Lubrication of Joints - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 302

over time. As we have stated before, this is not a typical biofluid flow problem, but instead
it relies more on convection and diffusion of species out of a particular compartment.

Example
Calculate the time rate of change of proteoglycans within the synovial fluid, if their diffusion
is only restricted by the presence of glycosaminoglycans (GAGs) within the synovial space.
Assume that the molecular weight for the proteoglycan of interest is 300 kDa and its density is
1.45 g/mL. The effective radius for the GAGs is 0.5 nm and the volume fraction is 0.008. The free
diffusion of proteoglycan in water is 1.1 E 2

7cm 2 /s. The permeability of the proteoglycan

through the synovial membrane is 1 E 2 6 cm/s. Assume that the degradation of the proteoglycan

is negligible. The formation rate of proteoglycans within the synovial membrane and the carti-
lage are 6 E 2

7 mg/(cm 2 s), respectively. The cross-sectional areas of the
synovial membrane and the cartilage are 29.5 mm 2 and 13.5 mm 2 , respectively. The concentration
gradient of the proteoglycan is 0.13 mg/mL.

9 mg/(cm 2 s) and 3 E 2

Solution
First it will be necessary to calculate the effective radius of the proteoglycan:

1 = 3

3 300 kDa

a 5 4 : 35 nm

5

4 π ð 1 : 45 g = mL Þð 6 : 02 E 23 Þ

As a side note, Avogadro's constant converts Daltons into grams. Now we can calculate the
restricted diffusion coefficient:

Þe ð 2

p

ð Þ 5

7cm 2

0 : 008 Þ 1

4 35 nm

0 : 5nm

:

8cm 2

1

D i 5 ð 1 : 1 E 2 = s 4 : 62 E 2 = s

To calculate the time rate of change of the proteoglycan, we need to know the flux of the
proteoglycans out of the synovial membrane. The flux can be formulated from

8cm 2

J 5 pΔcA D i

4 : 62 E 2 = s

5mm 2

D 5 ð

1 E 2 6cm = s Þð 0 : 13 mg = mL Þð 29 : Þ 0 : 000058 mg = hr

5

1 : 1 E 2 7cm 2 = s

Therefore, the time rate of change of proteoglycans within the synovial fluid will be

@ð Vc Þ
@t 5 r s A s 1 r c A c 2 d 2 J
5 ð

=

cm 2 s

Þð :

5mm 2

Þ 1 ð =

cm 2 s

Þð :

5mm 2

Þ 2 : =

6 E 2 9mg 29 3 E 2 7mg 13 0 2 0 000058 mg hr

0 : 000094 mg = hr

5

11.4 MECHANICAL FORCES WITHIN JOINTS

For this section, we will briefly review how to calculate the mechanical forces that may
arise in joints, because as we have discussed, under different types of loading conditions,
the formation rate of particular molecular species may increase or decrease within carti-
lage. This material may be familiar from a bio-solid mechanics course or the more