Measurement of Nanoparticle Uptake by Alveolar Macrophages - Safety of Nanomaterials along Their Lifecycle - page 149

d
dt

≈ Δ
Δ

N
t

C

(7.2)

Nt q () =⋅− ( 1 Nt ( )) .

C C

From this equation, which describes the sedimentation rate of the particles in the
control region ( N C ) and is primarily based on observation, a simple kinetic model
can be derived:

(7.3)

Nt

C ()

=− −⋅ 1 exp( qt )

This equation describes the increase in particles number in the control ROI
and is valid for the case that no overlap occurs between sedimented particles (see
Equation 7.1 case b ≈ 0). The value of q represents the so-called sedimentation
parameter. It describes how fast the maximum number of particles is reached at the
bottom of the observation chamber. The q -value derives either from Formula 7.3 or
from the initial increase in N C according to Formula 7.2:

In(1 − N (t)) dN ( t

dt

= 0 )

≈ Δ=
Δ

Nt
t

( 0 )

C C C

q = ≡

(7.4)

t

Active macrophages will take up particles and this will reduce the particle num-
ber N C within the macrophage area with increasing time. We supposed a monoexpo-
nential decrease resulting in the actual particle number N P :

t t

Nt Nt () = () exp( )1exp( ⋅ −=−−⋅ qt )) exp( ⋅ − ) (7. 5)

P C

τ τ

The time constant t is a value for the phagocytosis rate of the macrophage and
describes its “phagocytosis ability.” Its value derives from the maximum of the
kinetic model 7.4:

exp( qt

q

⋅ − *) 1

τ=

(7.6)

Figure 7.8 shows an experiment with AlOOH nanoparticles (22.5 µg/mL): The
development of particle numbers (normalized to the maximum) in a control ROI
and in an active ROI was calculated according to our model (7.6). It can be seen that
small fluctuations based on particle movement are eliminated using an arithmetic
average function. The quantity of particles inside the control ROI rises to a maximum
after 80 min. This point in time marks the end of the sedimentation process. The
parameters q = 0.04/min and t = 234 min were derived from the best fit of model
and measured AlOOH data. The second graph of Figure 7.9 represents the quantity
of particles inside an active ROI, which underscores the control ROI throughout the
experiment due to ongoing particle uptake. It can also be seen that after 80 min the
macrophage had phagocytized nearly 28% of the particles of the control region.