NACoM-2003 Extended Abstracts 1 – 4
Constraint Reasoning with Differential Equations
Jorge Cruz∗1 and Pedro Barahona∗∗1
1
Centro de Inteligência Artificial, Departamento de Informática, Faculdade de Ciências e Tecnologia da
Universidade Nova de Lisboa, 2829-516 Caparica, Portugal
Received 14 March 2003
System dynamics is naturally expressed by means of differential equations. Despite their expressive power,
they are difficult to reason about and make decisions, given their non-linearity and the important effects that
the uncertainty on data may cause. In contrast with traditional numerical simulations that may only provide
a likelihood of the results obtained, we propose a constraint reasoning framework to enable safe decision
support despite data uncertainty and illustrate the approach in the tuning of drug design.
1 Introduction
Parametric differential equations are general and expressive mathematical means to model system dynamics. Notwithstanding its expressive power, reasoning with such models may be quite difficult, given their
complexity. Analytical solutions are available only for the simplest models. Alternative numerical simulations require precise numerical values for the parameters involved, often impossible to gather given the
uncertainty on available data.
To overcome this limitation (given non-linearity, small differences on input values may cause important
differences on the output produced), Monte Carlo methods rely on a large number of simulations to estimate
the likelihood of the options under study. However, they cannot provide safe conclusions, given the various
sources of errors accumulated in the simulations (both input and round-of errors).
In contrast, constraint reasoning models the uncertainty of numerical variables within intervals of real
numbers and propagates them through a network of constraints on these variables, to decrease the underlying uncertainty (i.e. width of the intervals). To be effective, these methods rely on advanced methods to
constrain uncertainty sufficiently as to make safe decisions possible.
Interval analysis techniques (e.g. the interval Newton [8]) provide efficient and safe methods for solving
Continuous Constraint Satisfaction Problems [6] (CCSPs) where real variables are constrained by equalities and inequalities. These methods prune the variables domains to impose local consistency [1], guaranteedly loosing no solutions (value combinations satisfying all constraints). The results obtained may be
further improved with search techniques for imposing stronger consistency requirements such as global
hull consistency [3, 4].
In the context of differential equations, validated [9] and constraint based [7] approaches provide safe
methods for solving initial value problems which verify the existence of unique solutions and produce
guaranteed bounds for the true trajectory. We developed an approach [2, 5] that uses a validated method,
Interval Taylor Series (ITS), to include Ordinary Differential Equations (ODEs) in the CCSP framework.
An ODE system y 0 = f (y, t) is considered a restriction on the sequence of values that y can take over t.
Since it does not fully determine the sequence of values of y (but rather a family of such sequences), further
information is usually provided commonly in the form of initial or boundary conditions. An ODE system
together with related information is denoted a Constraint Satisfaction Differential Problem (CSDP).
∗
∗∗
Corresponding author: e-mail: jc@di.fct.unl.pt, Phone: +00 351 212 948 536, Fax: +00 351 212 948 541
Second author: e-mail: pb@di.fct.unl.pt, Phone: +00 351 212 948 536, Fax: +00 351 212 948 541
c 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
°
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J. Cruz and P. Barahona: Constraint Reasoning with Differential Equations
A brief introduction in section 2 shows the expressive power of the framework developed, and stresses
the active use of less common constraints on upper and lower values of the functions involved, and on
the time or the area under curve in which they exceed a certain threshold. The expressive power of the
approach is illustrated in the tuning of drug design, presented in section 3. We show how the active use of
constraints of the types above is sufficient to make safe decisions regarding the intended goals. The paper
ends with a summary of the main conclusions.
2 Constraint Satisfaction Differential Problems
In a CSDP a special variable (xODE ) is associated to an ODE system S for every t within the interval T
through a special constraint, ODES,T (xODE ). Variable xODE represents all functions that are solutions
of S (during T ) and satisfy all the additional restrictions. The other real valued variables of the CSDP,
denoted restriction variables, are used to model a number of constraints of interest in many applications.
Restriction V aluej,t (x), associates variable x with the value of a trajectory component j at a particular
time t, and can be used to model initial and boundary conditions. Restriction maximumV aluej,T (x)
associates x with the maximum value of a trajectory component j within a time interval T (minimum
restrictions are similar). Restriction T imej,T,≥θ (x) constrains x to the time within T when trajectory
component j exceeds a threshold θ. Similarly, restriction Areaj,T,≥θ (x) associates x with the area of a
trajectory component j, within time period T , above threshold θ.
The solving procedure for CSDPs that we developed maintains a safe enclosure for the set of possible
solutions based on an ITS method for initial value problems. The improvement of such enclosure is combined with the enforcement of the ODE restrictions through constraint propagation on a set of narrowing
functions associated with the CSDP. Some reduce the domain of a restriction variable given the current
trajectory enclosure. The safety of such pruning is guaranteed by identifying functions within the current
enclosure that maximise and minimise the restriction variable. For example, the area restriction variable is
maximised by a function that for every value of t associates the maximum possible trajectory value within
the current enclosure. Consequently, the upper bound of such restriction variable cannot exceed the area
computed for such extreme function. Other narrowing functions safely reduce the uncertainty of the trajectory according to the domain of a restriction variable. For example, the trajectory enclosure cannot exceed
the upper bound of a maximum restriction variable. Other narrowing functions reduce the uncertainty of
the trajectory by successive application of the ITS method between consecutive time points.
The full integration of a CSDP within an extended CCSP is accomplished by sharing the restriction
variables of the CSDP. The CSDP solving procedure is used as a safe narrowing procedure for reducing
the domains of the restriction variables.
3 A Differential Model for Drug Design
The gastro-intestinal absorption process subsequent to the oral administration of a therapeutic drug is
usually modeled by the following two-compartment model [10]:
dx(t)
= −p1 x(t) + D(t)
dt
dy(t)
= p1 x(t) − p2 y(t)
dt
(1)
x is the concentration of the drug in the gastro-intestinal tract;
y is the concentration of the drug in the blood stream;
D is the drug intake regimen; p1 and p2 are positive parameters.
The effect of the intake regimen D(t) on the concentrations of the drug in the blood stream during the
administration period is determined by the absorption and metabolic parameters, p1 and p2 . We assume that
the drug is taken in a periodic basis (every six hours), providing a unit dosage that is uniformly dissolved
into the gastro-intestinal tract during the first half hour. Maintaining such intake regimen, the solution
where
NACoM-2003 Extended Abstracts
3
of the ODE system asymptotically converges to a six hours periodic trajectory, the limit cycle, shown in
Figure 1 for specific values of the ODE parameters.
1
1.5
y(t)
x(t)
0.5
1
0
0.5
0
1
2
3
4
5
6
0
t
1
2
3
4
5
6
t
Fig. 1 The periodic limit cycle with p1 = 1.2 and p2 = ln(2)/5.
In designing a drug, it is necessary to adjust the ODE parameters to guarantee that the drug concentrations are effective, but causing no side effects. In general, it is sufficient to guarantee some constraints on
the drug blood concentrations during the limit cycle, namely, to impose bounds on its values, on the area
under the curve and on the total time it remains above some threshold.
We show below how the extended CCSP framework can be used for supporting the drug design process. We will focus on the absorption parameter, p1 , which may be adjusted by appropriate time release
mechanisms (the metabolic parameter p2 , tends to be characteristic of the drug itself and cannot be easily
modified). The tuning of p1 should satisfy the following requirements during the limit cycle: (i) the concentration in the blood bounded between 0.8 and 1.5; (ii) its area under the curve (and above 1.0) bounded
between 1.2 and 1.3; (iii) it cannot exceed 1.1 for more than 4 hours.
3.1 Using the Extended CCSP for Parameter Tuning
The limit cycle and the different requirements may be represented in the extended CCSP framework. Due to
the intake regimen definition D(t), the ODE system has a discontinuity at time t = 0.5, and is represented
by two CSDP constraints in sequence.
The first, PS1 , ranges from the beginning of the limit cycle (t = 0.0) to time t = 0.5, and the second
PS2 , is associated to the remaining trajectory of the limit cycle (until t = 6.0). Both CSDP constraints
include Value, Maximum Value, Minimum Value, Area and Time restrictions for associating variables
with different trajectory properties. Besides variables representing the ODE parameters, the initial trajectory values and the final trajectory values, there are variables representing the maximum and minimum
drug concentration values and respective area above 1.0 and time above 1.1 during the segment of time
associated with each constraint.
The extended CCSP P , shown below, connects in sequence the two ODE segments by assigning the
same variables to both the final values of PS1 and the initial values of PS2 (parameters p1 and p2 are shared
by both constraints). Moreover, the 6 hours period is guaranteed by the assignment of the same variables
to both the initial values of PS1 and the final values of PS2 . In addition to the restriction variables of each
ODE segment, new variables for the whole trajectory sum up the values in each segment.
CCSP P = (X, D, C) where:
X = < x0 , y0 , p1 , p2 , x05 , y05 , ymax1 , ymax2 , ymin1 , ymin2 , ya1 , ya2 , yarea , yt1 , yt2 , ytime >
D = <Dx0 ,Dy0 ,Dp1 ,Dp2 ,Dx05 ,Dy05 ,Dymax1 ,Dymax2 ,Dymin1 ,Dymin2 ,Dya1 ,Dya2 ,Dyarea ,Dyt1 ,Dyt2 ,Dytime >
C = { PS1 (x0 , y0 , p1 , p2 , x05 , y05 , ymax1 , ymin1 , ya1 , yt1 ),
yarea = ya1 + ya2 ,
PS2 (x05 , y05 , p1 , p2 , x0 , y0 , ymax2 , ymin2 , ya2 , yt2 ),
ytime = yt1 + yt2 , }
The tuning of drug design may be supported by solving P with the appropriate set of initial domains
for its variables. We will assume p2 to be fixed to a five-hour half live (Dp2 = [ln(2)/5]) and p1 to
be adjustable up to about ten-minutes half live (Dp1 = [0..4]). The initial value x0 , always very small,
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J. Cruz and P. Barahona: Constraint Reasoning with Differential Equations
is safely bounded in interval Dx0 = [0.0..0.5]. Additionally, the following bounds are imposed by the
previous drug requirements:
Dymin1 = [0.8..1.5],
Dymax1 = [0.8..1.5],
Dyarea = [1.2..1.3],
Dymin2 = [0.8..1.5],
Dymax2 = [0.8..1.5],
Dytime = [0.0..4.0]
Solving the extended CCSP P (enforcing global hull consistency), with a precision of 0.001, narrows
the original p1 interval to [1.191..1.543] in less than 3 minutes (the tests were executed in a Pentium 4
computer at 1.5 GHz with 128 Mbytes memory). Hence, for p1 outside this interval the set of requirements
cannot be satisfied.
This may help to adjust p1 but offers no guarantees on specific choices within the obtained interval.
However, guaranteed results may be obtained for particular choices of the p1 values. Solving P with
initial domains Dx0 = [0.0..0.5], Dy0 = [0.8..1.5], Dp1 = [1.3..1.4] and Dp2 = [ln(2)/5] narrows the
remaining unbounded domains to:
ymin1 ∈ [0.881..0.891],
ymax1 ∈ [1.090..1.102],
yarea ∈ [1.282..1.300],
ymin2 ∈ [0.884..0.894],
ymax2 ∈ [1.447..1.462],
ytime ∈ [3.908..3.967]
Notwithstanding the uncertainty, these results do prove that with p1 within [1.3..1.4] (an acceptable
uncertainty in the manufacturing process), all limit cycle requirements are safely guaranteed. Moreover,
they offer some insight on the requirements showing, for instance, the area to be the most critical constraint.
4 Conclusion
This paper presents a framework to make decisions with models expressed by differential equations, with
a constraint reasoning approach. In contrast to Monte Carlo and other stochastic techniques that can only
assign likelihoods to the different decision options, and despite the data uncertainty and approximation
errors during calculations, the enhanced propagation techniques developed (enforcing global hull consistency) allow safe decisions to be made. Whereas the traditional use of complex differential models for
which there are no analytical solutions is currently unsafe, the constraint reasoning framework extends
the possibility of practical introduction of this type of models in decision making, specially when safe
decisions are required.
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