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Journal of Non-Crystalline Solids 351 (2005) 2929–2934

Section 11. Dielectric relaxations in complex systems and biological materials

Using dielectric spectroscopy to detect thermal hysteresisin frog muscle

Francis Hart a,*, Eric Davila-Moriel b, Nancy Berner b, Rachel McMillen b

a The Department of Physics, The University of the South, Sewanee, TN 37383, USAb The Department of Biology, The University of the South, Sewanee, TN 37383, USA

Available online 25 July 2005

Abstract

Dielectric spectroscopy has been used to characterize a wide variety of physiological changes in biological systems. We show herehow dielectric spectra taken during a temperature cycle can indicate the presence of subtle physiological changes in the tissue. Wemeasured the impedance spectra of frog gastrocnemius muscle, in vivo, in the frequency range 1 Hz to 1 MHz for temperatures fromabout 10 �C up to a maximum value, Tmax, and back down to about 10 �C. The range of values for Tmax was from 20 �C to 42.5 �C.For frequencies below about 50 kHz we observed hysteresis for thermal cycling above Tmax = 35 �C. The impedance at 10 �C at theend of the run was significantly lower than the impedance at 10 �C at the start of the run, even after corrections had been made forthe change in impedance with time of the exposed muscle. For Tmax = 42.5 �C thermal hysteresis was apparent at all frequencies. Wespeculate that the impedance decrease was due to subtle changes in the membrane glycocalyx above 35 �C and to membrane leakageabove 40 �C. We conclude that thermal cycling of impedance spectra can provide a sensitive measure of structural change for anyphysical system, not just biological tissue.� 2005 Elsevier B.V. All rights reserved.

PACS: 77.22.Gm; 87.19.Nn

1. Introduction

The variation of the electrical properties of tissuewith temperature has been used to monitor the progressof hyperthermia treatment [1–3]. The low frequencyelectrical impedance of excised rat skeletal muscle andtumors decreased abruptly after hyperthermia tempera-ture was reached. Dissado et al. [4] analyzed these resultsin terms of self-similarity and a hierarchical circuit mod-el. We investigate here whether impedance spectroscopyis capable of identifying subtler changes in the tissueoccurring at a lower temperature.

0022-3093/$ - see front matter � 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.jnoncrysol.2005.05.040

* Corresponding author. Tel.: +1 931 598 1549; fax: +1 931 5981145.

E-mail address: fhart@sewanee.edu (F. Hart).

For many materials the electrical impedance at a gi-ven temperature has a unique value. Its value at theend of a thermal cycle is the same as at the start. Forother materials, however, the impedance depends onthe history of the material. If the material�s structure ischanged during the cycle by, for example, the break-down of complex organic compounds or an irreversiblephase change, then thermal hysteresis occurs. Theimpedance at the end of the cycle differs from its valueat the start.

We measured the impedance of frog gastrocnemiusmuscle, in vivo, during thermal cycling to determinewhether thermal hysteresis would occur if the maximumtemperature during the cycle exceeded a particular va-lue. If so, then some physiological change had occurredin the muscle during the cycle. Thermal cycling may pro-vide, in some cases, information regarding relatively

2930 F. Hart et al. / Journal of Non-Crystalline Solids 351 (2005) 2929–2934

subtle changes in the physiological state of a biologicalsystem.

2. Experimental procedures

We purchased bullfrogs, Rana catesbeiana, fromWards Scientific and kept them in glass aquaria at roomtemperature (22–25 �C). There they had free access tofresh water and were fed crickets three to four timesper week. Immediately prior to the measurements wecranially and spinally pithed a frog to allow continualblood flow to the gastrocnemius muscle while destroyingsensory and motor pathways. After the frog was pithed,we skinned both legs from mid-thigh to the Achilles ten-don. We isolated the gastrocnemius muscles, but main-tained them intact by placing a piece of flat plasticbetween the muscle and the rest of the lower leg. In thismanner, we were able to make measurements on themuscles while they were intact in the living animal.

Fig. 1 is a schematic diagram of the probes insertedinto the frog gastrocnemius. A pair of nickel-platedbrass needles, diameter 0.60 mm and separation18.0 mm, served as the electrodes. We embedded thenon-pointed ends of the needles in a small Plexiglasblock to maintain mechanical rigidity. The pointed endsof the needles extended 11.0 mm beyond the block to en-sure complete penetration of the muscle. A thin, flexiblewire connected the other end of each needle to theimpedance spectrometer. Frog gastrocnemius muscle isanisotropic in its electrical properties [5]. The electrodeswere inserted longitudinally, parallel to the long axis ofthe muscle, for all impedance measurements.

Fig. 1. Schematic diagram of the frog gastrocnemius muscle intowhich various probes are inserted.

We connected the electrodes to a Solartron 1260Impedance Analyzer, which was controlled by a Dellcomputer using the program Z60 from Scribner Associ-ates. The data were first analyzed using the programZVIEW from Scribner Associates and then transferredto a Power Macintosh 8600 computer for further mod-eling using the spreadsheet program Microsoft Excel.We confirmed the accuracy of the system by measure-ments of the impedance spectra of various parallel resis-tor and capacitor combinations. For resistance andcapacitance values comparable to those of frog muscle(R between 200 X and 2000 X, C between 0.01 lF and0.5 lF) the calculated and measured impedance spectraagreed to within about 2–3% over the frequency range100 Hz to 1 MHz.

Fisher Amber Natural Rubber Latex Tubing (innerdiameter 1/16 in., outer diameter 1/8 in.) was wrappedseveral times around each gastrocnemius. Because thistubing was very flexible, it made good thermal contactwith the muscle without mechanically stressing it. Weconnected the tubing around the right gastrocnemiusto a Fisher Isotemp Refrigerator Circulator Model900. The tubing around the left gastrocnemius was notattached to a circulator.

Varying the temperature of water that flowedthrough the tubing controlled the temperature of theright gastrocnemius, as measured by a Type-T thermo-couple probe. We inserted the probe into the musclewithin the region about which the tubing was wrappedand distal to the second electrode. Preliminary imped-ance spectra demonstrated that placement of the probein this position did not affect the impedance spectrameasured between the electrodes. The thermocoupleprobe was connected to a Cole-Parmer Digisense Ther-mocouple Thermometer unit to measure the tempera-ture of the muscle. We measured the temperature ofthe left gastrocnemius in a similar manner. Due to thecirculation of the blood from the right gastrocnemius,the temperature of the left gastrocnemius varied a littleabove and below room temperature during the runs.

We measured the variation of the temperature alongthe muscle to determine the uncertainty in the tempera-tures used in the analysis. The temperature measured atthe position of the thermocouple probe differed from thetemperature of the inter-electrode region by at most2–3 �C.

A few drops of saline solution were periodicallyplaced on the muscle to prevent its drying. During themeasurements a plastic cover was placed over the frogto prevent desiccation and to isolate the frog thermallyfrom the surroundings. Preliminary measurementsshowed that approximately 30 min were required forthe impedance to decrease to a steady value after inser-tion of the electrodes.

Following insertion of the electrodes the frog musclewas cooled to a temperature of about 10 �C. After

F. Hart et al. / Journal of Non-Crystalline Solids 351 (2005) 2929–2934 2931

electrical equilibrium had been reached, an impedancespectrum was taken over the frequency range 1 Hz to1 MHz. Preliminary measurements described later indi-cated that inter-lead inductance, which produced achange in sign of ImZ, became important beyond1 MHz. For this reason data collection was stopped at1 MHz. The temperature of the muscle was then in-creased approximately 2 �C and another impedancespectrum was taken. This process was repeated for a ser-ies of temperatures up to Tmax. After Tmax had beenreached, the muscle was cooled in 2 �C increments toabout 10 �C while another series of impedance measure-ments was taken. We performed similar thermal cycleswith the following values for Tmax: 30.0 �C, 33.0 �C,35.0 �C, 37.2 �C, 40.0 �C, 42.5 �C. We used a differentfrog for each run.

3. Results

Previous research on frog gastrocnemius, measuredin vivo [6], has shown that electrode effects tend to dom-inate the measured electrical properties of bulk tissue forfrequencies below about 1 kHz. In order to determinethe effects of artifacts such as the electrode/tissue inter-face and inter-lead inductance in the present experiment,we measured the impedance spectra of the electrodes im-mersed in physiological saline and determined that theimpedance spectra between 2 kHz and 1 MHz representthe bulk muscle material.

Fig. 2 is a plot of the impedance spectra for a froggastrocnemius muscle measured at three temperatures.At each temperature the imaginary part of the imped-ance initially decreases as the frequency increases. Fromabout 1 kHz to 20 kHz the imaginary part decreasesslowly and then resumes a more rapid decrease as the

1

2

3

4

2 3 4 5 6Log frequency, f (Hz)

Log

Impe

danc

e, Z

* (O

hms)

Fig. 2. Impedance spectra of a frog gastrocnemius muscle at threetemperatures. Solid symbols represent ReZ; open symbols, ImZ.Squares represent 8.4 �C; triangles, 25.8 �C, circles, 39.5 �C.

frequency is increased further. The region of slow de-crease in the imaginary part corresponds to a knee inthe real part of the impedance. As the temperature ofthe muscle increases, the real part of the impedance,ReZ, and the imaginary part, ImZ, both decrease inamplitude. The rise in impedance below 1 kHz is dueto the onset of electrode interface effects; above 1 kHzthe impedance spectra are due to the muscle. When acircuit model analysis is performed and the electrodeimpedance subtracted, the region of slow decrease inImZ is revealed to be a peak.

Fig. 3 shows the variation of ReZ for four selectedfrequencies during a cycle that starts and ends at about10 �C and reaches Tmax = 40 �C. We performed similaranalyses for frequencies of 4.75 kHz, 47.5 kHz and475 kHz, but do not show the results on the graphsfor the sake of clarity. The conductivity of muscle typi-cally increases at a rate of 2%/�C [3,7]. Hence, thereshould be a decrease of about 60% in ReZ during theheating phase, but no net change over a complete ther-mal cycle. At 1.19 kHz and 11.9 kHz the ReZ valuesat the end of the cycle are significantly lower than atthe start; hence, thermal hysteresis is apparent at thesefrequencies. In contrast, there is little difference in theReZ values over the course of the cycle at 119 kHzand 1.19 MHz and there is, therefore, no hysteresis indi-cated at these frequencies.

We also analyzed the ImZ data and found similar re-sults. Although the ImZ values appeared to be some-what more sensitive to hysteresis, they are lower inmagnitude than the ReZ values and exhibited muchgreater scatter. We show only the ReZ results here be-cause they display the hysteresis phenomena moreclearly.

Some of the hysteresis observed in Fig. 3 could be theresult of muscle deterioration caused by the insertion ofthe electrodes and the removal of the skin. To determine

0

200

400

600

800

1000

1200

1400

1600

1800

2000

5 10 15 20 25 30 35 40 45

Temperature (ºC)

Re

Impe

danc

e, Z

(Ohm

s)

Fig. 3. Changes in ReZ at four frequencies during a thermal cycle withTmax = 40.0 �C. Diamonds represent 1.19 kHz; squares, 11.9 kHz;circles, 119 kHz; triangles, 1.19 MHz.

Table 1Results of linear fits ReZ change with time for various frequencies atconstant temperature

Frequency(kHz)

Slope(X/min)

Standarderror (X/min)

1.19 �0.74 �+/�� 0.164.75 �0.25 �+/�� 0.1111.9 �0.008 �+/�� 0.0747.5 0.10 �+/�� 0.04119 0.07 �+/�� 0.03475 0.05 �+/�� 0.021190 0.003 �+/�� 0.02

0.00

0.20

0.40

0.60

0.80

1.00

1.20

20 25 30 35 40 45

Maximum Temperature, Tmax (ºC)

Fina

l Im

peda

nce

Rat

io R

e Z f

/Re

Z f'

Fig. 5. The ratio of the measured final impedance to the finalimpedance caused solely by deterioration versus the maximumtemperature in the cycle. Solid diamonds represent 1.19 kHz; solidsquares, 4.75 kHz; solid circles, 11.9 kHz; open triangles, 47.5 kHz;open diamonds, 119 kHz; open squares, 475 kHz; open circles,1.19 MHz.

2932 F. Hart et al. / Journal of Non-Crystalline Solids 351 (2005) 2929–2934

the significance of these effects, we measured the imped-ances of the two gastrocnemius muscles for over fivehours, a typical duration of a thermal cycle, at a con-stant temperature of 20.3 �C.

Fig. 4 is a plot of the measured ReZ values versustime after electrode insertion in the right leg. Also shownare the corresponding linear fits to the data. Preliminarymeasurements indicated that approximately 30 min isrequired after electrode insertion for electrical equilib-rium to be established at the needle/electrode interface.For this reason we always waited for this length of timebefore commencing cycling.

Table 1 presents the results of the linear fits to thevariation of ReZ with time data. Only at 1.19 kHzand 4.75 kHz, there is significant decrease of impedancewith time. ReZ did not vary significantly with time forthe left gastrocnemius, which experienced the same skin-ning and electrode insertion as the right leg. As the onlydifference in the two legs was the flow of water throughthe tubing surrounding the right leg, we conclude thatthe deterioration with time at the low frequencies wascaused by mechanical disturbances of the muscle dueto the flow.

For each run for each of the seven analyzed frequen-cies we calculated Z 0

f , the final impedance that wouldhave resulted from any tissue deterioration with time.We compared this value to Zf, the measured final imped-ance. If there were no thermal hysteresis, the ratio Zf=Z 0

f

should be equal to 1; that is, the impedance–temperaturedata for decreasing temperatures in Fig. 4 should coin-cide with the data for increasing temperatures, exceptfor the effects of deterioration.

The ratio ReZf=ReZ 0f is plotted in Fig. 5 against the

various maximum temperatures reached during the cy-cles. The point at 20 �C corresponds to the data taken

0

200

400

600

800

1000

1200

1400

1600

1800

0 30 60 90 120 150 180 210 240 270 300 330Time (min)

Re

Impe

danc

e, Z

(Ohm

s)

Fig. 4. Changes in ReZ at four frequencies at various times followingelectrode insertion. The muscle temperature remains constant at20.3 �C. Diamonds represent 1.19 kHz; squares, 11.9 kHz; circles,119 kHz; triangles, 1.19 MHz. The dashed lines represent linear fits tothe data. Table 1 presents the fit results.

at constant temperature. Between 30 �C and 35 �C theratio falls significantly below 1 for the lower frequencies.The onset of thermal hysteresis appears to occur be-tween these temperatures at the lower frequencies. Amajor change in the muscle appears to take place be-tween 40 �C and 42.5 �C where major hysteresis isapparent at all frequencies.

To identify more closely the temperature at whichhysteresis commences at low frequencies, we performedcycles for Tmax = 33.0 �C and 35.3 �C, but with the mus-cle held at Tmax for 45 min. If physiological change takesplace above a certain temperature, then the longer themuscle is held above that transition temperature, thegreater should be the change. We compared the hyster-esis results with the muscle held at Tmax to those whenthe temperature was immediately decreased when Tmax

was reached.Fig. 6 compares the thermal cycle results for

Tmax = 35 �C for the regular cycle and for the cycle inwhich the temperature was held at Tmax for 45 min.

0

500

1000

1500

2000

2500

5 10 15 20 25 30 35 40Temperature (ºC)

Re

Impe

danc

e, Z

(Ohm

s)

Fig. 6. Thermal cycles for Tmax = 35 �C. Open symbols represent acycle for which the muscle was held at Tmax for 45 min. Solid symbolsrepresent a cycle in which the temperature was immediately reducedafter reaching Tmax. Diamonds represent 1.19 kHz; squares, 11.9 kHz.

F. Hart et al. / Journal of Non-Crystalline Solids 351 (2005) 2929–2934 2933

Holding at Tmax clearly increases the hysteresis effect at1.19 kHz and 11.9 kHz. The results are not shown forother frequencies for the sake of clarity. A clear increaseof hysteresis is also apparent at 4.75 kHz. There is stillno hysteresis indicated at 119 kHz, 475 kHz and1.19 MHz. A slight hysteresis is present at 47.5 kHz. Asimilar analysis was performed for Tmax = 33 �C andno significant hysteresis was apparent at any frequency.We conclude that some physiological change occurs inthe muscle between 33 �C and 35 �C.

Figs. 5 and 6 indicate that hysteresis occurs for Tmax

values around 35 �C for frequencies below about50 kHz. Only for Tmax above 40 �C does hysteresis ap-pear at higher frequencies. Comparison in Fig. 7 ofReZ spectra taken at 12 �C at the start and at the endof cycling for Tmax = 37.2 �C and 42.5 �C illustrates thispoint. For Tmax = 37.2 �C there is a significant difference

2

3

4

2 3 4 5 6Log frequency (Hz)

Log

Re

Impe

danc

e, Z

(Ohm

s)

Fig. 7. Impedance spectra taken at 12 �C. Solid symbols represent thestart of a cycle; open symbols, the end of a cycle. Squares representTmax = 42.5 �C; circles, Tmax = 37.2 �C.

in the spectra only for frequencies below about 50 kHzwhereas for Tmax = 42.5 �C the spectra differ at allfrequencies.

4. Discussion

Foster and Schwan [7] ascribe a low frequencydispersion in muscle to counterion polarization alongthe muscle membrane and/or polarization of the sarco-tubular system and a radio frequency dispersion to thecharging of the cellular membranes through Maxwell–Wagner polarization. The significant decrease in imped-ance at high frequencies for Tmax = 42.5 �C may then beattributed to a deterioration of membrane integrity atthis temperature with a leakage of cytosol into the extra-cellular fluid.

The impedance decrease below 50 kHz may be relatedto changes produced in the counterion polarization inthe glycocalyx along the membrane surface. Alterna-tively, the low frequency increase could be an artifactproduced by a major hysteresis in the impedance ofthe electrode interface. To test this possibility we per-formed thermal cycling of the electrodes immersed inphysiological saline for Tmax = 37 �C and did not ob-serve significant hysteresis except at 1.19 kHz. We con-clude that the hysteresis observed below 50 kHz is acharacteristic of the tissue and not the electrode inter-face. The impedance hysteresis may be due to an in-crease in the charge carrier density produced by adeterioration of the glycocalyx for T > 35 �C and ofthe membrane integrity at 42.5 �C.

Finally, although there is some indication in Fig. 2that a change in material properties has occurred be-cause of the drop in ImZ at the highest temperature,that change is much more apparent in the hysteresis re-sults. Thermal cycling could provide a useful method todetect the occurrence of small changes in the structure ofother, non-biological materials as the temperature isincreased.

5. Conclusions

Thermal cycling of impedance spectra can detectchanges in muscle physiology that might be missed bya unidirectional sequence of impedance–temperaturemeasurements. This method appears most sensitive forfrequencies below about 50 kHz and could be appliedusefully to a wide range of materials.

References

[1] D.A. McRae, M.A. Esrick, Phys. Med. Biol. 37 (1992) 2045.[2] D.A. McRae, M.A. Esrick, Int. J. Hyperthermia 9 (1993) 247.

2934 F. Hart et al. / Journal of Non-Crystalline Solids 351 (2005) 2929–2934

[3] E. Gersing, in: P.J. Riu, J. Rosell, R. Bragos, O. Casas (Eds.),Electrical Bioimpedance Methods: Applications to Medicine andBiotechnology, The New York Academy of Sciences, New York,NY, 1999, p. 13.

[4] L.A. Dissado, J.M. Alison, R.M. Hill, D.A. McRae, M.A. Esrick,Phys. Med. Biol. 40 (1995) 1067.

[5] F.X. Hart, N.J. Berner, R.L. McMillen, Phys. Med. Biol. 44 (1999)413.

[6] F.X. Hart, W.R. Dunfee, Phys. Med. Biol. 38 (1993) 1099.[7] K.R. Foster, H.P. Schwan, Crit. Rev. Biomed. Eng. 17 (1989)

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