Nitric Oxide Delivery in Stagnant Systems via Nitric Oxide Donors: A

As a small biological molecule, nitric oxide (NO), plays a key role in diverse functions including smooth muscle cell regulation, neurotransmission, i...
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Chem. Res. Toxicol. 2003, 16, 7-14

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Articles Nitric Oxide Delivery in Stagnant Systems via Nitric Oxide Donors: A Mathematical Model Mahendra Kavdia† and Randy S. Lewis* School of Chemical Engineering, Oklahoma State University, Stillwater, Oklahoma 74078, and Department of Biomedical Engineering, Johns Hopkins University, Baltimore, Maryland 21205 Received March 4, 2002

As a small biological molecule, nitric oxide (NO), plays a key role in diverse functions including smooth muscle cell regulation, neurotransmission, inhibition of platelet aggregation, and cytotoxic actions. The assessment of NO effects in biological systems has extensively been studied using NO donor compounds that often have differing NO release mechanisms and kinetic rates. Due to the differing kinetic rates and release mechanisms, in addition to reactions involving NO (such as autoxidation of NO), the NO concentrations to which biological systems are exposed may vary significantly depending upon the NO donor compound. Thus, quantifying the effects of NO using different NO donors is difficult unless the NO concentration profile in the experimental system is predicted or measured. In this study, the spatial and temporal NO concentration in a stagnant system (such as a culture plate or micro-well) is modeled following the addition of an NO donor characterized with first-order NO release kinetics. Two NO donors were utilized: diethylamine NONOate (DEA/NO) and spermine NONOate (SPER/NO). The use of a mathematical model can eliminate the need of complex in situ NO measurements and be useful for predicting the physical loss of NO from the experimental system. In addition, properly scaling the NO concentration can be useful in estimating the maximum NO concentration that will exist in solution. The results show that under widely used in vitro experimental conditions, including varying NO donor concentrations, cellular oxygen consumption rates, and aqueous phase heights, the spatial and temporal NO concentration range can vary significantly. In addition, hypoxic conditions can occur in the vicinity of cells, and in some situations, the physical loss of NO from the experimental system may be significant.

Introduction Nitric oxide (NO), a small biological molecule, plays a key role in diverse physiological functions including regulation of smooth muscle tone, neurotransmission, and inhibition of platelet aggregation (1). Nitric oxide donor compounds have been utilized to assess the effects of NO in physiological systems (2-5). NO donors release NO and other related redox species and hold great potential as therapeutic agents for many conditions, such as vasospasm, restenosis, and impotence, where physiological NO levels are diminished. A wide range of NO donors is available, and they are categorized based on their chemical structure, such as organic nitrates/nitrites, NONOates, sydnonimines, and s-nitrosothiols (6, 7). The mechanisms leading to NO release differ significantly among the individual donor categories. The kinetics of NO release are often more important than the total NO released by the NO donor since the kinetics affect the spatial and temporal NO exposure levels to cells in many experimental systems. Due to the autoxidation of NO, the same total amounts of NO released over different time ranges may lead to different NO exposure levels. * To whom correspondence should be addressed at Oklahoma State University. † Johns Hopkins University.

Thus, it is difficult to quantitatively assess the effects of NO in experimental systems using different NO donors unless the NO concentration is measured or predicted. Wink et al. (8) have reported the effects of various NO donors on the hydrogen peroxide (H2O2)-mediated cytotoxicity in Chinese hamster lung fibroblasts. The NO donors used for the study were s-nitrosoglutathione (GSNO), s-nitroso-n-acetylpenicillamine (SNAP), diethylamine NONOate (DEA/NO), sodium nitroprusside (SNP), and sulfite NONOate (Sulfi/NO). The measured release rates of NO are first order for many of these NO donors. The study reported that DEA/NO, SNAP, and GSNO at concentrations of 0.1, 1.0, and 1.0 mM, respectively, protected cells against H2O2 cytotoxicity, but 1.0 mM SNP enhanced H2O2 cytotoxicity. In addition, Sulfi/NO had no effects on the protection against H2O2-mediated toxicity. The contradictory results between NO donors have led the authors to state that caution should be exercised when using the NO donor agents and correlating their effects. Some of the reasons for the differing results may be varying time-dependent release rates of NO donors and release of additional species, such as cyanide (CN-) in the case of SNP. In addition, the experiments were conducted in a stagnant solution contained in a Petri dish leading to a potential NO concentration gradient in the experimental system.

10.1021/tx025528r CCC: $25.00 © 2003 American Chemical Society Published on Web 12/03/2002

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Kavdia and Lewis

Figure 1. Stagnant experimental system. The cells are at Z ) 0. At T ) 0, the O2 concentration is 185 µM and NO concentration is 0 in the aqueous phase (culture media). At all times, the concentration of O2 is 185 µM and NO is 0 at the gas/culture media interface.

Many other studies have also assessed the effects of NO on various biological systems using NO donors (912). These studies have several of the aforementioned problems of varying NO release rates, release of other species, and stagnant systems, therefore limiting the quantitative interpretation of NO effects on the system. In view of the importance of knowing the NO concentration in an experimental system, Schmidt et al. (13) developed a mathematical model to estimate the NO concentration in a well-stirred system following the addition of an NO donor. The model successfully predicted the experimental NO concentration incorporating the first-order decomposition rate of an NO donor together with the autoxidation of NO. However, the model assumed a constant O2 concentration in a well-stirred solution, which is not applicable to in vitro studies usually performed in stagnant solutions, especially when O2 consuming cells are present. The loss of NO to the headspace was also not considered, although the NO loss depends strongly on the solution dynamics of the experimental system. Ramamurthi and Lewis (14) included the loss of NO to the headspace to predict the temporal NO concentration in a well-stirred system containing NO donors. Since many experimental systems used to assess the effects of NO on cellular components are stagnant (i.e., culture plate), a mathematical model was developed to quantify the spatial and temporal NO concentration in stagnant systems, such as culture plates or micro-wells, following the addition of an NO donor characterized with first-order release kinetics. The model takes into account the diffusion of NO and O2 in the culture medium, the kinetics of NO autoxidation in aqueous solutions, and the NO and O2 consumption by cells, thus eliminating several assumptions of previous mathematical studies.

Mathematical Model Model Description. The modeled system is a culture plate or micro-well plate containing adherent cells at the bottom as shown in Figure 1. Dimensionless values for the height, time, and concentrations are utilized as described below. NO is assumed to be uniformly released into the media after the addition of an NO donor. The assumption of NO donor uniformity is valid if the diffusion time of the NO donor within the medium is much shorter than the characteristic times of NO release or if the NO donor is well mixed in solution prior to adding the solution to the culture plate. For this study, the NO donor release kinetics were assumed to be first order (typical of NONOates) and have been described according to

NO release rate ) R(t) ) ENO kNOCNOie-kNOt

(1)

where CNOi is the concentration of the NO donor at the initial time. The first-order decomposition rate constant of the NO donor is represented as kNO, and the moles of NO released per mole of NO donor decomposed is represented as ENO (14). Integration of both sides of eq 1 with time from 0 to ∞ (i.e., complete release of NO) shows that the total NO delivered to the media following complete decomposition of the NO donor is ENOCNOi. However, some of the NO may be lost to the gas space above the solution as described later. Following the release of NO, NO can diffuse through the media and react with O2 to form nitrite (NO2-) according to the following reaction scheme k1

2NO + O2 98 2NO2 k2

NO2 + NO {\ } N2O3 k -2

k3

N2O3 + H2O 98 2NO2- + 2H+

(2) (3) (4)

The model for predicting the spatio-temporal distribution of NO and O2 in the media involves the NO and O2 continuity equations, which include the above reactions, and are represented as

∂cNO ∂2cNO ) DNO 2 - 4k1cNO2cO2 + R(t) ∂t ∂z

(5)

∂2cO2 ) DO2 2 - k1cNO2cO2 ∂t ∂z

(6)

∂cO2

where R(t) is given by eq 1. NO donors DEA/NO and SPER/NO were used for the simulations since DEA/NO and SPER/NO are commonly used NO donors of the NONOate class and have widely differing first-order decomposition rates (13-16). The decomposition rate constants (kNO) are 7.8 × 10-3 and 0.30 × 10-3 s-1 and the ENO values are 1.5 and 1.7 for DEA/NO and SPER/NO, respectively, at 37 °C and pH 7.4 (14). Although first order release rate kinetics for the NO donor are used to solve the model, other known NO release rate kinetics can be easily applied by substitution into eq 5. The diffusivities of NO (DNO) and O2 (DO2) in the aqueous phase are assumed similar to that in water which are 5.1 × 10-5 and 3.0 × 10-5 cm2/s, respectively at 37 °C (17). The autoxidation rate constant of NO (k1) is 2.4 × 106 M-2 s-1 at 37 °C (18). Note that only the autoxidation of NO (eqs 2-4) was included for the reactions involving NO and O2. If other species react with NO, these reactions would need to be included in eq 5, and additional continuity equations for the reacting species would need to be solved simultaneously with eqs 5 and 6. Dimensionless Model. The use of dimensionless equations in place of eqs 5 and 6 is often beneficial for analyzing the results. Therefore, the dimensionless terms are as follows. The dimensionless concentration of NO (CNO) is cNO/C/NO where C/NO is defined as

C /NO )

x

R(0) 4k1CO2,s

(7)

where R(0) is the initial release rate (t ) 0) evaluated from eq 1. The discussion section explains the reasoning for choosing the above C/NO as the scaling factor. When C/NO is properly chosen, C/NO provides an estimate of the maximum order-ofmagnitude for the NO concentration. For O2, the dimensionless concentration (CO2) is cO2/CO2,s where CO2,s is the aqueous O2 concentration in equilibrium with the O2 concentration in the gas headspace. The value of CO2,s is 185 µM at 37 °C (13). The dimensionless time T is t × kNO such that a unit value of T represents the inverse time of kNO (characteristic release time

Modeling NO Delivery from NO Donors

Chem. Res. Toxicol., Vol. 16, No. 1, 2003 9

for a given NONOate). Thus, for a given T, the fraction of available NO released is the same for all NO donors although the time to release the amount differs. The dimensionless height Z is z/L where the z coordinate represents the liquid height (z ) 0 at the bottom) and L is the total depth of the aqueous phase. Thus, substituting the dimensionless definitions and eq 1 into eqs 5 and 6, the following dimensionless NO and O2 continuity equations are obtained:

( )

(

)

4k1C /NOCO2,s ∂CNO DNO ∂2CNO ) CNO2CO2 + ∂T kNO kNOL2 ∂Z2

( )

)

ENOCNOi

∂CO2 ∂T

)

( )

DO2 ∂2CO2

kNOL2 ∂Z2

-

(

C/NO

e-T (8)

k1(C/NO)2 2 CNOCO2 kNO

(9)

For eq 8 and 9, the term on the left represents the dimensionless temporal change in concentration. The first term on the right represents the diffusion of the species. The reaction of NO with O2 is represented by the second term on the right. The last term on the right in eq 8 represents the production of NO via the decomposition of the NO donor. An important component of this study is to predict the NO concentration changes during the time when NO is being released by the NO donor; this is when the NO concentration is the highest in the experimental system. According to the time parameter of the NO production term (e-T), NO release is significant for 0 e T e 2 but then drops off rapidly. Thus, this study will only look at the model predictions of NO and O2 for T e 2. Obviously, the model can still be solved for T > 2. As the NO production term becomes negligible, it would be necessary to use a different time scale other than T ) t × kNO (such as incorporating the kinetic rate constant or diffusivity) to assess the contributions of diffusion and reaction to the changing NO concentration. Initial and Boundary Conditions. The solution to eqs 8 and 9 requires one initial and two boundary conditions for each equation. Assuming at T ) 0 that NO is not present in the medium and the medium is saturated with O2, the initial conditions at T ) 0 are CNO ) 0 and CO2 ) 1 for all Z. Assuming the headspace contains air and does not contain NO [see Chen and Deen (19) for validation] and the headspace is in equilibrium with the aqueous phase interface, the boundary conditions at Z ) 1 for NO and O2 are CNO ) 0 and CO2 ) 1, respectively, for all T. The boundary conditions for NO and O2 at the bottom (Z ) 0) are

[

CO2 dCO2 dCNO ) FNO, ) FO2 dZ dZ (km/CO2,s) + CO2

]

Z)0

with Fi )

( ) LNcellvi C/i Di

(10)

The half-maximum oxygen uptake concentration is km, L, and Di were previously defined, Ncell is the number of adherent cells per unit cross-sectional area, vi is the maximum cellular consumption rate of species i per unit cell, and C/i is the scaling parameter. For NO, C/i is defined by eq 7 and C/i is CO2,s for O2. The value of km is assumed to be 0.01 mM as the oxygen uptake rate does not depend on the dissolved O2 concentration as low as 0.015 mM (20). The dimensionless parameter Fi accounts for cumulative effect of changes in cell density, liquid height, and the cellular consumption rate of species i. Adjustable Parameters. The effects of change in adjustable experimental parameters on spatial and temporal profiles of NO and O2 were simulated by varying the NO donor decomposition kinetics (kNO) and initial concentrations (CNOi), the aqueous phase height (L), and Fi. As mentioned before, DEA/NO and

SPER/NO were used for the simulations since they have widely differing first-order decomposition rates. In addition to the NO donor release rates, the amount of NO donor used in reported experiments varied from nM to mM concentrations (12). Thus, two different initial NO donor concentrations (CNOi) of 10 and 100 µM were used. The height L was assumed to be either 3 or 6 mm based on typical culture plates (d ) 35 mm) containing 2.5 mL of culture media (L ) 3 mm) or micro-wells containing 0.2 mL of culture media (L ) 6 mm). For assessing the effects of adjustable parameters on the NO and O2 concentration predictions, a base-case with L ) 3 mm and CNOi )100 µM was used. The range of the dimensionless parameter Fi was obtained from typical values as follows. The number of cells per unit area (Ncell) varies between 11 and 280 × 106 cells/m2 (based on 1-10 × 104 cells per culture plate). Gardner et al. (21) has shown that NO consumption rates (vNO) vary among different cell lines. Using vNO ) 0.043 nmol min-1 106 cells-1, along with the maximum values of Ncell ) 280 × 106 cells/m2 and L ) 6 mm, an upper estimate of FNO is 0.04 and 0.01 for SPER/NO and DEA/NO, respectively, at CNOi ) 100 µM. A lower estimate is FNO ) 0 (i.e., cells do not consume NO) and will be used for the base-case since the NO concentration in solution will be highest for the lowest cellular NO consumption rate. For O2, the maximum oxygen uptake rate (vO2) for mammalian cells varies between 1 and 20 nmol min-1 106 cells-1 (22). The value of vO2 is also affected by NO as studies have shown that NO has an inhibitory effect on the oxygen uptake rate (23). With the maximum values of vO2 ) 20 nmol min-1 106 cells-1, Ncell ) 280 × 106 cells/m2, and L ) 6 mm, an upper estimate of FO2 is 1.0 and will be used as the base-case since a higher cellular O2 consumption rate will lead to a greater NO concentration in solution. Using the minimum values of vO2 ) 1 nmol min-1 106 cells-1, Ncell ) 11 × 106 cells/m2, and L ) 3 mm, a lower estimate of FO2 is 0.001. Numerical Solution. The coupled system of eqs 8 and 9 were solved using FlexPDE software package (PDESolutions, Inc., Antioch, CA). The time integration error tolerance had a relative error of 1 × 10-5.

Results Spatial and Temporal Distribution of NO and O2. The dimensionless spatial and temporal distributions of NO and O2 are shown in Figure 2 for CNOi )100 µM DEA/ NO or SPER/NO, L ) 3 mm, FNO ) 0, and FO2 ) 1.0 (basecase). Since the NO concentrations were nondimensionalized according to eq 7, C/NO corresponds to 5.4 and 25.7 µM for SPER/NO and DEA/NO, respectively. In addition, T ) 1 corresponds to 55.6 and 2.1 min for SPER/NO and DEA/NO, respectively. As shown, the spatial distribution of NO and O2 varies significantly with time for both NO donors. With increasing time (T > 0.5), CNO decreased at all Z for both NO donors as expected. Note that DEA/NO has a larger NO concentration gradient at Z ) 1 as compared to SPER/ NO and DEA/NO results in a more uniform NO concentration between Z ) 0 and 0.75 as compared to SPER/ NO. At Z ) 0 and T ) 0.5, cNO (the actual NO concentration) is 22.2 and 6.3 µM for DEA/NO and SPER/ NO, respectively. Furthermore, for the same initial NO donor concentration, cNO was always higher for DEA/NO as compared to SPER/NO at the same time and height. This is due to the higher NO release rate for DEA/NO as compared to SPER/NO. As also observed in Figure 2 for both NO donors, CO2 reduced over time and there was a significant variation in CO2 at all Z. For both NO donors, the O2 levels reached steady-state when no significant NO remained in the system (data not shown). The steady-state value of 0.20

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Figure 2. Dimensionless NO and O2 concentration profiles for NO donors using the base-case parameters of L ) 3 mm (represents Z ) 1), FO2 ) 1, FNO ) 0, and CNOi ) 100 µM. CO2,s is 185 µM and C/NO, as defined by eq 7, is 5.4 and 25.7 µM for SPER/NO and DEA/NO, respectively. The cells are at Z ) 0 and the gas-liquid interface is at Z ) 1. At T ) 0, CNO is zero and CO2 is 1.0. T ) 1 is 55.6 and 2.1 min for SPER/NO and DEA/ NO, respectively.

for CO2 at Z ) 0 was achieved at T ) 30.1 (63 min) and T ) 1.0 (55.6 min) for DEA/NO and SPER/NO, respectively. For such a scenario, cells will observe O2 concentrations much lower than the saturated O2 conditions. Effect of Aqueous Phase Height. The spatial and temporal profiles of NO and O2 for L ) 6 mm (with FNO ) 0, FO2 ) 1.0, and CNOi ) 100 µM) were also evaluated for both NO donors, since the height of the aqueous phase can vary depending on the experimental system. The effect of height (L ) 6 mm) on the spatial distribution of NO and O2 relative to the base-case (L ) 3 mm) is shown in Figure 3 for DEA/NO at T ) 1. C/NO is 25.7 µM for both the base-case and L ) 6 mm. NO was more evenly distributed at L ) 6 mm as compared to L ) 3 mm. However, CNO was similar at Z ) 0 for both heights. Thus, cells would be exposed to a similar NO concentration irrespective of the depth of the aqueous phase. For Z between 0.5 and 1.0, a higher CNO was predicted for L ) 6 mm compared to L ) 3 mm. This agrees with the lower CO2 observed in the same region for L ) 6 mm since more NO could react with O2. The converse was true for Z between 0 and 0.5. As is also evident, CO2 at Z ) 0 was higher for L ) 6 mm due to less NO available for reaction. For SPER/NO shown in Figure 4, the spatial distribution profiles of NO were similar for both heights with a more even distribution of NO for L ) 6 mm. In Figure 4, C/NO is 5.4 µM for both the base-case and L ) 6 mm. The O2 concentration was higher for L ) 6 mm. Effect of O2 Consumption. The cell number, cellular O2 consumption rate, and the height of the system vary largely among experiment studies. Thus, FO2 can vary significantly. The NO and O2 spatial and temporal profiles were simulated for a lower FO2 of 0.001 and compared with those of the base-case (FO2 ) 1.0) for both NO donors. In the case of DEA/NO with FO2 ) 0.001, CNO and CO2 were essentially constant (0.73 and 0.90, respec-

Kavdia and Lewis

Figure 3. Dimensionless NO and O2 concentration profiles for DEA/NO at T ) 1 (2.1 min). The base-case adjustable parameters are L ) 3 mm (represents Z ) 1), FO2 ) 1, FNO ) 0, and CNOi ) 100 µM. Profiles based on changes in one adjustable parameter are also shown. CO2,s is 185 µM and C/NO, as defined by eq 7, is 25.7 µM at CNOi ) 100 µM and 8.1 µM at CNOi ) 10 µM. The cells are at Z ) 0 and the gas-liquid interface is at Z ) 1.

Figure 4. Dimensionless NO and O2 concentration profiles for SPER/NO at T ) 1 (55.6 min). The base-case adjustable parameters are L ) 3 mm (represents Z ) 1), FO2 ) 1, FNO ) 0, and CNOi ) 100 µM. Profiles based on changes in one adjustable parameter are also shown. CO2,s is 185 µM and C/NO, as defined by eq 7, is 5.4 µM at CNOi ) 100 µM and 1.7 µM at CNOi ) 10 µM. The cells are at Z ) 0 and the gas-liquid interface is at Z ) 1.

tively) between Z values of 0 and 0.75, as shown in Figure 3. C/NO is 25.7 µM for both the base-case and FO2 ) 0.001. CNO was not affected by varying FO2 relative to the basecase. CO2 at Z ) 0 was higher for FO2 ) 0.001 since cells consumed less O2. For SPER/NO (Figure 4), CO2 increased and CNO decreased at all Z for a lower FO2. This is a result of less O2 consumption by the cells. For Figure 4, C/NO is 5.4 µM for both the base-case and FO2 ) 0.001.

Modeling NO Delivery from NO Donors

Effect of NO Donor Concentration. Since the amount of NO donor used in experiments can vary, a lower value of CNOi (10 µM) was used for simulation for both NO donors with L ) 3 mm, FNO ) 0, and FO2 ) 1.0. The dimensionless spatial profiles for a lower CNOi relative to the base-case are shown for NO and O2 at T ) 1 in Figure 3 for DEA/NO and Figure 4 for SPER/NO. It should be noted that the base-case value of C/NO is 25.7 and 5.4 µM for DEA/NO and SPER/NO, respectively, whereas C/NO is 8.1 and 1.7 µM for DEA/NO and SPER/ NO, respectively, when CNOi ) 10 µM. According to Figure 3, the actual NO concentration (cNO) at Z ) 0 is 19.0 µM for the base-case (CNOi ) 100 µM), which represents 13% of the maximum possible NO concentration (ENOCNOi) that may exist. For CNOi ) 10 µM, cNO is 5.5 µM at Z ) 0, which represents 32% of the maximum possible NO concentration. Thus, a greater fraction of the NO donor remains in solution for a lower CNOi, although the NO donor releases NO at a lower rate (see eq 1). The greater fraction occurs since the rate of NO reacting with O2 decreases as CNOi decreases. Similar conclusions occurred with SPER/NO. With regards to O2, the low NO concentrations due to CNOi ) 10 µM increased the amount of available O2 in the media for only DEA/NO (compare O2 profiles in Figure 3 for DEA/NO with Figure 4 for SPER/ NO). For SPER/NO, changes in the O2 profile were not observed for changing CNOi since the O2 consumption by cells dominates O2 consumption by reaction with NO at the specified conditions. Effect of Cellular NO Metabolism. All of the above results assumed NO consumption by cells did not exist (FNO ) 0). Recent studies have shown that some cells consume NO (21). To assess the effect of cellular NO metabolism on the NO profiles, the previously estimated values of FNO ) 0.04 or 0.01 for SPER/NO or DEA/NO, respectively, were used to calculate the concentration profiles. Compared to the base-case, the NO profiles for both NO donors did not significantly change. Increasing FNO another 5-fold still did not have a significant effect on the profiles. Thus, the cellular metabolism of NO does not affect the NO profile predictions for both NO donors. Exposure of Cells to NO. An important component of studies involving NO donor compounds is the NO concentration to which cells are exposed. For the model described, the cells are located at Z ) 0. In Figure 5, the temporal NO concentration is shown for Z ) 0. For DEA/ NO, C/NO is 25.7 µM for all cases except CNOi ) 10 µM, in which C/NO is 8.1 µM. Thus, for the base-case of DEA/ NO, the maximum NO concentration (cNO) to which cells are exposed is 22.6 µM and occurs around T ) 0.4 (0.8 min). The liquid depth and oxygen consumption rate do not affect the cellular NO exposure as compared to the base-case. For CNOi ) 10 µM, the maximum NO concentration (cNO) is 5.6 µM and occurs around T ) 0.8 (1.6 min). The time-averaged CNO from T ) 0 to T ) 4 is 0.52, 0.50, 0.47, and 0.47 for the base-case, L ) 6 mm, FO2 ) 0.001, and CNOi ) 10 µM, respectively. Thus, for all cases, the time-averaged actual concentration (cNO) CNOC/NO) to which cells have been exposed is approximately onehalf of C/NO at a point when a majority of the NO has been released (at T ) 4 or 8.4 min). For SPER/NO, C/NO is 5.4 µM for all cases except CNOi ) 10 µM, in which C/NO is 1.7 µM. As seen for SPER/NO, the maximum CNO to which cells are exposed occurs at T < 0.5 for all cases and is close to the value of C/NO. The time averaged CNO

Chem. Res. Toxicol., Vol. 16, No. 1, 2003 11

Figure 5. Dimensionless NO concentration predictions for NO donors at Z ) 0 (i.e., the location of cells) as a function of dimensionless time (T). The base-case adjustable parameters are L ) 3 mm, FO2 ) 1, FNO ) 0, and CNOi ) 100 µM. Profiles based on changes in one adjustable parameter are also shown. For DEA/NO, T ) 1 is 2.1 min and C/NO, as defined by eq 7, is 25.7 µM at CNOi ) 100 and 8.1 µM at CNOi ) 10 µM. For SPER/ NO, T ) 1 is 55.6 min and C/NO, as defined by eq 7, is 5.4 µM at CNOi ) 100 µM and 1.7 µM at CNOi ) 10 µM.

from T ) 0 to T ) 4 is 0.60, 0.63, 0.41, and 0.41 for the base-case, L ) 6 mm, FO2 ) 0.001, and CNOi ) 10 µM, respectively. Similar to DEA/NO, the time-averaged actual concentration (cNO ) CNOC/NO) to which cells have been exposed is also approximately one-half of C/NO at a point when a majority of the NO has been released (T ) 4 is 222 min).

Discussion This study estimates the spatial and temporal distributions of NO and O2 following the addition of an NO donor to a stagnant media. The model presented here is the first attempt that incorporates the diffusion and autoxidation of NO, as well as the O2 consumption of the adherent cells, for NO donor compounds. Previously, a model was developed to predict spatial and temporal concentrations of NO and O2 in stagnant solutions except the cells, rather than NO donor compounds, produced NO (19). Thus, this study adds additional light to systems where NO donors are utilized since the conditions are different between the two studies. The results show that the spatial and temporal profiles of NO and O2 can be significantly affected by the experimental conditions. There can be a significant variation in the NO and O2 concentration in different regions of the stagnant media at a given time. A fast releasing NO donor distributes NO more evenly in the media at a given time as seen for DEA/NO (see Figure 3) than a slow releasing NO donor as seen for SPER/NO (see Figure 4). The assumption of constant O2 concentration also requires careful study of media conditions as in some cases the O2 concentration dropped as low as 35 µM. The consumption of NO by cells appeared to have no effect on the model predictions for the cases studied.

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For both NO donors at T ) 4, most of the NO has been released so cellular exposure to NO begins to rapidly diminish (see Figure 5). As is notable in Figure 2, the cellular exposure (Z ) 0) to O2 is less for SPER/NO than DEA/NO at the same T. However, since T ) 1 represents 55.6 min for SPER/NO and 2.1 min for DEA/NO, the near-linear O2 concentration profiles for SPER/NO demonstrate that sufficient time has occurred for the O2 profile to reach steady state. At steady state, the O2 profile is controlled by cellular consumption. For DEA/ NO, the O2 profiles are dominated by both cellular consumption and reaction, although at longer T the O2 profile will also approach steady state. On the basis of a well-stirred model with a constant O2 concentration (13), CNO was 0.81 and 0.41 for DEA/ NO and 0.79 and 0.38 for SPER/NO, at T ) 0.5 and 2.0, respectively. However, the stagnant model predictions for CNO at Z ) 0 were 0.86 and 0.51 for DEA/NO and 1.17 and 0.54 for SPER/NO, at T ) 0.5 and 2.0, respectively. This shows that the NO concentrations obtained from a well-stirred model give approximate (6-32% error), but not accurate predictions of cellular exposure levels. The difference between the two models is especially apparent for slow releasing NO donors (SPER/NO) at all times and for the fast releasing NO donors (DEA/NO) at later times. Another important consideration in assessing the effects of NO on biological systems is the amount of total NO delivered to the system. The total NO delivered is calculated from the NO donor decomposition rate given in eq 1. However, for many experimental systems, there may be a significant loss of NO to the headspace, which would reduce the amount of NO available to the biological system. On the basis of the calculated flux of NO at the gas-liquid interface (Z ) 1), the fraction of NO lost (flost) to the headspace relative to the total NO delivered is

(

)

C/NODNO NOlost flost ) ) NOdelivered ENOCNOikNOL2

dC

∫0T dZNO|Z)1dT ∫0T e-TdT (11)

For parameter values of CNOi ) 100 µM, L ) 3 mm, FO2 ) 1 and FNO ) 0 (base-case values), the term in parentheses is 0.06 and 0.012 for SPER/NO and DEA/ NO, respectively. From Figure 2, the dimensionless slope (dCNO/dZ) at Z ) 1 for SPER/NO is approximately onehalf the dimensionless slope for DEA/NO at all times. Thus, the ratio of flost for SPER/NO to flost for DEA/NO is estimated to be (0.06/0.012 × 1/2) ) 2.5. The calculated flost at T ) 2.0 (87% of available NO delivered) from the solution to the model is 0.36 for SPER/NO and 0.15 for DEA/NO for a ratio of 2.4 which is consistent with the estimation. Therefore, 36 and 15% of the total NO delivered leaves the experimental system for SPER/NO and DEA/NO, respectively, signifying a significant loss of NO to the headspace. Thus, quantifying NO exposure based on theoretical NO delivered would be in error. As to the absolute amount of NO lost to the headspace, the NO flux at Z ) 1 [(DNOC/NO/L)dCNO/dZ] represents the molar rate of NO lost per unit area. Since C/NO is 25.7 µM for DEA/NO and 5.4 µM for SPER/NO, and the DEA/ NO dimensionless slope at Z ) 1 is greater than for SPER/NO, more moles of NO are lost using DEA/NO although, as noted above, the fraction of NO lost relative to the total NO delivered is less than for SPER/NO.

As for choosing C/NO, eq 8 is properly scaled to obtain a dimensionless equation when all the terms not shown in parentheses are of the order-of-magnitude of 1 (represented as ∼1). For eq 8, the dimensionless terms in parentheses,

A)

( ) DNO

kNOL2

B)

(

)

4k1C/NOCO2,s kNO

C)

(

)

ENOCNOi C/NO

(12) provide an assessment as to the importance of the diffusion, reaction, and NO production terms, respectively. When properly choosing C/NO, solving the model will result in values of CNO ranging between 0 and ∼1. Thus, Figures 2-5 show that scaling NO with eq 7 was a correct choice and that the value of C/NO gave a good prediction as to the maximum NO concentration (cNO) to which cells would be exposed. Comparing the terms A, B, and C in eq 12 helps determine the proper choice for C/NO. If the entire NO production term of eq 8 is of the same order as the time derivative (which is ∼1), then ENOCNOi/C/NO is ∼1 since e-T is ∼1. Thus, a possible choice for C/NO is ENOCNOi with B ) 4k1ENOCNOiCO2,s/kNO and C ) 1. For the base case of this study (CNOi ) 100 µM, L ) 3 mm), the values of A and B are 0.07 and 34 for DEA/NO and 1.9 and 1006 for SPER/NO, respectively. The large value for B signifies an incorrect scaling parameter for C/NO since the entire NO production term (assumed ∼1) appears negligible compared to the entire reaction term (.∼1), although the NO production term is significant during T e 2. It should be noted that for extremely fast-releasing NO donors (kNO . than values in this study such that B , 1), choosing C/NO as ENOCNOi would be appropriate since the NO production term would be dominant as required. Under these circumstances, the NO donor would rapidly release the maximum amount of NO (ENOCNOi) before the reaction or diffusion began to play a significant role. A second approach for determining C/NO would be scaling the NO production and diffusion terms to the same order (i.e., A ≈ C) such that

C/NO )

ENOCNOikNOL2 DNO

(13)

For the base case, the values of A, B, and C are 0.07, 470, and 0.07 for DEA/NO and 1.9, 533, and 1.9 for SPER/ NO, respectively. Again, the large value of B compared to C signifies an incorrect scaling parameter for C/NO. For an extremely slow-releasing NO donor (e.g., kNO ) 10-3 x kNO for SPER/NO with all other parameters the same), the values of A, B, and C would be 1900, 533, and 1900, respectively. For this scenario, eq 13 would be the proper scaling for C/NO since the NO production term would be a dominant term. Note that diffusion would be more dominant than reaction for the disappearance of NO. A final approach for determining C/NO would be scaling the NO production and reaction terms to the same order (i.e., B ≈ C) such that

C/NO )

x

kNOENOCNOi 4k1CO2,s

)

x

R(0) 4k1CO2,s

(14)

Modeling NO Delivery from NO Donors

For the base case, the values of A, B, and C are 0.07, 5.8, and 5.8 for DEA/NO and 1.9, 32, and 32 for SPER/ NO, respectively. Note that the reaction (B) is more dominant than diffusion (A) for the disappearance of NO. Since the NO production term remains dominant for the NO donors modeled in this study, eq 14 is the proper scaling parameter for C/NO used in eqs 8 and 9. For DEA/ NO, the ratio of C/A is 83, whereas the ratio is 17 for SPER/NO. The higher the ratio, the more uniform the NO concentration will be throughout the entire solution. Figure 2 demonstrates this principle. If the value for kNO becomes too small (i.e., a very slow NO-releasing donor), the value for A would become greater than C and the scaling would not be appropriate (eq 13 should be used instead). As is evident, the scaling parameter C/NO depends greatly on the values of many parameters. An important point is that the NO production term must be a dominant term when assessing the NO concentration during significant NO release (T e 2). The values of A (representing diffusion) and B (representing reaction), as compared to C (representing NO production), will provide an estimate as to the importance of the diffusion and reaction processes relative to the NO production process. When C/NO is properly chosen, C/NO provides an estimate of the maximum order of magnitude for the NO concentration. A brief mention on the scaling parameter L is of necessity. The value of A in eq 12 represents the ratio of NO diffusion to the NO release rate. The media appears well-mixed over the majority of the height when A , 1 and the media is essentially stagnant when A . 1. As noted above for the base case, A is 0.07 for DEA/NO and 1.9 for SPER/NO when the liquid height (3 mm) is used for the value of L. When A < 1, forcing A ) DNO/(kNOL2) to be ∼1 [i.e., L is ≈(DNO/kNOD)0.5] provides an estimate as to the distance (L) to which spatial NO concentration changes would primarily occur in the media (the changes would be near the liquid/gas interface). For the base case, the predicted L is 0.8 and 4.2 mm for DEA/NO and SPER/ NO, respectively. Since the value of L cannot physically be greater than 3 mm, spatial concentration changes of NO for SPER/NO would occur over the entire liquid height. This is consistent with Figure 4. However, spatial concentration changes for DEA/NO would primarily occur over 27% (0.8/3) of the liquid height, which is consistent with Figure 3. In summary, the controlled and constant delivery of NO through chemical methods using NO donors can lead to concentration gradients, NO loss to the headspace, and changing exposure concentrations in an experimental system. In such systems, care must be taken in the interpretation of NO effects on biological systems. The mathematical model described above can be used to estimate the NO concentration gradients with time and the NO loss to the headspace in various regions of a system and thus eliminate the need of complex in situ measurements of the NO concentration. The proper choice of C/NO provides an estimate of the maximum NO concentration that will exist in the solution. Even though the present model was applied to NONOates, the spatiotemporal profiles can be estimated using eqs 5 and 6 for other NO donors if the NO release rate kinetics of NO donors are known. However, care must be taken as the NO release rate kinetics of NO donors may vary depending on the experimental conditions, such as pH and

Chem. Res. Toxicol., Vol. 16, No. 1, 2003 13

temperature (24). The model presented in this study is expected to overestimate the NO concentrations in systems where the consumption of NO is not solely due to the reaction with O2. Example of these conditions include the presence of heme proteins and superoxide, the latter which is released by SIN-1, a widely used NO donor compound. In addition, biological systems may not be completely stagnant which could lead to a more homogeneous concentration of NO and O2 in the system than the estimated concentrations in this study. As demonstrated, however, the potential for concentration gradients and loss of NO to the headspace may exist in many experimental studies involving NO donors in stagnant solutions and care must be taken when applying a quantitative evaluation to the results.

Acknowledgment. This work was supported by a grant from the National Institute of Health (R15DK51327).

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