Page 213 - Computational Modeling in Biomedical Engineering and Medical Physics

202 Computational Modeling in Biomedical Engineering and Medical Physics

transfer through them is negligibly small during the time span on the study. In the
absence of the magnetic field by the diffusive convection model
@c 2
1 uUrÞc 5 Dr c;
@t ð ð6:24Þ
where c is the species (MNPs), and D is the diffusivity of MNPs in the blood. The
suggested order of magnitude balance is

1 U D
; B ;
t MNP L L 2 ð6:25Þ
where t MNP is the time scale of the mass transfer process, U is the velocity scale, and L
is the streamwise length scale of capillary. Apparently two time scales are competing to
lead the mass transfer: (1) the velocity time scale, t U,MNP B U/L and (2) the diffusion
2
time scale, t D,MNP B L /D.
In vitro studies (Lücker, 2018a,b) and experiments conducted on phantoms
aimed at investigating Doppler ultrasound procedure for noninvasively measuring
pulsatile capillary speed velocities for human and artificial blood (Law et al., 1989;
Ting et al., 1992; Li et al., 1993; Razavi et al., 2018), and other studies indicate
velocities of the order U B 1 mm/s. Although the no-slip assumption is arguable
(Zeeshan et al., 2018), this reference is the leading transport quantity. For L B
2
100 μm(Almaça et al., 2018, Lücker, 2018a)and D B 10 29 m /s (Li et al., 2008),
3 3
it yields t U,MNP B 10 sand t D,MNP B 10 s, meaning that the two mechanisms
(transport and diffusion) have the same time scale.
The contribution to the MD of the transport magnetic body forces produced by
the magnetic field may be assessed use the stationary form of the momentum equation
2
ρ uUrÞuŠ 52 rp 1 μr u 1 f mg ; ð6:26Þ
ð ½
which indicates the order of magnitude balance

ρU 2 P 0 μU 0
0 B ; ; F mg;0 ;
d d d 2 ð6:27Þ
2
where d is the vessel diameter (space scale), U 0 is the velocity scale, P 0 5 ρU the pres-
0
2
sure scale, W mg;0 5 B μ =d the magnetic energy density scale, B 0 the magnetic flux
0 mg
2
density scale, μ mg the magnetic susceptibility, and F mg;0 5 B μ =dμ 5 W mg;0 =d the
mg
0 mg
magnetic body forces scale. It should be noted that B 0 andd are not B rem . Relation
Eq. (6.27) then yields
1 W mg;0
1B1; ; 2 ; ð6:28Þ
Re d ρU
0