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

Abstract

To find in vivo behaviors of an antioxidant when used as an adjuvant cancer therapy, a more detailed integrated pharmacokinetic scheme is needed. Major reaction parameters associated with the sequential routes from ingestion to decay of an antioxidant were used in mathematical analysis, which included absorption rate coefficient k_a, quenching rate coefficient of the antioxidant k_q1 and tissue quenching rate coefficient k_r. The model was then treated with computer simulation using cited decay rate coefficients and some assumed parameters. When intestinal absorption rate coefficient k_a becomes larger, retention time of antioxidant in plasma would be prolonged. moreover, k_a had no effect on either quenching ability of antioxidants or tissue recovering capability. in quenching plasma ROS, the larger the quenching coefficient k_q1, the shorter peak- and the life-times would be for the secondary free radicals that are formed in primary quenching. Conclusively, it is suggestive to prescribe an antioxidant therapy with an appropriate values of k_a and larger values of k_q1.

Keywords::

• Therapeutic behaviours
• quenching
• complementary therapy
• cancer

Introduction

Antioxidants are often utilized to inactivate in vivo reactive oxygen species (ROS). Generally, ROS include superoxide anion, hydrogen peroxide, hydroxyl free radicals, singlet oxygen molecule, lipid hydroperoxide, and RO₁•, RO₂•, etc. These reactive species are apt to exert damage to tissues even though they may be acting as a protective. Some antioxidants such as glutathione and uric acid are naturally present within human cells, while vitamin E (α-tocopherol), vitamin C (ascorbic acid), polyphenolics and carotenoids are available from foods. The role of dietary antioxidants including vitamin C, vitamin E, carotenoids and polyphenolics in disease prevention has received much attention in recent years (Feri, 1994; Halliwell et al., 1995; Sies & Stahl, 1995; Jeffery et al., 2003). Lycopene had been reported to be the most potent antioxidant, with a singlet oxygen quenching ability twice as high as that of  β-carotene and 10-times higher than that of α-tocopherol (Di Mascio et al., 1989).

Current dietary guidelines to combat chronic diseases, such as cancer and coronary artery disease (CAD), recommend intake of plant foods that are rich in antioxidants (DHHS, 2005; USDA, 2004). These observations with the epidemiological data indicate that diets rich in fruits and vegetables are able to reduce risk of numerous chronic diseases (Block et al., 1992; Gaziano et al., 1995; Steinmetz & Potter, 1996; WCRF, 1997; Hollman & Katan, 1999).

Recently, the use of Chinese medicines and herbs has expanded in the Western medical world. Most of their constituents possess antioxidative properties. More importantly, their specific polysaccharides or characteristic proteoglycans have been successfully applied to cancer treatment through stimulating or enhancing the immune activity of the patients (Mains, 1958; Kim et al., 2003; Mau et al., 2003). Recently, many other singlet oxygen scavengers as well as antioxidants are continuously under investigation (Lapenna et al., 2002; Egashira et al., 2003; Kim et al., 2003; Kuzkaya et al., 2003; Leonard et al., 2003; Mancuso et al., 2003; Matsuzaka et al., 2003; Nakanishi et al., 2003).

Patients afflicted with malignant tumors are always suffering from severe attacks by high levels of ROS inherently released in their bodies. The species of ROS concerned depends upon the pathological status of the tumors (Jackson & Loeb, 2001; Haklar et al., 2001). As a rule, superoxide anions are normally found as the major ROS in many patients at the initial stages, while some have merely tremendously high hydrogen peroxide levels (Sauer et al., 1997). Nevertheless, in the last clinical stages hydrogen peroxides are the only predominating ROS species (Szatrowski & Nathan, 1991). Some hematological neoplasms are more or less improvable at the initial stages if antioxidants are properly administered in combination with chemotherapy (Drisko et al., 2003; Weijl et al., 2004). Conventionally, the antioxidants often used are α-tocopherol and ascorbic acid (Wolf et al., 1998; Bjelakovic et al., 2004). However, the prognosis was found to vary, depending upon both the dosages and duration of treatment (Jurkiewicz & Granger, 1998; Young & Lowe, 2001). And even more often, higher dosages and/or prolonged therapy always show much adverse effect in the long run (Tzeng-H Lin, personal communication).

Up to the present, kinetic studies related to antioxidants (Aliaga et al., 2003; Cevallos-Casals & Cisneros-Zevallos, 2003; Egashira et al., 2003; Goldstein et al., 2003; Kim et al., 2003; Latkar et al., 2003; Reddan et al., 2003; Udilova et al., 2003) were mostly associated only with the quenching ability of the specific component or some extract fraction of phytochemicals. In fact, the prognosis of any cancer therapeutics with antioxidants is still unclear if only by considering the quenching capability (Taubert et al., 2003; Zhao et al., 2003). To our knowledge, an integrated concept or guidance associated with the selection rule of an antioxidant has never been documented. Thus, within a safe range of an antioxidant on prescription, this present study first tried to develop a selection rule through mathematical analysis with the aim to give a more practical and solid guidance to such an adjuvant therapy.

Materials and methods

Since the Stern-Volmer Equation as often cited in the literature is obviously inapplicable to the selection rule of antioxidants, we first tried to develop a novel integrated mathematical model to describe the in vivo pharmacokinetic aspects of an antioxidant. In this modeling, some selected antioxidants were screened by computer simulation for its relevant parameters in order to achieve a more precise selection rule of an antioxidant.

Derivation of the model

Notations

Table 1 lists all the notations used in the mathematical modeling.

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•

The descriptive model

The model illustrating an antioxidant uptake from ingestion to the focus (tumor) site can be described by Figure 1.

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

Briefly, when an amount of antioxidant Q_o is ingested per os, it passes through the intestine by intestinal absorption to reach a primary serum concentration of Q₁. During circulation and transportation, the antioxidant (Q₁) is partially attenuated by the in vivo free radicals (RO•) co-existing in plasma, whereupon it is transformed into less active secondary free radical Q₂•, which in turn is decayed by two mechanisms: one through tissue quenching (TH1), and the other by water quenching (H₂O). The former is accompanied with the formation of slightly damaged tissue (T•) which soon recovers to a pseudo-healthy state TH′1. In the latter, the secondary free radicals Q₂• are quenched in vivo on reacting with the water molecule. As a consequence, the tertiary free radicals are released. On reacting with the hydroxyl free radicals (OH•), a concurrent formation of inactivated Q₂H is achieved. Normally, in healthy subjects, all the hydroxyl free radicals (OH•) produced through such a mechanism are first coupled to form hydrogen peroxide (H₂O₂). Subsequent and rapid degradation is followed by enzyme catalase action in vivo, yielding 1 mole of H₂O and 1/2 moles of oxygen from decomposition of per mole of H₂O₂ (Figure 1). However, in patients with overwhelming production of free radicals, an adjuvant antioxidant therapy would be preferentially recommended.

Assumptions

Assumption 1

The parameters concerned in any medicine to measure its efficacy, if administered per os, involve the solubility, the absorbability, the transportability, and the signal transduction. Presuming equal solubility, the absorbability would play the primary limiting factor. Thus it is assumed that the intestinal absorption (Figure 1) of all antioxidants into the circulatory system is a primary limiting factor regardless of their species and quantity.

Assumption 2

All reaction steps involved are considered to proceed by elementary first order kinetics. The transient serum amount of antioxidant Q₁ is limited compared with the huge amount of in vivo ROS radicals (symbolized as RO• in Figure 1) that are constantly “pumped out” by the damaged tissues, hence first order kinetics with respect to the antioxidant concentration is applicable.

Assumption 3

With exception of the small hydroxyl free radicals, coupling reactions occur neither among the ROS free radicals RO• nor within the secondary free radicals Q₂• (Figure 1) formed upon quenching.

The prohibiting rules are:

1. The mass action law: further self-coupling reactions are unlikely according to the mass action law (Zumdahl, 2004) when extensively diluted by tissue fluids which co-exist with the secondary free radicals Q₂•.

2. The steric hindrance: the complexity of their original main structures and the possible huge chemical structures of the products, in case resulting from the coupled reactions. Theoretically, the termination of Q₂• (Figure 1) by self-coupling reaction is likely to occur; however, in reality the drawback of its huge molecular steric hindrance (such as with formation of a tannin molecule) (Nawar, 1996) and dilution effect by the tissue fluids would actually exclude such probability. Instead, the overwhelming large population of tissue cells nearby can more easily take the advantage to combine with Q₂• (Figure 1).

Assumption 4

Both the ROS and the intermediate antioxidant Q₂•, the latter even already having been greatly attenuated to less active radicals, are considered still to be harmful to the tissues and capable of damaging the cells, a phenomenon usually evidenced by the abuse of over-dosage of antioxidants (Tzeng-H Lin, personal communication).

Assumption 5

The newly formed free radicals Q₂• (Figure 1) are eliminated through two possible mechanisms, i.e. either by autodecay (to trap electrons from H₂O) or reacting with the cytoplasmic components inside the tissues (TH1) (Figure 1), both of which co-exist in tremendous amounts compared to any other reacting species. Consequently, their changes due to free radical reactions are negligible, i.e. practically a zero order kinetics with respect to tissues or tissue fluids nearby.

Assumption 6

The reaction of the intermediate antioxidant free radicals Q₂• with H₂O is a minor one in vivo compared to that reacting with the tissues (TH1) (Figure 1), because the oxidative reactions of all antioxidants are always irreversible, whether in the atmosphere or in aqueous solution (Di Mascio et al., 1989; Nawar, 1996; Matsuzaka et al., 2003).

Assumption 7

Normally, the decay of the hydrogen peroxide produced from the hydroxyl free radical coupling can be sufficiently accomplished by enzyme catalase in vivo. However, in cases of overwhelming production, such as in patients with carcinoma, the use of antioxidants as an adjuvant therapy would be required.

The mathematical description

By following the descriptive model (Figure 1), the net accumulation rate of an antioxidant to reach a concentration of Q₁ in serum is

On quenching by the reactive oxygen species [RO•] co-existing in plasma, we have the disappearance rate for [Q₂•]

practically,

hence, assuming that the tissue and water quenching can be lumped together, the net production rate of the secondary free radicals [Q₂•] resulting from the quenching process is

Mathematical model derivation

The quenching rate equation for the in vivo reactive oxygen species free radicals RO• (Figure 1) is

Because each reaction obeys first order kinetics (Assumption 2), the rate laws for each species, Q₁, Q₂•, and Q₂H, respectively, are

and

Alternatively, Equation 7 is expressed as

These coupled differential equations (Equations 1, 6, and 7) can be solved analytically. Assuming that the initial concentrations at time t = 0, we have

From Equation 1 we have the instantaneous accumulation of antioxidant by intestinal absorption:

Integration of Eq. 10 gives

and similarly, we have

Thus, according to our hypothesis, Equations 11 to 13 are the mathematical equations that will be used to solve the graphical problem of chemical reactions (Equations 15 to 18).

Parameter collection and estimation

The focus of interest is to analyze the optimum kinetics starting from the effective concentration of antioxidant (Q₁) through the intermediate free radicals Q₂• and its subsequent decay reactions (Figure 1). Thus, the related free radical—antioxidant interaction in vivo can be described by the following three stages.

Initiation and generation

This step (Figure 1) can be initiated by many factors, either internal or external, such as carcinoma, chemotherapy, etc.

Propagation and retardation

The retardation by the antioxidants (Q₁) in plasma on the in vivo free radicals (RO•) (Figure 1) is mainly through

The secondary free radicals Q₂• produced are in turn decayed through tissue (TH1) quenching:

Thereupon damaged tissues (T•) may soon recover, provided the patients are in good, healthy condition, to a less damaged status TH′1 with simultaneous production of tertiary free hydroxyl radicals OH• (Figure 1).

Alternatively, the free radicals Q₂• (Equation 15), on reaction with water molecules, also yield tertiary hydroxyl free radicals (Figure 1).

Termination

The probable overall termination reactions may involve 1) the self-coupling of free radicals Q₂•:

a mechanism which is prohibited by Assumption 3, and 2) the self-coupling of hydroxyl free radicals OH•’:

Thus, only Equation 20 is considered to be significant. Normally, all the hydroxyl free radicals coupled to produce H₂O₂ (Equation 20) molecules could be in turn completely decomposed by enzyme catalase in healthy subjects (Equation 21). However, for patients with carcinoma or some inflammatory responses, the overwhelming production of free radicals in vivo would require an adjuvant antioxidant therapy to suppress such huge amounts of free radicals constantly released from tissues (Table 2).

A collection of the relevantly cited and/or assumed parameters are listed in Table 2.

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.

Computer simulation

The parameters used for computer simulations using Equations 11 to 13 are shown in Table 3, some of which were assumed values.

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.

Computer simulation was performed using three representative categories of antioxidants as listed in Table 3, i.e., the low quenchers as presented by model compound α-tocopherol, the moderate quencher by mannitol, and the high quencher by lycopene.

Results and discussion

Figure 2 (A to F) shows the outcome behavior resulting from the computer simulation of different antioxidant therapeutic settings (Table 3).

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.

In addition to the parameter reported by Nawar (1996), we took advantage of α-tocopherol, mannitol, and lycopene as the authentic model compounds, representing low quencher (k_q1 0.3 × 10¹⁵ μM⁻¹s⁻¹), moderate quencher (k_q1 1.9 × 10¹⁵ μM⁻¹s⁻¹), and high quencher (k_q1 31 × 10¹⁵ μM⁻¹s⁻¹), respectively (Table 3). Methionine (k_r, 5.6 x 10¹⁵ μM⁻¹s⁻¹), being abundant in all tissues where there is protein, was uniquely used to serve as the tissue quencher by which the secondary damage caused by the antioxidant free radicals can be eliminated. In fact, the quenching capability for ROS by mannitol has attracted much attention and is extensively used to treat patients with brain edema (Peter Ott, personal communication). According to Ott, since the baseline mannitol concentration in plasma is almost negligible (similar results as shown in Figures 2C and 2D), and the regular dosage commonly given to patients is 500 mL mannitol (15%), being actually equivalent to 75 g (or 411 mmol) of mannitol to yield a final concentration of 130 mM, provided it is completely and evenly distributed in 3 L of plasma. Thus, in this paper the initial concentration, 130 mM (i.v.) or 823410 μmol/L mannitol (per os), was chosen as the authentic clinical moderate antioxidant for model simulation.

In the case of a low quencher, such as α-tocopherol at 2000 μmol/L (category 1 in Table 3), results showed that absorption was almost instantaneously effected (Figures 2A and B) and relapsed within 10 s and 12 × 10⁻² s for slow (k_a = 1 μM⁻¹s⁻¹) and rapid (k_a = 50 μM⁻¹s⁻¹) absorptions, respectively (abscissa in Figures 2A and B inserts). For both cases, and as can be seen in Figures 2A and B, the peak concentrations of the intermediate free radicals [Q₂•] all reached a peak of 64 × 10⁻⁶ mol/L at 2 × 10⁻¹⁵ s, tailing off to nil at 20 × 10⁻¹⁵ s and 18 × 10⁻¹⁵ s, while the recovery of the stable α-tocopherol (generalized as Q₂H) took a period of 18 s and 18 × 10⁻² s, respectively. In the case of the moderate quencher, mannitol, which at a concentration of 823410 × 10⁻⁶ mol/L (category 2 of Table 3), behaved as a more powerful quencher than α-tocopherol, has suppressed the intermediate free radicals (Q₂•) to a peak of 140000 × 10⁻⁶ mol/L at 5 × 10⁻¹⁶ s with a life time of 32 × 10⁻¹⁶ s and 32 × 10⁻¹⁶ s, and correspondingly, with the stable recovered form of mannitol (generalized as Q₂H) maximized at 40 × 10⁻¹⁶ s and 35 × 10⁻¹⁶ s for the slow (k_a = 5 μM⁻¹s⁻¹) and the rapid (k_a = 5000 μM⁻¹s⁻¹) absorptions, respectively (Figures 2C and D). Correspondingly, the serum mannitol levels (expressed in general formulas Q₁) relapsed within 12 × 10⁻¹ s and 12 × 10⁻⁴ s (abscissa of Figures 2C and D inserts), respectively. In contrast, for the most powerful quencher lycopene (category 3 in Table 3), its serum levels (expressed in general form as Q₁) as effected by absorption were found to have been completed within 120 s and 6 s for slow and rapid absorptions, respectively (abscissa of Figures 2E and 2F inserts). Comparing the two species of quenchers as discussed above, the peak time of the intermediate free radicals (Q₂•) has been greatly reduced to 2 × 10⁻¹⁶ s (Figures 2E and 2F) with a life time of 12 × 10⁻¹⁶ s, regardless of the slow (k_a = 0.05 μM⁻¹s⁻¹) or rapid (k_a = 0.5 μM⁻¹s⁻¹) absorption. Most prominently, the stable recovered lycopene (Q₂H) reached a saturated phase within 12 × 10⁻¹⁶ s despite the slow or rapid absorption (Figures 2E and 2F). The implication indicates that lycopene can be an excellent adjuvant antioxidant therapy, because of being characteristic of persisting serum levels and with extremely shortened life time (12 × 10⁻¹⁶ s) for its secondary free radicals. As well cited, whose secondary damaging effect is also currently being concerned. A comparison of the outcomes from different settings by computer simulation is listed in Table 4.

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).

In summary, all results obtained from simulation implicitly indicated that with the same quencher, more rapid absorption actually could have shortened the lifetime as well as the secondary damaging effect of Q₂•. As the overall damaging effect of an antioxidant is proportional to the total area covered by the peak, which in turn is a function of the intensity (y-axis) multiplying with the life time (x-axis), such a parameter has been used to evaluate the damaging effect of radiotherapy. Thus, lycopene can exhibit a better choice for antioxidant therapy, whose attacking capability resulting from the intermediate free radicals (Q₂•) could have been greatly eliminated within an extremely short period, i.e., a choice of selective condition with the least damaging effect.

In the above simulated results, obviously, the overall effects can be seen to be inter-dependent on the kinetic constants including the absorption rate coefficient K_a, the quenching activity K_q1, and the tissue recovering ability K_r (Figures 2A to F, and Table 4). Yet for patients with dyspepsia, their absorption rates can be reduced to a great extent or even totally to indigestion, as usually found in patients with late stage carcinoma or lymphoma, whose absorption capability thus may significantly become the limiting step in the whole protecting sequence of reactions. Moreover, the predicted selection rule as shown in Figure 3 has revealed that the quenching would be extremely affected to a great extent in which the lifetime of the antioxidant free radical can be significantly shortened, provided a ratio of K_r /K_q1 is maintained at values below 0.1. When the ratio is raised to a value above 10, the life time of the secondary free radical could be extended and lengthened to a rather longer period, even with the peak having been suppressed to less than 15 × 10⁻⁶ M with a flattened curve, still resulting in a larger area to be covered by the curve of Q₂•, as discussed above, a corresponding status of increased risk of secondary damage. However, when the ratio is raised to values larger than 1000, the vanishing effect on the secondary free radicals is still also appreciated (Figure 3).

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.

Hence, to serve as an antioxidant therapy, other than the toxicity, the conditions with proper values of K_a and as large as possible the values of K_q1, and if applicable, with the ratio K_r/K_q1 below 0.1 or higher than 1000, are suggested to be taken into consideration (Figure 3).

Comparing with other species of antioxidants which are usually used clinically as complementary medicines, results indicated that both lycopene and mannitol are indeed good to excellent antioxidants within the modeling concern, besides, they are characteristic of revealing moderate to high quenching capability, being easy to administer either per os (or mannitol by i.v.), non-toxic, and most importantly, prescribeable in larger dosages that are usually needed for a more prolonged and a safer protection of the patients.

Conclusion

Using antioxidants as a complementary therapy in treating cancers may be feasible, yet the clinical outcomes and prognosis have been rather varying and crucial. Our present mathematical analysis indicates that the absorption rate coefficient K_a could only affect the retention of the ingested antioxidant plasma levels. Higher quenching rate coefficient K_q1 (such as that with lycopene) is able to shorten the peak and the lifetimes of the antioxidant secondary free radicals to within 2 × 10⁻¹⁶ s and 14 × 10⁻¹⁶s, compared to 2 × 10⁻¹⁵ s and 18 × 10⁻¹⁵s, respectively by low quencher α-tocopherol.

We conclude that in considering an antioxidant therapy, other than the toxicity, it is suggestive to prescribe an antioxidant therapy with a proper value of K_a and larger values of K_q1. Following such a rule, we also confirmed that both mannitol and lycopene are excellent antioxidants and suggestive of use in the complementary therapy.

Declaration of interest

The authors are grateful to the financial support from NSC-96-2320-B-241-006-MY3 and 94TMU-TMUH-02. The authors also acknowledge the partial financial supports of the grants: SKH-TMU-93-37 from the Shin Kong Wu Ho-Su Memorial Hospital and TMU96-AE1-B08 from the Taipei Medical University, Taiwan.