Pairing Food and Drink: a Mathematical Model of Blood Ethanol Concentration

Abstract

We present a blood ethanol concentration compartment model which utilizes an animal’s ethanol intake, food intake, and weight to predict the animal’s blood ethanol concentration at any given time. By incorporating the food digestion process into the model we can predict blood ethanol concentration levels over time for a variety of drinking and eating scenarios. The model is calibrated and validated using data from cohorts of male monkeys, and is able to capture blood ethanol concentration kinetics of the monkeys from a variety of drinking behavior classifications.

1. Introduction

The long-term goal of this work is to develop a mathematical model of alcohol consumption that would predict drinking behavior. A useful predictive model requires extensive, time-dependent data, which we have obtained from laboratory studies of drinking in monkeys. Experiments with animals can provide critical insight into complex, human disorders. In particular, studying how Non-Human Primates (NHP) orally self-administer alcohol provides a remarkably stable, reproducible and physiologically accurate model for alcoholic drinking phenotypes seen in humans [1]. Rhesus macaques in our model closely mimic behavioral drinking phenomena associated with a range of Alcohol Use Disorders (AUD), including neuroanatomical, physiological, and behavioral attributes [2, 3]. Rigorous longitudinal and cross cohort analysis of these data has revealed the interactions of stress and drinking[4], identified attributes associated with future drinking [5], epigenetic characteristics [6] and brain transcriptomics [7], among numerous other physiological and behavioral studies [8, 9, 10, 11].

In addition, advanced monitoring devices, data collection techniques, and rigorous reproducible experimental paradigms allow for cross-species comparisons of highly granular data sets. Our previous work using these deep data sets provided by the Monkey Alcohol and Tissue Research Resource (MATRR, www.matrr.com) [12] allowed us to construct a mathematical model that describes drinking behavior [13]. While this model is an important first step in understanding the attributes associated with the behavior of drinking, it does not precisely describe the metabolic impact of real time Blood Ethanol Concentration (BEC) on the NHP model. Sampling BEC directly from venous puncture in laboratory animals or human subjects is costly and invasive and may consequently lead to behavioral and physiological modification of the subject. Alternatives to direct BEC measurement may provide a reasonable substitution; however, these studies are limited to humans and continue to require physical measurement, such as expired breath [14], saliva [15], transdermal devices [16], or vitreous humour [17], and rely heavily on correlating the non-blood measurement to a BEC estimation.

Approaches that rely on mathematical models provide another potential method for determining BEC based on ethanol consumption. Typical mathematical models for BEC are human-based compartment models that lead to systems of differential equations, as in [18]. In these models each variable represents the concentration of ethanol in a designated organ or system such as the vascular or digestive system. The rate at which ethanol is ingested, moves to another compartment, or is metabolized is described by a differential equation. Models range in size from three compartments, as described in [19, 20], to more complicated multi-compartment models [21]. Models that rely on fractional approaches have also been employed [22], but rely heavily on human subjects under constant levels of ethanol consumption.

Since the ultimate goal of these models is to predict the BEC at any given time after a fluctuating rate of ethanol intake, a successful model must also account for ethanol metabolism rate within the context of contributing conditions such as food intake or weight. Studies in human pharmacokinetics of ethanol provide a precedent for our understanding of rates of metabolic metabolism [16], but there is limited evidence in the literature to reconcile pharmacokinetic processes with systems models.

Here we develop a detailed compartment model for ethanol consumption in the NHP, which extends our previous work [13]. The previous model describes drinking behavior over time as a Markov process, with transition probabilities and rates assumed to be functions of physiological features of the individual monkeys. Once a monkey’s drinking is simulated, the average daily alcohol consumption is calculated to classify the monkey as a low drinker (LD), binge drinker (BD), heavy drinker (HD), or very heavy drinker (VHD) (as described in section 3.1). In particular, the previous model did not explicitly use BEC to assign drinking categories, which is critical to the definition of a “binge drinker” (BD). Our new model overcomes the omission of BEC for the prediction of BD animals by simulating a monkey’s ethanol metabolism. More generally, a validated BEC model provides the critical component required to extend existing ethanol consumption models to enhance their predictive capabilities. Understanding the interplay between ethanol consumption, food, animal physical characteristics, and ethanol metabolism moves us towards our goal of creating a whole animal model of alcohol consumption that is parameterized using each individual’s physical characteristics.

We describe the model development in Section 2, focusing on the novel aspects. The model was developed in close collaboration with experimentalists, and the data and experimental methods are described in Section 3. Details on parameter fitting and results of simulations are described in Section 4, followed by a discussion section outlining the next steps in the project.

2. Model Development

In this section we describe in detail a continuous-time model of BEC dynamics that incorporates an arbitrary input of drinks and the effect of food on the kinetics of BEC. Numerical solutions of the parameterized model can then be used to simulate a variety of drinking behaviors, and model outputs can be used to characterize these behaviors. The model consists of a system of differential equations, with time dependencies that are prescribed by the amounts and timings of food and alcohol inputs. The model is a four-compartment model that builds on the three-compartment model described in [18]. In that work, the model described the kinetics of alcohol through the system after one intravenous dose, without including the effects of food. Since our goal is to predict BEC levels during prolonged periods of drinking and eating, we add functions that describe the flux of alcohol from the gut to the vasculature which depend on the amount of food ingested, as well as the volume and timings of each drink. A schematic of the model is given in Figure 1. Additions to the previous model are the elements in the top box, where the intake and influence of food is described. Details of the new model functions and assumptions are given in this section. In contrast to [18], who fit their model to human data where one individual is given one intravenous “drink”, we fit the model to data collected from monkeys who eat and drink in controlled experiments, described in Section 3. Results from parameter fitting and simulations are discussed in Section 4.

Figure 1:

The model diagram shows the four compartments of the model: liver (μ_L), vasculature system (μ_V ), peripheral tissue (μ_T ), and gut along with the mass flux rates of alcohol between compartments i and j (denoted by M_i,j). Model elements that describe the effect of the food on the blood-ethanol concentration are shown in the upper box. The multiplier, $\mathcal{M}$ is a function of the food in the gut. The parameter r is the rate of dose delivery and the parameter δ is the digestive rate. The BEC measurement is taken directly from the vein as shown in the diagram.

2.1. Model Overview

The four compartments are liver (μ_L), peripheral tissue (μ_T), vascular system (μ_V ), and gut. Each variable represents the amount of ethanol in the corresponding compartment, while mass flux rates between compartments i and j are denoted by the time-varying functions M_i,j(t). In particular two rates, M_P,L and M_H,L from [18], indicate two separate blood supplies to the liver. H stands for hepatic artery and carries oxygen to the liver. P stands for portal vein which carries deoxygenized blood from other organs such as those we are modeling with M_gut but does carry nutrients to the liver. Ethanol enters the system orally at rate r, goes to the gut, and is absorbed in the small intestine. We assume that the time from mouth to gut is negligible and, following [18], we describe the rate at which ethanol enters the vasculature from the gut by the function M_G,V (t). We also keep track of how much food is present in the gut compartment, denoted by the variable F. Once the ethanol enters the vasculature, we assume that it is only eliminated in the liver. The novelty in this model is that we use the input rate function, M_G,V (t), to describe the effect of multiple drinks on blood ethanol content, with and without food. In the model equations, the function M_G,V is the product of two functions: $M_{G,V}^{nf}$ and $\mathcal{M}$, where $M_{G,V}^{nf}$ is given by a differential equation that depends on the amount, length, and timing of each drink, and $\mathcal{M}$ depends on the amount of food in the gut, F(t). The amount of food in the gut at time t is given by the variable F. Thus, our model consists of a system of five differential equations. The first three govern the mass of ethanol in the vasculature, liver and peripheral tissue, the fourth governs the flux rate from the gut to the vasculature, and the fifth determines the amount of food in the gut:

$$\frac{d{\mu }_V}{dt}=-M_{H,L}-M_{P,L}+M_{L,H}+M_{T,V}-M_{V,T}+M_{G,V}$$ (1)

$$\frac{d{\mu }_L}{dt}=M_{H,L}+M_{P,L}-M_{L,H}-M_{\text{metab}}$$ (2)

$$\frac{d{\mu }_T}{dt}=M_{V,T}-M_{T,V}$$ (3)

$$\frac{d{M}_{G,V}^{nf}}{dt}=Q(\overrightarrow{D},\overrightarrow{L},\overrightarrow{S})$$ (4)

$$\frac{dF}{dt}=-\delta F$$ (5)

The rate functions that appear on the right-hand sides of Equations (1), (2) and (3) are the same as those in [18], and are listed in Appendix B. The function Q which appears in the differential equation for $M_{G,V}^{nf}$ depends on the drinking data, where the i^th drink has a volume, D_i, a length, L_i and a start time, S_i. In equation (4) the vector notation indicates that the function Q depends on the entire sequence of volumes or “doses,” lengths, and start-times: $\overrightarrow{D}=\left(D_1,D_2,\dots D_n\right)$; $\overrightarrow{L}=\left(L_1,L_2,\dots L_n\right)$; $\overrightarrow{S}=\left(S_1,S_2,\dots ,S_n\right)$. Details of the derivation of this function are given in Section 2.2.

The rate of mass transfer of ethanol from the gut to the vasculature is also affected by the amount of food in the gut, represented in our model by the state variable F(t). The presence of food slows down the rate of transfer of ethanol; we model this effect by introducing the multiplier, $\mathcal{M}$, which is a function of F, so that the actual rate of transfer of ethanol from gut to vasculature is given by:

$$M_{G,V}(t)=\mathcal{M}(F(t))\cdot M_{G,V}^{nf}(t).$$ (6)

The derivation of the function $\mathcal{M}$ is given in Section 2.3.1.

2.2. Modeling ethanol intake.

In this section, we derive a differential equation for the function $M_{G,V}^{nf}$, the rate of transfer of ethanol from the gut to the vasculature in the absence of food (hence the superscript “nf,” which stands for “no food”). To incorporate the effect of multiple drinks over time, we develop a differential equation for $M_{G,V}^{nf}$.

We start with an empirically derived model suggested in a footnote in [18] which describes what happens with one drink in the absence of food. Let M_oral(t) be the rate at which ethanol is entering the system orally. The rate at which the ethanol is transferred from the gut to the vasculature can then be written as the convolution:

$$M_{G,V}^{nf}(t)=M_{\mathit{\text{oral}}}(t)*h(t)={\int }_{-\infty }^{\infty }{M}_{\mathit{\text{oral}}}(\tau )h(t-\tau )d\tau$$ (7)

where h(t) is derived experimentally as:

$$h(t)={\beta }^{2}t{e}^{-\beta t}u(t).$$ (8)

In this equation, u(t) indicates when ethanol is in the gut, which we model by a step function. So, for a given drink

$$u(t)=\{\begin{array}{ll}1\hfill & t\ge t_0\hfill \\ 0\hfill & \text{else}\hfill \end{array}$$

where t₀ is the time at which the drink starts.

In the data, ethanol intake is given by a series of discrete drinks, so we model M_oral(t) as a series of step functions. Suppose we assume that the rate of ethanol intake during one drink of length L is described as:

so M_oral(t) = r for t₀ ≤ t ≤ t₀ + L, and is zero otherwise.

Using Equation 7 with the specific functions for M_oral(t), h(t) and u(t), we get:

$$M_{G,V}^{nf}(t)=\{\begin{array}{ll}r\left(1+e^{\beta \left(t_0-t\right)}\left(\beta \left(t_0-t\right)-1\right)\right)\hfill & t_0\le t<t_0+L\hfill \\ r{e}^{-\beta t}[\beta t{e}^{\beta t_0}\left(e^{\beta L}-1\right)-\beta \left(t_0+L\right)e^{\beta \left(t_0+L\right)}\hfill & \hfill \\ +\beta t_0{e}^{\beta t_0}+e^{\beta \left(t_0+L\right)}-e^{\beta t_0}]\hfill & t_0+L\le t\hfill \\ 0\hfill & \text{else}\hfill \end{array}$$ (9)

If there are multiple drinks, then $M_{G,V}^{nf}(t)$ is the sum of the rate functions derived from each drink, under our modeling assumption that drinks do not overlap. Each drink, then, is parametrized by its rate, r, the start-time t₀, and the length of the drink, L.

Since the function for $M_{G,V}^{nf}$ described in Equation (9) also depends on the start time, t₀, of the drink, and since we will be concatenating many drinks in one simulation, it becomes computationally more convenient to describe the time-varying rate function, $M_{G,V}^{nf}(t)$, using a differential equation. We do this by taking the derivative of Equation (9), and rewriting it in terms of the function itself to get, for a drink that takes place in the time interval [t₀, t₀ + L = T]:

$$\frac{d{M}_{G,V}^{nf}}{dt}=\{\begin{array}{ll}r\beta \left(1-e^{-\beta \left(t-t_0\right)}\right)-\beta M_{G,V}^{nf}(t)\hfill & t_0\le t<T\hfill \\ -\beta M_{G,V}^{nf}(t)+r\beta \left[e^{-\beta t}\left(e^{\beta T}-e^{\beta t_0}\right)\right]\hfill & t\ge T\hfill \end{array}$$

We can then model a sequence of drinks during the day by summing the rates for individual drinks. Recall that the drinking data consists of sequences of D_i, L_i and S_i denoting the dose (volume), length, and start-time of the i^th drink, respectively. The end time, E_i, and intake rate, r_i, of the i^th drink is therefore derived from the data:

• E_i = end time of drink i, so E_i = S_i + L_i

• r_i = intake rate of drink i, so $r_i=\frac{D_i}{L_i}$.

For a given value of t, we consider two cases.

1. If t is during a drink interval, so that S_i ≤ t < E_i for some i, then:

   $$\frac{d{M}_{G,V}^{nf}}{dt}=-\beta M_{G,V}^{nf}(t)+r_i\beta \left(1-e^{-\beta \left(t-S_i\right)}\right)+\sum _{k<i}{r}_k\beta \left(e^{-\beta \left(t-E_k\right)}-e^{-\beta \left(t-S_k\right)}\right)$$ (10)
2. If t is not during a drink interval: E_i ≤ t < S_i+1 for some i, then:

   $$\frac{d{M}_{G,V}^{nf}}{dt}=-\beta M_{G,V}^{nf}(t)+\sum _{k\le i}{r}_k\beta e^{-\beta t}\left(e^{\beta E_k}-e^{\beta S_k}\right)$$ (11)

   Thus, in the simulation, we need to keep track of all the drink end-times (E_k), drink start-times (S_k) and drink rates (r_k) for drinks that occurred prior to the current time, t. Representing the data as vectors of drink volume (“dose”), $\overrightarrow{D}$, drink lengths, $\overrightarrow{L}$, and drink start times, $\overrightarrow{S}$, we see that equations (10) and (11) give an explicit formulation for the function Q that appears in Equation (4).

2.3. Modeling the effect of food on blood ethanol content.

In the previous subsection we derived a model for the rate of flow of ethanol from the gut to the vasculature, $M_{G,V}^{nf}$, which takes as inputs the lengths and volumes of individual drinks, where the superscript “nf” indicates that we are describing the rate when no food is present. The presence of food in the gut has the overall effect of decreasing the rate at which ethanol moves from the gut to the vasculature.

2.3.1. The effect of food on the rate of flow to the vasculature.

To capture this effect, we use a sigmoid function that depends on the amount of food in the gut as a multiplier of the rate function $M_{G,V}^{nf}$, shown in Figure 2. We refer to the the rate that is diminished by the presence of food as $M_{G,V}=\mathcal{M}\cdot M_{G,V}^{nf}$, where:

$$\mathcal{M}(t)=\frac{1-m}{\pi }\left(-\arctan (F(t)-s)+\frac{\pi }{2}\right)+m.$$ (12)

This scaling function has two parameters: m, which determines the maximum scaling factor, and s, which determines the steepness of the sigmoid function, i.e. how sensitive the rate, M_G,V, is to the presence of food in the gut. As individuals consume food, the value of F increases, $\mathcal{M}$ decreases, and the rate M_G,V decreases. As food is digested, $\mathcal{M}$ increases to nearly one, and M_G,V returns to its maximum value of $M_{G,V}^{nf}$ (without food), which in turn depends on the amount of drinks taken, as given by the differential equations (10) and (11).

Figure 2:

Here the M_G,V multiplier (i.e. the function $\mathcal{M}$) is plotted as a function of the amount of food in the gut, F, using parameter values estimated from the data. This function gives the fraction of the maximum absorption rate at which ethanol moves from the gut to the vasculature. When there is no food, the function is close to 1, so this rate is close to its maximum. As F increases, $\mathcal{M}$ decreases to a minimum of m = 0.75. The function $\mathcal{M}$ scales the rate, M_G,V, at which ethanol enters the vasculature: the greater the value of F, the slower the rate, M_G,V. The parameter s gives the value of F at which $\mathcal{M}$ is halfway between its maximum value of 1, and its minimum, m.

2.3.2. A simplified model of digestion.

The digestion of food is a complicated process, and the rate of digestion varies according to the type of food, the gut microbiome, and other individual and environmental factors. A general differential equation for the amount of food in the gut between eating events is:

$$\frac{dF}{dt}=-k(t,\overrightarrow{p})F$$

where k is a function of time, t, and a set of parameters, $\overrightarrow{p}$, that may change from individual to individual, and will depend on the type of food eaten as well as the internal and external environment. The simplest description of digestion is when k is a constant parameter, so that F(t) obeys first-order kinetics. A careful study of the rate of digestion as a function of time after eating is described in [23], where they note that gastric emptying does not, in fact, follow first-order kinetics, i.e. the fractional rate of gastric emptying is not constant. Rather, by taking detailed measurements on a group of Rhesus monkeys, they discovered that the fractional rate of gastric emptying is highest immediately after a meal, and the rate first decreases and then increases over the next 50 minutes. We compared results using this time-varying rate versus one with a constant rate of emptying in our model, and found no significant difference in BEC levels predicted by the model (results not shown). Some potential variability was eliminated in our simulations, since only one type of food was given (banana pellets). Furthermore, F(t) has only an indirect effect on the BEC, which is the quantity of interest, so small variations in its levels have minor consequences. Therefore, we choose to approximate the rate of gastric emptying using a constant fractional rate, denoted by the parameter δ.

New model equations.

In summary, we add three equations to the model to describe the effect of drinking and eating on BEC levels: two differential equations, one for M_G,V and one for F(t), as well as the multiplier equation for $\mathcal{M}(t)$ that relates F(t) to the rate of transfer of ethanol into the bloodstream from the gut.

2.4. The full model system.

The full model consists of a system of five differential equations, Equations (1) through (5), along with the multiplier function, Equation (12). As described in Section 2.3, Equation (4) is determined by the data, which gives a sequence of drink volumes, lengths and start times.

In addition to drinking events, the data also gives food events, each with a start time and an amount of food ingested. These food events are modeled as instantaneous increases in the amount of food in the gut. In other words, when a food event occurs, the amount of food eaten is immediately added to the amount of food in the gut, F(t).

The full system of equations, including all time-varying rates and parameter values, is given in Appendix B and Appendix C.

3. Methods

3.1. Monkeys

The Monkey Alcohol Tissue Research Resource (MATRR) is a combined tissue repository and analytics platform designed to support alcohol research in the primate model [12], www.matrr.com. It aims to provide experimental reproduciblility by using a standard core alcohol exposure protocol, and to reduce the overall burden on NHP resources by curating, storing and distributing tissues to requesting researchers. Researchers are required to deposit resulting data and results back into the analytics platform. MATRR contains behavioral, physiological, genetic, and molecular data on 188 Rhesus Macaque and Cynomolgus monkeys under rigorous alcohol exposure experiments.

A total of 30 Rhesus (Macaca mullata) male monkeys were used for this study, representing cohorts 4, 5, and 14 from the MATRR data [12]. Data from an additional five late adolescent male rhesus monkeys, indicated here as cohort D and described previously [24], were used to calibrate food intake for analysis. The average male rhesus monkey in our study is 5–6 years old (full adult) and weighs about 9kg.

In [3], four drinking classifications are established: very heavy drinkers, heavy drinkers, binge drinkers, and light drinkers. Very heavy drinkers (VHD) have an average daily intake greater than 3 g/kg and more than 10% of their daily intakes are greater than 4 g/kg. Heavy drinkers (HD) maintain a daily intake greater than 3 g/kg for more than 20% of drinking days. Animals that are classified as binge drinkers (BD) have a daily intake greater than 2 g/kg for more than 55% of the days and one BEC level greater than 80 mg/dl. The remainder of the monkeys are classified as light drinkers (LD).

Five monkeys from cohort D were used to fit the four flow parameters: V_max, K_m, k_TV, and β. One monkey from each category, taken from cohorts 4 and 5 were used to fit the food parameters s and m, namely LD: 10055; BD: 10054; HD: 10067; VHD: 10066. Six additional monkeys from cohort 4 were used to test the model (see Table 1), and data from ten of the monkeys in cohort 14 were used to estimate liver volumes (see Table 2). All animals used in this study were born into a pedigreed population and housed at the Oregon National Primate Research Center (ONPRC) and remained with their mothers until weaning. Approximately three months prior to the onset of ethanol self-administration the animals entered into a single cage laboratory-controlled housing environment.

Table 1:

Four cohorts of Rhesus monkeys were used in this study.

ID	N	Sex	Age (yrs)	Weight (kg)	LD	BD	HD	VHD
4	10	M	8.24	9.4	5	4	1	0
5	8	M	5.63	8.31	0	1	3	4
14	12	M	5.29	10.998	1	2	3	3
D	5	M	6.3–6.6	7.8–10.2	-	-	-	-

Age and Weight values represent cohort averages. Cohorts enrolled in open access drinking protocols are further subdivided into four drinking categories: light drinkers (LD), binge drinkers (BD), heavy drinkers (HD) and very heavy drinkers (VHD).

Table 2:

For this study, liver data from cohort 14’s necropsy were used to estimate V_L.

MATRR ID	Drinking Category	Body Weight (g)	Liver Weight (g)	Liver % of Body Weight	Liver Displ. Volume (ml)	Density (g/ml)
10245	control	11440	155.94	1.36	120	1.3
10242	VHD	10740	141.16	1.31	132	1.07
10249	HD	10220	120.39	1.18	109	1.1
10251	BD	10740	142.15	1.32	128	1.11
10252	VHD	9840	151.01	1.53	140	1.08
10243	BD	11020	128.65	1.17	112	1.15
10246	LD	11560	141.68	1.23	118	1.2
10247	HD	10060	131.39	1.31	110	1.19
10250	control	11680	151.51	1.30	155	0.98
10244	HD	10460	149.41	1.43	132	1.13

Each monkey’s drinking category, body weight, liver weight, percentage of body weight that is the liver, liver volume, and liver density is displayed in the table.

3.2. SIP Protocol and Data Collection

All animals involved in the open access drinking protocol followed a well-established self-induced polydipsia (SIP) protocol [2] where animals receive a 1g (3.351 calories) food pellet at a fixed interval of 300 seconds until they reach their preset dose of ethanol. Ethanol (4% w/v) self-administration is first established in an induction phase where daily intakes are limited to 0.5 g/kg. 1.0 g/kg, or 1.5 g/kg. Each intake phase is maintained for a duration of 30 consecutive sessions. Following the induction phase, animals enter an open-access phase where they are allowed their choice of drinking either ethanol or water 22 hrs/day, 7 days/week, from 6 – 12 months. Behavioral and metabolic data derived during open access is the basis for classifying categorical levels of drinking.

During the drinking protocol, all animals are fed a nutritionally complete diet, body weights are acquired weekly, and blood-associated physiological markers, including BEC information, are collected at cage-side as part of the established protocol [2]. Drinking behavioral data was collected using protocols described previously [2, 25, 13]. Briefly, custom interfaces capture ethanol self-administration and food intake patterns on a second-by-second basis at a granularity equivalent to 4.0 mg of ethanol.

3.3. Numerical Methods

Numerical solutions of the model were implemented using the open source platform, R. For solving the system of differential equations, the lsoda solver from the deSolve package [26] was used. The FME package [27] was used for parameter estimation, applying both Nelder-Mead and Newton’s method. Code is available from the authors on request.

In addition to physiological data on each monkey such as weight and data describing drinking and eating events over time, as well as BEC levels was used to fit model parameters, (see Section 4.1). Data used for parameter fitting was first “cleaned” so that:

• Drinks of one second or less are ignored.

• Successive food events that happened at the same time (multiple pellets eaten at once) were combined into one “food” event.

• If a food event occurred during a drink, its start time was moved to the end of that drink.

These adjustments to the data were necessary since the differential equations solver is restarted at the beginning of every drink, and each time food is consumed.

4. Results

4.1. Parameter Fitting

Model parameters were estimated in several steps. First, model parameters that do not involve food were estimated. Four of these parameters are estimated from the literature: the flow rate through the aorta, R_A, the fraction of cardiac output that goes to the liver, F_L, and the fraction of this liver-directed output that goes through the portal vein and gut, rather than through the hepatic vein, F_P . These parameters affect all of the rates of blood flow between the model compartments: M_V,T, M_T,V, M_H,L, M_L,H and M_P,L, M_H,L. (See Figure 1 and Appendix B.)

4.1.1. Volume parameters

The volumes of the three model compartments: vasculature (V_V ), liver (V_L) and peripheral tissue (V_T ) are estimated from our data.

To estimate the liver volume, we used data from a subset of monkeys in cohort 14, intentionally including drinkers from all drinking categories as well as non-drinkers. Liver volumes were calculated by measuring the volume of water displaced by the liver at necropsy of 10 monkeys from cohort 14. The liver weight and density were also calculated. We then used the average density 1.13 g/ml and the average percentage of body weight 1.48% to estimate the liver volume of each simulated monkey. The data is shown in Table 2. Note that, while the monkeys shown in Table 2 are all male rhesus monkeys, the liver is approximately the same size in males and females. Since females are smaller the female liver will be a larger percentage of body weight.

The blood volume, V_V, is estimated using the blood volume calculator in [28], which gives an estimate of approximately 64 ml of blood per kg of body weight for the average male monkey. Thus, in our simulations, V_V = 0.064 × (body weight). We note that a female 6 kg monkey would have about 61.5ml of blood per kg.

To estimate the volume of the peripheral tissue we used information about monkeys from [29] and scaled. In particular, we estimate the weight of bone as 12% of body weight. Thus, the volume of peripheral tissue, V_T, is estimated as:

$$V_T=W_M(1-.064-0.12-0.0148)\cdot \frac{1}{1.01}$$

where W_M is the monkey’s weight in kg, and we use 1.01 kg/L as the tissue volume to weight conversion factor [30].

4.1.2. Vascular Flow Parameters

We use five naive rhesus monkeys to fit the flow rate parameters between the model compartments (k_V,T, k_T,V, k_H,L, k_L,H, and k_P,L), the rate of passage through the gut (β), the maximum metabolism rate (V_max), and Michaelis-Menten constant (K_m). The flow rates are given in the full set of model equations, listed in Appendix B.

One dose of 0.5 g/kg or 1 g/kg is given at time 0 to each monkey. At the time of the ethanol dose, the monkeys have an empty stomach; thus, for the 0.5 g/kg dose it takes about L = 30 seconds for the alcohol to reach the gut and for the 1 g/kg dose it takes about 60–90 seconds for the alcohol to reach the gut (we use L = 60 seconds). Thus, for these simulations, we use 0.5 g/kg and 1.0 g/kg as the dose, and 30 or 60 seconds as the length of the drinks, respectively. This gives us the rate, r, that appears in the rate equation for $M_{G,V}^{nf}$, the flow rate from the gut to the vasculature. See Appendix A for more details.

Although ethanol stays in blood and muscle longer (vasculature and peripheral tissue compartments, respectively) and it is easier for ethanol to get into water compartments than fat compartments, the transfer rates are empirically the same, [18]. Therefore, we fit one parameter, k_TV, and used this value for all the k_i,j.

We used the FME package in R [27] to fit the four parameters: k_TV, β, V_max and K_M to the data from each individual monkey in the training set, and then used the median of the fit values in the model. The cost function was a weighted sum of squared differences from the model simulations to the data, where weights were an increasing function of the data, giving larger BEC values more weight. The fitting was done in two stages: first using a Nelder-Mead algorithm, and using those results as the initial values for Newton’s method. More details are given in Appendix A. The results are shown in Figure 4, Figures A.10 – A.12. Table C.5 gives all parameter values used in the model, including standard errors for the fitted parameters. Using these parameter values, the model is able to capture the kinetics of the blood ethanol concentration in both doses for all five subjects.

Figure 4:

This plot shows the simulated BEC using the final fitted parameters compared to the data points for each monkey. The final parameters are the median of the results from fitting parameters to each monkey as seen in Figure A.11. In other words, the same set of parameters was used for all simulations. These values are given in Table C.4. Note that there was no data for Monkey ID 5, dose 0.5 g/kg.

These single drink simulations highlight one difference between this study and the one described in [18]. Their simulations describe the BEC after a single intravenous injection, and thus their model does not include the rate function, $M_{G,V}^{nf}$ which we are fitting here.

4.1.3. Digestion Parameters

Once the flow rates between compartments were parameterized, we fit the parameters used in the gut multiplier function, $\mathcal{M}$, given in Equation (12). The goal was to capture the range of BEC kinetics observed in the data using one set of parameter values.

As described in Section 2.3, the digestion of food is adequately described for our purposes by first-order kinetics parametrized by δ. Based on empircal evidence that 90% of food is digested in two hours, we use the value δ = 0.00032sec⁻¹ in all simulations [31]. In this phase of parameter-fitting, we fit two other parameters, m and s, which control the mininum and steepness, respectively, of the multiplier function $\mathcal{M}$ (see Figure 2). We fit these to the data from four “prototypical” monkeys from cohorts 4 and 5 and induction drinking data from MATRR.

The induction data consists of a list of events, either a drink or a food pellet. Each event has a start time and a volume, and - if it is a drink event - the volume of the drink. The data also shows the daily weight of each monkey, which we use to estimate the liver, vasculature and tissue volumes, and the BEC at the time that blood is drawn. For the fitting, we only consider data up until the time of the blood draw. We first fit the induction data for each monkey, and then selected values that best captured the spread of behavior for the entire set of monkeys.

As shown in Figure 6, the BEC values range from close to 0 to 97.1 mg/dl. The BEC model with the fitted parameters is able to simulate the spread of behavior for the entire set of prototypical monkeys, representing a range of drinkers and thus various inputs.

Figure 6:

Here we show a prototypical monkey in each drinking category, namely 10055 (LD), 10054 (BD), 10067 (HD), and 10066 (VHD). We plot the measured BEC values during the third induction period (red data points) against the simulated BEC values (black lines) using m=0.75 and s=1.5. For each monkey, BECs are taken from approximately 6 different sessions for each induction phase.

All parameter values are given in Table C.5 in Appendix C. Further simulations are presented in Section 4.2.

4.2. Model Validation

We used a test set of monkeys and two training sets of monkeys, one set of 5 to estimate the flow parameters, and one set of 4 used to estimate the two food parameters. This last set is described in Table 3. Note that we used one test monkey from each drinking category. The test set of monkeys was taken from cohort 4, and the represent individuals from different drinking categories. Using the fitted parameter values, we now validate the model by comparing model simulations to data taken in a variety of circumstances.

Table 3:

Training set of monkeys used to estimate the two food parameters: m and s.

Training Set - Food Parameters
Cohort Number	Monkey ID	Drinking Category	BEC Detail
4	10055	LD	one BEC over 80 mg/dl during open access and on most days had consumed greater than 75% of it’s allotted intake at the time of BEC blood draw
4	10054	BD	had binge day s and about 75% of intake consumed at the time of BEC blood draw with most BECs over 80 mg/dl
5	10067	HD	stable intake and about 75% of ethanol consumed by time of BEC blood draw with he majority of BEC values over 80 mg/dl. However, this animal did have some difficulty during the 1.0 g/kg induction period.
5	10066	VHD	nearly all BECs over 150 mg/dl during open access and most of it’s allotted ethanol consumed by time of BEC, particularly in the last few months.

The four prototypical monkeys were chosen based on their drinking behavior and BEC consumption patterns during induction. An animal was marked as prototypical if it had consumed most of its ethanol by the time its BEC was collected, which was less important for the VHD monkeys, and had stable intakes and BECs that were consistent with their category descriptions.

In Figure 7 we simulate monkey 10066’s BECs for all three induction periods, Phase 1: 0.5 g/kg of 4% w/v ethanol administered, Phase 2: 1.0 g/kg of 4% w/v ethanol administered, and Phase 3: 1.5 g/kg of 4% w/v ethanol administered, using parameter values from Table C.4. Each induction phase lasts 30 sessions (days) and BECs are taken from approximately 6 different sessions for each induction phase. BECs are typically taken once every 5 days during induction. See Section 3.2 for more details. The monkey’s drinking pattern changes as it goes through induction, and we see that the model has the ability to capture all three induction periods, demonstrating the model’s ability to capture various drinking habits and various levels of intoxication. For Phase 3, we felt like the spread of of the data was captured well even though the specific data wasn’t hit exactly.

Figure 7:

Here we plot the BEC values of monkey 10066, a very heavy drinker, for the induction period to show the flexibility of the model, and its ability to capture BEC kinetics under a range of drinking behaviors. The top left plot shows the measured BECs (red data points) during the first induction phase(0.5 g/kg of 4% w/v ethanol administered) compared to the simulated BEC values (black lines). Each data point represents a different BEC sample. The samples were taken on different session dates. A monkey’s BEC is typically taken once every 5 days during induction. Similarly the top right plot shows the second induction phase (1.0 g/kg of 4% w/v ethanol administered) and the bottom plot shows the third induction phase (1.5 g/kg of 4% w/v ethanol administered). Even though the data points were not hit exactly, the model does a good job capturing the spread of the data. Note that some data points overlap each other: e.g. in Phase 1, the BEC was measured as 8 mg/dl on two different days, and multiple measurements gave a BEC of 0 mg/dl.

To further validate the new BEC model, we test it using six monkeys from cohort 4 who were not used in the fitting process. These test cases illustrate that drinking behavior varies between individuals in the same cohort, and between different sessions for the same individual. Figure 8 shows the results from the experiments in which the largest amount of alcohol is given (1.5 g/kg of 4% alcohol) over the longest time period (BEC is measured at 90 minutes). For all six monkeys, the parameters given in Table C.4, were used, with the individual’s weight used to estimate volumes. For each monkey, six experiments were run at this higher drink level, resulting in six curves and six data points in each panel. The sequence of drinks and eating events were input into the model, as described in Section 2. The model captures the BEC time course fairly well, even when there is variability between sessions.

Figure 8:

This plot shows the simulated BECs compared to the experimental values during the third phase of induction (1.5g/kg induction period) for six test monkeys. These test monkeys were taken from cohort 4. For each monkey, six sessions were run in the induction phase at this higher drink level, resulting in six curves and six data points in each panel. The model parameters are the same for all six test monkeys, with only the monkey’s weight used to estimate compartment volumes. For each session, a sequence of drink and food events is used as input to the model, and the BEC kinetics are simulated (black lines). BEC was measured from blood samples take at 90 minutes after the beginning of the sessions - these data are shown as red dots. In some cases, multiple BEC measurements had the same value, so these data are plotted on top of each other.

5. Discussion

The evolution of our understanding of how to characterize a model organism’s drinking phenotype has progressed in large part due to advanced monitoring devices and rigorous reproducible experimental paradigms. However, even highly granular methodologies fail to provide a continuous systems level perspective that can act as a surrogate for the clinical phenomena, or mimic components in the complex multi-system interactions that drive AUD associated pathology. While the pathophysiology of complex trait diseases prohibit the development of an exhaustive mathematical representation, focused model development has the potential to expand our experimental understanding of these diseases. For example, the BEC model that we present here has the potential to mitigate the need for invasive BEC measurements while maintaining high fidelity between the predicted and actual BEC values.

By elucidating the relationship between food intake, ethanol consumption, and time, we are able to accurately apply our understanding of ethanol metabolism to predict BEC. Importantly, the rigorous NHP ethanol induction model makes it possible to calibrate our data model across known ethanol consumption rates, e.g. 0.5 g/kg, 1.0 g/kg, and 1.5 g/kg, with animals spanning a known pattern of future drinking, and apply that knowledge to predict real-time BECs. Importantly, we can leverage these models to enhance the precision of our understanding of the impact of BEC on additional behavioral, metabolic, immunological, and physiological data. A clear understanding of these inter-dependencies will allow us to further develop models identifying the potential impact of perturbing these systems on drinking behavior, such as the introduction of pharmacology on the drinking patterns.

Furthermore, a reproducible prediction strategy for instantaneous BEC is an important step in aggregating drinking models which will eventually allow the creation of synthetic whole system models of ethanol intake, behavior, and metabolic consequences.

6. Future Work

One of the motivations for this model was to enable us to more accurately classify drinkers by expanding our previous drinking model [13] to include BECs. Since the work presented here provides a reliable approximation of BEC over time, our existing ethanol consumption model will now be able to include the BD category requirement that binge drinkers have at least one binge (BEC greater than 80 mg/dl). By combining this BEC model with the stochastic behavioral model we will be able to more accurately classify animals and also relate behavioral data to ethanol intake. In Figure 9, we show how the two models will be combined resulting in a more reliable classification of monkeys as light, binge, heavy, or very heavy drinkers. Our first step in combining the two models is to compare our induction results to open access results. The monkeys have different drinking habits in the two scenarios and drinking functions need to be adjusted to reflect these differing behaviors.

Figure 9:

In this figure, we show how our previous drinking model as described in [13] (labeled Behavioral Model: calculates drinks over time) will be combined with the model described in this paper (labeled Physiological Model: predicts BEC levels). A monkey’s drink data and weight are input into the physiological model which produces second by second BEC levels. This BEC value along with drinking behavior from our previous model is passed into the classification function which outputs the monkey’s drinking category.

In addition to creating a more robust drinking model by combining our previous work with this model, we will also incorporate refinements to this model. One possible refinement is to divide the gut compartment into two compartments, the gut and intestine compartments. We would build on previous work by Pieters [20] where he calculates the amount of ethanol entering the blood stream by dividing the gut compartment into the gut and intestine compartments. Replacing our single gut compartment with two compartments may serve to increase overall accuracy, particularly in cases where gut to intestine diffusion changes due to the rate of ethanol consumption.

Another area of future work is exploring some additional data fitting refinements. One possibility is to make dosed-based parameters responsive to our fits. When considering the parameter distributions, we found that β seems to have a different distribution when comparing a low dose to a high dose. For example, β ranges from 0.0006 to 0.00085 for high doses, but for low doses the range goes up to 0.0014. Since β is in the M_gut function, the elimination rates in the gut could depend highly on gut volume. In particular, the low dose might be eliminated more quickly than a high dose. Another possibility is further investigating the elimination rate from the liver. Two of the fitted parameters, V_max and K_m are an expression for the elimination rate from the liver. Because the rate is very large at 0 it would not typically operate as a biological “switch” function. Though these parameters correspond to [18], further investigation is necessary to determine if they need to be adjusted to better reflect the biology. We could also fit the food parameters, m and s, for each induction period separately. Since the monkeys are learning to drink during the first induction period and they also get a low dose of ethanol, it makes sense to consider a different absorption rate. Drinking patterns also change as the monkey learns to consume his allotted ethanol. In particular, during the third induction period, the time to consumption is less.

Further enhancements to the model would also include modeling female monkeys and social drinking. Female monkeys’ consumption levels are highly correlated with hormone levels [1]. By including follicular and luteal phase information in our model, we hope to be able to closely approximate the complexity of NHP drinking in female animals. Social drinking influences are observed in rodent studies [32] and among humans [33]. Our model would incorporate social drinking by accounting for the influence of the monkeys in neighboring cages. We plan to explore the social effects on the drinking behavior of monkeys by modeling a network of monkeys that reflects cage locations and observed interactions.

Another potential refinement to our model would be to adjust some of our parameters (such as K_m, m, or V_max) to account for differences in individual characteristics instead of having universal constants. Cederbaum [34] provides a review of factors that effect changes in metabolism of alcohol for each individual. Many of these factors, such as biological rhythms, tolerance, and liver injury, would serve well for future consideration. The model in [35] accounts for several ways that alcohol is cleared from the system for various types of drinkers: male and female as well as “normal” and “misuser”. The authors note that female misusers have significantly faster metabolism of alcohol than any of the other drinkers. They suggest this could be due to the enzymes that metabolize the alcohol or due to change in rate of the absorption of alcohol into the peripheral tissues. Adjusting our parameters for individual characteristics would account for these variations.

Ultimately, we would like to extend our work to humans and our model lends itself to direct correlation between humans and the NHP model. For example, the volume that corresponds to 4% ethanol is proportional to the individual’s weight, and the concentration is normalized to an individual’s weight by expressing intake in terms of g/kg. Likewise, blood ethanol concentrations are already scaled (mg ethanol/100ml blood) and that measure can be directly translated to humans. In terms of blood ethanol concentration needed for intoxication, the macaque shows the same outcomes at the same BEC. For example, monkeys and humans can detect 30 mg/dl BEC reliably, become impaired on cognitive tasks at about 50 mg/dl BEC, show motor impairments at BECs of 50–100 mg/dl, and lose consciousness at BECs greater than 200 mg/dl. Due to the lack of required scaling, the model presented here may be applied directly to humans without modifications to fundamental concepts.

Figure 3:

This plot shows the food variable, M_G,V, and the no food variable, $M_{G,V}^{nf}$, for monkey 10057 during the three different phases of induction (note the different scales on the horizontal axis). Food is given in 1g pellets, giving the food line a stair step look. The downward slope between pellets shows the food being digested. The black line represents the rate at which ethanol enters the vasculature from the gut when there is no food. The red line looks similar to the black line as it is the rate at which ethanol enters the vasculature from the gut when there is food. Notice that as there is more food, there is a larger gap between the black and red lines.

Figure 5:

When fitting parameters m and s, we ultimately choose m = 0.75 and s = 1.5 as those values gave the best spread of the data, see the bottom right plot. Here we plot various choices of m and s for one monkey, id 10054, to compare BEC fits. The plots in the top row show the simulated BEC misses the upmost data point while the first two plots on the bottom row show the simulated BEC misses the lowest two data points. During Phase 3 induction, monkey 10054’s BEC was taken during 6 different sessions. A couple of the BEC values are very close together, so they look like the same data point.

Highlights:

Novel mathematical model of monkey alcohol consumption that includes food effects Captures the kinetics of blood ethanol concentration for various drinking behaviors Model parameters are fit to data with and without food across physiological traits

Appendix A. Parameter Fitting Details

Parameters describing the flow rates between the vascular, liver, and peripheral tissue compartments were fit to data from five monkeys, namely, V_max, K_m, k_V,T, k_T,V, k_H,L, k_L,H, k_P,L, and β. We used the following bounds on the parameters based on the literature, or biological feasibility: V_max : [0.5, 10], K_m : [1, 200], k_{V T} : [0.000001, .8], β : [.0000001, .0005]. For each dose and each monkey, our fitting process was the following: First we used the Nelder-Mead fitting method in the FME package in R to fit all four parameters (see Figure A.10). The median of the individual Nelder-Mead results were used as starting values for Newton’s method. We looped through one parameter at a time using Newton’s Method. The results for each individual monkey are plotted in Figure A.11. Finally, we took the median of the results to give us values for those four parameters. The results are compared to the data in Figure 4.

The distribution of estimated parameter values is shown in Figure A.12. Note that we fit to the high dose (1g/kg) and low dose (0.5g/kg) experiments separately for each individual. The distributions shown here are from the final phase of fitting, using Newton’s method with starting values determined by using a Nelder-Mead algorithm in the four-dimensional parameter space. To fit the four parameters using Newton’s method, we looped through the parameters in a random order, ending up at different local minima. The distributions shown in Figure A.12 are the results of 10 fitting loops for each individual monkey, and for each of the two doses. We then looked at the parameter combinations that minimized the cost function for each monkey, and took the median of the combined estimates as the parameter to use in the model simulations.

Figure A.10:

Five naive monkeys were administered one ethanol dose on an empty stomach via gavage and each monkey’s BEC was taken at 15, 30, 45, 60, 75, 90, 120, 180, 240, and 300 minute time intervals. The test was repeated for two ethanol doses: 0.5 g/kg and 1.0 g/kg. Here we show a plot for each monkey and each dose, where the red data points represent the actual BEC and the black lines represent the simulated BEC. We simulated the BEC using the median of the individual Nelder-Mead results.

Figure A.11:

These plots show the second step in the fitting routine for the vascular flow parameters. We used the median of the Nelder-Mead fitting results (shown in A.10) as a starting point for Newton’s method. We then used Newton’s Method to fit one parameter at a time for each monkey. The results for each monkey is shown in this figure.

Figure A.12:

Distributions of the parameter estimates for the flow parameters: V_max, K_m, k_{V T} and β after fitting to the five monkeys in the training set. Each monkey was given a high dose (1g/kg) and a low dose (0.5g/kg) of alcohol by gavage. The panels show distributions of ten estimates of each parameter set for each dose and each monkey, for a total of 90 estimates (one monkey was only given one dose). The median of parameter values that gave the best fit for each experiment, shown in red, were used to simulate BEC levels from individuals in all drinking categories, as shown in Figure 4. These final parameters values, along with standard errors, are given in Table C.4.

Appendix B. Full Model System

The full model system, given by a system of differential equations, seven state-dependent rate equations and two functions of time, is given below. Parameter values are given in Appendix C.

$$\frac{d{\mu }_V}{dt}=-M_{H,L}-M_{P,L}+M_{L,H}+M_{T,V}-M_{V,T}+M_{G,V}$$

$$\frac{d{\mu }_L}{dt}=M_{H,L}+M_{P,L}-M_{L,H}-M_{\text{metab}}$$

$$\frac{d{\mu }_T}{dt}=M_{V,T}-M_{T,V}$$

$$\frac{d{M}_{G,V}^{nf}}{dt}=\{\begin{array}{ll}-\beta M_{G,V}^{nf}+r_i\beta \left(1-e^{-\beta \left(t-S_i\right)}\right)+\sum _{k<i}{r}_k\beta \left(e^{-\beta \left(t-E_k\right)}-e^{-\beta \left(t-S_k\right)}\right)\hfill & S_i\le t<E_i\hfill \\ -\beta M_{G,V}^{nf}+\sum _{k\le i}{r}_k\beta e^{-\beta t}\left(e^{\beta E_k}-e^{\beta S_k}\right)\hfill & E_i\le t<S_{i+1}\hfill \end{array}$$

$$\frac{dF}{dt}=-\delta F$$

$$\mathcal{M}=\frac{1-m}{\pi }\ast \left(-\arctan (F(t)-s)+\frac{\pi }{2}\right)+m$$

$$M_{G,V}=\mathcal{M}(t)M_{G,V}^{nf}(t)$$

$$M_{V,T}=k_{V,T}{R}_A\left(1-F_L\right)\mathcal{H}\left(\frac{{\mu }_V(t)}{V_V}-\frac{{\mu }_T(t)}{V_T}\right)$$

$$M_{T,V}=k_{T,V}{R}_A\left(1-F_L\right)\mathcal{H}\left(\frac{{\mu }_T(t)}{V_T}-\frac{{\mu }_V(t)}{V_V}\right)$$

$$M_{H,L}=k_{H,L}{R}_A{F}_L\left(1-F_P\right)\mathcal{H}\left(\frac{{\mu }_V(t)}{V_V}-\frac{{\mu }_L(t)}{V_L}\right)$$

$$M_{L,H}=k_{L,H}{R}_A{F}_L\mathcal{H}\left(\frac{{\mu }_L(t)}{V_L}-C_{HV}(t)\right)$$

$$M_{P,L}=k_{P,L}{R}_A{F}_L{F}_P\mathcal{H}\left(\frac{{\mu }_V(t)}{V_V}+\frac{M_{G,V}}{R_A{F}_L{F}_P}-\frac{{\mu }_L(t)}{V_L}\right)$$

$$M_{\text{metab}}=V_L{V}_{\max }\frac{{\mu }_L(t)/V_L}{K_m+{\mu }_L(t)/V_L}$$

$$C_{HV}(t)=\{\begin{array}{cc}\frac{1}{1+k_{L,H}}\left(\alpha (t)+k_{L,H}\frac{{\mu }_L(t)}{V_L}\right)& \alpha (t)<\frac{{\mu }_L(t)}{V_L}\\ \alpha (t)& \alpha (t)\ge \frac{{\mu }_L(t)}{V_L}\end{array}$$

$$\begin{gathered} \alpha (t)=\left(1-F_P\right)\frac{{\mu }_V(t)}{V_V}-k_{H,L}\left(1-F_P\right)\mathcal{H}\left(\frac{{\mu }_V(t)}{V_V}-\frac{{\mu }_L(t)}{V_L}\right) \\ +\frac{{\mu }_V(t)}{V_V}{F}_P+\left(1-k_{P,L}\right)M_{\text{gut}}-k_{P,L}{F}_P\mathcal{H}\left(\frac{{\mu }_V(t)}{V_V}-\frac{{\mu }_L(t)}{V_L}\right) \end{gathered}$$

$$\mathcal{H}(x)=\{\begin{array}{ll}0\hfill & x\le 0\hfill \\ x\hfill & x>0\hfill \end{array}$$

Appendix C. Parameter Values

The following two tables list all of the parameters used in the model simulations, along with their sources. In the first table, we list volumes of the three main model compartments, as well as typical overall vascular flow rates (see Section 4.1.1). In the second table, we list parameters used to describe flow rates between compartments, including those that control the effect of food. Except for the rate of gastric emptying, these parameters were either taken directly from the data, or fit to data, as described in Section 4.1.

Table C.4:

Parameter values used to describe flow rates between compartments.

Name	Value	Units	Description	Source
VL	function of monkey’s weight	L	volume of the liver	Table 2
VT	function of monkey’s weight	L	volume of peripheral tissue	[29]
VV	0.64	L	volume of vascular compartment	[28]
RA	0.071	$\frac{\mathrm{L}}{\min }$	volume flow rate of aorta	[18]
FL	0.26	none	fraction of cardiac output directed to the liver	[18]
FP	0.75	none	fraction of liver-directed cardiac output that goes through the gut and the portal vein	[18]

These are primarily taken from literature or estimated from liver data. Details are given in Section 4.1.1.

Table C.5:

Parameter values from Monkey Alcohol and Tissue Research Resource (MATRR, www.matrr.com) and fitting routines.

Name	Value(± sd)	Units	Description	Source
Vmax	3.03 (±0.23)	$\frac{\mathrm{m}\mathrm{g}}{\mathrm{L}\times \mathrm{m}\mathrm{i}\mathrm{n}}$	maximal metabolism rate	fit see Sec. 4.1.2
Km	16.55 (±6.12)	$\frac{\mathrm{m}\mathrm{g}}{\mathrm{L}}$	MM constant or concentration of the drug at which metabolism is one-half the maximal rate	fit see Sec. 4.1.2
Ki,j	0.79 (±0.14)	none	fraction of available etoh that is actually transported from the ith compartment to the jth compartment	fit see Sec. 4.1.2
β	7.4 (±1.3) ⋅ 10−4	none	affects the transfer rate of ttoh from gut to vasculature	fit see Sec. 4.1.2
Li	depends on each drink	seconds	length of ith drink	MATRR
ri	depends on each drink	$\frac{\mathrm{L}}{\sec }$	rate of ith dose delivery: $\text{rate}=\frac{\text{volume}}{\text{drink}\text{length}}$	MATRR
Si	depends on each drink	seconds	Start time of ith drink. End time: Ei = Si + Li	MATRR
m	.75 (±0.14)	none	fraction of maximum absorption rate when stomach is completely full	fit see Sec. 4.1.3
s	1.5 (±0.18*)	none	mass of food (in grams) where absorption fraction is halfway to minimum	fit see Sec. 4.1.3
δ	3.1 × 10−4	sec−1	digestive rate	[31]

See Section 4.1.2 and Section 4.1.3 for details on the fitting procedures. One of our assumptions is that all k_i,j are the same.Standard errors for fitted parameters are given in parentheses.

^*The parameter s was fit using only two of the monkeys in the training set, since the other two did not eat enough for the food to have any effect.