Modeling of the in vivo kinetics of antioxidants delineates suitable parameters for selecting potential antioxidant adjuvants for cancer therapy

Figures & data

Table 1.  Notations used in mathematical descriptions.

Notation	Unit	Definition
OH•	μmol/L or μM	The hydroxyl free radicals
Qo	μmol/L or μM	The initial pre-ingestion concentration of the antioxidants
Q1	μmol/L or μM	The post absorptive concentration of antioxidants
Q2•	μmol/L or μM	The concentration of secondary free radicals of antioxidants after quenched by RO• in blood plasma
RO•	μmol/L or μM	The concentration of reactive oxygen species in blood
TH1	μmol/L or μM	The exposed effective native tissue concentrations
TH′1	μmol/L or μM	The concentration of the recovered tissue components from free radical attacking by Q2•
T•	μmol/L or μM	The concentration of the transiently attacked tissue free radicals in carcinoma tissues
ka	μM^−1S^−1	The overall absorption rate coefficient for antioxidant Qo
kq1	μM^−1S^−1	The overall quenching rate coefficient by free radicals RO•
kq2	μM^−1S^−1	The quenching rate coefficient of Q2• by native tissues
kq2’	μM^−1S^−1	= kq2 [TH1]
kq3	μM^−1S^−1	The overall quenching rate coefficient of the intermediate antioxidant free radical Q2• with water molecules
kq3’	μM^−1S^−1	kq3 [H2O]
kr	μM^−1S^−1	The overall decay rate coefficient of the intermediate antioxidant free radicals Q2•
d[Qo]/dt	μM-S^−1	The absorption rate of antioxidant Qo by intestine
d[Q1 ]/dt	μM-S^−1	The net accumulation rate of antioxidant in blood plasma immediately post absorption
d[Q2• ]/dt	μM-S^−1	The formation rate of the intermediate antioxidant free radical in blood plasma immediately post attacked by RO•

Figure 1.  The descriptive model for an antioxidant in vivo.

Table 2.  The cited parameters of antioxidants, free radicals in healthy and cancer subjects.

Items	Dimension	Feasibility	References
Qo
Ascorbic acid	1500 μmol/L	Being feasible for all kinds	Assumed
Lycopene	1000 μmol/L	of tumors	Assumed
Mannitol	823410 μmol/L (per os)		**
Q1
Ascorbic acid	51 μmol/L, plasma	–	Podmore et al., 1998
	72 ± 14μmol/L, plasma	–	Rehman et al., 1998
	31.2 ± 14, plasma	–	Davison et al., 2002
	130 × 10^3μmol/L (i.v.)	–	**
Mannitol
RO•	10.19 ± 3.66 μmol/L	Control/healthy people	Sandhya Devi et al., 2002^#
	19.7 ± 0.9 μmol/L	In S180 solid tumor	Kriska et al., 1996
	16.6 ± 1.3 μmol/L	In P388 solid tumor	Kriska et al., 1996
	140.71 ± 54.55 μmol/L	In leukemias	Sandhya Devi et al., 2002^#
ka, L/μmol-s	0.05-5.0 μM^−1s^−1 (slow absorption) 0.5-5000 μM^−1s^−1 (rapid absorption)		Assumed; depending upon the characteristics of each individual antioxidant
kq1, L/μmol-s
Mannitol	1.9 × 10^15 μM^−1s^−1	(serving as antioxidant)	Zaho et al., 1994
Lycopene	31 × 10^15 μM^−1s^−1	(serving as antioxidant)	Di Mascio et al., 1989
α-tocopherol	0.3 × 10^15 μM^−1s^−1	(serving as antioxidant)	Matsuzaka et al., 2003
			Stryker et al., 1988
			Gey et al., 1987
kr, L/μmol-s
Methionine	5.6 × 10^15 μM^−1s^−1	(serving as tissue quencher)	**Zaho et al., 1994; Di Mascio et al., 1989; Matsuzaka et al., 2003

^#Reconstructed by this paper.

**Personal communication with Peter Ott in Department of Hepatology, University of Copenhagen, Denmark.

Table 3.  Different settings considered for computer simulations.

Status/parameter	Qo (μmol/L)	kq1 (μM^−1s^−1)	ka * (μM^−1s^−1)	Kr (μM^−1s^−1)
Slow absorption	2000 *	0.3 × 10^15	1.0	5.6 ×
+ low quencher	(α-tocopherol)	(α-tocopherol)		10^15
Rapid absorption	2000*	0.3 × 10^15	50	5.6 ×
+ low quencher	(α-tocopherol)	(α-tocopherol)		10^15
Slow absorption	823410	1.9 × 10^15	5	5.6 ×
+ moderate quencher	(mannitol)	(mannitol)		10^15
Rapid absorption	823410	1.9 × 10^15	5000	5.6 ×
+ moderate quencher	(mannitol)	(mannitol)		10^15
Slow absorption	200*	31 × 10^15	0.05	5.6 ×
+ high quencher	(lycopene)	(lycopene)		10^15
Rapid absorption	200*	31 × 10^15	0.5	5.6 ×
+ high quencher	(lycopene)	(lycopene)		10^15

*Estimated values; other parameters cited in literature were obtained from Table 2.

Figure 2.  Simulated behaviors of different antioxidant settings. (A) Category 1: The low quencher α-tocopherol with slow absorption. [Q₂•] is shown with the second Y-axis and [Q1] is simultaneously shown in the insert with the abscissa from 0 to 20 sec. (B) Category 1: The low quencher α-tocopherol with rapid absorption. [Q₂•] is shown with the second Y-axis and [Q1] is simultaneously shown in the insert with the abscissa from 0 to 20 x 10⁻² sec. (C) Category 2: The moderate quencher mannitol with slow absorption. [Q₂•] is shown with the second Y-axis and [Q1] is simultaneously shown in the insert with the abscissa from 0 to 20 x 10⁻¹ sec. (D) Category 2: The moderate quencher mannitol with rapid absorption. [Q₂•] is shown with the second Y-axis and [Q1] is simultaneously shown in the insert with the abscissa from 0 to 20 x 10⁻⁴ sec. (E) Category 3: The high quencher lycopene with slow absorption. [Q₂•] is shown with the second Y-axis and [Q1] is simultaneously shown in the insert with the abscissa from 0 to 200 sec. (F) Category 3: The high quencher lycopene with rapid absorption. [Q₂•] is shown with the second Y-axis and [Q1] is simultaneously shown in the insert with the abscissa from 0 to 10 sec.

Table 4.  A comparison of the outcomes from computer simulation of different settings*.

Category	Q1 × 10^6 mol/L (peak time: s)	Q2• × 10^6 mol/L, (peak time; life time: s)	Q2H × 10^6 mol/L (saturation: s)	Absorption relapse (ka, μM^−1s^−1; time, sec)	Relevant coefficient/Figure no.
Low Q’er
Slow abs	2000 (0.00)	2000 (2 × 10^−15s; 18 × 10^−15s) 2000	2000 (20 × 10^−15S)	(1; 10 s)	2A
Rapid abs	2000 (0)	(2 × 10^−15s 18 × 10^−15s)	2000 (18 × 10^−15S)	(50; 12 × 10^−2 s)	2B
Mod Q’er
Slow abs	823410 (0)	823410 (5 × 10^−16s; 32 × 10^−16s) 823410	2000 (32 × 10^−16s)	(5; 12 × 10^−1s)	2C
Rapid abs	823410 (0)	(5 × 10^−16s; 32 × 10^−16s)	823410 (32 × 10^−16s)	(5000; 12 × 10^−4 s)	2D
High Q’er
Slow abs	200 (0)	200 (2 × 10^−16s; 14 × 10^−16s)	200 (14 × 10^−16s)	(0.05; 120 s)	2E
Rapid abs	200 (0)	200 (2 × 10^−16s; 14 × 10^−16s)	200 (14 × 10^−16s)	(0.5; 6 s)	2F

*Q’er, quencher; mod, moderate; abs, absorption; Mod, moderate; Quenching coefficient, for low quencher, k_q1 0.3 × 10¹⁵(μM⁻¹s⁻¹); moderate quencher, k_q1 1.9 × 10¹⁵(μM⁻¹s⁻¹); high quencher, k_q1 31 × 10¹⁵(μM⁻¹s⁻¹); Tissue quencher methionine (kr, 5.6 × 10¹⁵ μM⁻¹s⁻¹).

Figure 3.  To find the optimum kinetic constant ratio by prediction of the behavior of the secondary free radical, Q₂• with respect to the ratio of K_q2/K_q1.

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