This document discusses the rheology and composition of blood and how it leads to non-Newtonian fluid behavior. It states that blood is composed primarily of plasma and formed elements, mainly red blood cells. The presence of red blood cells causes two effects - rouleaux formation, where stacks of red blood cells form networks that increase viscosity, and alignment of cells along streamlines under shear, which decreases viscosity. Together, these effects make blood a shear-thinning fluid whose viscosity decreases with increasing shear rates.

1. BIOPHYSICS
(MLS 103)
(MODULE 4: HAEMODYNAMICS
1ST SEMESTER (2024)
J. Owusu

2. Haemodynamics
• The term hemodynamics comes from the
Greek words haima (blood) and dunamis
(power).
• It refers to the movement and deformation
(i.e., flow) of blood, and the forces that
produce that flow

3. Haemodynamics
• Blood as a transport medium:
1. it carries oxygen and nutrients to
metabolically active tissues, returns carbon
dioxide to the lungs,
2. delivers metabolic end-products to the
kidneys, etc.

4. Other Functions of Blood
• provides a buffering reservoir to control the
pH of bodily fluids
• serves as an important locus of the immune
system
• transports heat, usually from centrally
located tissues to distal ones, in order to
help maintain a suitable temperature
distribution throughout the body.

5. Blood Rheology
• Rheology is the study of how materials
deform and/or flow in response to
applied forces. The applied forces are
quantified by a quantity known as the
stress.
• Stress is defined as the applied force per
unit area.
• The applied force is a vector quantity: it
has magnitude and direction.

6. Blood Rheology
• The surface to which the force is applied
also has direction (=orientation)
• This surface orientation is characterized by
its normal vector.
• The orientations of both the force and the
surface must be accounted for when
computing the stress.
• Therefore the resulting quantity is a second-
order tensor whose diagonal elements are
normal stresses and whose off-diagonal
elements are shear stresses.

7. Blood Rheology
• For a solid, the deformation is quantified in terms
of the fractional change in the dimensions of a
small material element of the solid. This quantity,
known as the Strain.
• Strain depends on both the direction of the
deformation and the orientation of the material
element that is being deformed.
• Strain is therefore also a second-order tensor.

8. Stress-Strain Relationship
• For a fluid, a similar situation obtains,
except that we replace the deformation by
the rate of deformation to obtain a rate-of-
strain tensor.
• Rheological knowledge of a material can
be expressed (in part) by a constitutive
relationship between the applied stress
and the resulting strain or rate-of-strain.

9. Stress-Strain Relationship
• For a Newtonian fluid, the simple
constitutive relationship between stress
and strain is given as:
Where τ is the applied shear stress, γ is the
rate of strain, and μ is a material constant
known as the dynamic viscosity.
𝝉 = 𝝁γ (1)

10. Blood composition
• To understand why blood is a non-
Newtonian fluid, we consider blood
composition.
• There are approximately 5 liters of blood
in an average human being.
• This complex fluid is essentially a
suspension of particles (the formed
elements) floating in an aqueous medium
(the plasma).

11. Blood composition
• Fluid (water) = 50%
• Formed elements = 46%
– Erythrocytes = 45%
– Leukocytes < 1%
– Platelets < 1%
• Proteins ≈ 4%
• Ions < 1%
• Other < 1%

12. Blood Composition
• All blood constituents are important
physiologically.
• For rheological purposes we can make
some simplifications. For example, white
cells (leukocytes) and platelets play a
major role in the immune response and in
blood clotting, respectively.

13. Blood Composition
• However, there are relatively few white
cells and platelets compared with the
number of red cells (erythrocytes).
• Consequently, the mechanical behavior of
the formed elements is usually dominated
by the red cells
• the volume fraction of red cells is so
important to the rheological and
physiological characteristics of blood that a
specific term, known as the hematocrit, H

14. Blood Composition
• Hematocrit is given by the equation
• In conclusion, we approximate blood as a
suspension of red cells in a Newtonian
fluid and focus on the behavior of the red
cells to explain blood’s non-Newtonian
rheology.
𝐻 = 𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑟𝑒𝑑 𝑏𝑙𝑜𝑜𝑑 𝑐𝑒𝑙𝑙𝑠
Total blood volume
(2)

15. Blood Composition
• Individual red cells are shaped like “biconcave
disks,” with a diameter of ≈8 μm and a
maximal thickness of ≈ 3 μm
• There are ≈ 5 billion red cells per milliliter of
blood, so the total red cell surface area in a
normal human adult is ≈ 3000m2.
• The cytoplasm contains large amounts of the
iron-containing protein hemoglobin.
• Hemoglobin is highly efficient at binding
oxygen; hence, the presence of large amounts of
hemoglobin sequestered inside red cells greatly
increases the oxygen transport capacity of
blood

16. Relationship between blood
composition and rheology
• Rheologically, there are two major effects
from the presence of red cells.
Rouleaux formation. If blood is allowed to sit
for several seconds, stacks of red cells
(rouleaux) begin to form.
• If allowed to grow, these rouleaux can
become quite large and will eventually form
an interconnected network extending
throughout the blood.

17. Relationship between blood composition
and rheology
• Rouleaux provide mechanical coupling between
different fluid regions, and thereby increase the
resistance to deformation of fluid elements.
• This implies that the effective viscosity μeff is
increased by the presence of rouleaux.
• Rouleaux are broken up by imposed shearing of
the blood. this will cause a relative decrease in
μeff.
• Thus, the action of rouleaux formation is to
make blood a shear-thinning fluid.

18. Relationship between blood composition and
rheology
• Rouleaux forming in
stagnant blood.
• (A) Rouleaux formation
under stagnant conditions.
• (B) Normal erythrocytes,
showing the classical
“biconcave disk” shape
characterized by a thick rim
and a central depression.
• (C) Under certain
conditions, erythrocytes can
adopt a spiny configuration,
in which case they are
known as echinocytes.

19. Relationship between blood composition and
rheology
Red cell alignment.
• Because individual red cells are disc shaped, their
hydrodynamic influence depends on their
orientation.
• There are two forces that compete in orienting
red cells. First, Brownian motion, always present,
tries to randomize the orientation of red cells.
• Second, fluid shearing forces cause red cells to
align their long axes with streamlines.

20. Relationship between blood composition and
rheology
• These forces lead to two possible extrema.
• When red cells are randomly or near-randomly
oriented, then individual cells will “bridge”
between different streamlines,
• thereby providing mechanical coupling between
two different fluid regions having potentially
different velocities.
• This will tend to increase μeff.
• The magnitude of this effect will decrease as red
cells become progressively more oriented, that is,
as ˙ γ increases.

21. Relationship between blood composition and
rheology
• Two limiting orientations for red cells. On the left, the fluid shear rate is very low so Brownian forces
predominate and the red cells are randomly oriented.
• On the right, the fluid shear rate is large, causing the red cells to line up along streamlines with very little
randomization of orientation.