Bioelectromagnetics 22:66^70 (2001)
Ho w M i ght Spatial No nu niform ity of Dose
in a Homogeneous Bio logical System
Affect itsTotal Response?
William F. P ickard
Department of Electrical Engineering,Washington University, St. Louis, MO
The total response of a homogeneous biological system to a ®xed total dose of a biological agent is
modeled by dividing the system into N cubical voxels, each of which can be associated with an
individual dose D
n
and an individual response R
n
FD
n
. Among the results shown are the
following:
A. (Voxel Theorem). Let the average dose D
avg
be held ®xed as the dose distribution is shifted from
uniform u to arbitrary a. Then, if F
0
0overD
min
; D
max
and R
P
N
n1
R
n
, a suf®cient condition
that
NFD
avg
RuRa
is that F be a concave-upwards function of dose; that is, F
00
0overD
min
; D
max
.
B. If F
0
is constant over D
min
; D
max
, then RaRu. That is, the total response is a function of
D
avg
only. The applications of these (and other) results are illustrated by examples from
bioelectromagnetics. Bioelectromagnetics 22:66±70, 2001.
# 2001 Wiley-Liss, Inc.
Key words: nonuniform dose; voxel theorem; ultra high frequency
INTRODUCTION
A frequent problem which confronts that prac-
tical biologist who wishes to apply some agent to a
biological system (®eld, organ, culture ¯ask, etc.) is
that it is easier to estimate the total dose to the system
than to ensure that this total dose is uniformly
distributed over the individual elements of the system.
This is especially true in the ®eld of bioelectromag-
netics, particularly at microwave frequencies.
The dose metric of choice for exposure to
microwave radiation is the speci®c absorption rate
(SAR) in watts per kilogram. Because this frequently is
dif®cult to measure accurately in cases of practical
interest, one often resorts to the prediction of SAR by
using the electromagnetic constitutive parameters of
the system and one or another numerical technique
[Burkhardt et al., 1996; Chou et al., 1996; Popovic
Â
,
Hagness, Ta¯ove, 1998; Guy, Chou, and McDougall,
1999; Pickard, Straube, Moros, and Fan, 1999;
Pickard, Straube, and Moros, 1999]. Hence, it is
frequently the case that the delivered dose is available
in the form of voxel-by-voxel estimates over the
volume of the system; and the nature of microwave
exposure systems is such that some subregions of the
system may receive signi®cantly higher doses than
other subregions.
Nevertheless, it is an overall response of the
system to such a nonuniform dose which is most
commonly reported, and then generally in terms of the
average dose. This raises an interesting question: how
does the nonuniformity of dose distribution affect the
overall response which is ascribed to the nominal
average dose? And should it be standard practice to
report not only the average of the dose distribution but
also its variance and/or extrema?
This article reports initial investigations of these
questions for an idealized model system and presents
several potentially useful results. It should be borne in
mind:
(i) Although the investigation was motivated by
problems relating to microwave exposure, the
results are applicable to any combination of
biological system and biological agent that
satis®es the simplifying assumptions of the model.
(ii) The putative utility of the theorems does not
require a study of their proofs. It suf®ces merely to
ß 2001Wiley-Liss,Inc.
ÐÐÐÐÐÐ
*Correspondence to: William F. Pickard, Department of Electrical
Engineering, Washington University, One Brookings Drive, St.
Louis, MO 63130. E-mail: wfp@ee.wustl.edu
Received 5 November 1999; Final revision received 20 March
2000
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