Physics question

Separation of particles with centrifugation

Why analyze blood? • Informative about general health

status: – there is a capillary <10 cells away from

every cell in the body – blood transports everything

• Accessibility: – easy to obtain (venipuncture, finger

prick) – sampling blood is culturally acceptable

(vs. stool, urine) • Abundance of analytical assays:

– about 200 diagnostic tests are done on blood (out of 700 available)

What is blood? • Composition:

– highly concentrated suspension of blood cells (mostly red blood cells) in blood plasma (a concentrated protein solution)

– circulates throughout the body in vessels of cardiovascular system (in balance with interstitial fluid and lymphatics)

• Physiologic function(s): – transport of nutrients, respiratory gases, metabolic waste

products, hormones & other regulatory molecules, drugs – traffic of blood & other circulating cells, microparticles,

infectious agents, parasites

– helps regulate things (temperature, balance of water) – first response to infections (compliment system, leukocytes)

Blood cells:

Functions of blood cell: • Red Blood Cells (~45% of blood, ~5×106/µL)

– delivery of O2 / removal of CO2 – sensing pO2 in tissues / regulation of local blood flow

• Leukocytes (5-10×103/µL) – neutrophils (phagocytosis of bacterial microorganisms) – lymphocytes (chronic bacterial / acute viral infections)

• T cells (cellular-type immune reactions) • B cells (antibody production)

– monocytes (become macrophages) – eosinophils (modulate inflammation, kill helminths) – basophils (immediate hypersensitivity / allergy)

• Platelets (150—400×103/µL) – blood clotting (coagulation)

Examples of other things in blood: • Other cells:

– stem cells – circulating tumor cells, circulating endothelial cells – fetal cells (nucleated RBCs)

• Nutrients: – dissolved nutrients and byproducts of metabolism, fat globules

• Regulatory molecules: – hormones, drugs, NO, ATP…

• Harmful mediators: – cytokines, microparticles, free iron…

• Artificial particles: – drug delivery vehicles (e.g., magnetic nanoparticles) – contrast agents (e.g., microbubbles)

• Pathogens: – viruses, bacteria, cysts, ameba, worms, eggs…

Blood sample collection: Finger prick

Venipuncture

Separation of blood cells from plasma:

Why and how centrifugation separates blood cells from plasma?

Centrifugation:

centrifuge rotor

tube with blood

plasma leukocytes,

platelets

red blood cells

Q: How long do we need to centrifuge the tube with blood to yield the maximum amount of plasma?

Q: How long would it take for a particle to move from the top of the tube (point closest to the axis of rotation) to the bottom of the tube (point farthest from the axis of rotation)?

𝑥𝑥0

𝜔𝜔

𝑙𝑙

x

yz

Motion of a particle in a rotating vessel:

x*

z*

𝑥𝑥0

𝜔𝜔

𝑙𝑙

• Initial simplifying assumptions: – Red blood cells are spherical particles (R = 3.5 μm) – Cell-cell interactions are negligible

�⃗�𝐹𝑔𝑔�⃗�𝐹𝑆𝑆

x

yz

inertial frame

rotating frame 𝑚𝑚�⃗�𝑎 = �⃗�𝐹𝑆𝑆 + �⃗�𝐹𝑔𝑔

𝑚𝑚�⃗�𝑎∗ = −6𝜋𝜋𝜋𝜋𝜋𝜋�⃗�𝑣∗ + 4𝜋𝜋 3

𝜌𝜌𝑝𝑝 − 𝜌𝜌𝑚𝑚 𝜋𝜋3�⃗�𝑔 − 4𝜋𝜋 3

𝜌𝜌𝑝𝑝 − 𝜌𝜌𝑚𝑚 𝜋𝜋3 2𝜔𝜔 × �⃗�𝑣∗ + 𝜔𝜔 × 𝜔𝜔 × 𝑟𝑟∗ + 𝑑𝑑𝜔𝜔 𝑑𝑑𝑑𝑑

× 𝑟𝑟∗

(Inertial frame)

(Non-inertial frame rotating with the vessel)

�⃗�𝐹𝑆𝑆 �⃗�𝐹𝑔𝑔

Mathematical correction due to use of non-inertial, accelerating reference frame

centrifugal Coriolis Euler

(vector sum of three fictitious inertial forces)

�⃗�𝑎∗ = �⃗�𝑎 − 𝐴𝐴 − (2𝜔𝜔 × �⃗�𝑣∗ + 𝜔𝜔 × 𝜔𝜔 × 𝑟𝑟∗ + 𝑑𝑑𝜔𝜔 𝑑𝑑𝑑𝑑

× 𝑟𝑟∗)

0

(Inertial and rotating frames share the origin)

1

Ev an

s 1 98

9

(Symon, K.R. Mechanics, 2nd edition, Addison-Wesley, 1960; Chapter 7, pp. 269-280)

𝑚𝑚�⃗�𝑎∗ = −6𝜋𝜋𝜋𝜋𝜋𝜋�⃗�𝑣∗ + 4𝜋𝜋 3

𝜌𝜌𝑝𝑝 − 𝜌𝜌𝑚𝑚 𝜋𝜋3�⃗�𝑔 − 4𝜋𝜋 3

𝜌𝜌𝑝𝑝 − 𝜌𝜌𝑚𝑚 𝜋𝜋3 2𝜔𝜔 × �⃗�𝑣∗ + 𝜔𝜔 × 𝜔𝜔 × 𝑟𝑟∗ + 𝑑𝑑𝜔𝜔 𝑑𝑑𝑑𝑑

× 𝑟𝑟∗

x*

z*

𝑥𝑥0

𝜔𝜔

𝑙𝑙

�⃗�𝐹𝑔𝑔�⃗�𝐹𝑆𝑆

x

yz

inertial frame

rotating frame

• More simplifying assumptions: – Assume that the particle is moving with terminal velocity, �⃗�𝑎∗ = 0 – Angular velocity does not change with time, 𝑑𝑑𝜔𝜔

𝑑𝑑𝑑𝑑 = 0

0 0

(Symon, K.R. Mechanics, 2nd edition, Addison-Wesley, 1960; Chapter 7, pp. 269-280)

Motion of a particle in a rotating vessel:

centrifugal Coriolis Euler

−6𝜋𝜋𝜋𝜋𝜋𝜋�⃗�𝑣∗ + 4𝜋𝜋 3 𝜌𝜌𝑝𝑝 − 𝜌𝜌𝑚𝑚 𝜋𝜋

3�⃗�𝑔 − 4𝜋𝜋 3 𝜌𝜌𝑝𝑝 − 𝜌𝜌𝑚𝑚 𝜋𝜋

3 2𝜔𝜔 × �⃗�𝑣∗ + 𝜔𝜔 × 𝜔𝜔 × 𝑟𝑟∗ = 0

−6𝜋𝜋𝜋𝜋𝜋𝜋𝑣𝑣𝑥𝑥∗ ∗ −

4𝜋𝜋 3

𝜌𝜌𝑝𝑝 − 𝜌𝜌𝑚𝑚 𝜋𝜋3 2𝜔𝜔𝑣𝑣𝑦𝑦∗ ∗ − 𝜔𝜔2𝑥𝑥∗

−6𝜋𝜋𝜋𝜋𝜋𝜋𝑣𝑣𝑦𝑦∗ ∗ −

4𝜋𝜋 3

𝜌𝜌𝑝𝑝 − 𝜌𝜌𝑚𝑚 𝜋𝜋3 −2𝜔𝜔𝑣𝑣𝑥𝑥∗ ∗ − 𝜔𝜔2𝑦𝑦∗

−6𝜋𝜋𝜋𝜋𝜋𝜋𝑣𝑣𝑧𝑧∗ ∗ +

4𝜋𝜋 3 𝜌𝜌𝑝𝑝 − 𝜌𝜌𝑚𝑚 𝜋𝜋

3𝑔𝑔

= 0

𝑣𝑣𝑥𝑥∗ ∗ =

𝛼𝛼𝜔𝜔 2 1 + 𝛼𝛼2 𝑥𝑥

∗ − 𝛼𝛼2𝜔𝜔

2 1 + 𝛼𝛼2 𝑦𝑦 ∗ 𝛼𝛼 =

4 9𝜋𝜋

𝜌𝜌𝑝𝑝 − 𝜌𝜌𝑚𝑚 𝜋𝜋3𝜔𝜔

η = 4.02 ⋅ 10−3 [kg⋅m−1⋅s−1], R = 3.5 ⋅ 10-6 [m], ρp = 1097 [kg⋅m−3], ρm = 1023 [kg⋅m−3], ω = 126 [rad⋅s−1]

𝛼𝛼 ≈ 6.4 × 10−6

, where

Motion of a particle in a rotating vessel:

2 3 HW#2

x*

z*

𝑥𝑥0

𝜔𝜔

𝑙𝑙

�⃗�𝐹𝑔𝑔�⃗�𝐹𝑆𝑆

x

yz

inertial frame

rotating frame

𝑣𝑣𝑥𝑥∗ ∗ =

𝛼𝛼𝜔𝜔 2 1 + 𝛼𝛼2 𝑥𝑥

∗ − 𝛼𝛼2𝜔𝜔

2 1 + 𝛼𝛼2 𝑦𝑦 ∗ ≈

𝛼𝛼𝜔𝜔 2 𝑥𝑥

∗

𝛼𝛼 = 4

9𝜋𝜋 𝜌𝜌𝑝𝑝 − 𝜌𝜌𝑚𝑚 𝜋𝜋3𝜔𝜔 ≈ 6.4 × 10−6

� 𝑥𝑥0

𝑥𝑥0+𝑙𝑙 1 𝑥𝑥∗ 𝑑𝑑𝑥𝑥

∗ = 𝛼𝛼𝜔𝜔 2 �

0

𝑑𝑑0

𝑑𝑑𝑑𝑑 𝑙𝑙 = 𝑥𝑥0 exp 𝛼𝛼𝜔𝜔 2 𝑑𝑑0 − 1

𝑉𝑉𝑝𝑝𝑙𝑙𝑝𝑝𝑝𝑝𝑚𝑚𝑝𝑝 = 𝜋𝜋 4 𝑑𝑑

2𝑥𝑥0 exp 𝛼𝛼𝜔𝜔 2 𝑑𝑑0 − 1

𝛼𝛼 = 4

9𝜋𝜋 𝜌𝜌𝑝𝑝 − 𝜌𝜌𝑚𝑚 𝜋𝜋 3𝜔𝜔

Motion of a particle in a rotating vessel:

4

Comparison with experimental data:

Wong et al. Lab Chip, 8, 2032–2037, 2008

Homework assignment:

• Supporting reading: – Wong, Egg beater as centrifuge, 2008

– Symon, K.R. Mechanics, 2nd edition, Addison- Wesley, 1960; Chapter 7, pp. 269-280

• Complete Homework #2 – due today (Mon, 9/17, 11:59pm)