Application in Bioterrorism
H.T. Banks 1
, David Bortz 2
, Gabriella Pinter 3
and Laura Potter 1;4
1
Center for Research in Scientic Computation, North Carolina State
University, Raleigh, NC 27695-8205; 2
Department of Mathematics,
University of Michigan, Ann Arbor, MI 48109; 3
Department of
Mathematical Sciences, University of Wisconsin-Milwaukee,
Milwaukee, WI 53201
1 Introduction
In this paperwe present a survey of several recent and emerging ideas and eorts on
mod-eling and system interrogation in the presence of uncertainty that we feel have signicant
potentialfor applications relatedto bioterrorism. The rst focuses onphysiologically based
pharmacokinetic (PBPK) type models and the eects of drugs, toxins and viruses on
tis-sue, organs,individualsand populations wherein both intra- and inter-individualvariability
are present when one attempts to determine kinetic rates, susceptibility, eÆcacy of toxins,
antitoxins, etc., in aggregate populations. Methods combining deterministic and stochastic
concepts are necessary to formulate and computationally solve the associated estimation
problems. Similar issues arise inthe HIV infectious models wealso present below.
A secondeort concernsthe use ofremoteelectromagnetic interrogationpulseslinkedto
dielectricpropertiesofmaterialstocarryoutmacroscopicstructuralimagingofbulkpackages
(drugs, explosives, etc.) aswell as test for presence and levels of toxic chemical compounds
intissue. Thesetechniquesalsomay beuseful infunctionalimaging(e.g.,of brainandCNS
activitylevels) todeterminelevelsof threatinpotentialadversariesviachanges indielectric
propertiesand conductivity.
The PBPK and cellular level virus infectious models we discuss are special examples
of a much wider class of population models that one might utilize to investigate potential
agents for use in attacks, such as viruses, bacteria, fungi and other chemical, biochemical
or radiologicalagents. These include general epidemiologicalmodels such as SIR infectious
4
Currentaddress: ScienticComputingandMathematicalModeling,GlaxoSmithKline,ResearchT
private transport; residence times inexposure; subnetworks of populations) as wellas more
general population models with heterogeneities and/or behavioral structures (e.g., social
interaction,age/sizedependency,spatial/temporaldependency,adaptivetransientbehavior,
etc.). Thesemayinvolvegeneraldynamicalsystems,bothdiscreteandcontinuous,including
ordinary and/orpartial dierentialequations and delay dierentialequations. Included are
well known structured population models, such as those of Sinko-Streifer and
McKendrick-VonFoerster. Thesedeterministicmodelsoftenmustbeaugmentedwithprobabilisticand/or
statistical structures such as mixing distributions, random eects, etc. (see [20, 22] for
discussions and references). Such models combine ideas from continuum populationmodels
with aspects of agent based models incorporating individual level eects. The results are
populationmodels encompassingintra-individual and/or inter-individual variabilitythat in
somecasesdescribe(predict) continuouspopulationevolutionthatisdriven bydistributions
of individual level mechanisms and behaviors. The models described in Section 2 below,
where the parameters are viewed asrandom variables, or realizations thereof, are examples
of these.
The use of models such as those outlined above ultimatelylead to estimation or inverse
problems containing both a mathematicalmodeland a statisticalmodel. These are treated
in a t-to-data formulation using either experimental data or synthetic \data" simulating
a desired response. The latter arises, for example, in design of a drug or therapy that will
result in a sought-after response of an individual or a population to a threat. However,
the rationale to support elaborate models with structures does not lie simply in the desire
to better t a data set, but rather to aid in understanding basic mechanisms, pathways,
behavior, etc. and to better frame population as well as individual responses to a
chal-lenge or to a prophylactic. But, it is not just inverse problems that arise in the context of
these models (although that is the focus in this chapter); indeed, ideas from control
the-ory and system optimization are alsoimportant. In almost every instance, including those
discussed inthe examplesbelow, fundamentalmathematics,especiallymodeling,theoretical
and computationalanalysis, probability and statistics,play a signicant role.
TheelectromagneticinterrogationandimagingideasdiscussedinSection3could
conceiv-ablybeapartofasurveillancetechnologyinarstlineofdefenseagainstbioterrorism. More
precisely, physical detection and identicationof hidden substances and agents (whether in
food and water supplies, luggage, mailand packages, etc.) as a part of biodefense depends
not only onthe electromagnetictechniques discussed below, but alsooncharacterization of
dielectric properties of specic molecules and compounds. Although we present only
deter-ministicaspects of theproblems here, itcan beexpectedthat asuccessful methodology will
also involve probabilistic and statistical formulations as well as tools from computational
Inthesediscussionsweshallconsiderinverseorestimationproblemsinvolvingaggregatedata
for populations which may be described by two dierent types of \parameter dependent"
dynamics; for the lack of better terms we shall refer to these as \individualdynamics" and
\aggregatedynamics". Inboth casesthedata andpopulationsinherentlycontainvariability
of parameters; this variabilitymay beintra-individual, inter-individual orboth.
The problems for individual dynamics can be used to treat many examples of practical
interestincludingphysiologicallybasedpharmacokinetic(PBPK)models,biologicallybased
dose response(BBDR) models, and susceptible-infectious-recovered(SIR) modelsof disease
spread. The aggregate dynamics problems include cellular levelvirus models such as those
for human immunodeciency virus(HIV) growth.
In the rst type of problem we consider below, one has a mathematical model at what
we shall term (in perhaps something of a misnomer) the \individual" level. That is, the
population count or density is described by a parameter dependent system. To facilitate
our discussions here weuse, withoutlossof generality,ordinary dierentialequation(ODE)
models of the form
_
x(t)=f(t;x(t);q); q 2Q; (2.1)
where the parameters q (e.g., growth, mortality, fecundity, etc.) in the model vary from
individual toindividual across the population according tosome probabilitydistribution P
on a set of admissible parameters Q. More precisely, we suppose that the population is
made up of subpopulations distinguishedby common values of the parameters q and whose
time course is described by the solutionx(t;q) of (2.1) for the shared value of q. The total
population count or density is then given by a weighted sum of these solutions over all
possible q2Qsothat the counts ordensitiesone expects toobserve atany timet are given
by
x (t;P) = E[x(t;q)jP]
Z
Q
x(t;q)dP(q): (2.2)
Experimental observations or data f ^
d
i
g corresponding to times ft
i
g are then given by the
expected values x(t
i
;P)of (2.2) plus some error "
i
so that
^
d
i =x (t
i
;P)+"
i :
Assumptions about the error f"
i
g in the observation process constitute the basis of an
associatedstatisticalmodelfor theinverse problems. Fordiscussions in this chapter,we will
simply(andperhapsnaively)assume thatthe errors are independent identicallydistributed
(i.i.d.) Gaussian and will use an ordinary least squares (OLS) formulation for our inverse
minimize
J(P)= n
X
i=1 jE[x(t
i ;q)jP]
^
d
i j
2
(2.3)
over P in the set P(Q) of probability measures on Q subject to t ! x(t;q) satisfying (2.1)
for agiven q 2Q.
The second type of problem involves aggregate dynamics wherein one has ODEs that
describe the expected values of the population counts or densities. Essentially one has
dy-namics which already have been summed over the variability in parameters resulting in
measure dependent dynamics (as opposed to parameter dependentdynamics) given by
_
x(t)=g(t;x(t); P); P 2 P(Q); (2.4)
where now x (t; P) is the average or expected value of the population count or density at
time t. In this case the OLS formulation takes the formof minimizing
J(P)= n
X
i=1 jx (t
i ;P)
^
d
i j
2
(2.5)
over P 2 P(Q) subject to the aggregate dynamics (2.4). As we shall note in the examples
below, models such as (2.4) occur naturally and may not be readily formulated in terms of
dynamics of the form(2.1) and viceversa.
InSection2.1weoutlineatheoreticalandcomputationalframeworkforproblems
involv-ing(2.1), (2.3)andillustratetheapproachwithaPBPKmodelfortrichloroethylene(TCE).
We follow this by discussing a framework for problems based on (2.4), (2.5) in the context
of aninverse problemfor virusdynamics (HIV in this case).
2.1 Inverse Problems for Individual Dynamics
Our goal is to estimate q 2 Q R m
from solutions of x (t)_ = f(t;x(t);q). To do this
we visualize parameters as realizations of a random variable and attempt to estimate the
probabilitydistributionfunction (PDF)P 2P(Q)whereP(Q)isthe setof allPDFsonthe
Borel subsetsof Q. We then attemptto estimateP fromgiven data ^
d
i
;i=1;:::;n where
^
d
i
E[x(t
i ;q)jP]
= Z
Q x(t
i
;q)dP(q);
whichin the case of a discreteprobability measure can bewritten as
^
d
i
M
X
j=1 x(t
i ;q
j )p
for P a discretePDF with atoms at fq
j g
j=1
Qand associated probabilitiesfp
j g
j=1 .
We can then, asnoted above, denethe OLS estimation problemof minimizing
J(P)= n
X
i=1 jE[x(t
i ;q)jP]
^
d
i j
2
(2.6)
overP 2P(Q). Toconsider atheoretical and computationalfoundationfor such problems,
one needs the following items:
(i.) A topologyon P(Q);
(ii.) Continuity of P !J(P);
(iii.) Compatible compactnessresults (well-posedness);
(iv.) Computationaltools(approximations, etc.).
Fortunately, probabilitytheory oers agreat starttoward apossible complete, tractable
computationalmethodology[16]. The mostimportanttoolistheProhorovmetric,whichwe
proceed to dene. Suppose (Q;d) is acomplete metricspace. For any closed subset F Q
and ">0; dene
F "
=fq 2Q:d(~q;q)<";q~2Fg:
We thendene the Prohorov metric :P(Q)P(Q)!R +
by
(P
1 ;P
2
) inff">0:P
1
[F]P
2 [F
"
]+"; F closed; F Qg:
This can be shown to be a metric on P(Q) and has a number of well known properties
including
(a.) (P(Q);) isa complete metric space;
(b.) IfQ is compact, then (P(Q);) is acompact metric space.
Wenote thatthedenitionof isnot intuitive. Forexample,whatdoesP
k
!P inmean?
Wehave the followingimportantcharacterizations [16].
Theorem 2.1 Given P
k
;P 2P(Q); the followingconvergence statements are equivalent:
(i.) (P
k
;P)!0;
(ii.) R
Q fdP
k (q)!
R
Q
fdP(q)for all bounded, uniformly continuous f :Q!R 1
;
(iii.) P
k
Convergence inthe metricis equivalent to convergence indistribution;
Let C
B
(Q) denote the topological dual of C
B
(Q), where C
B
(Q) is the usual space of
bounded continuous functions on Q with the supremum norm. If we view P(Q)
C
B
(Q); convergence inthe topology isequivalent toweak
convergence in P(Q).
More importantly,
(P
k
;P)!0 is equivalent to Z
Q x(t
i ;q)dP
k (q)!
Z
Q x(t
i
;q)dP(q);
and P
k
!P in metric ishence equivalent to
E[x(t
i ;q)jP
k
]!E[x(t
i ;q)jP]
or\convergence in expectation." This yieldsthat
P !J(P)= n
X
i=1 jE[x(t
i ;q)jP]
^
d
i j
2
is continuous in the topology. Continuity of P ! J(P) and compactness of P(Q) (each
with respect to the metric) allows one to assert the existence of a solution to minJ(P)
over P 2P(Q).
2.1.1 Computational issues and approximation ideas
Werstnotethat(P(Q);)isinnite-dimensionalandhenceonemustusenite-dimensional
approximations to obtain tractable computationalalgorithms. To this end, one may prove
(see [5])
Theorem 2.2 Let Q be a complete, separable metric space with metric d;S the class of all
Borel subsets of Q and P(Q) the space of probability measures on (Q;S). Let Q
0 =fq
j g
1
j=1
be a countable, dense subset of Q. Then the set of P 2P(Q) such that P has nite support
in Q
0
and rational masses is dense in P(Q) in the metric. That is,
P
0
(Q)fP 2P(Q):P = k
X
j=1 p
j Æ
q
j
;k 2N +
;q
j 2Q
0 ;p
j
rational; k
X
j=1 p
j =1g
is dense in P(Q) relative to , where Æ
q
j
is the Dirac measure with atom at q
Given Q d = 1 M=1 Q M with Q M =fq M j g M j=1
chosen so that Q
d
isdense in Q, dene
P M
(Q)=fP 2P(Q):P = M X j=1 p j Æ q M j ;q M j 2Q M ;p j rational; k X j=1 p j =1g:
Then we nd
P M
(Q)is a compact subset of (P(Q););
P M
(Q)6P M+1
(Q);
\P M
(Q)!P(Q)" in the topology;that is,elements inP(Q) may be approximated
arbitrarilyclosely inthe metric by elements inP M
(Q)for M suÆciently large.
These ideas and results can then be used to establish a type of \stability" of the
in-verse problem (see [5, 13]). We rst dene a series of approximate problems consisting of
minimizing J(P M )= n X x=1 jE[x(t i ;q)jP M ] ^ d i j 2 over P M 2P M
(Q). Then we have
Theorem 2.3 Let Q be a compact metric space and assume solutions x(t;q) of x(t)_ =
f(t;x(t);q) are continuous in q on Q. Let P(Q) be the set of all probability measures on Q
andletQ
d
be acountabledensesubsetofQas denedpreviouslywithQ
M =fq M j g M j=1 . Dene P M
(Q) asabove. Suppose P M ( ^ d k
) isthe set of minimizersfor J(P)over P 2P M
(Q)
corre-spondingtothedataf ^
d k
gandP
( ^
d)isthesetof minimizersoverP 2P(Q)correspondingto
^ d, where ^ d k ; ^
d 2R n
are the observed data such that ^
d k
! ^
d. Then dist(P M ( ^ d k );P ( ^
d))!0
as M ! 1 and ^
d k
! ^
d. Thus the solutions depend continuously on the data and the
approximate problems are method stable.
To illustrate the above methodology with a relevant example, we present here a brief
description of aPBPK-hybrid modelfor trichloroethylene (TCE) and indicatehow one
for-mulatesand implementsthe correspondingestimationproblems. TCE isametaldegreasing
agentthatis awidespreadenvironmentalcontaminant,and has been linked toseveral types
of cancer in laboratory animals and humans. This compound is highly soluble in lipids
and is known to accumulate within the fat tissue. Therefore, in order to accurately predict
toxicity-related measures such as the net clearance rate of TCE and the eective dose of
TCE delivered totarget tissues, itisimportantto accurately capture the transportof TCE
within the fat tissue.
Physiologically based pharmacokinetic (PBPK) models are used to describe the
transportofthecompound,aswellascompartmentsfortissuesthataretargetsofthe
chemi-cal'stoxiceects. See[24]foradetaileddescriptionofstandardPBPKmodelingtechniques.
As discussed in [1, 27], the standard perfusion-limited and diusion-limited
compart-mentalmodelsused inPBPKmodelingare notable todescribe thedynamics ofTCE infat
tissueasseeninexperimentaldata,andthe assumptionsfortheseODE-basedmodelsdonot
matchwellwith theheterogeneous physiologyof fattissue. This motivatedthe development
ofaspecializedcompartmentalmodel forthefattissue,whichisthencoupledwith standard
compartments forthe remaining non-fat compartmentsto produce aPBPK-hybrid model.
The resulting compartmental model for the fat tissue is based on an axial dispersion
modeloriginally developed by Roberts and Rowland [28] for the transport of solutes in the
liver. The underlying assumptions for the dispersion model matchwell with the physiology
of fat tissue (see [1, 27] for details), and the geometry for the PDE-based fat model is
based specically onthe known geometry of fat cells and their accompanying capillaries. A
key feature of the dispersion model is its aggregate nature, using a representative \cell" to
capture the transport behaviorof the compoundina collectionofmany similar\cells" that
have varying properties.
In this particular case, the representative \cell" is a unit containing three
subcompart-ments: asingleadipocyte(fatcell)togetherwithanadjoiningcapillary,andthesurrounding
interstitialuid. Inthemodel,theadipocyteisrepresented byasphereandthecapillaryisa
cylindricaltubewithcircularcross-section;the interstitialuidllsinthespacesurrounding
theothertworegions. Themodel geometryandequationsaregiveninsphericalcoordinates.
See [1, 27] for acomplete description of the model.
Itisassumed thatTCEentersthecapillaryregionofthefatcompartmentalongwiththe
arterialblood. Thecapillaryequation(2.7)includesaone-dimensionalconvection-dispersion
termtogether withatermbasedonFick'srst lawofdiusionfortheexchange between the
capillaryandtheothertwosubcompartments. Theaccompanyingboundaryconditions(2.8)
and (2.9) connect the capillary with the arterial and venous blood systems, and are based
on ux balance. The adipocyte and interstitial equations (2.10) and (2.15) each contain
two-dimensional diusion terms together with terms for the exchange of TCE between the
subcompartments. The boundary conditions (2.11){(2.14) and (2.16){(2.19) are based on
standard periodicand niteness conditions that are appropriate for diusionon a spherical
domain.
In addition to the fat compartment, there are perfusion-limited tissue compartments
used inthe PBPK-hybrid modeltorepresent the brain,kidney, liver, muscle and remaining
tissues. Uptake of TCE is via inhalation in the lungs, which is modeled using a standard
steady-state assumption. Metabolism of TCE is described with aMichaelis-Menten termin
the liver with parameters v
max
(mg/hour) and k
M
(mg/liter). The resulting equations for
V B @C B @t = V B r 2 sin @ @ sin D B r 2 @C B @ vC B + I BI (f I C I ( 0 ) f B C B ) + A BA (f A C A ( 0 ) f B C B ) (2.7) D B r 2 @C B @
(t;)+vC
B (t;) ="1 = Q c 1000A B C a (t) (2.8) D B r 2 @C B @
(t;)+vC
B (t;) = "2 = Q c 1000A B C v (t) (2.9) V I @C I @t = V I D I r 2 1 1 sin 2 @ 2 C I @ 2 + 1 sin @ @ sin @C I @ + Æ 0 () B () I BI (f B C B f I C I )+ IA (f A C A f I C I ) (2.10) C I
(t;;) = C
I
(t;+2;) (2.11)
@C
I
@
(t;;) = @C
I
@
(t;+2;) (2.12)
C
I
(t;;0) < 1 (2.13)
C
I
(t;;) < 1 (2.14)
V A @C A @t = V A D A r 2 0 1 sin 2 @ 2 C A @ 2 + 1 sin @ @ sin @C A @ + Æ 0 () B () A BA (f B C B f A C A )+ IA (f I C I f A C A ) (2.15) C A
(t;;) = C
A
(t;+2;) (2.16)
@C
A
@
(t;;) = @C
A
@
(t;+2;) (2.17)
C
A
(t;;0) < 1 (2.18)
C
A
(t;;) < 1 (2.19)
V l l dt = Q l (C a C l =P l )
max l l
k M +C l =P l (2.25) V k dC k dt = Q k (C a C k =P k ): (2.26)
The variables in the model are the concentrations of TCE (in mg/liter) in each of the
compartments/subcompartments, and are denoted by C with subscripts corresponding to
the respective tissue/region. Model parameters include tissue volumes V in liters, blood
ow rates Q in liters/hour and partition coeÆcients P, each with the appropriate tissue
subscripts. Parameters specic to the dispersion model include the dispersion coeÆcient
D
B
and the diusion coeÆcients D
I
and D
A in m
2
/hour; unbound fractions f
B , f I , f A ; permeability coeÆcients BA , IA , BI
in liters/hour; blood ow parameters v (m/hour)
and F; and inter-region transport parameters
I
and
A
. A complete discussion of the
modelequations and parameters ispresented in [1,27].
Here weutilizetheTCE model(2.7){(2.26)toillustrateparameterestimationtechniques
for models with individual-leveldynamics that have realization-dependent derivatives. We
present results for both parametric and nonparametric parameter estimation approaches,
wheretheparameterofinterestistheprobabilitydistributionofthefatdispersioncoeÆcient
D
B
inthe capillary. This parameter is animportant measure of the degree of heterogeneity
within anindividual'sfat tissue.
The parametric and nonparametric approaches each t into the general framework
pre-sented earlier inthis chapter formodels with individualdynamics. We assumethat the
pa-rameter q D
B
2Q is distributed across the populationwith distribution P 2 P(Q),
whereisasetofadmissibleprobabilitydistributionfunctions(possiblyallofP(Q)). Then
the general objective function for the standard least squares parameter estimation problem
is given by
J(P)= n X i=1 E[x(t i ;q)jP]
^ d i 2 ; (2.27)
where,inthiscase, ^
d
i
representsameasurementofthespatialmeanconcentrationofTCEin
thefatcellsattimet
i
,andx(t
i
;q)isthe spatialmeanconcentrationofTCEintheadipocyte
region of the fat compartment that isobtained by solving (2.7) {(2.26) with parameter q.
For the parametric approach, we assume that the probability distribution P for q is of
a particular form with parameterization q~2 R Nq
(e.g., a normal distribution N(;) with
parameterization q~ = (;)), so that the set of admissible probability distributions is
dened as the set of alldistributionsP
~ q
of that given form. The estimationproblemis then
reduced to the N
q
-dimensionalproblem of minimizing
J(~q)= n X i=1 E[x(t i ;q)jP ~ q ] ^ d i 2 (2.28) over P ~ q
2 foradmissible parameterizationsq~2 ~
QR Nq
objectivefunction(2.27) toa moretractable N
q
-dimensionalproblem. Whenthere isahigh
degree of condence about the specic form of the probability distribution P, this method
can be expected to perform reasonably well. In many cases, however, the exact form of
P is unknown, making it diÆcult to choose the proper restriction for the set and the
corresponding parameterization q.~ If an incorrect form and parameterization are chosen
for the distribution function, the parametric approach is likely to provide a poor t to the
data since the \true" underlying distribution may not correspond to a distribution in the
admissible set . Even more alarming are situations where a reasonable t is found even
though an incorrect parameteric form has been assumed (see [15] for examples). In this
situation,a nonparametric approachis oftenmore appropriate.
Instead of using a specic form for the distribution P with a nite-dimensional
param-eterization q,~ the nonparametric parameter estimation approach utilizes a discretization of
the admissible parameter set Q toachieve a nite-dimensional approximation for the
origi-nalobjectivefunction(2.27). Theresultingfamilyof nite-dimensionalestimation problems
can be solved in a straightforward manner using quadratic programming, and theoretical
results established in [5, 15] guarantee that the minimizersconverge toa minimizer for the
innite-dimensionalproblem(e.g., see Theorem 2.3above).
As described earlier in this chapter, we utilize the set Q
d =
S
Q
M
, a dense, countable
subsetofQ,togetherwithconvexcombinationsofDiracdeltadistributionsdened overQ
M ,
todene the following familyof objective functions over P M
(Q):
J(P
M )=
N
X
i=1
E[x(t
i ;q)jP
M ]
^
d
i
2
; (2.29)
where ^
d
i
are observations corresponding to the expected value, and P
M
is a probability
distribution inP M
(Q) asdened inTheorem 2.3 above.
Note that (2.29) can be rewritten as
J(P
M )=
N
X
i=1
M
X
j=1 x(t
i ;q
M
j )p
j ^
d
i
2
; (2.30)
sothat the minimizationof (2.30)is equivalent tosolving aconstrained quadratic
program-ming problemfor fp
1 ;:::;p
M
g with constraintsp
j
0and P
M
j=1 p
j =1.
Example results for the parametric and nonparametric methods are given in Figures 1
and2respectively. Ineachcase,the observationsusedinthe parameterestimationproblems
were generated by solving the TCE model (2.7){(2.26) with a xed parameter set q
. In
this case, the solutionx(t
i
;q)isthe spatial meanadipocyteconcentrationof TCE given the
parameter q = D
B
. The probability distributions obtained by the estimation methods are
presented in Figures 1 and 2. In Figure 1, the solid line represents the true distribution
corresponding to q
, q
0
denotes the initialiterate used inthe optimizationprocedure and q
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
0
0.1
0.2
0.3
Probability distribution 0.4
0.5
0.6
0.7
0.8
0.9
1
Normal distributions
q
*
q
q
0
Figure 1: Example solution for the parametric method applied to the TCE PBPK-hybrid
0
0.5
1
1.5
2
2.5
3
3.5
4
0
0.2
0.4
0.6
0.8
1
q
Probability distribution
Bimodal, qM = 32, tf = 2, Nt = 6, Ns = 50 (Type I)
P
*
P
prob
Figure2: Examplesolutionforthe nonparametricmethodappliedtotheTCEPBPK-hybrid
model.
For the parametric case, the data-generating probability distribution we chose is a
bi-truncatednormaldistributionforq
withmean
=1,standard deviation
=0:0833,and
support over the interval [
3
;+3
]. The objective function (2.28) was minimized
over the set of bitruncated normal distributions with parameterizations (;) and with
nite support in[ 3;+3]. See [27] forcomplete details and additionalexamples.
Forthe nonparametric case, we useda bimodalgaussiandistribution with means
1 =1
and
2
= 3 and standard deviations
1
= 0:1667 and
2
= 0:3333. The objective
func-tion (2.30) was minimized using the quadratic programming routine quadprog in Matlab.
More details and examples for the nonparametric approach applied to the TCE model are
given in[15].
2.2 Aggregate Dynamics
Weturn next tothe problems with aggregate dynamics(2.4) and OLS functional(2.5). For
these problems one can also develop a general theoretical framework. We rst outline the
reverse transcriptase
RNA
capsid
envelope
single-strand RNA
lentivirus
infection
cell membrane
cytosol
loss of
envelope
loss of
viral capsid
cellular
DNA
viral DNA
altered
cellular DNA
integration
transcription into multiple RNA copies
translation
translation
assembly
viral budding
24 hour
mean delay
Figure3: HIV infection pathway.
Given the system dynamics
dx
dt
=g(t;x(t); P); P 2P(Q); (2.31)
one rst argues that (t;x ; P)!g(t;x; P)is continuous from[0;T]R n
P(Q) toR n
, and
locallyLipschitzinx . Thenbyextensionofstandardcontinuousdependenceon\parameters"
results for ODEs, one obtains that P ! x(t; P) is continuous from P(Q) to R n
for each t.
This again yields P ! J(P) = P
i jx(t
i ;P)
^
d
i j
2
is continuous from P(Q) to R 1
, where
P(Q), with the Prohorov metric,is compact for Q compact.
Then the general theory of Banks-Bihari [5] can be followed to obtain existence and
stability for inverse problems (continuous dependence with respect to data of solutions of
the inverse problem)as inTheorems2.2and 2.3 above. Moreover, anapproximation theory
as abasis for computationalmethodsis obtained.
We illustrate the ideas in the situation where the underlying ODE system (2.31) is
re-placedbya nonlinear functionaldierentialequation(FDE) system. This example arisesin
modelingprogression of HIV forwhicha schematicof the cellular levelinfection pathway is
_
V(t) = cV(t)+n
A Z
0
r
A(t+)dP
1
()+n
C
C(t) pV(t)T(t) (2.32)
_
A(t) = (r
v Æ
A
ÆX(t))A(t) Z
0
r
A(t+)dP
2
()+pV(t)T(t) (2.33)
_
C(t) = (r
v Æ
C
ÆX(t))C(t)+ Z
0
r
A(t+)dP
2
() (2.34)
_
T(t) = (r
u Æ
u
ÆX(t) pV(t))(t)+S(t); (2.35)
where X = A+C +T and V(t) is the expected value of the population count (number)
of virus cells, A(t) is the number of acutely infected cells, C(t) is the expected value of the
number of chronically infected cells, and T(t) is the total number of target or uninfected
cells,eachattimetrespectively. The probabilitymeasuresP
1 ;P
2
inthemodel arisebecause
there are delays
1 and
1 +
2
fromthe timeof acutecellular infection untila cellbecomes
productively infected and from the time of acute infection until chronic infection,
respec-tively (see Appendix A of [6] or Chapter 2 of [17] for a careful and detailed derivation).
Biologically, these delay times must vary across the population and this variability is
de-scribed by the PDFsP
1
and P
2
inthe system (2.32)-(2.35). More specically,the variables
V(t)and C(t)have substructures (classesV(t;);C(t;)grouped accordingtotheirortheir
\mothers" delay times) which are averaged across the populations using the distributions
P
1 , P
2
,respectively,so that
V(t) = E[V(t;)jP
1 ]=
Z
0
r
V(t;)dP
1 ();
C(t) = E[C(t;)jP
2 ]=
Z
0
r
C(t;)dP
2 ():
Thisyieldsthesystem(2.32)-(2.35)withvectorvaluedmeasuredependent(P =(P
1 ;P
2 ))
dynamics as formulated in (2.31) wherein the \state" variables are expected values of
pop-ulation counts. A careful consideration of the derivation of this system reveals that it does
not arise from aparameter dependent system for
x(t;q)=(V(t;q);A(t;q);C(t;q);T(t;q))
with parameters q = (
1 ;
2
) and thus the associated inverse problems for the estimation of
P
1 ;P
2
are fundamentallydierent fromthose in the PBPK examplesof Section 2.1above.
The dynamicalsystem (2.32)-(2.35)for given P
1 ;P
2
isitselfaninnite-dimensionalstate
system (similar to apartial dierential equation(PDE) system inthis regard). Tosee this,
we note that (2.32)-(2.35)can bewritten (see [6, 17] for details)in the form
_
x(t)=L(x (t);x
t )+f
1
(x(t))+f
2
0
5
10
15
20
25
0
0.5
1
1.5
2
2.5
3
x 10
7
Total Cells vs. Time
Days
A+C+T
AEE Optimized Solution
Experimental Data
where !x
t
()x(t+0); r 0,isafunctionfrom[ r;0]toR . Thissystemrequires
initialdata (x (0);x
0
) in the state space ~
Z = R 4
C( r;0;R 4
) which is readily recognized
as being innite dimensional. For such systems one needs an approximation theory and
resultingcomputationalmethodology (e.g., nite element methods similarto those popular
inPDEtheoryand implementation)eventocarry outforward simulations(an integralpart,
ofcourse,ofmostinverseproblemmethodologies). Fortunately,suchatheoryexists[4,9,10]
in the contextof abstract evolution equations
_
z(t)=Az(t)+(f
2 (t);0)
in astate space Z =R 4
L
2
( r;0;R 4
) where
D(A)=f((0);)2Z :2H 1
( r;0;R 4
)g
and A:D(A)Z !Z is given by
A((0);)=(L((0);)+f
1
((0)); d
d )
for !() inH 1
( r;0;R 4
).
This theory can be used as a foundation to develop a theoretical and computational
framework for inverse problems similar to that outlined for parameter dependent systems
such as the PBPK example in Section 2.1. While the resulting wellposedness and method
stability (see Chapter 3 of [17]) statements are similar inspirit to the Banks-Bihari results
given in Section 2.1, the technical details are quite dierent and rely heavily on the FDE
theory in [4,9, 10]. Detailsare given in [6,17].
The methodology outlined here (along with an ANOVA type statistical methodology)
was successfully used to analyze in vitro data [29] from the experiments of Dr. Michael
Emerman of the Fred Hutchinson Cancer Research Center in Seattle. A comparison of the
simulation of the modelwith minimizingP
= (P
1 ;P
2
) obtained fromthe inverse problem
(i.e.,(2.5) withsystem (2.32)-(2.35))toaset ofEmerman'sexperimentaldataisdepictedin
Figure4. Wenote that the measures P
1 ;P
2
used for the simulationdepicted here consisted
of Dirac measures with single atoms at
1
and
1 +
2
, respectively, where
1
=22:8 hours
and
2
=3:2hours.
3 Electromagnetic imaging of hidden substances
Inthissectionwesummarizeoureortsinmodelingtheuseofelectromagneticpulsedsignals
toremotelyextractinformationabout geometricandchemicalpropertiesofsubstances. Our
in which this theory is currently being extended.
The interactionofveryhighfrequency electromagneticwaves, X-rays,withmaterialshas
longbeenexploitedforimagingpurposesinmedicaldiagnostics. Manynoveltechniqueshave
been developed during the past several years to extend the capabilities of traditionalX-ray
methods. Moreover, waves at dierent frequency ranges of the electromagnetic spectrum
have been utilized. A close inspection of the interaction of materials with electromagnetic
radiation at dierent frequency ranges reveals dierent underlyingmechanisms which need
to be correctly captured in the appropriate models. At the same time, the diversity of this
interaction makes possible a variety of applications from laser surgery to the detection of
environmental contaminants. Some of these techniques have great potentialto play an
im-portantroleinthecurrenteortsinprovidingamoresecureenvironmentfromdierentforms
ofterroristactivities. AsstatedinSection1,interrogationofmaterialswithelectromagnetic
waves couldbeuseful inlook-downsurveillance, imagingof structures, identicationof
con-taminants, airport security devices, detection of hidden substances, explosives, chemicals,
toxins and bioagents.
The successful use of these techniques is wrought with many technical and theoretical
challenges. While portable lasers and X-ray machines are widely available, other ranges of
the EM spectrumare not aswellrepresented. Terahertz signalgenerators and detectors are
currently being developed and exhibit a great promise for providingnovel imaging devices.
Terahertz radiationhasseveraladvantagesovertraditionalX-raymethodsand iswell-suited
for imaging applications. T-rays have low photon energies and are non-ionizing, thus they
are thought to be safer than X-rays. Recently developed devices can generate very short
(sub-ps) burstsofTHzradiationconsisting ofonlyafewcyclesofthe electriceld, yet
span-ning a broad bandwidth. THz waveforms passing through, or reected from an object can
berecordedinthetimedomainwith veryhighsignal-to-noiseratio. Manyorganicmolecules
show strong absorption and dispersion inthis frequency range. These eects constitute the
polarization mechanism of the molecules which has an inuence on the electric eld and
the propagationof the electromagnetic wave insidethe material. Since these transitions are
characteristic to the particular molecules, detection of the temporal distortions produced
thus yieldsinformation about the composition of the material inreal time. For example,it
is known that cancerous and benign tumors have dierent electromagnetic characteristics.
Therefore animagingdevice basedonTHz wavescould not onlygiveinformationabout the
structure of an object (geometrical properties) but could help in determining their
compo-sition and electromagneticproperties aswell ina non-invasiveway. As shown in[25],T-ray
imagingcan beuseful bothby sendinga pulse through the materialand detecting iton the
other side orby sending apulse toward the materialand recording the reections from the
interface(s) (reection imaging). This latter procedure is especially important when
detec-tors cannot be placed on the other side of the object, or when only slices of an object need
tobeevaluated. Potentialapplicationsrangefrommedicalanddentaldiagnosticstoquality
control in food processing, semiconductor and chip manufacturing and to the detection of
The technicaladvances ingeneratingelectromagnetic radiationindierent rangesof the
spectrumandtheir emergingapplicationsinboth medicaland generalimagingeldscallfor
better theoretical understanding and accurate models of the interaction of electromagnetic
signalsandvarioussubstances. Indevelopingthesemodelsspecialattentionhastobeplaced
onthespecicfrequencyrangeandintensityoftheelectromagneticradiation,thetypeofthe
interrogating signal that is used and the type of materialthat it encounters. For example,
inthe highopticalrangeone generallyassumesanonlinearrelationshipbetween theelectric
eld and the polarization, and uses the slowly-moving envelope assumption to derive the
nonlinear Schrodinger equation for the propagation of wave-packets in a dielectric medium
from Maxwell's equations. While the latter is a reasonable assumption for pulses that are
\long"comparedtoacharacteristicfrequency,itmaybeinadequatetoaccountforultrashort
pulses. Inthat casea dierent,full-wavederivation isnecessary tocapture transient eects.
In the microwave range of the electromagnetic spectrum one can assume that the
rela-tionship between the electric eld and the polarization is linear for most materials. In the
following we will summarize a model developed in [8] for the propagation of windowed
mi-crowave (3-100 GHz)pulses in a dielectric medium. In that work the basic question, which
was answered in the aÆrmative, was whether a variationalformulation of Maxwell's
equa-tions foraspecic 1-Dsituationcouldsuccessfully beused inthe identicationof geometric
and dielectric properties of a material slab that is interrogated by microwave pulses from
antenna sources.
3.1 Variational approach for microwave pulse propagation
In this 1-D model an innite slab of material is placed in the interval = [z
1 ;z
2 ] with
faces parallel to the xy plane. The interrogating signal is assumed to be a short planar
electromagnetic pulse normally incident on the material and the electric eld is polarized
with oscillationsinthe xz plane only.
Thus the electric eld is parallel to the^{ axis atall points in
0
and the magnetic eld
~
H is parallel to the ^| axis. Since the material properties are assumed to be homogeneous
in the x and y variables,it can be shown that the propagatingwaves in are alsoreduced
to one nontrivial component [8]. This makes it possible torepresent the elds in and
0
withthe scalarfunctionsE(t;z)andH(t;z). Undertheseassumptions,Maxwell'sequations
reduce to
@E
@z
=
o @H
@t
(3.37)
@H
@z =
@D
@t
+E+J
s
(3.38)
for the scalar elds E and H. The magnetic eld can be eliminated from the equations by
z
H(t;z)
E(t;z)
z
1
z
2
y
Figure 5: Geometry of the physicalproblem.
usingtheequationforelectricuxdensityD=E+P where=
0 (1+(
r 1)I
)toobtain
0
E+
0
P +
0
_
E E
00
=
0 _
J
s
: (3.39)
Ageneralintegralequationmodelcanbeemployedtodescribethebehaviorofthemedia's
macroscopic electricpolarizationP:
P(t;x)= Z
t
0
g(t s;x)E(s;x)ds: (3.40)
This constitutive law is given in terms of a susceptibility kernel g, and expresses the fact
that the materialresponds tothe electriceldin nitetime. This formulationissuÆciently
generaltoincludemicroscopicpolarizationmechanismssuchasdipoleororientational
polar-izationaswellasionic andelectronic polarization(see later)[3,21]. Wenotethat P(0;x)is
assumedtobe0. Toallowforacomponentofthepolarizationwhichdependsinstantaneously
onthe electriceld one can includea term
0
E inD. Hence,
D =
0
(1+)E+P (3.41)
=
0
r
E+P; (3.42)
where
r
=1+1isarelativepermittivitywhichcanbetreatedasaspatiallydependent
1
locationoftheoriginalbackboundaryatz =z
2
,i.e.,the depthof theslab,isunknown. The
unknown boundarycreates computationaldiÆculties in the inverse problemsince changing
domains would involve changing discretization grids in the usual nite element schemes.
Thus the method of mappings [11, 12, 26] is applied to transform the problem to a known
reference domain. The domain of the computation is dened to be the interval ~
= [0;1].
Anabsorbing boundary conditionis placedatthe z =0boundaryof the intervaltoprevent
the reection of waves. This can beexpressed by
1 c @E @t @E @z z=0
=0 (3.43)
wherec 2 1= 0 0
. Asupraconductivebackingisplacedontheslabatz =z
2
. Theboundary
conditions onthis supraconductive reector (after mappingz
2
to the reference point z =1)
are given by E(t;1)=0. Substituting an expression for
P derived from equation (3.40) we
obtain the strong formof the equation
~
r
E(t;z)+ 1
0 I
(z)((z)+g(0;z)) _ E(t;z) + 1 0 I
(z)g(0;_ z)E(t;z)+ Z t 0 I (z) 1 0
g(t s;z)E(s;z)ds (3.44)
c 2
E 00
(t;z)= 1 0 _ J s (t;z);
where indicator functions I
have been added to explicitly enforce the restriction of
po-larization and conductivity to the interior of the transformed medium = [z
1
;1] and
~
r
==
0
=1+(
r 1)I
1 throughout [0;1].
Due to the form of the interrogatinginputs, the dielectrically discontinuous medium
in-terfaces,andthepossiblelackofsmoothnessinmappingtheoriginaldomain
0 S
=[0;z
2 ]
to the reference domain ~
= [0;1], one should not expect classical solutions to Maxwell's
equationsinstrong form. Thusitisdesirabletowrite the system equationsinweakor
vari-ational form. Using the spaces H =L
2
(0;1) and V = H 1
R
(0;1) =f 2 H 1
(0;1)j(1) =0g
and the boundaryconditions (3.43), the equation (3.44) can be writtenin weak formas
h ~
r
E;i+h _
E;i+hE;i+h Z
t
0
(t s;)E(s;)ds;i
+hc 2 E 0 ; 0
i+c _
E(t;0)(0)=hJ(t;);i (3.45)
with initialconditions
E(0;z)=(z) _
E(0;z)= (z);
where the coeÆcients are given by
(t;z)= 1
0 I
(z)g(t;z); (z)= 1
0 I
(z)g(0;_ z);
(z)= 1 0 I
and h;i is the L inner product. The functions ; and are dependent on parameters
which must be identied. These functions are assumed to be in L 1
but may lack any
additionalregularity.
Existence,uniquenessandregularityofsolutionsisestablishedin[8],andacomprehensive
approximation framework is developed for the forward as well as the inverse problems. It
is shown computationally that it is possible to simulate and identify Debye and Lorentz
polarization mechanisms in media using rst reected pulses. The thickness of a layered
slab using reected signals from a supraconductive back boundary can also be accurately
estimated. Itisdemonstratedcomputationallythatthismodelcapturestransienteectsand
shows the formationof Brillouinprecursors inside the material[8].
0
0.2
0.4
0.6
0.8
1
−150
−100
−50
0
50
100
150
z
electric field
0
0.2
0.4
0.6
0.8
1
−15
−10
−5
0
5
10
z
electric field
0
0.2
0.4
0.6
0.8
1
−10
−5
0
5
z
electric field
0
0.2
0.4
0.6
0.8
1
−6
−4
−2
0
2
4
z
electric field
Formation of Brillouin precursors in the material [0.33,1]
Time=0.6 ns
Time=3.7 ns
Time= 5 ns
Time=6.8 ns
Figure6: Formationof Brillouinprecursors using alinear Debye model.
Insummary,thisapproachisamenabletoultrashortinputpulsesandprovidesacomplete
theoreticaland computationalframeworkfor the directand the inverse probleminthis
one-dimensional model.
This work has been extended in dierent directions. A corresponding analysis with
acousticreectorsatthebackoftheslabofmaterialandpressuredependentMaxwellsystem
coeÆcientsis developedin [2]. Itisshown that insteadofa supraconductive backing(which
is not practical in many medical or remote imaging applications), an acoustic wave can be
employed toreectthe electromagneticsignal. Moreover, thesereectionscan againbeused
toidentify geometric and dielectric properties of the material.
To develop and use a similar methodology for terahertz signals we must capture the
response of materials to higher frequencies. Thus, we need to represent the absorption and
dispersion properties of the material by accurately modeling the underlying polarization
mechanisms. As interrogating frequencies increase, it is not unreasonable to expect that
Polarization, the general macroscopic response of a material to an electric eld, is an
im-portant dielectric characteristic specic to a given material and hence is important to any
interrogation methodology. It depends heavily on the molecularstructure of the material.
Dielectricmaterialscontainboundnegativeandpositivechargesthatarenotfreetomove
aschargesdoinconductors. Thesechargesare kept inplaceby atomicand molecularforces.
When subjected to anexternal electric eld, dipole moments are induced inthe atoms and
molecules. Theelectricpolarizationvector isdened asthedipolemomentperunitvolume.
Themechanismbywhichthesedipolemomentsarecreatedisdierentindierentmaterials,
whethergases,liquids,orsolids. Moleculesofcertaingases(e.g.,oxygen)containasymmetric
pair of atoms in each moleculeand thus have no inherent dipolemoments. Such molecules
arecallednonpolar. Inothers,(e.g.,water vapor)the center ofgravityofthe positivecharge
(inthiscase onthehydrogenatom)andthenegativecharge(onthe oxygen)donotcoincide,
andthe total chargedistributiononthe moleculehas adipolemoment. Thesemolecules are
called polar.
First we consider nonpolar molecules. When an electric eld is applied to the atoms of
such molecules the electrons are forced in one direction, while the nucleus is forced in the
opposite direction by the eld. Thus there is a net displacement of the centers of charge,
and a dipole moment is created. This displacement of the electron distribution is called
electronic polarization. In a changing electric eld the displacementof the center of charge
oftheelectrons isusuallymodeledbyaharmonicoscillatorandthisgivesrise tothe Lorentz
modelfor electronic polarization:
~
P + 1
_
~
P +! 2
0 ~
P =
0 !
2
p ~
E;
where
0
is the dielectric constant, and !
p
is the so called plasma frequency given by !
p =
p
s
1
; with
s and
1
being the relativepermittivitiesof the materialinthe limitof the
static and very high frequencies,respectively.
The same mechanism can be observed in polar molecules. However, in additionto this
eect, the electric eld forces a portion of the originally randomly oriented internal dipoles
to line up with the applied eld, producing a net moment per unit volume. This is called
dipoleororientationalpolarization,and isdescribedby theDebyemodelwhichcaptures the
relaxationof the molecules once the electric eld isturned o:
_
~
P + ~
P =
0 (
s
1 )
~
E: (3.46)
It takes timefor the molecules tolineup becauseof their momentof inertia,so this
mecha-nism becomes less pronouncedif the materialis subjected tovery high frequencies. In that
case the molecules simply cannot follow the changing electric eld suÆciently fast and at
some level appear to\freeze."
Polarizationin denser materials, liquidsand solids, is even more complicated. Here the
tronic polarization). In solids that are made up of ionic crystals, e.g., NaCl, the positive
and negative ions are displaced as a result of an applied eld, which is called ionic
polar-ization. In certain crystals thereis apermanentinternal polarizationinthe sense that each
unit cell of the lattice has a permanent dipole moment. If the relative position of the
lat-tice points change, e.g., by heating orstressing the material,external eldsappear creating
pyroelectricity and piezoelectricity, respectively. For anideal dielectric, orientational
polar-izationdominates forlowerfrequencies givingway tovibrationalandelectronic polarization
as the frequency increases. At very high frequencies (X-rays, gamma rays) there is almost
nopolarizationsince the materialsimplycannot \followthe wave" due to inertialeects.
Inallofthesemodelssofarweassumedthattherelationshipbetweentheappliedelectric
eld and the polarization is linear, given by, in general, an integral convolution. However,
it is known that in the optical range this relationship becomes nonlinear (more so for non
innitesimal elds), as evidenced by nonlinear optical eects like solitons, second harmonic
generation and self-focusing [31]. For some materials this transition starts to take place in
the IR range. For example, while for microwaves a linear model is appropriate (indeed a
Debyemodelprovidesagoodtforwater), nonlineareects, especiallyfornon-innitesimal
amplitudes,needtobetakenintoaccountforhigherfrequencyranges. Thereisexperimental
evidenceforsmallbutsignicantdeparturefromstrictlinearity athighvaluesoftheelectric
eld [30] (p. 245). An example is the Kerr eect, in which insulating liquids, containing
anisotropic molecules, become doubly refracting when subjected to very strong elds. As
suggested in [30], this could be modeled by the constitutive relation ~
P = ~
E + sj ~
Ej 2
~
E:
However, we have already seen that inertial eects, i.e., the nite time response of the
material may be important, so instead we will consider a Debye model where the electric
eld providesnonlinear forcing. For acentrosymmetric medium wemight assume
_
~
P + ~
P = ~
f( ~
E);
where ~
f( ~
E) = c
1 ~
E +c
2 j
~
Ej 2
~
E; for j ~
Ej < M and 0 otherwise, i.e., ~
f is a saturated cubic
nonlinearity. In integral formwe obtain the relationship
~
P(t;~x)= Z
t
0
g(t s;~x) ~
f( ~
E(s;~x))ds; (3.47)
where g(t;~x)=e t
:We note that anonlinearlydriven Lorentzmodel,
~
P + 1
_
~
P +! 2
0 ~
P =
0 !
2
p ~
f( ~
E);
leads to a similar integral representation with kernel function g(t;~x) =
0 !
2
p
0 e
1
2 t
sin(
0 t);
where
0 =
q
! 2
0 1
4 2
:As a rst step, we considered ageneral modelwith nonlinear
tion
We consider a polarization mechanism of the form (3.47) with ~
f = E + f(E) together
with the one-dimensional model outlined above. As before, an innite slab of material
with supraconductive backing is interrogated by a normally incident polarized plane wave
windowed pulse originatingat an antenna source z =0 in free space
0
=[0;z
1
]: The slab
of material in = [z
1 ;z
2
] is assumed to be homogeneous in the directions orthogonal to
the direction z of propagation of the plane wave. As we have already noted, under these
assumptions it is possible torepresent the strength of the electric and magnetic elds in
and
0
by the scalar functions E(t;z) and H(t;z), respectively. One can readily eliminate
the magnetic eld fromthe full Maxwellequations and substitute the assumed constitutive
lawforthe polarizationtoarriveatthestrongformulationoftheproblemwithsimilarinitial
and boundary conditions as inSection 3.1:
^ "
r
E(t;z) + 1
"
0 I
(z)((z)+g(0;z)) _ E(t;z) + 1 " 0 I
(z)g(0;_ z)E(t;z)+ Z t 0 1 " 0 I
(z)g(t s;z)E(s;z)ds
+ 1 " 0 I
(z)g(0;_ z)f(E(t;z))+ Z t 0 1 " 0 I
(z)g(t s;z)f(E(s;z))ds
+ 1 " 0 I (z)g(0;z) d dt
f(E(t;z)) c 2 E 00 (t;z) = 1 " 0 _ J s
(t;z); t >0; 0<z <z
2 ; (3.48) 1 c @E @t @E @z z=0
=0 t>0; (3.49)
E(t;z
2
)=0 t >0; (3.50)
E(0;z)=(z); _
E(0;z)= (z) 0<z <z
2
: (3.51)
Inthe physicalproblemz
2
isassumed tobeunknown, andit isdesirabletoestimateitfrom
given data. Since the theoretical analysis is constructive in the sense that the numerical
method we use to solve this problem (for both forward and inverse problems) follows the
theoretical arguments,it is desirabletoconvert the problemto axed spatialdomain, e.g.,
[0;1]; as explained above and in [8]. Thus we use the method of maps and subsequently
formulate the variationalproblemasfollows.
We let H = L 2
(0;1); V = H 1
R
(0;1) = f 2 H 1
(0;1)j(1)= 0g leading to the Gelfand
triple ([23, 33]) V ,! H ,! V
: We say that E 2 L 1
(0;T;V) with _
E 2 L 2
(0;T;H);
E 2
L 2
(0;T;V
); is aweak solutionif itsatises for every '2V
h" r E;'i V ;V +h _
E;'i+hE;'i+h Z
t
0
+hf(E);'i+h t
0
(t s;)f(E(s;))ds;'i+h^ d
dt
f(E);'i
+hc 2
h 0
E 0
;' 0
i+c _
E(t;0)'(0)=hJ(t;);'i
V
;V
(3.52)
and
E(0;z)=(z); _
E(0;z)= (z): (3.53)
Using a Galerkin type approach and special considerations for the nonlinear terms we
were abletoshow[14]that,underfairlygeneralassumptionsonthenonlinearityf,aunique
global weak solutionexists and itdepends continuously oninitialdata.
Thusthe one dimensionalproblemwithnonlinearlyforced dynamicsforthe polarization
iswell-posed. Thissystemcanalsobethoughtofasatypeofapproximation(usingtruncated
Taylor expansions) to the nonlinear polarizationdynamics:
_
~
P +f( ~
P)=k ~
E (3.54)
and
~
P + _
~
P +f( ~
P)=k ~
E; (3.55)
which represent nonlinear Debye and Lorentz mechanisms and are suggested in [18].
Cur-rently a study is underway to compare these dierent systems theoretically and
computa-tionally.
3.4 Extension to higher dimensions
To extend the above methodology to more realistic situations one needs to formulate the
problems in higher(two or three)dimensions and demonstrate the applicabilityof the
vari-ational framework in that setting. The work on microwave interrogating signals has been
extended totwodimensionscomputationally[7]foradiagonallyanisotropicslabofmaterial.
The extensions to higher dimensions and higher frequencies are closely related and several
new challenges arise.
Theoretically, the one-dimensional model formulated above depends on the tacit
as-sumption that the polarization eld ~
P in the dielectric remains parallel to the electric
eld ~
E: Even then, the usual Maxwell equation r ~
D = 0 along with the constitutive
law ~
D =
0
r ~
E+f
1 (
~
P) ~
P need not result in r ~
E = 0: This is important in deriving the
second orderform of Maxwell'sequation where the identity rr ~
E =r(r ~
E) r
2
~
E
results in the simple Laplacian only if r ~
E = 0 or if one assumes this term is negligible
as often done in nonlinear optics ([18], p. 54-60). We believe that it may be important to
consider the fullsystem to capture the dynamics of the propagated electromagnetic signal.
Experimentally it is known that birefringence occurs in anisotropic dielectrics as a
eldare couplednonlinearly. It ispresent inlivingorganismseven atmicrowavefrequencies,
but its eect is small at 1-3 GHz. At frequencies higher than 10 GHz the eect cannot
be neglected and anisotropy needs to be taken into account even if linear polarization
dy-namics are assumed. Anisotropic eects and the tensor nature of the dielectric constant is
especiallyimportantforthe detection ofaerosols,suspended particlesinuids,and bacteria
(e.g., anthrax) with membranes of complex geometries. At even higher frequencies where
nonlinearitiesinthe polarizationdynamicsbecomepronounced, itisexpectedthat thereare
strong interactions between the nonlinear and anisotropic eects, so their correct modeling
is crucialfor the accurate representation of the propagation and reection dynamics.
In the computational treatment of the two- or three-dimensional interrogation problem
oneencounters severaldiÆculties. Naturally,thehigherspatialdimensioninvolvesincreased
computationalcomplexities,especiallywithnonlinearpolarizationdynamics. However,there
are additionalinherentchallenges. Asdescribed in[7], theinterrogatingsignalsfromanite
antenna produce oblique incident waves on a planar medium, and they must be treated in
reectionsaswell. Thusonecannotusetheuniformityassumptionasintheone-dimensional
modeltoreducetheproblemtoanitecomputationaldomain. Inthiscasetheinnitespatial
domainmust be approximatedby a nitecomputational domainwith articial boundaries.
Attheseboundariessometypeofboundarydampingmustbeemployed toremoveunwanted
numerical reections. In [7] perfectly matched layers (PML-s) along with Enquist-Majda
absorbing boundary conditions are used to successfully control these reections. Another
possibility that is currently being explored is to enlarge the computational domain so that
reections from the sampleand fromthe articialboundaries mightbe separated in time.
In summary, we believe that the variationalframework for the interrogation problem is
suitable for capturing important dynamic eects associated with the propagation of
elec-tromagnetic pulses in dierent materials. Although it is challenging both theoretically and
computationally, it has a great potential for providing a rm foundation for novel imaging
methods which can contribute to the current eorts for greater security against terrorist
activities.
4 Concluding Remarks
The atmosphere of the real threat of terrorismathome and abroadhas unfortunately
initi-ated a new environment and urgency for scientic and technological research. While some
inour communitysuggest[19] \forthe mostpart wedonot neednew methods," ourviewis
somewhat dierent. While it istrue that we in the mathematicalsciences community have
techniques andapproaches that maybeextremelyimportantinthe newproblems arisingin
the war on bioterrorism,as we enjoin this ght we willnd muchwork to doto pursue our
ideas in a relevant manner. It is not true that we have all the tools we need nor are those
we do have in the needed form for immediate application. Our strong belief is that more
willbe requiredof mathematics and statisticsthan collectingof existing toolsand applying
multidisciplinaryaswellasinterdisciplinaryapproachbeyondthatofthisvolumeandbeyond
thatwhichthe community has embracedto date. Thereis avirtualcatalogue offarranging
topics from the engineering, physical, mathematical and biological sciences: data mining,
networkanalysis,biomathematics,genomics,operationsresearch(gametheory,riskanalysis,
logistics),etc.,whichmustbecombinedwiththesocialandpsychologicalsciences(individual
andgroupbehavior,e.g.,fanaticism,cognition,etc.) inwaysandonascaleunprecedentedin
the historyofscience. Andthis mustbedonewithanewsense ofurgency. Forexample,the
developmentofagent-specicbiosensors,sometimes inthe contextof \smart"materials,has
forsometime been apriorityatseveral ofour nationallabs;the needshavebeen heightened
by events of the past several years.
Lest our viewappear toopessimistic,we hastentoadd thatwhile we donot have ready
\solutions"toquestionsandproblems thatperhapsareonlynowbeingpreciselyformulated,
the mathematical and statisticalsciences do havea rich history of modeldevelopment with
associated toolsand techniques. This willundoubtedly provide a solid foundationthat will
prove extremely valuable in the pursuit of many specic problems related to terrorism in
general and bioterrorism in particular. We are optimistic about the value we can bring to
society in this essential eort.
Acknowledgments
Research reported here was supported in part by the U.S.Air Force OÆce of Scientic
Re-search undergrantAFOSR F49620-01-1-0026and inpartbythe JointDMS/NIGMS
Initia-tivetoSupportReserachintheAreaofMathematicalBiologyundergrant1R01GM67299-01.
The authors are grateful to Dr. Richard Albanese, Dr. Carlos Castillo-Chavez and Dr.
MarieDavidianforseveral informativediscussions. Partofthischapterwascompletedwhile
H.T.B.wasavisitortotheMittagLeerInstituteoftheRoyalSwedishAcademyofSciences,
Djursholm,Sweden. Collaborationwas alsofacilitatedwhile allauthorswere visitors to the
Statisticaland Applied Mathematical Sciences Institute (SAMSI), Research Triangle Park,
NC.
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