Using mathematical modeling to infer the valence state of arsenicals in tissues: a PBPK model for Dimethylarsinic Acid (DMA^V) and Dimethylarsinous Acid (DMA^III) in Mice

Abstract

Chronic exposure to inorganic arsenic (iAs), a contaminant of water and food supplies, is associated with many adverse health effects. A notable feature of iAs metabolism is sequential methylation reactions which produce mono- and di-methylated arsenicals that can contain arsenic in either the trivalent (III) or pentavalent (V) valence states. Because methylated arsenicals containing trivalent arsenic are more potent toxicants than their pentavalent counterparts, the ability to distinguish between the +3 and +5 valence states is a crucial property for physiologically based pharmacokinetic (PBPK) models of arsenicals to possess if they are to be of use in risk assessment. Unfortunately, current analytic techniques for quantifying arsenicals in tissues disrupt the valence state; hence, pharmacokinetic studies in animals, used for model calibration, only reliably provide data on the sum of the +3 and +5 valence forms of a given metabolite. In this paper we show how mathematical modeling can be used to overcome this obstacle and present a PBPK model for the dimethylated metabolite of iAs, which exists as either dimethylarsinous acid, (CH₃)₂As^IIIOH (abbreviated DMA^III) or dimethylarsinic acid, (CH₃)₂As^V(O)OH (abbreviated DMA^V). The model distinguishes these two forms and sets a lower bound on how much of an organ’s DMA burden is present in the more reactive and toxic trivalent valence state. We conjoin the PBPK model to a simple model for DMA^III-induced oxidative stress in liver and use this extended model to predict cytotoxicity in liver in response to the high oral dose of DMA^V. The model incorporates mechanistic details derived from in vitro studies and is iteratively calibrated with lumped-valence-state PK data for intravenous or oral dosing with DMA^V. Model formulation leads us to predict that orally administered DMA^V undergoes extensive reduction in the gastrointestinal (GI) tract to the more toxic trivalent DMA^III.

INTRODUCTION

Although best known for its role at acute doses as a metabolic poison, chronic sub-acute exposure to inorganic arsenic (iAs) increases the risk of skin, bladder, and lung cancers in humans (IARC, 2004), as well as the risk of diabetes and other pathologies (Abernathy et al., 2003). A common feature of arsenic metabolism in organisms as diverse as bacteria and humans is the conversion of inorganic arsenic (iAs) into methylated metabolites. In mammals, reactions that convert iAs into methylated metabolites are catalyzed by an S-adenosylmethionine-dependent arsenic methyltransferase, arsenic (+3 oxidation state) methyltransferase (As3mt) (Thomas and Rosen, 2013). The profile of arsenicals in urine from humans exposed to iAs reflects the extensive conversion of iAs to methylated metabolites: in a U.S. adult population exposed to iAs in drinking water, iAs accounted for 4.4 to 37.9% (1^st and 99^th percentiles), monomethylated arsenic for 5.9 to 29.3%, and dimethylated arsenic for 47.4 to 86.0% of the arsenic in urine (Hudgens et al., 2016).

In both inorganic and methylated species, arsenic can occur in a trivalent (As^III) or pentavalent (As^V) state. Urine from individuals exposed to iAs contains arsenate (iAs^V), arsenite (iAs^III), monomethylarsonic acid (MMA^V), monomethylarsonous acid (MMA^III), dimethylarsinic acid (DMA^V), and dimethylarsinous acid (DMA^III) (Aposhian et al., 2000; Le et al., 2000; Mandal et al., 2001; Wang et al., 2004; Valenzuela et al., 2005; Xie et al., 2006).

Trivalent methylated arsenicals have been found to be particularly toxic. In some cases, dimethylated arsenicals containing As^III exceed both iAs^III and DMA^V in potency as cytotoxins, genotoxins, and enzyme inhibitors (Styblo et al., 1997, 2000; Lin et al., 1999; Mass et al., 2001; Dopp et al., 2004, 2005; Naranmandura et al., 2011a, 2012; _Rehman et al., 2012a, b). In vitro studies have shown DMA^III to kill rat and human hepatocytes even when present in medium at concentrations less than 10 μM (Naranmandura et al., 2011b; Dopp et al., 2008), via the production of excessive levels of reactive oxygen species (ROS). In contrast, DMA^V has been found to be non-cytotoxic even at 5000 μM (Dopp et al., 2008). Hence, it is crucial that a physiologically based pharmacokinetic (PBPK) model treat the two valence states separately. Unfortunately, experimental techniques available to quantify arsenicals in tissues disturb the valence state, limiting data available for model calibration.

In the current work, we show how mathematical modeling can be used to partially overcome this obstacle for DMA. We present a PBPK model in mouse which provides a lower bound on the concentration of DMA^III and an upper bound on the concentration of DMA^V in liver, kidney, urinary bladder, and lung after oral or intravenous dosing with DMA^III or DMA^V. Our model can be extended to pharmacodynamics applications. To illustrate this, we formulate a simple model for baseline and DMA^III-induced generation of hydrogen peroxide (H₂O₂), a major ROS, in hepatocytes, using the results of in vitro experiments investigating DMA^III cytotoxicity (Dopp et al., 2008). We then conjoin the H₂O₂ model to our PBPK model and predict H₂O₂ levels and cell death in mouse liver after the high oral dose of DMA^V.

Although the model distinguishes DMA valence state, it was calibrated on lumped-valence-state PK data for arsenicals measured in liver, kidney, lung, urinary bladder, red blood cells, plasma, urine, and feces of mice that received DMA^V intravenously (Hughes and Kenyon, 1998; Hughes et al., 2000) or orally (Hughes et al., 2008). We briefly discuss how our modeling approach differs from that taken by Evans et al., who also used these data sets to construct a PBPK model in mouse for orally and intravenously administered DMA^V (Evans et al., 2008).

Evans et al. (2008) assumed no reduction of DMA^V in any compartment. Hence, intravenously or orally administered DMA^V remains in this valence form. A single compartment was used for the GI tract and it was assumed that all of an oral dose of DMA^V is absorbed. First-order kinetics were used for all terms and biliary excretion to feces was neglected.

The data sets indicate that less than 5% of intravenously (IV) administered DMA^V is excreted into feces, hence it is acceptable to neglect biliary excretion. However, the fecal data for oral gavage show a greater than 50% excretion of DMA^V-derived species into feces, indicating incomplete absorption. Furthermore, the greater the oral dose, the larger the percentage found in feces, indicating saturable absorption kinetics.

Our model uses two compartments to model the GI tract. Doses begin in the first compartment and, if not absorbed, proceed to the second compartment; material not absorbed from the second compartment undergoes fecal excretion. Saturable (Michaelis-Menten) kinetics are used to describe absorption of DMA^V from both GI compartments. The GI tract is modeled as it is using two compartments because orally ingested DMA is not immediately subject to fecal excretion, as it needs to progress through the GI tract.

Including fecal excretion led to the hypothesis, during fitting to the oral data sets, that a significant portion of orally administered DMA^V undergoes reduction in the GI tract to the more toxic and rapidly absorbed trivalent form, DMA^III. Such reduction is likely in light of experimental work (Calatayud et al., 2013). We model this reduction using Michaelis-Menten kinetics, as this is reasonable for a process which may be catalyzed by gut bacteria (Rowland and Davies, 1981). Hence, our model predicts that an oral dose of DMA^V results in both DMA^V and DMA^III entering the bloodstream.

MATERIALS AND METHODS

Pharmacokinetic studies in mice

Three pharmacokinetic studies reported by Hughes and coworkers were used for model development and calibration. A 2000 study (Hughes et al., 2000) investigated the distribution of radiolabeled dimethylarsinic acid ([¹⁴C]-DMA^V) administered intravenously to adult female B6C3F1 mice at two dosage levels (0.6 and 60 mg elemental As/kg). Mice were killed via cardiac puncture at intervals up to 2 hours post dosing. Concentration–time profiles of DMA^V-derived radioactivity in liver, kidney, lung, red blood cells, and blood plasma were reported. Because neither methylated nor demethylated products of DMA^V were detected in this study, tissue concentration-time course data represent solely dimethylated arsenicals.

A 1998 study (Hughes and Kenyon, 1998) using the same dosing protocol in adult female B6C3F1 mice reported cumulative percent dose of DMA^V-derived radioactivity in urine at various time points over 24 hours post dosing. Because less than 5% of the dose was excreted in feces within 24 hours of dosing, we chose to neglect fecal excretion for intravenously administered DMA^V in model construction.

Hughes et al. (2008) investigated the distribution of orally administered (gavage) radiolabeled dimethylarsinic acid ([¹⁴C]-DMA^V) in adult female B6C3F1 mice at dosage levels of 0.6 and 60 mg elemental As/kg. Mice were housed in individual metabolism cages to collect urine and feces and killed by cardiac puncture at intervals up to 24 hours. Concentration–time profiles of DMA^V-derived radioactivity in liver, kidney, lung, urinary bladder, and whole blood were reported, as were cumulative percent dose in urine and feces. In the same study, arsenic speciation analyses were performed in 24-hour urine collections from mice that received unlabeled DMA^V at the 0.6 and 60 mg elemental As/kg dosage levels. For both dosage levels, urine contained DMA^V, DMA^III, and trimethylarsine oxide (TMAO); TMAO is a trimethylated metabolite which is only produced in minor quantity in mice.

PBPK model for DMA

The PBPK model consists of ten compartments for both DMA^V and DMA^III: a compartment representing the stomach and upper small intestine, a compartment representing the lower small intestine and the cecum, and compartments representing the blood plasma, the red blood cells, the liver, the kidneys, the lungs, the urinary bladder, the richly perfused tissues, and the slowly perfused tissues. In our model, the arterial and venous blood are combined. In addition to the compartments for DMA^V and DMA^III, there is a plasma compartment for TMAO. A schematic of the PBPK model is shown in Figure 1 and model parameters are given in Tables 1 and 2.

Figure 1.

Schematic diagram for the DMA^III/DMA^V PBPK model.

Table 1.

Physiological model parameters.

Parameters	Values	References
Body weight, BW (kg)	0.025	mean weight in Hughes et al. experiments
Cardiac output, QCC (L/h/kg0.75)	16.5	Brown et al., 1997, p. 454
Cardiac plasma output, QCPC (L/h/kg0.75)	9.0750	Equal to QCC(1-hem)
Hematocrit (hem)	0.45	Hedrich, 2004
Blood flows (fraction of cardiac output, dimensionless)
Liver (Qliverf)	0.161	Brown et al., 1997, Table 23
Kidneys (Qkidneyf)	0.091	Brown et al., 1997, Table 23
Lungs (Qlungf)	0.005	Brown et al., 1997, Table 23
Urinary bladder (QUBf)	0.0033	Stott et al., 1983, Table 1
Richly perfused (QRf)	0.4997	Calculated to balance
Slowly perfused (QSf)	0.24	Arms & Travis, 1988, 4–35
Tissue volumes (fraction of body weight, dimensionless)
Blood (Vbloodf)	0.049	Brown et al., 1997, Table 21
Liver (Vliverf)	0.055	Brown et al., 1997, Table 21
Kidneys (Vkidneyf)	0.017	Brown et al., 1997, Table 21
Lungs (Vlungf)	0.007	Brown et al., 1997, Table 21
Urinary bladder (VUBf)	0.0009	Stott et al., 1983, Table 2
Richly perfused (VRf)	0.0101	Arms & Travis, 1988, 4–11 (sum of liver and richly perfused with liver, kidneys, lungs, and urinary bladder subtracted out)
Slowly perfused (VSf)	0.771	Calculated to balance after subtracting out 9% for skeletal tissue; close to Arms & Travis, 1988, 4–11
Capillary volumes in tissues (fraction of tissue volume, dimensionless)
Fraction of liver volume that is blood (Vcapliverf)	0.31	Brown et al., 1997, Table 30
Fraction of kidney volume that is blood (Vcapkidneyf)	0.24	Brown et al., 1997, Table 30
Fraction of lung volume that is blood (Vcaplungf)	0.5	Brown et al., 1997, Table 30
Fraction of urinary bladder volume that is blood (VcapUBf)	0.03	Brown et al., 1997, Table 30 (assumed equal to adrenal)
Fraction of slowly perfused volume that is blood (VcapSf)	0.03	Brown et al., 1997, Table 30 (assumed equal to skin)
Fraction of richly perfused volume that is blood (VcapRf)	0.31	assumed equal to liver

Table 2.

Chemical-specific model parameters (see text). Tissue-to-plasma distribution ratios for DMA^V are consistent with Dong et al. (2016).

Parameters	Values
Tissue-to-plasma distribution ratios for DMAV (unitless)
Liver (Pliver5)	1
Kidneys (Pkidney5)	6
Lungs (Plung5)	2
Urinary bladder (PUB5)	3
Richly perfused tissues (PR5)	2
Slowly perfused tissues (PS5)	5
Tissue-to-plasma distribution ratios for DMAIII (unitless)
Liver (Pliver3)	100
Kidneys (Pkidney3)	600
Lungs (Plung3)	200
Urinary bladder (PUB3)	300
Richly perfused tissues (PR3)	200
Slowly perfused tissues (PS3)	500
Permeability coefficients for DMAV (L/h/kg0.75)
Liver (πliverC5)	0.02
Kidneys (πkidneyC5)	0.2
Lungs (πlungC5)	0.001
Urinary bladder (πUBC5)	0.001
Richly perfused tissues (πRC5)	0.1
Slowly perfused tissues (πSC5)	0.3
Permeability coefficients for DMAIII (L/h/kg0.75)
Liver (πliverC3)	0.25
Kidneys (πkidneyC3)	2
Lungs (πlungC3)	0.5
Urinary bladder (πUBC3)	0.01
Richly perfused tissues (πRC3)	1
Slowly perfused tissues (πSC3)	3
Oral dosing parameters
Scaling factor for maximal rate of absorption of DMAV from stomach/upper small intestine and lower small intestine/cecum compartments (Vmax,abs5C) (μmol/h/kg0.75)	50
Scaling factor for Km for absorption of DMAV from stomach/upper small intestine and lower small intestine/cecum compartments (Km,abs5C) (μmol/kg)	2000
Scaling factor for maximal rate of reduction of DMAV to DMAIII in stomach/upper small intestine and lower small intestine/cecum compartments (Vmax,redC) (μmol/h/kg0.75)	120
Scaling factor for Km for reduction of DMAV to DMAIII in stomach/upper small intestine and lower small intestine/cecum compartments (Km,redC) (μmol/kg)	800
First-order rate constant for absorption of DMAIII from stomach/upper small intestine and lower small intestine/cecum compartments (kabs3C) (h−1·kg0.25)	1
First-order rate constant for methylation of DMAIII in lower small intestine/cecum compartment to TMAO appearing in plasma (k3→TMAOC) (h−1·kg0.25)	1
First-order rate constant for passage of DMAV and DMAIII from stomach/upper small intestine compartment to lower small intestine/cecum compartment (ktoGI2C) (h−1·kg0.25)	0.25
First-order rate constant for removal of DMAV and DMAIII from lower small intestine/cecum compartment to feces (ktofecesC) (h−1·kg0.25)	0.4
First-order rate constants for oxidation of DMAIII to DMAV in tissues (L/h/kg0.75)
Liver (kliver3→5C)	0.1
Kidneys (kkidney3→5C)	0.03
Lungs (klung3→5C)	0.01
Urinary bladder (kUB3→5C)	0.002
Richly perfused tissues (kR3→5C)	0.02
Slowly perfused tissues (kS3→5C)	2
Red blood cells (kRBC3→5C)	0.04
First-order rate constants for exchange of DMAV and DMAIII between red blood cells and plasma (L/h/kg0.75)
Uptake of DMAV in plasma by red blood cells (kplasmatorbc5C)	0.008
Release of DMAV in red blood cells to plasma (krbctoplasma5C)	0.02
Uptake of DMAIII in plasma by red blood cells (kplasmatorbc3C)	0.2
Release of DMAIII in red blood cells to plasma (krbctoplasma3C)	0.03
First-order rate constants for urinary elimination (L/h/kg0.75)
Excretion of TMAO from plasma to urine (kTMAO,urineC)	0.3
Excretion of DMAIII from kidney capillaries to urine (kDMA3,urineC)	1
Excretion of DMAV from kidney capillaries to urine (kDMA5,urineC)	1

For the two compartments modeling the gastrointestinal (GI) tract, the PBPK model equations specify the time rate of change in the amount of DMA^V or DMA^III in the compartment (units of μmol per hour). For all other compartments, the equations specify the time rate of change in the concentrations (μM per hour). All rate equations are statements of mass balance. The rate of increase in the amount or concentration of a chemical in a compartment (or capillary or cellular portion of a compartment, as described later) is the sum of the rates at which it is taken up by the compartment and produced within it, less the sum of the rates at which it is released or lost internally by conversion to another species. Model equations were coded in MATLAB and solutions computed using the ode15s solver (MATLAB R2015b, The MathWorks Inc., Natick, MA, 2015).

The model was developed for the mouse and parameterized to enable allometric scaling to rats or humans. Hence, in the model equations which follow, where a parameter x appears in an equation but Tables 1 or 2 only indicate a parameter xC, xC must be scaled by the organism body weight (0.025 kg for the mouse) to yield the parameter value x featured in the equation.

Compartments modeling the stomach/upper small intestine and lower small intestine/cecum

DMA^V administered orally begins in the compartment modeling the stomach/upper small intestine, where it (1) is absorbed at a rate governed by Michaelis-Menten kinetics, (2) undergoes reduction to DMA^III at a rate governed by Michaelis-Menten kinetics, and (3) is transferred from this compartment to the lower small intestine/cecum compartment by a process with first-order kinetics. In the lower small intestine/cecum compartment, we assume that DMA^V not only continues to be absorbed and reduced at the same rates as in the stomach/upper small intestine compartment, but is also excreted into feces.

Our choice of kinetics for DMA^V absorption is based on two observations of the mouse data sets. First, less than 5% of intravenously administered DMA^V was excreted in feces, indicating that most DMA^V found in feces after oral administration was non-absorbed material. Second, fecal excretion of orally administered DMA^V showed dose dependency, with a higher fractional excretion at the higher dose level (Hughes et al., 2008). This behavior can be modeled using saturating (Michaelis-Menten) absorption kinetics. We obtained by optimization an effective K_m and V_max for DMA^V absorption across the intestinal epithelium. We emphasize that here we take a phenomenological approach and do not make mechanistic claims. There is experimental evidence that the multidrug resistance-associated protein 4 (MRP4) functions in DMA^V efflux across the basolateral membrane of enterocytes (Roggenbeck et al., 2016), but there is insufficient data to model this transporter explicitly. A similar approach underlies our handling of DMA^III, as little is known about epithelial transporters involved in the movement of this species (Roggenbeck et al., 2016).

Bacteria in the rat gastrointestinal (GI) tract have been found to reduce arsenate (As^V) to arsenite (As^III) (Rowland and Davies, 1981), suggesting that a similar pathway may exist for DMA^V reduction. Studies in in vitro systems that mimic the functioning of the GI tract indicate that DMA^V is reduced to DMA^III (Calatayud et al., 2013). Notably, the in vitro system replicated the pH changes associated with digestive processes but lacked the microbiota of the GI tract. In light of these findings, we assume some reduction of DMA^V to DMA^III in the GI tract, as discussed below.

DMA^III appears via reduction of DMA^V in the compartment modeling the stomach/upper small intestine, where it (1) is absorbed with first-order kinetics and (2) enters the lower small intestine/cecum compartment with the same kinetics as DMA^V (i.e., first-order kinetics with the same rate constant). DMA^III in the lower small intestine/cecum compartment (1) is absorbed at the same rate as in the compartment modeling the stomach/upper small intestine, (2) undergoes methylation to TMAO with first-order kinetics, and (3) is excreted into feces with the same kinetics as DMA^V (first-order kinetics with the same rate constant). Evidence exists for the methylation of DMA^V to TMAO by the bacteria of mouse cecum (Kubachka et al., 2009). This is likely a two-step process in which reduction of DMA^V to trivalency precedes methylation. For simplicity in modeling, we do not track TMAO in the lower small intestine/cecum compartment but rather have it appear directly in the blood plasma. Therefore, the associated rate constant is a lumped parameter representing both methylation and absorption.

The model equations for the two GI tract compartments are given below. GI1₅ and GI1₃ are the quantities (in μmol) of DMA^V and DMA^III, respectively, in the compartment modeling the stomach/upper small intestine. GI2₅ and GI2₃ are the quantities of DMA^V and DMA^III in the compartment modeling the lower small intestine/cecum. Parameter definitions and values are given in Table 2. Notably, the two Km values associated with absorption and reduction of DMA^V actually have units of amount (μmol) rather than concentration. This discrepancy reflects the lack of volume estimates for the two compartments modeling the GI tract, as these compartments are phenomenological, rather than physiologically defined. Thus, the two Km values scale linearly with body volume (body weight) rather than remain constant, with scaling constants given by K_m,redC and K_m,abs5C (see Table 2).

$${GI1}_5^{\mathrm{\prime }}=\frac{{-V}_{max,abs5}{GI1}_5}{{GI1}_5+K_{m,abs5}}-\frac{V_{max,red}{GI1}_5}{{GI1}_5+K_{m,red}}-k_{toGI2}{GI1}_5$$ (1)

$${GI1}_3^{\mathrm{\prime }}=\frac{V_{max,red}{GI1}_5}{{GI1}_5+K_{m,red}}-k_{abs3}{GI1}_3-k_{toGI2}{GI1}_3$$ (2)

$${GI2}_5^{\mathrm{\prime }}=k_{toGI2}{GI1}_5-\frac{V_{max,abs5}{GI2}_5}{{GI2}_5+K_{m,abs5}}-\frac{V_{max,red}{GI2}_5}{{GI2}_5+K_{m,red}}-k_{tofeces}{GI2}_5$$ (3)

$${GI2}_3^{\mathrm{\prime }}=k_{toGI2}{GI1}_3+\frac{V_{max,red}{GI2}_5}{{GI2}_5+K_{m,red}}-k_{abs3}{GI2}_3-k_{tofeces}{GI2}_3-k_{3\to TMAO}{GI2}_3$$ (4)

Liver, kidneys, lungs, urinary bladder, and slowly and richly perfused compartments

Because DMA^V is poorly membrane-permeable (Dopp et al., 2004, 2005; Naranmandura et al., 2011a), we model its pharmacokinetics as diffusion-limited rather than flow-limited. This requires us to model separately the cellular and capillary portions of each organ compartment (including the slowly and richly perfused compartments), and to have a separate equation for each dictating the time rate of change in DMA^V concentration. Cellular uptake of DMA^III is much more rapid (Dopp et al., 2004, 2005; Naranmandura et al., 2011a) and in the absence of oxidation, flow-limited kinetics would probably be appropriate. However, Hippler et al. (2011) found that MMA^III taken up by cells is rapidly oxidized to MMA^V, likely by lysosomal degradation of MMA^III-containing complexes. We assume the same process could oxidize DMA^III in cells. We model the intracellular oxidation of DMA^III to DMA^V with a first-order term, with rate constant k_3→5. We do not model a reduction of DMA^V inside cells because there is little information on the quantitative significance of this reaction. By neglecting reduction, we create a model that provides a lower bound on the concentration of DMA^III and an upper bound on the concentration of DMA^V in an organ. For ease of coupling the equations for DMA^III with those for DMA^V, we also use diffusion-limited kinetics for DMA^III.

DMA^V and DMA^III enter the capillary portion of an organ from the plasma compartment (a lumped arterial/venous compartment) and exit via venous drainage, with exchange occurring between the capillary and cellular portions of that organ. For a given organ with volume V=BW∙V_f, where BW is body weight (0.025 kg for the mouse) and V_f is the fraction of body weight constituted by the organ, the volume of the capillary portion is V_cap=V·V_capf and the volume of the cellular portion is V_cell=V·(1-V_capf). Within the capillaries, only species dissolved in the plasma (rather than associated with the red blood cells) can be immediately taken up by the cells of an organ. Hence, it is actually the capillary plasma concentration that is modeled. We model the plasma portion of the capillaries as having volume (1-hem)V_cap, where hem is the hematocrit value (0.45 for mouse). The plasma flow Q into an organ is Q=Qf∙Q_CP, where Qf is the fraction of cardiac output received by the organ and Q_CP is the plasma portion of the cardiac output, related to the whole-blood cardiac output Q_C via Q_CP =Q_C(1-hem).

With these considerations in mind, the liver, kidneys, urinary bladder, lungs, and slowly and richly perfused compartments each have four equations of the form given below governing the time rates of change in the concentrations of DMA^III and DMA^V, with the following organ-specific modifications: the terms describing loss of DMA^III and DMA^V from the two GI tract compartments to systemic absorption appear as additional input terms to the liver capillary equations, and the kidney capillary equations feature additional terms describing loss of DMA^III and DMA^V to urine. We model the urinary loss using first-order kinetics and the same rate constant for both valence states.

$${Cap}_5^{\mathrm{\prime }}=\frac{({Pl}_5-{Cap}_5)Q-{\pi }_5\left({Cap}_5-\frac{{Cell}_5}{P_5}\right)}{(1-hem)V_{cap}}$$ (5)

$${Cell}_5^{\mathrm{\prime }}=\frac{{\pi }_5\left({Cap}_5-\frac{{Cell}_5}{P_5}\right)+k_{3\to 5}{Cell}_3}{V_{cell}}$$ (6)

$${Cap}_3^{\mathrm{\prime }}=\frac{({Pl}_3-{Cap}_3)Q-{\pi }_3\left({Cap}_3-\frac{{Cell}_3}{P_3}\right)}{(1-hem)V_{cap}}$$ (7)

$${Cell}_3^{\mathrm{\prime }}=\frac{{\pi }_3\left({Cap}_3-\frac{{Cell}_3}{P_3}\right)-k_{3\to 5}{Cell}_3}{V_{cell}}$$ (8)

Cap₅ and Cell₅ are the concentrations of DMA^V in the capillary plasma and cellular portions of an organ, respectively; Cap₃ and Cell₃ are the concentrations of DMA^III in the capillary plasma and cellular portions of an organ, respectively. Pl₅ and Pl₃ are the concentrations of DMA^V and DMA^III in the plasma compartment. Permeability coefficients are denoted by π and partition coefficients by P, with subscript indicating valence state. We note that, with the exception of hem, the parameters featured in the above equations are different for each organ compartment (see Tables 1 and 2).

TMAO is present in very small quantities and is taken up even more slowly than DMA^V (Dopp et al., 2004); hence, we do not model its uptake or release by organs. Rather, TMAO created in the lower small intestine/cecum compartment appears directly in the blood plasma, where it is subject to urinary loss (see equations in the next section). Unlike the rat enzyme, mouse liver As3mt does not efficiently metabolize DMA^III to TMAO (Fomenko et al., 2007) and we neglect this reaction in our model.

Plasma and red blood cell compartments

The equations governing arsenical concentrations in the blood plasma and red blood cells are given below. RBC₃ and RBC₅ are the concentrations of DMA^III and DMA^V, respectively, in the red blood cell compartment. TMAO is the concentration of trimethylarsine oxide in the plasma compartment.

$$\begin{gathered} {Pl}_5^{\mathrm{\prime }}=\frac{{-k}_{plasmatorbc5}{Pl}_5+k_{rbctoplasma5}{RBC}_5{+Q}_{liver}{Cap}_{liver5}{+Q}_{kidney}{Cap}_{kidney5}}{\left(1-hem\right)V_{blood}}+\cdots \\ \frac{Q_{lung}{Cap}_{lung5}+Q_{UB}{Cap}_{UB5}+{Q_{slowly}Cap}_{slowly5}+Q_{richly}{Cap}_{richly5}-Q_{CP}{Pl}_5}{\left(1-hem\right)V_{blood}} \end{gathered}$$ (9)

$${RBC}_5^{\mathrm{\prime }}=\frac{{k_{RBC3\to 5}{RBC}_3+k}_{plasmatorbc5}{Pl}_5-k_{rbctoplasma5}{RBC}_5}{hem{\bullet V}_{blood}}$$ (10)

$$\begin{gathered} {Pl}_3^{\mathrm{\prime }}=\frac{{-k}_{plasmatorbc3}{Pl}_3+k_{rbctoplasma3}{RBC}_3+Q_{liver}{Cap}_{liver3}{+Q}_{kidney}{Cap}_{kidney3}}{\left(1-hem\right)V_{blood}}+\cdots \\ \frac{Q_{lung}{Cap}_{lung3}+Q_{UB}{Cap}_{UB3}+{Q_{slowly}Cap}_{slowly3}+Q_{richly}{Cap}_{richly3}-Q_{CP}{Pl}_3}{\left(1-hem\right)V_{blood}} \end{gathered}$$ (11)

$${RBC}_3^{\mathrm{\prime }}=\frac{k_{plasmatorbc3}{Pl}_3-k_{rbctoplasma3}{RBC}_3{-k}_{RBC3\to 5}{RBC}_3}{hem\bullet V_{blood}}$$ (12)

$$TMA{O}^{\mathrm{\prime }}=\frac{k_{3\to TMAO}{GI2}_3-k_{TMAO,urine}TMAO}{\left(1-hem\right)V_{blood}}$$ (13)

The volume of the plasma compartment is given by (1-hem)V_blood and that of the red blood cells by hem∙V_blood, where V_blood is the volume of whole blood. DMA^V and DMA^III leaving organ capillaries via venous drainage enter the plasma compartment (a lumped venous/arterial compartment). DMA^V and DMA^III leave the plasma compartment and enter organ capillaries via arterial flow. We model diffusion of DMA^V and DMA^III between plasma and red blood cells. In addition, oxidation of DMA^III to DMA^V in red blood cells (Shiobara et al., 2001) is modeled. TMAO created in the lower small intestine/cecum compartment appears in the blood plasma and is subject to urinary excretion. Due to its slow cellular uptake and lack of pharmacokinetic data, we do not model its cellular uptake or exchange with red blood cells.

Parameterization and calibration

Values and references for the physiological parameters are given in Table 1. Chemical-specific parameter values were determined by fitting of model solutions to the PK data sets, under the assumption of mouse weight 0.025 kg. We term this fitting process “model calibration.” Chemical-specific parameter values are given in Table 2.

In the mouse studies, levels of DMA^V-derived species in organs were reported as μg elemental As per liter. In our model, concentrations are computed in units of μmol per liter. Therefore, model solutions for concentrations in organs were scaled by 75 g/mol, the molecular weight of elemental arsenic. To determine the concentration of DMA^V-derived species in an organ for comparison with experimental data, we summed the amounts of DMA^V and DMA^III in the cellular and capillary portions of the organ and divided by the total organ volume. That is, we did not assume that all arsenicals measured to be in an organ were intracellular, but that some portion of the total organ As burden resides in the capillaries. This approach is reasonable for a chemical with diffusion-limited kinetics, such as DMA^V, and was consistent with the use of non-perfused mouse tissues for arsenical analysis.

In the mouse studies, DMA^V-derived species measured in cumulative urine and feces at various time points were expressed as percent administered dose. We converted these data to units of μg elemental As, using average mouse body weight of 0.025 kg and administered dosage levels of 0.6 or 60 mg As/kg. In our model, DMA^V-derived species (DMA^V, DMA^III, and TMAO) in cumulative urine and feces are computed in units of μmol; these were scaled by the molecular weight of elemental As for comparison with experimental data. In the urine speciation portion of the mouse studies, concentrations of DMA^V, DMA^III, and TMAO in urine were reported in units of μg As per ml. For comparison with model output, we scaled these data points by the average 24-hour urinary output of a mouse of 1.6 ml (Green, 1966).

In determining chemical-specific parameter values by optimization, we adhered to the following constraints, informed by in vitro studies. First, for each organ, the permeability coefficient for DMA^III was required to be at least an order of magnitude greater than that for DMA^V. This is consistent with studies of cellular uptake of trivalent and pentavalent arsenicals (Dopp et al., 2004). Second, for each organ, the distribution ratio for DMA^III was required to be approximately 100 times greater than that for DMA^V. 99% of intracellular iAs^III is bound to sulfhydryl groups (Kitchin, 2011) and if we assume a similar figure holds for DMA^III, only 1% of DMA^III in a cell would be available for release at any given time. This assumption results in a DMA^III distribution ratio that is 100 times greater than that for DMA^V, which is assumed to be mainly free. Third, the rate of absorption k_abs3 of DMA^III from the GI compartments was required to be an order of magnitude greater than V_max,abs5/K_m,abs5, consistent with in vitro studies aimed at characterizing the intestinal absorption of DMA^III and DMA^V (Calatayud et al., 2010; Calatayud et al., 2011).

The model was calibrated iteratively. In the first step, permeability coefficients and distribution ratios for DMA^V in the compartments, rate constants for the diffusion of DMA^V between plasma and red blood cells, and the urinary excretion rate constant for DMA^V were determined by fitting to the intravenous data sets. We found that an excellent fit of the intravenous data sets could be obtained by assuming no reduction of DMA^V occurs outside of the gastrointestinal tract, thus reducing the intravenous case to a DMA^V-only submodel. There is a paucity of reliable data on the relative quantities of DMA^III and DMA^V reached at equilibrium inside of cells, and making this assumption yields a useful model in that the model provides a lower bound on the DMA^III concentration and an upper bound on the DMA^V concentration in an organ. After obtaining these parameters, remaining parameters were obtained through fitting to the oral data sets.

Sensitivity analysis

MATLAB code was written to perform a local sensitivity analysis. For each chemical-specific model parameter, the normalized sensitivity coefficient (NSC) was calculated using the following equation, while all other parameters remained at their original values.

$$NSC=\frac{\frac{M_i-O}{O}}{\frac{P_i-P}{P}}$$ (14)

Here, P is the original parameter value, O is the model output resulting from the original parameter value, P_i is the parameter value increased by 1%, and M_i is the model output resulting from the new parameter value. For bolus dosing protocols, such as those considered here, model outputs are not at steady state; hence, the NSC for each parameter is actually a function of time. Parameters with NSCs exceeding 0.1 in absolute value in some compartment at some time point are classified as sensitive.

In performing the sensitivity analysis for the DMA^V distribution ratios, DMA^V permeability coefficients, rate constants for the diffusion of DMA^V between plasma and red blood cells, and the urinary excretion rate constant for DMA^V, the model outputs considered were the concentration-time courses in plasma, red blood cells, lungs, liver, kidneys, and cumulative amount in urine of total DMA^V-derived species resulting from intravenous dosing with the low and high doses of DMA^V. An important function of a sensitivity analysis is to determine which data would be most helpful in identifying the value of a given model parameter (e.g., concentration at a given time point in a given organ). This process is as effective as the model is accurate; hence, we chose to do the sensitivity analysis for these parameters using only intravenous dosing with DMA^V as this approach effectively reduces the PBPK model to a DMA^V-only submodel, minimizing inaccuracy caused by errors in the model’s handling of DMA^III as well as its absorption from gut.

For the remaining parameters, the model outputs used for the sensitivity analysis were the concentration-time courses in whole blood, liver, kidneys, urinary bladder, lungs, and cumulative amount in urine of total DMA^V-derived species resulting from oral dosing with the high and low doses of DMA^V. Because these parameters have no impact on model output for intravenously administered DMA^V, only oral doses were used for this part of the sensitivity analysis.

Pharmacodynamic application: DMA^III-induced oxidative stress

We model the baseline production and removal of hydrogen peroxide in liver cells with a simple linear equation, with parameters Vh2o2prod and k_h2o2. Sies (2014) gives a value of 50 nnol/min·g tissue for the rate of production of H₂O₂ in liver. We assume that the liver is 70% water by weight and convert Sies’s value to units of μM/hr to obtain a value of 4286 μM/hr for Vh2o2prod. Sies states that the normal concentration of H₂O₂ in liver is 0.01 μM, and k_h2o2 is chosen to yield this value for the steady-state concentration. The presence of DMA^III is assumed to increase the rate of H₂O₂ production above baseline, with the strength of the effect depending upon a parameter dmaprod. Our equation for H₂O₂ is given below.

$$H_2{O}_2\mathrm{\prime }=Vh2o2prod(1+dmaprod\bullet {DMA}^{III})-k_{h2o2}{H}_2{O}_2$$ (15)

To determine the parameter value dmaprod, we created a model of an in vitro system studied by Dopp et al. (2008) in which human hepatocytes were exposed to DMA^III in medium and the concentration causing 50% cell death within 24 hours determined, referred to as the LC 50 dose; analogous studies for mouse have not been done, to the best of our knowledge. To model this system, we conjoined Equation (15) with five additional equations, four modeling DMA^III and DMA^V pharmacokinetics, and one modeling hepatocyte survival. Hepatocytes are assumed in the model to have a death rate which scales linearly with H₂O₂ concentration. The equations are given below.

$${Med}_5^{\prime }=\frac{-{\pi }_5\left({Med}_5-\frac{{Cell}_5}{P_5}\right)}{vmed}$$ (16)

$${Cell}_5^{\mathrm{\prime }}=\frac{{\pi }_5\left({Med}_5-\frac{{Cell}_5}{P_5}\right)+k_{3\to 5}{Cell}_3}{vcell}$$ (17)

$${Med}_3^{\mathrm{\prime }}=\frac{-{\pi }_3\left({Med}_3-\frac{{Cell}_3}{P_3}\right)}{vmed}$$ (18)

$${Cell}_3^{\mathrm{\prime }}=\frac{{\pi }_3\left({Med}_3-\frac{{Cell}_3}{P_3}\right)-k_{3\to 5}{Cell}_3}{vcell}$$ (19)

$$Sur{v}^{\prime }=-death·H_2{O}_2·Surv$$ (20)

Parameter values and descriptions are given in Table 3. Partition coefficients remain the same as in the PBPK model, but permeability coefficients as well as the rate constant for the oxidation of DMA^III to DMA^V are scaled by the ratio of vcell to the mouse liver tissue volume for the PBPK model. The value of the parameter death was chosen by holding H₂O₂ concentration constant at 3000 μM and finding a parameter value which results in 80% cell death after 4 hours (Surv=0.2), consistent with experiments conducted by Kanno et al. (1999) investigating the ability of H₂O₂ to induce cell death. Finally, dmaprod was chosen so that an initial DMA^III concentration in medium of 8.6 μM results in Surv=0.5 after 24 hours (Dopp et al., 2008). With the value of dmaprod determined, equations (15) and (20) were appended to the DMA PBPK mouse model and the impact on liver of the high oral dose of DMA^V investigated.

Table 3.

In vitro model parameters (see text).

Parameter	Description	Value
π3	Permeability coefficient for DMAIII	1.29429e-5 L/h
π5	Permeability coefficient for DMAV	1.03544e-6 L/h
k3→5	Rate constant for oxidation of DMAIII to DMAV	5.17718e-6 L/h
vcell	Volume of cells in culture	0.78125e-6 L
vmed	Volume of medium	499.21875e-6 L
death	Hepatocyte death rate	0.00013 μM−1h−1
dmaprod	Parameter governing increase in rate of H2O2 production over baseline, due to DMAIII	1500 μM−1
Vh2o2prod	Baseline rate of production of H2O2 in hepatocytes	4.286e3 μM h−1
kh2o2	Rate constant for H2O2 removal	4.286e5 h−1

RESULTS

Calibration data

Using the chemical-specific parameter values given in Table 2, model solutions were computed for intravenous and oral dosing of a 0.025 kg mouse with 0.6 and 60 mg As/kg [¹⁴C]-DMA^V. Figure 2 compares the total radioactivity in urine, kidneys, liver, lungs, blood plasma, and red blood cells as computed by the model, for intravenous dosing, with experimental data. Model solutions fall within a factor of 2 of most data points, taking into account error bars. Slightly more disagreement occurs for the early data points in plasma, especially for the lower dose. Here, total radioactivity in plasma computed by the model at the 0.25-hr mark is about 1/6 the observed value.

Figure 2.

Model solutions compared with experimental data (Hughes and Kenyon, 1998; Hughes et al., 2000) for intravenous dosing with [¹⁴C]-DMA^V. Total DMA^V-derived radioactivity is given in units of μ elemental As (in cumulative urine, Figure 2A) and micrograms elemental As per liter (all other graphs). High dose is 60 mg As/kg, low dose is 0.6 mg As/kg. Legend: low dose model solution (dashed line), low dose experimental data (solid dots), high dose model solution (solid line), high dose experimental data (asterisks).

Figure 3 compares model solutions for total radioactivity in urine, feces, kidneys, liver, lungs, urinary bladder, blood, and 24-hr cumulative urine speciation with experimental data from the oral dosing studies. Model solutions fall within a factor of 2 of most data points. For the lower dose, total radioactivity in urinary bladder at the 8-hr mark and kidneys at the 4-hr mark as computed by the model are both equal to about 0.25 the observed values. Early time points in feces disagree significantly with the model. However, we note that what the model actually computes is the quantity of DMA^III/V destined for fecal excretion at any given time. Given the time required for intestinal transit before excretion, this discrepancy between observed and predicted values is not unexpected.

Figure 3.

Model solutions compared with experimental data collected by Hughes et al. (2008) for oral dosing with [¹⁴C]-DMA^V (total radioactivity graphs, Fig. 3A-3G) or with unlabeled DMA^V (urine speciation graphs, Fig. 3H-3J). High dose is 60 mg As/kg, low dose is 0.6 mg As/kg. Legend: low dose model solution (dashed line), low dose experimental data (solid dots), high dose model solution (solid line), high dose experimental data (asterisks).

The fitting process provided insight into the fate of DMA^V in the GI tract. Michaelis-Menten kinetics were assumed for both the absorption of DMA^V into the bloodstream and its reduction in the GI tract to DMA^III. We determined that the reductive process is favored, having both a higher capacity (higher Vmax) and higher affinity (lower Km) than the absorptive process. This result indicates that orally ingested DMA^V is extensively reduced in the GI tract to the more toxic trivalent species before being absorbed. If we do not assume extensive reduction of DMA^V in the GI tract, then model solutions greatly under-predict experimental data in all organs. The assumption of no reduction is equivalent, in terms of model structure, to simplifying our approach by using a single state variable which represents the sum of both DMA^III and DMA^V. Figure 4 shows that making the trivalent/pentavalent distinction is crucial for obtaining a good fit to experimental data in liver; the situation in liver was representative of all organs.

Figure 4.

Model predictions for total radioactivity in liver after the high oral dose of DMA^V (60 mg As/kg). Fig. 4A was generated from a model using a single state variable to represent the sum of the +3 and +5 valence states. Fig. 4B was generated using our full model which separately tracks DMA^III and DMA^V and assumes reduction of DMA^V to DMA^III in the GI tract.

Valence-state specific tissue dosimetry predictions

A prime objective of the model is to make separate tissue dosimetry predictions for DMA^V and DMA^III. Figure 5 shows the predicted minimal amount of DMA^III and maximal amount of DMA^V in the cells of lung, urinary bladder, liver, and kidney for the larger oral dose of DMA^V (60 mg As/kg). We predict that lung cells could encounter a DMA^V concentration as high as 35,000 μg/L, while kidney cells are predicted to encounter a concentration of DMA^III exceeding 12,000 μg/L.

Figure 5.

Model predictions for DMA^V (solid line) and DMA^III (dashed line) in cellular portion of tissues (not including capillary bed) after the high oral dose of DMA^V (60 mg As/kg).

For each organ, the peak concentration for DMA^III occurs earlier than that for DMA^V. This temporal difference reflects the model assumptions that DMA^III has a higher permeability coefficient than DMA^V (hence, is taken up more rapidly) and is oxidized to DMA^V once inside the cell. The organs differ in their ratios of DMA^III to DMA^V, due to differences in permeability coefficients, distribution ratios, rates of blood flow through the organs, and organ volumes. For comparison, Fig. 6 shows predicted tissue dosimetry in liver and lung cells when the same dose of DMA^V is administered intravenously. Our model assumes no reduction of DMA^V outside of the GI tract, hence DMA^III concentration is zero. Oral and intravenous administration are predicted to yield a similar peak concentration of DMA (sum of DMA^III and DMA^V) in liver despite the fact that, in oral dosing, less than half as much DMA^V enters the system, the rest being excreted into feces. This is due to the much greater permeability of DMA^III versus DMA^V. In lung, even a larger peak concentration of DMA is achieved for oral versus IV administration. Furthermore, with oral dosing significant quantities of the more toxic DMA^III are produced. Hence, our model predicts that the oral route is significantly more toxic than the IV route.

Figure 6.

Model predictions for DMA^V (solid line) and DMA^III (dashed line) in cellular portion of liver (Fig. 6A) and lung (Fig. 6B) after intravenous injection with the high oral dose of DMA^V (60 mg As/kg).

Compare with Figures 5C and 5A, respectively, where the same dose was given orally.

Sensitivity analysis

For each chemical-specific model parameter, normalized sensitivity coefficients (as a function of time) were computed in response to either oral or intravenous dosing with DMA^V (see Methods), where the model outputs considered were total DMA^V-derived species in organ compartments and cumulative urine. When intravenous dosing was used for a given parameter, NSCs were computed for red blood cells, blood plasma, lungs, liver, kidneys, and urine. When oral dosing was used, NSCs were computed for blood, liver, kidneys, lungs, urinary bladder, and urine.

Sensitivity analyses found that the urinary excretion rate constant for DMA^V and the permeability coefficients and distribution ratios for DMA^V in liver, kidneys, and lungs were all sensitive, with NSCs exceeding 0.1 in magnitude for significant stretches of time in one or more compartments. Figure 7 provides NSCs for liver. The two parameters governing diffusion of DMA^V between red blood cells and plasma were very sensitive, with NSCs in the red blood cell compartment near 1 in magnitude for large time intervals falling within the first two hours after dosing.

Figure 7.

Normalized sensitivity coefficients in the liver compartment for the permeability coefficient (Fig. 7A) and distribution ratio (Fig. 7B) of DMA^V in liver. Computed for intravenous dosing with DMA^V (60 mg As/kg).

Sensitivity analyses found that the two parameters governing diffusion of DMA^III between red blood cells and plasma and the permeability coefficients for DMA^III in all organs except lung were sensitive. Oral dosing enabled us to identify the permeability coefficient and distribution ratio for DMA^V in urinary bladder (an organ for which concentration-time data were not reported for intravenous dosing): both these parameters had NSCs exceeding 0.1 in the urinary bladder compartment for most of the first 24 hours following oral dosing. None of the distribution ratios for DMA^III was sensitive. Parameters governing the rate of oxidation of DMA^III to DMA^V were only sensitive in kidney and liver. All parameters associated with the GI tract were sensitive, as were the rate constants for the urinary elimination of DMA^III and TMAO.

LC 50 of DMA^III in hepatocytes

The parameter governing excess H₂O₂ production induced by intracellular DMA^III (dmaprod, see Table 3) was chosen so that the in vitro model predicts the death of approximately half of hepatocytes in culture after 24 hours in response to an initial DMA^III concentration in medium of 8.6 μM, consistent with Dopp et al. (2008). Fig. 8 shows the intracellular DMA^III concentration, intracellular H₂O₂ concentration, and fraction of surviving cells computed by the in vitro model.

Figure 8.

Intracellular DMA^III concentration (left panel), intracellular H₂O₂ concentration (middle panel), and fraction of surviving cells (right panel) computed by the in vitro model for an initial DMA^III concentration in medium of 8.6 μM.

Predicted oxidative stress and cytotoxicity in mouse liver after the high oral dose of DMA^V

After determining dmaprod, Equations (15) and (20) were inserted into the DMA PBPK model, enabling us to make in vivo predictions of oxidative stress in liver and resultant liver injury in response to dosing with DMA. We predict that the high oral dose of DMA^V (60 mg As/kg) causes intracellular H₂O₂ levels to exceed 1000 μM (normal concentration is 0.01 μM) and causes the death of about 1/3 of the liver after 6 hours; see Fig. 9. Fig. 5C shows the predicted intracellular DMA^III concentration, assumed to be the causative agent. Studies of hepatectomy in mouse have determined that the liver can regenerate even after 70% removal (Hori et al., 2011). Hence, our extended PBPK/PD model is consistent with the fact that this was not a lethal dose, although it does predict significant liver cell death. We note that our value for dmaprod was determined using data on DMA^III toxicity in human hepatocytes. Model accuracy could be improved if analogous data for mouse hepatocytes became available.

Figure 9.

Intracellular H₂O₂ concentration in liver (Fig. 9A) and fraction of surviving hepatocytes (Fig. 9B) computed by the PBPK/PD model for the high oral dose of DMA^V (60 mg As/kg).

DISCUSSION

Model solutions for the sum of DMA^III and DMA^V in tissues are in excellent agreement with lumped-valence data on the distribution of arsenicals in mice after oral or intravenous dosing with DMA^V that was collected by Hughes and coworkers. Model solutions fall within a factor of two of most data points, taking into account error bars. One of the reasons we were able to achieve this level of agreement was our decision to posit that DMA^V-derived species in an organ include not only intracellular arsenicals but also arsenicals present in residual blood in the organ’s capillary bed. The assumption that arsenicals retained in an organ could be present in the capillary bed is appropriate because organs were not perfused to remove residual blood from the capillaries and DMA^V has diffusion-limited kinetics, unlike the small lipophilic molecules more typically dealt with in PBPK modeling. Hence, the model output that is compared with experimental data is the concentration of DMA^V-derived species averaged over the cellular and capillary portions of an organ.

Model formulation led to the prediction that there is extensive reduction of DMA^V to DMA^III in the GI tract. In our model, DMA^V can be reduced to DMA^III in both GI compartments, and one possible fate for the DMA^III so produced is absorption. Hence, some of an oral dose of DMA^V is ultimately taken up by tissues in the form of DMA^III_, for which the permeability coefficients are larger. We find that if we do not assume such reduction, the model greatly under-predicts the oral dosing PK data in all organs. Extensive reduction of DMA^V in the GI tract constitutes a model prediction which requires testing. It is very reasonable in light of the fact that cellular uptake of DMA^V is quite slow (Dopp et al., 2004, 2005; Naranmandura et al., 2011a) and yet DMA^V administered in water or the diet is rapidly excreted in urine. Although simulated digestion experiments (Calatayud et al., 2013) support the idea of a reduction of DMA^V in gut, additional in vitro experiments are needed, preferably ones in which the simulated digestion system not only emulates the pH changes associated with digestion, but also includes gut microbiota.

In vitro studies with different cell types to determine permeability coefficients in various organs for DMA^III and DMA^V, analogous to those conducted by Calatayud et al. (2010, 2011) for the study of intestinal transport, would do much to reduce model uncertainty. Although our values for DMA^V distribution ratios in organs were arrived at through optimization, the values we found are consistent with empirically measured partition coefficients determined in two cases of fatal human poisoning (Dong et al., 2016). This consistency supports the idea that over time most DMA^III in the body undergoes oxidation to DMA^V, and also lends support to our distribution ratio values for DMA^V. Regarding the interconversion of the two valence forms of DMA within cells, we assumed that the rapid intracellular oxidation of MMA^III to MMA^V reported by Hippler et al. (2011) also occurs for DMA^III. There are insufficient data to assess the quantitative significance of the reverse reaction, the reduction of DMA^V to DMA^III. Therefore, reduction was omitted, yielding a model which provides a lower bound on the DMA^III concentration and an upper bound on the DMA^V concentration inside cells.

We illustrated that our PBPK mouse model can be extended to pharmacodynamics applications, by conjoining it to a simple model for DMA^III-induced oxidative stress and resultant cytotoxicity in hepatocytes. Our prediction that the high oral dose of DMA^V (60 mg As/kg) caused the death of 1/3 of the liver is consistent with the facts that this was not a lethal dose and that the liver can regenerate even after 70% hepatectomy (Hori et al., 2011).

To summarize, we have created the first PBPK model for a methylated arsenical which makes separate tissue dosimetry predictions for its +3 and +5 valence forms, models systemic circulation of both valence forms, and accommodates dosing with both valence forms. This ability is crucial because trivalent methylated arsenicals are now believed to play a large role in toxicity resulting from chronic exposure to iAs. Creating a valence-state distinguishing PBPK model for an arsenical is not a straightforward task because current analytic techniques for quantifying arsenicals in tissues disturb the valence state, creating a shortage of data for model calibration. Nevertheless, by making reasonable assumptions about where reduction of DMA^V occurs in the body and utilizing the findings of in vitro studies we constructed a model which provides a lower bound on the concentration of DMA^III and an upper bound on the concentration of DMA^V in tissues of interest.