ex. #2 - In vitro: Inflammation – Leukocyte adhesion cascade - THP1 adhesion to VCAM-1 at 0.5 dyn/cm²

Cell responses on surface chemistry of channel walls: 1) surface hydrophobicity 2) protein adsorption 3) surface charge 4) surface roughness 5) surface softness and stiffness

Pinning fluid–fluid interfaces by chemically inhomogeneous surfaces in static (c) and flowing systems (d). Altering the wetting properties using chemically homogeneous, micro- and nanostructured surfaces: (e, f ). (Gűnther and Jensen, Lab on a Chip, 2006)

Cell responses on architecture of porous materials: 1) pore size 2) porosity 3) connectivity and tortuosity

Schematic of the different pore types found in tissue engineering scaffolds (Wang et al., Tissue Engineering Part C Methods, 2010).

Determination of tortuosity through a porous material using the arc-chord ratio (O'Connell et al., BioMedical Engineering , 2010).

Fluid dynamic approaches to cell suspensions • Navier-Stokes eq. for the sole carrier fluid • Lagrangian approach (dilute suspensions) • Two-phase flow • Non-Newtonian flow • Fluid-structure interaction

𝐻𝐻𝑑𝐻𝐻𝐻𝐻𝐻𝐻 𝑑𝐻𝐻𝑑𝑑𝜕𝑑𝐻 𝐷ℎ =4𝐴𝑡𝑝

𝑊𝐻𝐻𝐻 𝑠ℎ𝑑𝐻𝐻 𝑠𝜕𝐻𝑑𝑠𝑠 𝜏𝑤 = 𝜇�̇� =6𝑈𝜇𝑠

𝑀𝑑𝐻𝑀 𝑣𝑑𝐻𝑣𝐻𝐻𝜕𝐻 𝑈 =�̇�𝜌𝐴𝑡

𝐹𝐻𝐻𝐻𝜕𝐻𝑣𝑀 𝑓𝐻𝐻𝜕𝑣𝐻 𝑓 =𝐷ℎ∆𝜕

2𝜌𝑈2𝜇

Parameters from ‘macroscopic’ transport phenomena

𝐷𝐻𝑓𝑓𝐻𝑠𝐻𝑣𝐻𝜕𝐻 𝐷 =𝑘𝐵𝑇

6𝜋𝜇𝐻

𝜕𝑃𝐻𝐻𝑑𝜕 𝑁𝐻𝑑𝑁𝑑𝐻 𝜕𝑑 =𝑈𝐷ℎ𝐷

𝑆ℎ𝑑𝐻𝑤𝑣𝑣𝑑 𝑁𝐻𝑑𝑁𝑑𝐻 𝑆ℎ =ℎ𝐷ℎ𝐷

Diffusivity characteristic time vs convective characteristic time

Convective mass flux vs diffusive mass flux

Analyte D (m2/s) Pe

Na+ (100 pm) 10-9 10

Glucose 6×10-10 17

Albumine (BSA, 10 nm) 10-11 103

Viron (100 nm) 10-12 104

Bacterial Cell (1 µm) 10-13 105

Erythrocyte (10 µm) 10-14 106

Polystyrene Bead (100 µm) 10-15 107

Diffusivities and representative Péclet numbers for dilute analytes in water at 25 °C (100 µm wide channel, 100 µm/s mean velocity)

Smith et al., Electrophoresis, 2012

(a)-(d) Contours of fluorescent light intensity (FLI), which indicate bacterial concentration, plotted for RP437 E. coli at different time snapshots. (e)-(h) Bacteria collect in the vortex pair as shown by FLI contours overlaid on the flow streamlines (solid blue lines) (Yazdi and Ardekani, Biomicrofluidics, 2012).

Local fluid dynamics and cell adhesion


𝑆𝜕𝑣𝑘𝑑𝑠 𝑁𝐻𝑑𝑁𝑑𝐻 𝑆𝜕 =𝜌𝑝𝐷𝑝2𝑈18𝜇𝐷ℎ

Particle time scale vs flow time scale

PCTC: prostate circulating tumor cell

Possible ways to bring cells in contact to a wall • rely on a diffusive process to cause cells to randomly move

transverse to streamlines, • apply a body force (e.g., gravity or dielectrophoresis) to move the

cells transverse to streamlines, • create geometries in the flow so that flow is accelerated,

streamlines are compressed and the cells are effectively brought in proximity to the wall by motion along a streamline,

• make the wall permeable and allow the streamlines to cross the interface.

𝐶𝐻𝑝𝐻𝐻𝐻𝐻𝐻𝐻 𝑁𝐻𝑑𝑁𝑑𝐻 𝐶𝐻 =𝜇𝑈𝑑𝜎

𝐵𝑣𝑀𝑑 𝑁𝐻𝑑𝑁𝑑𝐻 𝐵𝑣 =∆𝜌 𝑔𝐷ℎ

2

𝜎

𝑊𝑑𝑁𝑑𝐻 𝑁𝐻𝑑𝑁𝑑𝐻 𝑊𝑑 =𝜇𝑈𝑑

2𝐷ℎ𝜎

Gravity vs interfacial forces

Viscous vs interfacial forces

Inertial vs interfacial forces

Presence of suspended cells multiphase microflows

Inertial, viscous and gravitational body forces, relative to interfacial forces, as a function of the channel size and characteristic velocity in microfluidic multiphase systems

Gűnther and Jensen, Lab on a Chip, 2006

Strain rates can be large in the microflows. In the simplest case, τ ≈ U/h, which can yield 103 - 104 s−1. Such values are sufficiently large to cause non-Newtonian rheological effects, if suspended deformable objects are present.

𝐷𝑑𝑁𝑣𝐻𝐻ℎ 𝑁𝐻𝑑𝑁𝑑𝐻 𝐷𝑑 =𝜕𝑐𝜕𝑝

Material stress relaxation time vs characteristic time scale

Presence of suspended cells non-Newtonian fluids

A well known effect - since 1929 - is the Fåhraeus effect for blood flowing in small tubes (I.D. < 0,3 mm).

A further issue: cell population dynamics

Galbusera et al., Biomed. Microdevices,2008 http://people.physics.anu.edu.au/~mak110/

Example 1: Shear-stress dependent leukocyte adhesion assays

Bianchi et al. Journal of Biomechanics, 2012

Schematics of a flow chamber (a), its computing model (b), a half computing model with active test region (c), micropatterned osteoblasts (d), and a unit with a single cell of computational model (e). In the current models, seven or fifteen units were placed in one row, and seven rows were used for simplicity of calculation (Cui et al. Ann. Biomed. Eng., 2011)

Kobel et al., Lab on a Chip, 2010

100 µm

10 µm

Example 2: Single cell trapping

Nason et al., COUPLED PROBLEMS 2013

Simulation of blood flow (Hct 30% and 95 ± 5 s-1shear rate; Dh = 19 µm (left) and Dh = 24 µm (right). The domains are cut at the centerplane of the vessels (Alizadehrad et al., Journal of Biomechanics, 2012).

Example 2a: Red blood cells in microvessels

Example 3: A microfluidic in vitro model for specificity of breast cancer metastasis to bone

Bersini et al., Biomaterials, 2014

Several reasons make microfluidic devices and systems interesting also for cell manipulation: • The increasing interest for living cells • The integration of several standard analytical operations • The possibility to manipulate large numbers of cells simultaneously • The possibility of manipulate single objects with cellular dimension by micromechanics device • The fast response of microenvironment to heat, chemical and electrical stimuli.

http://yoon.eecs.umich.edu/microfluidics.html

Cells are implicated in a regulatory network of inter-cellular signaling pathways that control homeostasis as well as the response to pathogenic stress. Cell-cell and cell-extracellular matrix is a prominent component of such pathways. Cell-cell interactions are essential for circulating cells to make contact with the vessel wall and eventually penetrate in the endothelial barrier.

Adhesion and extravasation processes are influenced by several factors: • properties and signaling of cellular adhesion • flow conditions • capillary geometry or confinement • cellular rheological properties • interactions with other cells.

Shear stress acts to deform cells in the direction of blood flow, and induces a structural remodeling and flattening to minimize a chronic stimulation.

Vascular endothelial cells in vivo are influenced by the hemodynamic mechanical stresses: • the transmural pressure, inducing

cyclical strain of the vessels • the frictional force generated by the

blood flow.

mediators

Blood flow Particles interactions

(platelets – erithrocytes)

τω

Fdrag

Ronen Alon (Immunity jan2007)

τm

Normal and Shear stresses on the cell membrane

Normal and Shear stresses on the

endothelial wall

• Multiple steps cascades controlled by integrated chemoattractant-dependent signals and adhesive events

• Endothelial ligands involved in that second step of firm adhesion are intercellular adhesion molecules (ICAMs) and vascular cell adhesion molecules (VCAMs).

inflammation

Leukocyte Shear dependent adhesion and transmigration

across vessel wall in inflammation

Thin flow chamber and thick flow chamber, referring to the ratio between cells and channel dimension.

Height of the channel (mm)

Design requirements:

SAMPLE 50 – 500 µl

Tunable shear stress conditions

Triplicated result for each tested

shear stress/assay (statistic consistence)

Allowing diapedesis of transmingrating cells

s vm

Several simultaneous

shear stress tests

s vm s v

m s vm

s vm

Suitable to be opened and re-usable

Several simultaneous assays

at the same shear stress ( different biomolecules configuration)

s vm s v

m s vm

High-throughput read-out of results

Integrability of a coated membrane eventually covered

by an endothelial activated monolayer

Parallel flow chamber for cell adhesion and migration

Sample is driven to 3 parallel group of 3

chambers (Syringe pump)

Each group of chamber offers a different wall shear stress condition τw :

different ∆P resistive pathways distribute flow among the groups

τw high τw medium τw low 1.5 mm

50 µm

Range 0.05<τw<4 Pa Ratio 1: 13.3 : 40

Design of a First prototype:

Flow rate 452 µL / min

Poiseuille relation – Hydraulic diameter

<5


System configuration

L1<L2<L3

3 Groups with different Shear Stress, each one with 3 chambers

Inlet (2)

Inlet (1)

Outlet (2)

Outlet (1a) Outlet (1a)

3 Groups with identical

Shear Stress, total of

9 chambers

Inlet (1)

Inlet (2)

Outlet

L3

L1 L2

3 chambers High τw 3 chambers Medium τw

3 chambers Low τw


Microfabrication

Powder blasting

70°

d d

d

Hydrofluoric Isotropic Etching

Thermic bonding

Thermic bonding

Borofloat glass


CFD simulations of the chambers: evaluation of the establishment length and the shear stress distribution on the membrane.

Fluent Ansys Hexaedrons dominated mesh


0

3

6

9

12

15

Pres

sure

dro

p [K

Pa]

Flow rate [µL/min]

Medium S.S. group // High S.S. group

Low S.S. group

Low S.S. // Medium S.S// High S.S.

Analytical estimation

Experimental data

452 inlet

P Complete configuration

P

P

Low Shear Stress path

Parallel Configuration

M//H

Design by analytic formulae

Channel with rectangular cross-section (CFD for

non standard geometries)

Shear stress on the membrane.


Experimental characterisation

Triple Layer chip

Experimental measurements MicroPIV: evaluation of the flow rate distribution among the parallel pathways

t’ t’ + ∆t

Frame A

Frame B

Statistical methods

TSI, Fluospheres 540/560- 1 µm Powerview cam

2048x2048- Nd:Yag 532nm – Ensemble

PIV – 7couples 760mmx760mm,

64x64px spot

MicroPIV system


Polycarbonate membrane

Pores - 8 μm h = 7-22 μm E = 70 MPa Millipore

(a) (b)

20X optic LD – inverted transmission microscope (Zeiss Axiovert 135 TV with QImaging Exi-Blue)

Suspension of murine neutrophils (RBL-2H3 rat basophilic leucemia cells,

mouse CXCR2 stably transfected), in HBSS (1mM Ca/Mg, 10 mM

HEPES, 0.1% BSA) 1.5 x 106 cells/ml

Polycarbonate membrane - 10μg/ml VCAM (Recombinant Mouse VCAM-1/Fc Chimera –

R&D Systems) 37°C – 1h,

Pores Cells

12 µl/min 0.108 Pa

Adhesion tests with cells (to assess light transmission, sealing efficacy and usability)


Conclusion

ご清聴ありがとうございました。

fluid dynamic design principles for flow … at ynus.pdfelements of fluid dynamics the micro / nano...

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