Page 24 - Mathematical Techniques of Fractional Order Systems

14 Mathematical Techniques of Fractional Order Systems

myeloma (MM), therapies include daily doses of PTH, endothelin, and pro-
teasome inhibitors, that act by targeting osteoblasts to recover bone mass
(Oyajobi et al., 2007). Of course, anticancer agents that target metastatic and
primary tumor cells directly (chemotherapy, as paclitaxel (Perez et al.,
2001), and hormone therapy) should be used in combination with the afore-
mentioned therapies in either case (Casimiro et al., 2009).

1.3.2 Pharmacokinetics and Pharmacodynamics (PK/PD)
PK models characterize drug absorption and disposition within the body
(Dhillon and Gill, 2006). The remaining drug concentration to be absorbed
(C g , in mg/L) and the effective drug concentration in the plasma (C p ,in
mg/L) are described by the system of differential equations, given by
Eq. (1.14). κ g and κ p are the absorption and elimination rate, respectively
(Mager et al., 2003).
1
D C g ðtÞ 52 κ g C g ðtÞ ð1:14aÞ
1
D C p ðtÞ 5 κ g C g ðtÞ 2 κ p C p ðtÞ ð1:14bÞ
For a subcutaneous drug administration, the initial concentration is
applied in the remaining drug to be absorbed ðC g ð0Þ 5 C 0 Þ; for the intrave-
nous case, the initial concentration goes directly to the plasma ðC p ð0Þ 5 C 0 Þ.
For a single dosage drug of initial concentration C 0 , the plasma concentra-
tion can be determined by Eq. (1.15a). For multiple doses, the plasma con-
centration of the n th dose is given by Eq. (1.15b), for initial conditions
C 0 5 D 0 F administrated at equally spaced intervals t 5 t 2 ðn 2 1Þτ, where
0
V d
D 0 is the dosage, F the bioavailability, and V d the volume distribution.
Multiple dosage is governed by the steady-state C pss 5 C 0 .
τκ p
κ g 2κ t κ t
C p 5 C 0 ðe p 2 e g Þ ð1:15aÞ
κ g 2 κ p
2nκ τ 2nκ g τ
κ g 1 2 e p 2κ p t 0 1 2 e 2κ g t 0
0 3 2
C p ðn; t Þ 5 C 0 2κ τ e 2κ g τ e ð1:15bÞ
κ g 2 κ p 1 2 e p 1 2 e
The PD consists in a drug effect. It can be described by a Hill function
that depends on the drug’s concentration, given by dðtÞ 5 C p ðtÞ (Pinheiro
C 50 ðtÞ 1 C p ðtÞ
p
50=base
et al., 2011). It varies between 0 and 1, where C p 50ðtÞ 5 fðtÞC p represents
50=base
the concentration at 50% of its maximum effect, C p is the initial value of
50
C ðtÞ, and resistance to a drug can be described by fðtÞ 5 1 1 K r
p
Ð t
0 max½0; L r 2 C p ðλފdλ (Pinheiro et al., 2011).
Different drugs can act in the same pathway and their combined action
may differ from the isolated administration of each drug. To quantify
the synergistic or antagonistic effect of the combination of drugs, the