New bioimpedance model accurately predicts lower limb muscle
volume: validation by magnetic resonance imaging
S.
Salinari1,
A.
Bertuzzi2,
G.
Mingrone3,
E.
Capristo3,
A.
Pietrobelli4,
P.
Campioni5,
A. V.
Greco3, and
S. B.
Heymsfield4
1 Dipartimento di Informatica e Sistemistica,
Università di Roma "La Sapienza," 00184 Rome;
2 Istituto di Analisi dei Sistemi ed Informatica del
CNR, 00185 Rome; 3 Istituto di Medicina Interna e
Geriatria and 5 Istituto di Radiologia,
Università Cattolica del Sacro Cuore, 00168 Rome,
Italy; and 4 Obesity Research Center,
St. Luke's-Roosevelt Hospital, Columbia University College of
Physicians and Surgeons, New York, New York 10025
 |
ABSTRACT |
Conventional
bioimpedance analysis (BIA) methods now simplify the representation of
lower limb geometry and electrical properties for body composition
estimation. In the present study, a three-dimensional model of the
lower limb was assembled by segmentation of magnetic resonance
cross-sectional images (MRI) for adipose tissue, skeletal muscle, and
bone. An electrical network was then associated with this model. BIA
and MRI measurements were made in six lean subjects (3 men and 3 women,
age 32.2 ± 6.9 yr). Assuming 0.85 S/m for the longitudinal
conductivity of the muscle, the model predicted in the examined
subjects an impedance profile that conformed well to the BIA impedance
profile; predicted and measured resistances were similar (261.3 ± 7.7 vs. 249 ± 9
; P = not significant). The
resistance profile provided, through a simpler model, muscle area
estimates along the lower limb and total leg muscle volume (mean 4,534 cm3 for men and 4,071 cm3 for women) with a
mean of the absolute value of relative error with respect to MRI of
6.2 ± 3.9. The new approach suggests that BIA can reasonably
estimate the distribution and volume of muscles in the lower
extremities of lean subjects.
bioimpedance analysis; body composition; nutritional assessment; magnetic resonance imaging
 |
INTRODUCTION |
THE STUDY OF HUMAN
BODY composition in health and disease is of increasing interest
in both research and clinical practice (13). The
availability of advanced techniques, such as computed tomography (CT)
and magnetic resonance imaging (MRI), allows accurate and reproducible
estimates of major tissue and organ compartments (10, 14).
However, CT and MRI can be performed only in specialized research
units, since the availability of instrument time is limited, the
evaluation cost is high, and technical expertise is required for image
analysis (20).
In the past decade, bioimpedance analysis (BIA) has been advocated as a
simple method for body composition estimation in humans because it is
noninvasive, inexpensive, and well suited for epidemiological studies
(3). However, current BIA electrical and geometric models
are oversimplified, and results are often inaccurate (3, 6). Typically, BIA estimates of total body water are based on
equations in which the fat-free compartment is assumed to be a cylinder
with uniform electrical characteristics, and between-individual differences in electrical path length are accounted for by measured height. The conventionally used electrical path is arm to leg, although
measurement of isolated limbs or trunk impedance is now gaining
recognition (2, 18, 19). Recently, a detailed analysis of
resistance data of the human thigh was reported by Aaron et al.
(1), and resistance data from the thigh and calf were
analyzed in association with MRI body composition estimates by Fuller
et al. (7).
The aim of the present study was to evaluate the feasibility of using
multiple BIA measurements for quantifying muscle volume and for
reconstructing the profile of the muscle cross-sectional area along the
lower limb. Our study was based on a mathematical model that describes
the electrical characteristics of the lower limb in healthy lean
subjects. The lower limb was selected as the study site since it has a
simple geometry and tissue composition compared with the trunk. The
mathematical model uses a realistic reconstruction of limb geometry and
tissue composition obtained from MRI measurements. The model provides
the distribution of electrical potential and the fluxes of current
induced in the lower limb by an externally applied current and thus
allows for a comparison with BIA measurements at multiple sites along
the lower limb. Furthermore, on the basis of the results obtained by
the model, a method for reconstructing muscle cross-sectional area
along the lower limb from BIA data is proposed.
 |
METHODS |
Subjects and anthropometry.
Six volunteers (3 men and 3 women, age 32.2 ± 6.9 yr, body mass
index 22.4 ± 1.4 kg/m2) were enrolled in the study.
All subjects were in good health, as assessed by clinical and
laboratory examinations, were not taking medications, and did not
participate in intensive physical activity. The women were studied in
the follicular phase of the menstrual cycle. Body weight was measured
to the nearest 0.1 kg by a beam scale and height to the nearest 0.5 cm
was measured using a stadiometer (Holatin, Crosswell, Wales, UK). The
protocol conformed to the directives given by the Ethical Committee of the Institutional Health Review Board of the Catholic University, School of Medicine, in Rome. Informed consent was obtained in all cases.
MRI.
Subjects completed an MRI scan of the lower limb using a 0.5-T scanner
(model Vectra 0.5; General Electric, Milwaukee, WI) with an axial
T1 weighted spin echo sequence. Axial views were acquired
with 10-mm slice thickness and a 50-mm interslice gap in five subjects
and 10 mm in one subject. The lower limb lengths corresponding to the
scan views are reported in Table 1. The area and volume of skeletal muscle within each slice were calculated by
a trained observer using the VECT image analysis software (Martel).
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Table 1.
Lower limb muscle volume measured by MRI and estimated from BIA; errors
on muscle volume and cross-sectional area
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|
To obtain the discretized three-dimensional geometric model and tissue
composition of the lower limb, as required by the BIA mathematical
model, the MRI were scanned (Epson scanner Perfection model 1200Photo)
with a resolution of 300 pixels/in. and then digitized. Each digitized
image was then processed to identify the regions corresponding to the
different tissues. Only three tissue types, namely bone, muscle, and
adipose tissue, were distinguished; cartilage and tendon were assigned
to bone, vascular tissue to muscle, and skin to adipose tissue. Each
tissue type was labeled by a different color. Each image was then
subdivided into 22 × 22 square cells of 1 cm side, and cells were
assigned to the prevailing tissue or to external air.
In the final step, a text file was formed as a stack of discretized
images. To have images at 1-cm intervals when the 50-mm interslice gap
was used in the MRI scan, each of the available images was replicated
four times (2 times the initial and the final image). The stack
provided a three-dimensional model of lower limb geometry and tissue
composition with a total, including the external air, of 22 × 22 × N cubic cells of 1 cm side, N
depending on the length of the subject's lower limb (N from
72 to 77; see Table 1).
Bioelectrical impedance analysis.
Resistance and capacitive reactance were determined using a
multifrequency BIA system (Human-IM DIP; DS-Medigroup, Milan, Italy)
with a delivered current of 800 µA at a frequency of 50 kHz. The
current-injection electrodes were positioned on the middorsum of
the right hand, just proximal to the metacarpal phalangeal joint line,
and on the middorsum of the right foot, just proximal to metatarsal
phalangeal joint line (3).
To determine the impedance profile along the lower limb, one of the
voltage electrodes was positioned on the middorsum of the right wrist
and kept fixed. The other electrode was positioned at various
contiguous levels along the lower limb at 2.5-cm intervals, starting
from the midanterior right ankle up to the midline of the anterior
surface of the right thigh at about the level of the inguinal crease.
By subtracting the measured impedances from whole body impedance, we
obtained the impedance profile along the lower limb, as would be
measured between an electrode located at different levels along lower
limb from the ankle to the hip and an electrode located at the ankle.
BIA mathematical model.
As seen in the APPENDIX, the
mathematical model for BIA consists of a set of equations of the form
of Eq. 7A, one for each cell of the discretized
three-dimensional model of the lower limb. These equations associate an
electrical network composed of admittances with the body region of
interest, and the admittance values depend on the electrical
characteristics of the various tissues.
The electrical characteristics of tissues were obtained from literature
data. For the conductivity of skeletal muscle in the longitudinal
direction (
mz), taken as the dominant direction of
muscle fibers in the lower limb and corresponding to the
z-axis in the model, we considered the values of 0.67 and
0.85 S/m according to the resistivity values reported by Fuller et al.
(7, 8). The muscle conductivity in the transverse
direction (
mx =
my) was set to 0.13 S/m, as reported in Ref. 1. Both adipose tissue (at) and
bone (b) were assumed to be isotropic tissues, with conductivity
at = 0.064 and
b = 0.013 S/m,
respectively (1, 5). The tissue permittivities were
assumed to depend on the frequency (f, in Hz) of the applied current
according to the following expressions (5)
where
is the permittivity of muscle in the transverse
direction (mx and my), in the longitudinal direction (mz), and of adipose tissue (at) and bone (b);
0 = 10
9/(36
)F/m is the permittivity of free space.
Conductivity and permittivity of external air were set equal to zero.
Concerning the externally applied currents (the term I in
Eq. 7A) that simulate the current delivered to the body
during BIA, we had to take into account that, in the present
measurements of lower limb bioimpedance, the current-injecting
electrodes were placed at locations that are far from the body region
considered (i.e., in the ipsilateral hand and foot) to minimize the
influence of current injection on the potential measurement. This
condition was simulated by replicating the terminal cross sections of
the three-dimensional model of the lower limb (one in the ankle and the
other in the thigh) to create 10-cm extra spaces below the ankle and
above the thigh. The external current was impressed in nodes of the
electrical network located at the extreme of these extra spaces, so the
region in which the distribution of electrical potential was calculated
was at least 10 cm away from the points where the external current was applied.
Under the applied current, the network generated the distribution of
the electrical potential in the body region considered and thus allowed
us to compute impedance values that were compared with data provided by
BIA. The set of equations composing the model was solved by the
technique of successive overrelaxation (5, 11).
Estimation of muscle cross-sectional area and volume.
A very simplified version of the model in the
APPENDIX can provide estimates of
muscle cross-sectional area and volume from BIA data, as detailed below.
Because the longitudinal conductivity of muscle is much larger than all
other conductivities, and the contribution of the reactive component of
impedance is small at 50 kHz, we assumed that the z-directed
current flowing in the lower limb during BIA is essentially carried by
the resistive component of muscle, at least in lean subjects. Moreover,
we assumed that, at locations remote from the current-injecting
electrodes, the total z-directed current in a cross section
is equal to the total current (
) delivered by the
current-injecting electrodes. Thus, denoting the cross-sectional area
by S and the muscle area by Sm (these areas change with the level z along the lower limb), and
considering the component in the z-direction of the real
part of the current density J (see the
APPENDIX), we may write
|
(1)
|
where ds is the element of the cross-sectional area.
Moreover, according to Eqs. 1A-4A of the
APPENDIX, we have
|
(2)
|
In a region far from current-injecting electrodes, the
electrical potential is likely to be approximately constant over the cross section; thus, V can be considered a function,
V(z), of z only. If the potential at
the ankle (z = 0) is set to zero, the quantity
V(z)/
can be interpreted as the
resistance between a point of the lower limb at distance z
from the ankle and a point on the ankle itself. Denoting this
resistance as R(z), we have
|
(3)
|
Thus Eqs. 1-3 give for any level z
along the lower limb the equation
|
(4)
|
and an estimate
[
m(z)] of muscle
cross-sectional area at the level z is obtained as
|
(5)
|
To evaluate the derivative in Eq. 5, the resistance
data obtained by BIA were approximated by a weighted sum of two
Gaussian cumulative functions plus a straight line according to
|
(6)
|
where F(z;µ,
) denotes the cumulative function of
a Gaussian with mean µ and SD
[note that it should be
(0) = 0]. The two cumulative
Gaussians represent the specific increase of resistance that is
observed in the regions of the ankle and of the knee. The parameters
c1, c2, c3,
1,
µ2, and
2 of the fitting function were
determined by minimization of a weighted least squares index, and
Eq. 5 was then applied with R(z) =
(z) and
mz = 0.85 S/m. The integral over
z of the estimated cross-sectional area of muscle provided
the total muscle volume of the lower limb.
 |
RESULTS |
Discretized model of the lower limb.
Although the discretization of MRI with cells of 1 cm side may seem
approximate, the three-dimensional reconstruction of the geometry and
tissue composition of the lower limb appeared in general to be rather
accurate, at least as the content of muscle tissue is concerned. Figure
1 presents the cross-sectional area of
the muscle (Sm) as computed on the original MRI
in the 78 cross sections of the lower limb of the subject in which the
MRI scan was performed with a 10-mm interslice gap. Moving from left to right (z = 0 corresponds to the ankle), the regions
corresponding to the ankle, the calf, the knee, and the thigh are
easily recognized. The total volume of muscle in the lower limb region
considered, as measured from the original MRI in this subject, was
equal to 5,751 cm3.

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Fig. 1.
Cross-sectional area of muscle along the lower limb of
subject 1 from the ankle to the thigh: experimental data
obtained from magnetic resonance imaging (MRI; triangles), areas of
muscle in the discretized three-dimensional model (continuous line),
and areas of muscle in the discretized model when cross sections at
5-cm intervals are used (dashed line).
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|
The muscle areas given by the discretized three-dimensional model are
also reported in Fig. 1. It is seen that the approximation becomes less
accurate in the regions of the ankle and of the knee, where
Sm is smaller. Considering all 78 cross
sections, the mean error of the area was 0.2 cm2. The total
volume of muscle in the discretized model was equal to 5,784 cm3. Figure 1 also shows the less accurate approximation
that was obtained for this subject when a reduced number of cross
sections at 5-cm intervals was used. Although the muscle area errors
are larger, the total volume of muscle given by the model in this case
was equal to 5,945 cm3, with an error smaller than 4% with
respect to the original MRI data. Similar error values are likely to be
found for the other subjects in which the discretized reconstruction of
the lower limb was obtained from MRI taken at 5-cm intervals.
The volumes of muscle in the part of the lower limb considered in this
study are reported in Table 1 (VMRI) for the six subjects. For subject 1, for whom MRI at 1-cm intervals were
available, the volume measured on the original images is given.
Mathematical model of BIA and analysis of bioimpedance data.
The distribution of the electrical potential provided by the BIA model
shows that, in the proximity of the current-injecting electrodes, the
potential varies considerably in the cross section from the points
closer to the electrode to the points far from it. This effect is
markedly reduced in a cross section at a distance of 10 cm. The
variability of the potential V in the cross section was
quantified by computing the ratio (Vmax
Vmin)/Vmean in all of the
cross sections, including those of the added extra spaces where the
external current was applied. In the cross sections containing the
electrodes, the above ratio was in the order of 10, whereas this ratio
decreased to values lower than 0.3 in the cross sections at least 10 cm
away from the electrodes. However, in the region of the knee where the
content of muscle tissue is lower, the ratio was larger, and values up
to 1.9 were achieved. When only the cells that form the border of a
cross section were considered, remarkable variations of the potential
were still found in the region of the knee. Thus the location of the
voltage electrode along the circumference of the limb at a given cross section is likely to affect the determination of the resistance.
The model also provided the currents that flow through the various
tissues at each location within the lower limb. As expected, the
currents flowing in the longitudinal direction (z direction) in the muscle were markedly larger than the currents flowing in the
other tissues. Far from the current-injecting electrodes, >90% of the
z-directed current was carried by the muscle, except in the
region of the knee where this percentage decreased to values of
~70%. These results show that the assumptions leading to Eq. 5 may be considered substantially fulfilled.
The total resistance RBIA and the capacitive
reactance XBIA of the part of the lower limb
considered, as measured by BIA, are shown in Table
2. The mean lower limb resistance was
243.7
for men and 255.3
for women. Reactances were in the range 31-38
. These data were compared with the resistance and
reactance computed by the mathematical model of BIA described in the
APPENDIX. Because this model
substantially utilizes the lower limb geometry observed by MRI, the
discrepancies between data and model predictions should be mainly
because of the interindividual variability of the electrical
parameters. As expected, the most influent parameter was the
conductivity
mz of skeletal muscle in the
longitudinal direction. When
mz was set at
0.85 S/m, the model slightly overestimated the total resistance of the
lower limb (see values of RMOD in Table 2), with
the relative errors ranging from ~0 to 10%. Even larger values of
RMOD were predicted with a muscle conductivity
of 0.67 S/m, so the value of 0.85 S/m was assumed in the subsequent
analysis. Moreover, the model underestimated the reactance of the lower
limb (see XMOD in Table 2), confirming, however,
the small contribution of the reactive component of the impedance at
the frequency of 50 kHz.
The resistance profile along the lower limb found by BIA in the subject
in whom the MRI were available at 1-cm intervals is shown in Fig.
2. The experimental values of the
resistance are plotted at 2.5-cm intervals. The data show that the
resistance increases more rapidly in the regions of the ankle and of
the knee, where the content of low resistivity muscle tissue is
smaller. Figure 2 also presents the profile of the resistance given by the model in the APPENDIX, computed
at each level z along the lower limb. The profile of the
resistance obtained from the model when using the coarser
discretization (i.e., MRI data at 5-cm intervals) is also reported in
Fig. 2 and shows that the general pattern of the profile is still
adequately reproduced by the BIA model when larger intervals are used
for MRI. Figure 3 shows the experimental and predicted profiles for the lower limb of a woman.

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Fig. 2.
Profile of resistance along the lower limb of
subject 1 calculated by bioimpedance analysis (BIA;
circles); profile of resistance predicted by the model with MRI cross
sections at 1-cm intervals (dashed line); and profile predicted by the
model when cross sections at 5-cm intervals are considered (continuous
line).
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Fig. 3.
Profile of resistance along the lower limb of
subject 6 by BIA (circles) and profile of resistance
predicted by the model (continuous line).
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|
Estimation of muscle cross-sectional area and volume.
The profile of muscle cross-sectional area along the lower limb and the
muscle volume were estimated on the basis of the model of Eqs.
5 and 6 (with
mz = 0.85 S/m), using the experimental BIA data. Means ± SD of the
estimated parameters of Eq. 6 (except µ2,
which depends on the lower limb length) were as follows:
c1 = 194.2 ± 46.0
,
1 = 10.4 ± 1.7 cm, c2 = 73.2 ± 7.6
,
2 = 8.9 ± 1.7 cm, c3 = 1.0 ± 0.2
/cm.
An example of the reconstruction of the profile of muscle
cross-sectional area along the limb is shown in Fig.
4. Figure 4 shows the experimental
profile of resistance together with the fitted profile
(z) and the estimated muscle
cross-sectional area
m(z)
obtained by Eq. 5 together with the areas given by MRI.
Although the estimated general pattern was reasonable, large errors may
be found at specific locations, such as the knee and the extreme
sections of the thigh. Table 1 shows, for each subject, the average
error on the cross-sectional area computed as
S = 100 ×
|SMRI
SBIA|/
SMRI,
where SMRI is the muscle cross-sectional area
measured by MRI, SBIA is the estimated area
m(z) evaluated at the same
level of the MRI, and the sum is carried over all the cross sections
considered. The errors on the cross-sectional areas did not cause a
large effect on the estimate of total lower limb muscle volume. Table 1
reports the estimates of muscle volume for all subjects with the
relative error
V = 100 × (VMRI
VBIA)/VMRI with
respect to the volume measured by MRI. Note that, by definition, it is
S
|
V|. The mean ± SD over the subjects of the absolute value of the error on volume was 6.2 ± 3.9%.

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Fig. 4.
Profile of resistance along the lower limb of
subject 1 by BIA (circles) and profile fitted by Eq. 6 [ (z), continuous line]. Muscle
cross-sectional area along the lower limb of the same subject obtained
by MRI (triangles) and profile of the muscle cross-sectional area
estimated by Eq. 5 [ (z),
continuous line].
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DISCUSSION |
In the past few years, a great deal of attention has focused on
the evaluation of skeletal muscle mass as a means of assessing the
nutritional status of individuals with various clinical conditions (9, 15, 16). For instance, protein-energy malnutrition is
a frequent feature in elderly patients in acute-care settings. Critical
illnesses and hospitalization result in catabolic stress, anorexia, and
immobilization, which worsen the nutritional status of elderly
subjects. Disuse of muscles because of bed rest or chair rest leads to
a loss of muscle mass in a range of 1.5%/day in adults
(4). In critically ill elderly patients, anorexia and
functional dependency, resulting in decreased energy and protein intake, could further contribute to decreased muscle mass. Conversely, adequate muscle mass appears to be the best predictor of good prognosis
in critical illness. Loss of skeletal muscle tissue also occurs in
patients after major accidents of the spinal cord, and it is important
and useful to monitor improvements in skeletal muscle mass and function
with rehabilitation therapy (17, 21, 24). Furthermore, the
possibility of obtaining a profile of the lower limb skeletal muscle
might be useful to follow training athletes during the time.
Skeletal muscle (SM) represents the largest fraction of body mass in
nonobese adults (22). Because the muscle mass of the limbs
accounts for an estimated 75-80% of total body muscle mass
(12), appendicular muscle mass has been endorsed as a
simple means of quantifying total body muscle mass with the exception
of body builders or weight lifters.
The mathematical model of BIA developed here, which used the
discretized reconstruction of lower limb geometry and tissue composition obtained by MRI, predicted the pattern of resistance along
the lower limb, as measured by BIA. Because the model showed that the
electrical current in the lower limb is essentially carried by muscle,
the pattern of resistance appears to be mainly dependent on the profile
of muscle cross-sectional area along the lower limb and appears to thus
contain the information needed for recovering the muscle volume from
BIA data. Thus an approach based on BIA data seems to be feasible,
allowing avoidance of the more sophisticated and expensive technique
based on MRI, at least in clinical practice and particularly when
several measurements are required, as in the present study.
Although some anatomical detail is lost with MRI with cells of
1-cm side, the gross structure appeared still to be represented in the
discretized three-dimensional model of the lower limb. Figure 1 shows
that the profile of muscle area along the limb was accurately
reproduced, even when MRI cross-sectional images at 5-cm intervals were
available. Thus the duration of the exposure during the MRI scans can
be reduced. We note, however, that, with cells of 1-cm side, the skin
cannot be represented, even if its contribution to the flow of current
across the limb is not negligible in the regions where the content of
muscle and adipose tissue is reduced.
The model of BIA reproduced the general behavior of impedance along the
lower limb, as shown in Figs. 2 and 3, at least when the value of the
longitudinal conductivity of muscle (
mz) was
set to a value of 0.85 S/m. It was found that the electrical potential
along the border of a cross section of the limb can show marked
changes, as also seen by BIA measurements (1). This effect
is particularly remarkable in the proximity of the current-injecting
electrodes and in the region of the knee, confirming that the location
of the voltage electrodes may be critical in bioimpedance analysis.
As shown in Figs. 2 and 3, the resistance profile given by the model
did not precisely follow the experimental BIA data in the regions of
the ankle and the knee, where the predicted resistance showed larger
variations than those experimentally observed. A number of factors can
cause the observed discrepancies: 1) the predicted profile
was obtained from the mean potential over the adipose tissue of the
cross section, whereas the experimental measurements were obtained by
voltage electrodes at specific locations; 2) because of the
discretization with cells of 1-cm side, anatomical structures such as
ligaments, tendons, and sinovial fluid were disregarded, and these
structures may present larger conductivities than those assumed in the
model, thus giving smaller increments in observed resistance;
3) the value of the longitudinal conductivity of muscle,
mz, which was assumed as a constant in the present model, may be partially dependent on interindividual
variability and on the arrangement and orientation of muscle fibers,
which are different in the thigh and in the calf (1).
With the use of the simplified mathematical model of Eqs. 5 and 6, the total muscle volume was estimated from the
experimental BIA data with an acceptable error with respect to the
reference value provided by MRI. The estimation of muscle
cross-sectional area from the profile of resistance along the lower
limb appears to be feasible, as shown by Fig. 4, although the average
estimation error on this area will be larger than the error on volume,
as shown in Table 1. Because the measured potential can change when the
voltage electrode is moved circumferentially at a given level of the
limb, it is suggested that the measurement of the potential at more
than a single point in a given cross section (e.g., in both the
anterior and the posterior aspects of the limb at the level of the
knee) might provide an improved determination of resistance and thus a
more accurate reconstruction of the muscle cross-sectional areas.
In conclusion, the mathematical model described in the present
paper allows for predicting with good accuracy the volume of lower limb
skeletal muscle mass by using BIA, thus representing a first step
toward advancing BIA as a replacement for CT and MRI when evaluating
the composition of the lower limb in lean subjects.
 |
APPENDIX |
Mathematical model of BIA.
Under the quasi-static approximation, valid at the frequency
f = 50 kHz used in the present bioimpedance
measurements, the electric field E (V/m) in the body region
considered is related to the potential V (V) by the
following equation
|
(1A)
|
Moreover, for the electric current density J
(A/m2), we can write the continuity equation
|
(2A)
|
where I (A/m3) is the externally applied
current per unit volume. Assuming harmonic fields, J is
related to E by
|
(3A)
|
where k (S/m) is the complex conductivity tensor
given by
|
(4A)
|
with
being the conductivity and
the permittivity of the media.
From Eqs. 1A-3A we obtain the following equation for
the complex potential
|
(5A)
|
to be solved in the region of interest with the appropriate
boundary conditions (11). If the electric field is
referred to Cartesian coordinates, all nondiagonal elements of the
tensor k(x,y,z) are zero
and k = diag(kx,ky,kz).
Thus Eq. 5A becomes
|
(6A)
|
The three components of the tensor k at a given point
(x, y, z) are possibly different
because of tissue anisotropy, but the complex conductivity of a given
tissue is assumed to be independent of the spatial position.
To solve Eq. 6A we used the finite difference method
(5, 23), with the nodal points taken coincidently with the
centers of the cells that form the three-dimensional model of the lower limb (see Fig. 5). The finite differences lead to the following equation (5)
|
(7A)
|
where V is the electrical potential in the node
located at the center (x, y, z) of a
cell, Vi are the potentials at the six neighboring nodes [(x
x, y, z),
(x +
x, y, z), and so on; see Fig. 5],
Yi are the complex admittances between the nodes (denoted as Yx
,
Yx+, and so on in Fig.
5), and I denotes the external
current impressed at the node (x, y,
z). It is I
0 only at the nodes representing the current-injecting electrodes. The values of the admittances depend
on the types of tissue associated with the various cells. For instance,
the admittance Yx+
between the nodes (x, y, z) and
(x +
x, y,
z), which may be associated with different tissues, has the
expression
|
(8A)
|
and similarly for the other admittances. The impedances to be
compared with BIA data were computed from the mean value of the
electrical potential of cells that represent the adipose tissue at a
given level z along the lower limb.

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Fig. 5.
Representation in the xy plane of the section
of electrical network associated with a cell and the neighboring cells
of the discretized three-dimensional representation of the lower limb
(V is potential and Y is admittance, as in
Eqs. 7A and 8A). The nodes at
(x,y,z z) and at
(x,y,z + z), with the impedances
Yz and
Yz+, are not
represented.
|
|
 |
ACKNOWLEDGEMENTS |
We thank Drs. A. Scarfone and P. Morini for technical assistance.
 |
FOOTNOTES |
Address for reprint requests and other correspondence:
S. Salinari, Dip. Informatica e Sistemistica, Università di
Roma "La Sapienza," Via Eudossiana, 18 00184 Rome, Italy (E-mail:
salinari{at}dis.uniroma1.it).
The costs of publication of this
article were defrayed in part by the
payment of page charges. The article
must therefore be hereby marked
"advertisement"
in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
First published November 20, 2001;10.1152/ajpendo.00109.2001
Received 9 March 2001; accepted in final form 8 November 2001.
 |
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