Knee bioelectric impedance assessment in healthy/with osteoarthritis subjects.
ABSTRACT The present study analyzes parameters estimated by bioelectric impedance spectroscopy (BIS) in subjects with healthy and with osteoarthritis (OA) knees. Thirty-two male volunteers, members of the Parachute Military Infantry Brigade of Rio de Janeiro, Brazil, participated in the study (62 knee joints). Clinical specialists used the Dejour scale for OA classification and divided the subjects into a control (without OA) and a pathological group (with different degrees of OA). BIS data were obtained in a standing position using a BIS technique based on the current response to a step voltage excitation. Differences between groups were measured by means of a Wilcoxon-Mann-Whitney test. Results indicate that raw bioimpedance parameters seem to be sensitive to the physiological changes associated with OA. Thus, data indicate that extra-cellular resistance (Re) and reactance of the equivalent capacitance (Xcx) increase according to the disease intensity (p < 0.001). In conclusion, the BIS technique seems to be able to provide the objective and non-invasive basis for helping the diagnosis of knee OA.
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ABSTRACT: BACKGROUND AND PURPOSE: Electrical bioimpedance spectroscopy (BIS) allows the evaluation of limb extracellular fluid (R0) and total fluid (Rinf). BIS could facilitate post-surgical oedema evaluation after total knee arthroplasty (TKA), as it is easily performed and is non-invasive. However, neither its applicability in this context nor the influence of metallic implants on measurement has been evaluated. The aim of this study was to evaluate the influence of TKA implants on the BIS R0 and Rinf variables used for oedema evaluation. METHOD: This was a prospective non-randomized comparative clinical trial. One oedema-free group of patients with TKA was compared with a group presenting similar characteristics except for the arthroplasty, to assess the influence of the implant on BIS measurement in the absence of oedema. The TKA group included 15 patients who had undergone surgery more than a year previously, and the control group included 19 patients awaiting TKA surgery. Volume and perimeter measurements served as reference criterions. The lower limb percentage differences for BIS, knee perimeter and volume were calculated. The significance of differences between groups was calculated for all measurement methods, using the Mann-Whitney test. The setting was a Department of Orthopedic Surgery and Traumatology in a university hospital. RESULTS: The differences between groups were not significant for R0, Rinf, volume and perimeter. R0 showed the smallest mean difference in limb percentage difference between groups [means (SD): TKA 3.98 (8.09), controls 3.97 (5.16)]. CONCLUSIONS: The lower-leg percentage difference in the TKA group is comparable with that of healthy subjects. R0 can be used for oedema evaluation following TKA surgery, as there was no sign of alteration from the metallic implant. These findings indicate the potential for early oedema evaluation after TKA. More research is warranted to extensively validate the application of BIS for oedema evaluation after TKA. Copyright © 2012 John Wiley & Sons, Ltd.Physiotherapy Research International 11/2012;
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ABSTRACT: Early detection of knee osteoarthritis (OA) is of great interest to orthopaedic surgeons, rheumatologists, radiologists, and researchers because it would allow physicians to provide patients with treatments and advice to slow the onset or progression of the disease. Early detection can be achieved by identifying early changes in selected features of degenerative articular cartilage (AC) using non-invasive imaging modalities. Magnetic resonance imaging (MRI) is becoming the standard for assessment of OA. The aim of this paper was to review the influence of MRI on the selection, detection, and measurement of AC features associated with early OA. Our review of the literature indicates that the changes associated with early OA are in cartilage thickness, cartilage volume, cartilage water content, and proteoglycan content that can be accurately, consistently, and non-invasively measured using MRI. Choosing an MR pulse sequence that provides the capability to assess cartilage physiology and morphology in a single acquisition and advanced multi-nuclei MRI is desirable. The results of the review indicate that using an ultra-high magnetic strength, MR imager does not affect early OA detection. In conclusion, MRI is currently the most suitable modality for early detection of knee OA, and future research should focus on the quantitative evaluation of early OA features using advances in MR hardware, software, and data processing with sophisticated image/pattern recognition techniques.Rheumatology International 01/2014; · 2.21 Impact Factor
Knee bioelectric impedance assessment in healthy/with osteoarthritis subjects
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Physiol. Meas. 31 (2010) 207–219
Knee bioelectric impedance assessment in
healthy/with osteoarthritis subjects
Eduardo Borba Neves1, Alexandre Visintainer Pino1,
Renan Moritz Varnier Rodrigues de Almeida1and
M´ arcio Nogueira de Souza1,2,3
1Biomedical Engineering Program, COPPE, Federal University of Rio de Janeiro, Centro de,
Tecnologia, Bloco H, sala 327, PO Box 68510, Rio de Janeiro, Brazil
2Electronic Engineering Department, Polytechnic School, Federal University of Rio de Janeiro
Received 29 August 2009, accepted for publication 11 November 2009
Published 16 December 2009
Online at stacks.iop.org/PM/31/207
The present study analyzes parameters estimated by bioelectric impedance
spectroscopy (BIS) in subjects with healthy and with osteoarthritis (OA)
knees. Thirty-twomalevolunteers, membersoftheParachuteMilitaryInfantry
Brigade of Rio de Janeiro, Brazil, participated in the study (62 knee joints).
Clinical specialists used the Dejour scale for OA classification and divided
the subjects into a control (without OA) and a pathological group (with
different degrees of OA). BIS data were obtained in a standing position using
a BIS technique based on the current response to a step voltage excitation.
Differences between groups were measured by means of a Wilcoxon–Mann–
Whitney test. Results indicate that raw bioimpedance parameters seem to be
sensitive to the physiological changes associated with OA. Thus, data indicate
that extra-cellular resistance (Re) and reactance of the equivalent capacitance
(Xcx) increase according to the disease intensity (p < 0.001). In conclusion,
the BIS technique seems to be able to provide the objective and non-invasive
basis for helping the diagnosis of knee OA.
Keywords: Bioelectrical impedance, osteoarthritis, knee
Musculoskeletal disorders can be considered the leading cause of disability around the world,
affecting one out of every four adults and accounting for billions of dollars in health costs
(Baldwin 2004). Osteoarthritis (OA), one of the most important of these disorders, is a joint
3Author to whom any correspondence should be addressed.
0967-3334/10/020207+13$30.00© 2010 Institute of Physics and Engineering in MedicinePrinted in the UK207
208E B Neves et al
disease characterized by the degeneration of the articular cartilage in combination with an
altered subchondral compartment (one of its key features) (Gobezie et al 2007).
Radiographic imaging is normally the method of choice for the diagnosis and monitoring
of knee OA (Brazilian Health Ministry 1994, Altman et al 1996). However, some authors
report that the radiography of the knee articulation can be normal in the early stages of the
and periarticular sclerosis is detectable as the disease evolves.
Another alternative diagnostic method used in clinical practice concerns psychometric
scales, in which the physician questions the patient about physical limitations and handicaps.
and McMaster Universities—WOMAC (Bellamy et al 1988) scale and the Lysholm knee
scoring scale (Lysholm and Gillquist 1982). However, both radiographic and psychometric
methods present disadvantages.The first demands a considerable infrastructure and the
exposition to ionizing radiation. The second implies a subjective assessment that actually may
not correctly indicate physiopathological changes in osteoarthritis.
Bioimpedance can be defined as the passive electrical properties of biological tissues.
It reflects the ability of biological tissues to oppose the flow of an electric current and is
associated with the conductivity and permittivity of the electrolyte inside and outside the cells
(the intra- and extra-cellular fluids) at frequencies below 1 MHz (Grimnes and Martinsen
2000). Thus, bioimpedance can be used to interrogate electrochemical processes that happen
in biological tissues and for monitoring physiological changes (pathological or not) associated
with such processes. Indeed, bioimpedance spectroscopy (BIS), a non-invasive, portable and
non-expensive technique, has been used in a large number of biomedical applications, such
as hydratation in hemodialysis patients (De Lorenzo et al 1991), diagnosis and control of
lymphedema after breast lumpectomy (Ward et al 1992), assessment of body liquid (Ferreira
and Souza 2004), detection of skin irritation (Ferreira et al 2007) and the differentiation of
inflamed and normal knees (Alvarenga and Souza 2003).
Physical-chemical processes involved in osteoarthritis have been widely studied (Levick
and Macdonald 1995, Doherty et al 1996, Nero et al 2006 and Gobezie et al 2007), but no
consensus has been reached about thechanges associated withthe differenttypes and stages of
osteoarthritis (Howell 1986). The present study hypothesizes that the major electrochemical
processes involved in it would result in detectable bioimpedance changes, and aims to analyze
and compare bioimpedance parameters observed in the knees of healthy/with osteoarthritis
subjects. The results could potentially lead to the establishment of a technique useful in the
assessment of knee osteoarthritis.
2. Material and methods
2.1. The subjects
Thirty-two male volunteers participated in the study. They were members of the Parachute
Military Infantry Brigade of Rio de Janeiro, Brazil. The inclusion criteria were aged between
20 and 60 years old, no physical exercises over the 24 hours prior to the exam and no traumatic
event in the knees over the prior 30 days. The exclusion criteria were the presence of chronic
discomfort or swelling in the knee. The study was performed according to the Declaration of
Helsinki principles and was approved by the Ethical Committee of the Airborne Instruction
Center in charge of the parachute unit.
Knee bioelectric impedance assessment in healthy/with osteoarthritis subjects209
Figure 1. Illustration of electrodes placement in the protocol of BIS data acquisition.
2.2. Experimental protocol
Before BIS data acquisition, the subjects were submitted to a clinical examination by two
orthopedists and one physiotherapist, who evaluated the radiographies of the volunteers’
knees that were obtained under load conditions (Brazilian Health Ministry 1994). After these
evaluations, the eventual knee osteoarthritis of the subjects was classified, according to the
Dejour classification (Dejour et al 1991), as
Stage I. No injury of the subchondral bone, abrasion of cartilage only. Few significant
symptoms and normal radiographies.
Stage II. This stage may be symptomatic or not. Radiographies show a clamping part
in the anteroposterior incidence and the lateral incidence shows a clamp effective in the
central part of the tibial plateau. The lateral incidence shows negligible clamping.
Stage III. This phase provides instability in unipodal support. The internal compartment
osteoarthritis of varus presents internal rotation of the tibia.
Stage IV. Degeneration of the anterior cruciate ligament (ACL). The patella, working
outside its normal route, also presents major injuries. This is the most serious stage of
osteoarthritis, with clear indication for surgical treatment.
results were considered as a reference for the present study in what concerns the osteoarthritis
Disposable 1.0 cm disk electrodes of Ag/AgCl (3M, Brazil) were used for BIS data
acquisition. The protocol to the electrode placement was established considering the best way
to maximize the pathway of the electric current within the synovial fluid and to minimize
the influence of variables out of main goal of the present study, i.e., vascularization and
muscle mass, among others. Considering these conditions and the small physical dimensions
involved, just a bipolar bioimpedance method, as that used in the present study, seems to
be suitable to obtain transversal data in the knee. In this sense the two electrodes were
placed in the medial and lateral sides of the interarticular line of the knee (figure 1), with the
subject seated and the knee flexed 90◦. All BIS measures were taken with the subjects in
a standing position, with the feet in contact with an electrically isolated floor. The subject
position adopted in the protocol was based on preliminary tests (not reported) performed on
210E B Neves et al
Figure 2. Electrical model used to interpret the current response to a step voltage excitation.
The dashed line indicates the impedance electric model of the biological tissue (see the text for
the first 14 knee joints, which were conducted in three different positions: sitting with the
knee flexed at 90◦, standing in the orthostatic position and standing with one leg support. The
results indicated that the adopted standing position was that with the smallest intra-subject
Data for the present study were obtained by a BIS technique (Neves and Souza 2000)
already used in previous studies (Alvarenga and Souza 2003, Ferreira and Souza 2004 and
Ferreira et al 2007). An important aspect of the technique is that it uses a dipolar arrangement
of electrodes, thus making the study of small body segments, such as the knee, easier. Another
important advantage is that it reduces the number of interrogating signals applied to the
implemented for the present application, a 500 mV step voltage was applied to the biological
system and the current response, obtained by the use of a transimpedance amplifier of gain
3300 ?, was digitized by a 16 bits acquisition card (DAQCard 6062E, National Instruments,
USA) at a sample rate of 500 kHz (see Neves and Souza 2000, for more details on the BIS
The electrical circuit depicted in figure 2 modeled the bioimpedance associated with the
experimental protocol illustrated in figure 1 and was used to estimate the current response
of the knee to a step voltage excitation. The resistance Rb and the capacitance Ce represent
the simplified model of the electrode–skin interface impedances; the resistances Re and Ri
model, respectively, the extracellular and intracellular liquids; and the capacitor Cm models
the capacitance of the cellular membrane.
For this electric model, with Vdbeing the amplitude of the voltage step, the current
response i(t) can be expressed by
i(t) = ip[(k1ep1t) + (k2ep2t)],
where ip, k1, p1, k2and p2are constants, dependent on the model parameters and on Vd.
Appendix A.1 shows the details concerning the constants and their relation to the electrical
From the theoretical expectation of i(t) (equation (1)) and its experimental counterpart, it
becomes possible to estimate the electric parameters of the bioimpedance model (Re, Ri, Cm,
Ce). In the present study, a multiparametric optimization procedure, based on the steepest
gradient method, was used in order to estimate the parameters that best fit the experimental
Knee bioelectric impedance assessment in healthy/with osteoarthritis subjects211
Figure 3. Experimental and fitted data of i(t).
data in a least-squares sense (figure 3). The value of Rb was estimated from the perimeter
of the electrode and the resistivity of the skin (Grimnes and Martinsen 2000) according
where ρ is the skin resistivity and r is the radius of the electrode.
After estimating the raw bioimpedance parameters (Re, Ri, Cm, Ce), three secondary
parameters(Rx, XcxandCx)werederived, representing, respectively, theresistiveandreactive
component of the equivalent impedance modeling the biological tissue (represented as the
dashed line in figure 2), as well as the correspondent capacitance associated with Xcx. Such
secondary parameters are calculated at the characteristic frequency, defined as the frequency
where the magnitude of the reactive component is maximal (see appendix B.2 for calculation
2.3. Data analysis and statistics
The behavior of raw (Re, Ri and Cm) and secondary bioimpedance parameters (Rx, Xcx
and Cx) was separately analyzed. The Wilcoxon–Mann–Whitney test was used to assess the
the knee joints classified in Dejour stage I) and in a pathological group (consisting of the knee
joints rated Dejour stage II or higher). This test was also used for assessing the differences
between the Dejour stages. In all statistical tests, p-values lower than 0.05 were considered
as statistically significant. Statistical tests were performed with the Statistical Package for the
Social Sciences 13.0 (SPSS Inc., USA).
In addition, the most sensitive resistive and reactive parameters were analyzed by plotting
the resultingbioimpedance vector and itsconfidence intervals withtheBIVAsoftware(Piccoli
and Pastori 2002); with the two-sample Hotelling’s T2test.
Anthropometric characteristics of the studied subjects can be summarized as follows: age =
42.03 ± 9.81 years; weight = 82.37 ± 12.00 kg; height = 1.75 ± 0.07 m; BMI = 26.76 ±
2.82 kg m−2. Following the selection criteria of the study, two subjects had just one knee joint
evaluated. Among the 62 evaluated joints, 34 joints were classified in stage I of Dejour, 11
in stage II and 17 in stage III. There were no cases of stage IV of osteoarthritis in the studied
sample. Raw BIS data are summarized and presented in table 1.
E B Neves et al
Table 1. Values of raw (Ri, Re, Cm) and secondary (Rx, Cx, and Xcx) bioimpedance parameters for the studied Dejour stages with the respective number (N) of subjects.
577.73 ± 224.19
491.55 ± 136.56
552.98 ± 153.69
525.61 ± 147.79
767.54 ± 110.03
845.16 ± 162.73
5.01 ± 1.95
7.10 ± 1.26
7.14 ± 1.03
391.09 ± 93.44
512.67 ± 67.96
574.12 ± 121.35
49.32 ± 32.82
37.00 ± 11.90
38.31 ± 12.03
134.98 ± 62.24
232.63 ± 42.93
253.02 ± 59.04
Knee bioelectric impedance assessment in healthy/with osteoarthritis subjects213
Figure 4. Box-plots of Re (a) and Xcx (b) for the Dejour classes.
Table 2. Wilcoxon–Mann–Whitney test between the healthy and the osteoarthritis groups for the
Asymp. Sig. (2-tailed)
and Xcx were recognized as the two most sensitive model components for the determination of
parameters against the Dejour stages).
Observing the behavior of such variables for the different stages of osteoarthritis, it is
possible to realize, in accordance with the Wilcoxon–Mann–Whitney U test and the statistic
Z (table 2), that these two parameters could be used to discriminate between the stages
of osteoarthritis. Considering Dejour stage I as associated with healthy knees and stages
II and III as osteoarthritic knees, bioimpedance assessment by Re and Xcx indicated good
discrimination between the healthy and osteoarthritic groups (table 2 and figure 4).
Figure 5 depicts that the knee joints presenting osteoarthritis are over the trend line,
relatively more distant from the origin (Re: 200 ?; Xcx: 0 ?). Thus, it is possible to establish
a rating index that could be used to help the OA diagnosis (equation (3)) (Bioelectric index
for knee osteoarthritis—BIKO) and that represents an overall distance to the origin in a plane
formed by (Re−200) and Xcx:
Table 3 presents the values of sensitivity and specificity according to the ROC curve
(receiver operating characteristic curve). The BIKO presented a predictive positive value
(PPV) of 0.867 and a predictive negative value (PNV) of 0.938 for the cut point of 476.9 ?.
When the BIVA software was applied to the parameters Re and Xcx, it indicated
(figure 6) statistically significant differences between healthy and osteoarthritis knees (T2=
60, p=0.0001, MahalanobisD=1.98); togetherwithanon-statisticallysignificantdifference
(Re − 200)2+ Xcx2.
214E B Neves et al
Figure 5. Scatter plot with parameters Re and Xcx for healthy and osteoarthritis knees.
Table 3. BIKO sensitivity and specificity values according to the ROC curve.
Bioelectric index for knee
osteoarthritis, BIKO (?)Sensitivity1 – Specificity
between the knees classified in Dejour stages II and III (T2= 2.2, p = 0.3638, Mahalanobis
D = 0.57).
Data on seven subjects of the sample that went through arthroscopic surgery for injury
treatment were analyzed in order to determine joint co-morbidity confusion. Three of these
seven cases presented meniscus injuries; two, rupture of the previous crossed ligament; one
presented patellar chondromalacia, and one, rupture of the PCL and a traumatic chondral
injury of 1.2 cm of diameter in the medial condole of the femur. In all of these cases, the
BIS technique presented 100% of agreement with the arthroscopic diagnosis. Only two knees
in the mentioned cases presented chondral degeneration (both presented meniscus injury).
The others presented parameters of bioimpedance compatible with Dejour class I despite the
Knee bioelectric impedance assessment in healthy/with osteoarthritis subjects 215
Figure 6. BIVA, with 95% confidence interval ellipses, when applied to Re and Xcx for the healthy
and osteoarthritic groups (a) and for the two osteoarthritic classes II and III of Dejour (b).
A recent study showed that, in Brazil, 21.2% of the total injuries during military parachute
jumps affected the knee joint (Neves et al 2009). Such a fact emphasizes the importance of
developing non-invasive techniques for the assessment of the OA in this population. In the
present study, osteoarthritis was stage classified by a group of experts, and the bioimpedance
parameters Re and Xcx were the best ones to differentiate the thus formed healthy/non-healthy
In fact, some pathophysiological characteristics of osteoarthritis support and help explain
these findings. The development of osteoarthritis promotes an increase of apatite crystals and
pyrophosphate of calcium in the synovial fluid (Nero et al 2006, Doherty et al 1996). Thus,
since Re is associated with the extracellular liquid and Xcx with the equivalent reactance of the
biological tissue, it is reasonable to suppose that the values of these variables would increase
with an increase in the seriousness of the disease. This would occur since the resistivity of
a fluid increases as the concentration of non-conductive particles in its suspension increases
(Kyle et al 2004).
Mohan et al (2006) reported a high negative correlation between the pH variations and the
quantity of intracellular liquid (ICW) estimated by bioimpedance spectroscopy, corroborating
the findings of some other authors (Damon et al 2002, Lindinger and Heigenhauser 1991).
The resistivity of the biological fluids seems to maintain a good correlation with the values of
pH. In the case of osteoarthritic knees, when the concentration of hyaluronic acid decreases,
the pH of the synovial fluid seems to exhibit a light increase, which is also signaled by the
increase in Re and Xcx.
216E B Neves et al
Other changes can also be observed in osteoarthritic knees: the volume of synovial liquid
surpasses 3.5 ml; there is a reduction of the concentration of hyaluronic acid and there is an
increase, up totentimes, inthenumber of itsleucocytes (Melo 2003). Ramos (2003)observed
that, in the time domain, within the interval 0–0.5 s, the conductivity decreases almost linearly
with the increase of the volumetric fraction occupied by the cells in an electrolytic suspension.
It is thus possible that the increase in the number of leucocytes (size of 10–20 μm) in the
synovial liquid also contributes to the increase of Re and Xcx.
Webb et al (1998) made a cross-study with cultures of articulated live cells obtained from
the knee joints of seven healthy volunteers (three men and four women) and six OA patients
(four men and two women), and observed an increase in the levels of cytokines IL-1 and IL-6
in the synovial fluid of OA patients (p < 0001). These inflammatory mediators unbalance
the articulation metabolism, promoting remodeling of bone superficies and contributing to the
appearance of synovitis and joint cartilage destruction (Penninx et al 2004).
In addition, in normal conditions, the resistivity of the extracellular environment is lower
than that of the intracellular. During hypoxia the cells are not able to guarantee the adequate
function of ionic pumps transport ions to the extracellular environment, resulting in more
water penetrating the cell. So, the increase of the resistivity of the extracellular environment
at low frequencies can be seen as an indicator of tissue ischemia (Ivorra 2003).
The results of the present work agree with those obtained by Gajre et al (2007) in a
study of electrical impedance plethysmography (EIP) of the knees. That study observed
that the subjects with OA knees presented bioimpedances higher than that of the subjects
without OA (control group). Regarding the difference in the magnitude of the values of knee
that the muscular tissue can be strongly anisotropic, with a ratio of transversal/longitudinal
in the longitudinal direction (as in the study of Gajre et al (2007)), one can expect lower values
than those transversally obtained in the same site. Moreover, the use of small electrodes (as in
the present work) implies an additional resistance due to the effect of a greater constriction of
the current density pattern. However, the effect of electrode resistance was considered in the
model of the experimental set-up. It should be noted that, as mentioned before, we decided to
obtain transversal BIS data in the knee trying to maximize the bioimpedance changes due to
the synovial liquid, and just a bipolar BIS technique could be used for such a purpose.
The small intersection observed for the values of bioelectric impedance regarding the
Dejour classification (figure 4) seems to be coherent with biological variability.
The monitoring of patients who went through arthroscopic surgery indicated that the
method seems not to be influenced by meniscus, ligaments or small cartilaginous injuries. The
proposed rating index presented good results for both sensitivity and specificity, particularly
for the cut-off point of 476.9 ? (92.9% and 88.2%, respectively), suggesting that the method
presents a good discrimination between healthy and with osteoarthritis knees.
Vector analysis using a confidence interval of 95% (figure 6) indicated that BIKO, or
equivalently Re and Xcx analysis, could be used as a clinical tool, given that they clearly
differentiate between healthy and osteoarthritic knee joints, even without the help of any
OA is, until now, a disease with a low correlation between clinical and radiographic
findings, and, in addition, no good definition is available that includes symptoms and
physiological/structural alterations (Sharma et al 2006). So, the present findings point to
an objective method that could be used in the clinical environment during the evaluation of
patients suspected of osteoarthritis.
Knee bioelectric impedance assessment in healthy/with osteoarthritis subjects 217
The present study estimated and presented values for knee bioimpedance parameters in a
group of healthy and in a group of OA-presenting subjects. The BIS technique, the electrical
modeling and the parameter estimates seemed to provide an objective and non-invasive tool
for assisting in the diagnosis of knee osteoarthritis. Despite the need for further research
(for instance, with increased sample sizes), the BIS method is a promising tool for objective
and long-term patient follow up. In the specific case of the population studied (parachuters),
this tool could contribute to the development of novel techniques for parachute-associated
This work was supported in part by Brazilian Agencies CNPq, CAPES and FAPERJ.
As mentioned, the current response i(t) of the model in figure 2 can be expressed by
i(t) = ip[(k1ep1t) + (k2ep2t)],
?1 − 4Q2
?1 − 4Q2
The quality factor (Q) and the frequencies of the pole (ωp) and zero (ωz) are expressed as
ωp[(Re+ Rb)Ce+ (Ri+ Re)Cm],
[RiRe+ Rb(Ri+ Re)]CmCe,
218E B Neves et al
It is possible to demonstrate that the total or equivalent impedance, Z(jω), for the electric
circuit modeling the biological tissue (dashed line in figure 2) is
where the angular frequencies, ωzand ωp, are written as
Re + Ri
??jω − ωz
jω − ωp
(Ri + Re)Cm.
After some algebraic manipulation it is possible to rewrite Z(jω) as
This impedance can be interpreted as a series of a resistance (real part) and a capacitive
reactance (imaginary part), leading to the equivalent series circuit R and C. The impedance of
this series circuit can then be written as
Re + Ri
p+ ω2? + j(ωz− ωp)ω
Z(jω) = R − jXc.
Thus, comparing equations (B.4)and (B.5)and considering thatRx andXcxarethe values
of R and Xc at the characteristic frequency, it can be demonstrated that
Re + Ri
Re + Ri
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