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Ordinary differential equation models for ethanol pharmacokinetic
based on anatomy and physiology
JaeJoon Han, Martin H. Plawecki, Peter C. Doerschuk, Vijay A. Ramchandani, and Sean O’Connor
Abstract—Physiologicallybased
models have been used to describe the distribution and elim
ination characteristics of intravenous ethanol administration.
Further, these models have been used to estimate the ethanol
infusion profile required to prescribe a specific breath ethanol
concentration time course in a specific human being, providing
a platform upon which other pharmacokinetic and pharmaco
dynamic investigations are based. In these PBPK models, the
equivalence of two different peripheral tissue models are shown
and issues concerning the mass flow into the liver in comparison
with ethanol metabolism in the liver are explained.
pharmacokinetic(PBPK)
I. INTRODUCTION
Nearly eight percent of people who use ethanol, will
become addicted during the course of their life, and about
a third of them will die of complications attributable to
ethanol [1], [2]. A major factor underlying the brain’s
exposure to ethanol1is its pharmacokinetics, that is, the
relationship between ethanol intake through various paths
(e.g., oral, intravenous) and its resulting concentration in
various tissues over time [3].
Administering ethanol and measuring the resulting tissue
concentrations of ethanol is a challenging problem in hu
man subjects, and controlling those concentrations is even
more difficult. First, the pharmacodynamic effects of ethanol
occur in the brain, but it cannot be directly administered
to the brain. Second, given the same administration, even
by infusion, the time course of flow and accumulation of
ethanol varies considerably in subjects whose size and body
composition differ. Finally, it is currently impossible to
measure brain ethanol concentration in humans directly.
Certain assumptions and methods make intermittent ap
proximate measurement of brain ethanol concentration pos
sible. It is reasonable to assume that brain and arterial
blood ethanol concentrations are equal, due to the high brain
blood flow rate relative to its small water volume. Further,
use of a breath test on endexpiratory air gives a good
This work was supported by NIH N01AA23102, P60 AA076111617,
and NIAA R01 AA1255505.
M. H. Plawecki is with the Weldon School of Biomedical Engineering,
Purdue University and the Indiana University School of Medicine. J.J. Han
and P. C. Doerschuk are with the School of Electrical and Computer Engi
neering, Purdue University. V. A. Ramchandani is with the National Institute
on Alcohol Abuse and Alcoholism and previously with the Department of
Medicine, Indiana University School of Medicine. S. O’Connor is with the
Department of Psychiatry, Indiana University School of Medicine and the
R. L. Roudebush VA Medical Center.
Corresponding author: P. C. Doerschuk, School of Electrical and Com
puter Engineering, Purdue University, 465 Northwestern Avenue, West
Lafayette, IN 479072035, doerschu@ecn.purdue.edu, +1 765 4941742.
1The pharmacokinetics of ethanol concerns the time course of ethanol
biochemistry while the pharmacodynamics concerns the time course of
ethanol’s cognitive effects.
approximation to arterial ethanol concentration. However, the
breath test can only be effectively used in a laboratory or a
forensic environment and the continuous measurement of the
breath alcohol concentration is not yet possible. Therefore,
mathematical models relating various input dose and various
concentration trajectories are critical to studying ethanol.
This paper studies and further develops one prominent
model, the socalled physiologicallybased pharmacokinetic
model (PBPK model) [4]–[11], developed at the Indiana
Alcohol Research Center of the Indiana University School
of Medicine. The PBPK model maintains a balance between
physiologic fidelity (the number of pathways and parameters)
and research utility (reliable prediction of brain exposure for
an individual) by making every parameter physiologic, but
minimizing their number.
The first contribution of this paper is to describe the PBPK
model as a differential equation with a nonlinear constraint
and then solve the constraint resulting in an unconstrained
differential equation. The second contribution is to provide a
graphical method for describing the PBPK model and related
models that is analogous to a circuit diagram except that
there are two “charges” flowing in the circuit, blood and
ethanol. The third contribution is to show the equivalence
among some different peripheral tissue models. The fourth
contribution is to demonstrate that the standard PBPK model
has anomalous behavior at low ethanol concentrations: the
mass flow of ethanol being degraded can be greater than the
total mass flow.
II. PBPK MODEL
We consider models that are interconnections of socalled
“well stirred” compartments; when ethanol enters a com
partment we assume that it is instantly spread uniformly
throughout the compartment and so there is a single ethanol
concentration for each compartment. The most complicated
model we consider has three compartments which approx
imate the vasculature, liver, and all other tissues which
we denote by periphery. Each compartment is described
by giving the mass of ethanol in the compartment and
the mathematical model for a compartment is a firstorder
nonlinear differential equation which describes the time rate
of change of the mass of ethanol in the compartment. The
interconnections between compartments transport ethanol
and blood. The interconnections mimic vascular anatomy
and physiology and include structures such as the portal
vein and hepatic vein. The flux of ethanol into and out of a
compartment depends on ethanol concentration differences.
Proceedings of the 28th IEEE
EMBS Annual International Conference
New York City, USA, Aug 30Sept 3, 2006
SaBP7.2
1424400333/06/$20.00 ©2006 IEEE.5033
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The vasculature compartment circulates ethanol through
the system at physiologically relevant concentrations until
elimination has been completed. The periphery compartment
acts as a storage reservoir from which ethanol may enter or
leave, based upon the gradients between its concentration and
the arterial and venous ethanol concentrations respectively.
Ethanol enters the system through two pathways. The
simpler pathway is by venous infusion, which occurs only in
a laboratory setting. In this case the mass flow rate of ethanol
is added directly to the wellstirred vascular compartment.
The more complicated pathway is by oral intake. In this
case the ethanol is transported through the proximal gut and
is absorbed in the small intestine from which it enters the
portal vein.
PBPK model elimination of ethanol occurs only in the
liver compartment, and emulates a single enzyme system
which follows MichaelisMenten (MM) kinetics. In fact,
three enzyme systems make up the major ethanol elimination
pathways within the liver. However, in vivo determination
of their individual characteristics is impossible and for that
reason they are lumped into one pathway in this model. In
addition, elimination from the other compartments is consid
ered negligible, which is supported by animal studies [12].
An important part of the model is the transport of ethanol.
The generic description of ethanol transport is
Mx,y(t)=
kx,yRxr(Cy(t) − Cx(t))
(1)
where Mx,y(t) [Mass]/[Time] is the mass flux of ethanol
from x to y at time t; Cx(t) and Cy(t) [Mass]/[Volume] are
the concentrations of ethanol in x and y, respectively; 0 ≤
kx,y≤ 1 is a dimensionless constant; Rx[Volume]/[Time] is
the volume flow of blood in x, r(·) is the unit ramp defined
by r(x) = xu(x) where u(·) is the unit step function. The
r(·) is used so that Mx,y(t) does not have a negative value,
which is not physiological.
The models described in this paper are roughly similar
to electrical circuits and a graphical description similar to a
circuit diagram is useful in understanding what is and what is
not included in the model. Such a circuit diagram for a three
compartment model is shown in Fig. 1(a) where the block
diagram fragments labeled A?and B?are to be ignored.
Each compartment in the block diagram of Fig. 1(a) is
labeled with its name, its volume (VX variable) and its
state variable [μX(t) variable] which is the mass of ethanol
in the compartment. In addition, CX(t)
is the concentration of ethanol in the compartment. The
compartment subscripts are V for the Vascular, T for the
peripheral Tissue, and L for the Liver parenchyma.
Each edge of the graph is labeled with a name, if it
clearly corresponds to a particular anatomical structure, and
is labeled with the ethanol mass flow [MX(t) variable]
and the volume flow [RX(t) variable] that occur along that
edge. The edges are directed, which indicates the positive
flux direction. In addition, CX(t)
concentration of ethanol in the edge. The edge subscripts
are A for Aorta, P for Peripheral artery, VC for Vena Cava,
.= μX(t)/VX(t)
.= MX(t)/RX(t) is the
PV for Portal Vein, HA for Hepatic Artery, HV for Hepatic
Vein, and Cap for peripheral Capillary bed.
The mass flows MGut(t) and MInfuse(t) are the external
inputs to the system and represent the flow of ethanol
from the gut and from a venous infusion, respectively. The
mass flow MMetab(t) is the ethanol sink created by liver
metabolism of ethanol.
There are three kinds of vertices in the graph. The first
type is the compartment, where the entering and exiting
edges form the right hand side of a differential equation
for the state variable. The second type are vertices where
only one edge enters (no symbol) which obey Kirchhoff’s
Current Law (KCL) and divide the input fluxes (ethanol mass
and volume) among the edges that exit. The FX constants
indicate the fractional of the input that exits on each of the
exiting edges. The subscripts are L for fraction of the cardiac
output directed to the liver and PV for the fraction of the
liverdirected cardiac output that is directed through the gut
and the portal vein. The third type are vertices where only
one edge exits (Σ symbol) which obey KCL and sum the
input fluxes (ethanol mass and volume) in order to determine
the fluxes (ethanol mass and volume) on the edge that exits.
Since the second and third type of vertices both obey KCL
they are really the same.
A. The 3state model
The basic threestate model is the model shown in Fig. 1(a)
and is a generalization of the standard PBPK model. The
state equations can be read directly from Fig. 1(a):
dμL/dt(t)=MHA,L(t)+MPV,L(t)−ML,HV(t)−MMetab(t)(2)
dμT/dt(t)=MP,T(t) − MT ,P(t)
dμV/dt(t)=MInfuse(t)+MGut(t)−MPV,L(t)
+MT ,P(t)−MP,T(t)−MHA,L(t)+ML,HV(t).(4)
(3)
Equations for the six unknown mass fluxes can be derived
in terms of the three state variables. The mass flux from the
liver parenchyma out of the system, which is denoted by
MMetab(t), is directly determined by MM kinetics:
MMetab(t) = VmaxμL(t)/(Km+ μL(t)/VL).
The remaining five unknown mass fluxes can be derived
following the general principle that results in (1).
(5)
MP,T(t)=kP,TRA(1 − FL)r(μV(t)/VV− μT(t)/VT) (6)
MT ,P(t)=kT ,PRA(1 − FL)r(μT(t)/VT − μV(t)/VV) (7)
MHA,L(t)=kHA,LRAFL(1−FPV)r(μV(t)/VV−μL(t)/VL) (8)
MPV,L(t)=kPV,LRAFLFPVr(μV(t)/VV
+MGut(t)/RAFLFPV− μL(t)/VL)
ML,HV(t)=kL,HVRAFLr(μL(t)/VL− CHV(t));
M(2)
(9)
(10)
PV(t) and MHA(t) ((11) and (12)),
M(2)
MHA(t)
PV(t)=
=
μV(t)/VVRAFLFPV+ MGut(t)
RAFL(1 − FPV)μV(t)/VV
(11)
(12)
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needed to evaluate
α(t)=(MHA(t)−MHA,L(t)+M(2)
γ(t)=
(α(t) + kL,HVγ(t))/(1 + kL,HV) for α(t)
CHV(t) = α(t) for α(t) > γ(t). The following constants are
required: the volumes of the three compartments, denoted
by VV, VL, and VT; the MM constants, denoted by Vmax
and Km; the cardiac output, denoted by RA; the fraction of
cardiac output that is delivered to the liver and the fraction
of the total liver supply that is delivered by the portal vein,
denoted by FL and FPV, respectively, and the fraction of
ethanol in a vessel or compartment that is transported to a
connected vessel or compartment, denoted by kP,T, kT ,P,
kHA,L, kPV,L, and kL,HV.
B. The 2state model
If we consider a second model which is the block diagram
shown in Fig. 1(a) with A replaced by A?, it has only two
states and does not require a value for VL. This model is the
standard PBPK model. The resulting equations are
PV(t)−MPV,L(t))/(RAFL);
and
CHV(t)
<
μL(t)/VL;
=
γ(t) and
dμT/dt(t) = MP,T(t) − MT ,P(t)
dμV/dt(t) = −RAFLμV(t)/VV+ MT ,P(t) − MP,T(t)
+ r[RAFLμV(t)/VV+ MGut(t)
− MMetab(t)] + MInfuse(t)
MMetab(t) = MmaxCL(t)/(Km+ CL(t))
CL(t) = μV(t)/VV+ MGut(t)/RAFL;
the expression MP,T(t) (6), and MT ,P(t) (7); and the two
external inputs MGut(t) and MInfuse(t).
(13)
(14)
(15)
(16)
III. EQUIVALENCE OF DIFFERENT PERIPHERAL TISSUE
MODELS
In this section we show that B and B?in Fig. 1(a) are
identical. MCap(t) and RCaprepresent the characteristics of
the capillary bed between the arterial side (MP(t) and RP)
and the venous side (MVC(t) and RVC). Note that RVC=
RCap= RPsince there is no volume flow into the periphery.
With B, MT ,P(t) is given in (7) which can also be written in
the form MT ,P(t) = kT ,PRPr(CT(t) − CP(t)). With B?,
MCap(t) = MP(t) − MP,T(t) which implies
CCap(t)=
MCap(t)/RCap
=
CP(t) − kP,Tr(CP(t) − CT(t))
since, by (6), MP,T(t) can be written in the form MP,T(t) =
kP,TRPr(CP(t) − CT(t)). Furthermore, for B?, MT ,P(t)
is modified to MT ,P(t) = kT ,PRPr(CT(t) − CCap(t)).
Therefore, CVC(t) = MVC(t)/RVC= [MP(t)+MT ,P(t)−
MP,T(t)]/RPhas one of two forms:
CVC(t)=
CP(t) − kP,Tr(CP(t) − CT(t))
+
kT ,Pr(CT(t) − CCap(t)),
B and B?are equivalent by the following argument. Insert
(17) into the B?case of (18). First consider the case where
(17)
?
kT ,Pr(CT(t) − CP(t)),B
B?(18)
CP(t) ≥ CT(t), simplify the r(·) functions in both cases
of (18), and find that the two cases are identical. Second,
consider the case where CP(t) < CT(t), again simplify the
r(·) functions in both cases of (18), and again find that the
two cases are identical. Therefore B and B?are identical.
IV. ANOMALOUS BEHAVIOR OF THE STANDARD MODEL
The anomalous behavior of the 2state model is the fact
that the model can indicate that the mass flow of ethanol
being degraded is greater than the mass flow of ethanol
entering the liver. Since the liver in the 2state model does
not store ethanol, this is nonphysiological. The lower limit on
the liver ethanol concentration can be computed as follows.
The mass flux of ethanol entering the liver is
Menter= MHA(t) + M(2)
PV(t).
(19)
From (16), (15) and (19) the fundamental requirement is
MMetab(t) ≤ Menter(t) which implies
Mmax/
RHA+ R(2)
PV
?
?
− Km≤ CL(t).
(20)
For a typical human subject, Mmax= 300mg/min, RHA+
R(2)
limit on CL(t) is 15.64mg/dL which is substantially below
the level at which a human is intoxicated or at which
data is recorded [9], [13]. For that reason, we have not
been concerned with this characteristic of the 2state model,
although we have included a ramp function at the hepatic
vein in order to guarantee that the mass flow of ethanol in
the hepatic vein is nonnegative.
PV= 11.7dL/min, and Km = 10mg/dL so the lower
V. NUMERICAL RESULTS
In the example the model is driven by an oral input
which is a rectangular pulse of amplitude 2.4 × 104mg/min
ethanol and duration 5 min. The resulting trajectories of
the two and threestate PBPK models are shown in Fig
ure 1(b).The major feature of this simulation is the fact that
the liver concentration exceeds the vascular concentration
by nearly a factor of two and that the two and threestate
models predict substantially different peak concentrations.
The reason that the liver concentrations are higher than the
vascular concentrations (which are what is measured by a
breath test) is that the ethanol from the gut travels via the
portal vein directly to the liver where some of the ethanol is
metabolized before entering the systemic vasculature. This
is an important feature of ethanol kinetics because the liver
toxicity of ethanol is related, at least in part, to the higher
concentrations of ethanol present in the liver. Therefore, the
threestate model is potentially more desirable than the two
state model because it allows a more accurate description of
this important effect.
VI. CONCLUSION
The standard PBPK model and generalizations are de
scribed as systems of nonlinear differential equations.We
show the equivalence of the different peripheral tissue models
and the anomalous behavior of the standard PBPK model.
The model demonstrates one of the causes of hepatic ethanol
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6

Hepatic vein
MHV(t)
RHV
Vasculature
μV(t), VV
?
Aorta
MA(t)
MP(t)
RP
?
?
ML(t)
RL
?
?
MHA,L(t)
0
Hepatic artery
MHA(t)
RHA
?
?
Σ
6
MHA(t) − MHA,L(t)
?
RHA+g
?

?
+
M(1)
PV(t)
R(1)
PV
Σ?
+g
?
MGut(t)
0

MP(t)
RP
MP , T(t)
Portal vein
M(2)
PV(t)
R(2)
?
PV
?
MPV,L(t)
0
Liver parenchyma
μL(t), VL
ML,HV(t)
0
?
?
+
+
?
0
MMetab(t)
?
A
M(2)
PV(t)−MPV , L(t)
R(2)
PV
?
(a)

?
0
Periphery
μT(t),VT
MP , T(t)

+g
Σ 
6
0
+
MT ,P(t)
6
?
Vena cava minus Hepatic vein
MVC(t)
RVC
?
MInfuse(t)
0
?
Portal
vein
?
−
6
0
g
Σ
++
?
?
MMetab(t)
A?
Hepatic
artery
??
?
r(·)
·
?
Hepatic
vein
B
?
Periphery
μT(t),VT
0

MCap(t)
RCap
+g
Σ
6
0
MVC(t)
RVC
+ MT ,P(t)
B?
050100150200250
0
20
40
60
80
100
120
time (min)
(b)
concentration (mg/dL)
2−State Blood
2−State Liver
2−State Periphery
3−State Blood
3−State Liver
3−State Periphery
Fig. 1.
(replace A by A?) is shown which reduces the model to a two compartment model. In addition, an alternative model for the connection of the periphery
(replace B by B?) is shown. Panel (b) shows PBPK variable trajectories for the pulse oral input. The three (two) state variables for the three (two) state
model are plotted.
Panel (a) shows Block diagrams for PBPK models. A complete three compartment model is shown. In addition, an alternative model for the liver
toxicity which is the elevated ethanol concentration which the
liver is subjected to due to the portal vein anatomy. Further
more, models in differential equation form and with a limited
number of parameters are key to system identification [14],
filtering [15], and pattern classification [15] applications of
these models to ethanol research studies in humans.
Modeling advances as described in this paper are central
to determining mathematical models for individual subjects
which will better predict the individual subjects’ responses
to ethanol. Using such models it will be possible to design
dose trajectories for individuals such that all individuals have
the same brain exposure to ethanol. If brain ethanol exposure
can be controlled safely with no measurements, then novel
experiments in imaging scanners can be performed that will
provide new forms of information concerning ethanol.
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