Japanese Journal of Applied Physics
Vol. 46, No. 7B, 2007, pp. 4793–4800#2007 The Japan Society of Applied Physics

Investigation of Frequency Characteristics in Cutting of Soft Tissue
Using Prototype Ultrasonic Knives
Kosuke EBINA, Hideyuki HASEGAWA, and Hiroshi K ANAI
Graduate School of Engineering, Tohoku University, Sendai 980-8579, Japan(Received November 24, 2006; revised March 21, 2007; accepted April 5, 2007; published online July 26, 2007)

Recently, ultrasonic surgical knives have been applied in a variety of surgical operations. In this paper, the operation
frequency of a surgical knife is focused on. Prototype ultrasonic knives operated at 24.3, 44.3, and 71.9 kHz were constructed.
Diﬀerences in the eﬀects on soft tissue depending on the operation frequency were investigated using these knives. Frequency
characteristics were measured using two parameters: coagulation ratio and coagulated depth. For the same vibration velocity,
at a lower frequency, the distribution of the coagulated tissue was deep and in a narrower region around the center of the tip.
For the same vibration amplitude, the coagulated depth at each frequency was similar for all these frequencies. Furthermore,
the dependences of tissue coagulation on the vibration velocity, pressure load, contact of the tip with tissue, and direction ofvibration were investigated. From these investigations, it was found that the mechanical eﬀect, rather than ultrasoundabsorption, is the dominant factor in tissue coagulation. [DOI: 10.1143/JJAP.46.4793]
KEYWORDS: ultrasonic surgical knife, frequency characteristic, coagulation ratio, coagulated depth, heat generation, stress–
strain relationship1.

Introduction

Recently, there are many studies on the medical treatment
technology using ultrasound in addition to those on thedevelopment of ultrasonic-diagnosis technology. For instance, focused ultrasound has been applied in extracorporeal shock wave lithotripsy1–5) and controlled ultrasoundpulses are expected to be used for making a perforation in
the atrial septum.6,7) In this paper, we focus on an ultrasonic
surgical knife, which is one of the medical treatment
technologies using ultrasound.8,9) It works in the frequency
range from 20 to 100 kHz and it fragments tissues by meansof various eﬀects, such as mechanical, cavitation, andthermal eﬀects. It is able to cut soft tissue using ultrasonicvibrations of the blade and at the same time, stop bleedingby coagulation. Moreover, it can coagulate proteins at acomparatively low temperature, resulting in less injury,
which is a great advantage in surgery. Thus, the existingultrasonic surgical knife (Johnson & Johnson, Harmonic
Scalpel , operating frequency = 55.5 kHz) has been appliedin a variety of surgical operations, e.g., cutting lungparenchyma,10) latissimus dorsi muscle mobilization,11) redo
cardiac surgery,12) thyroidectomy,13) tonsillectomy,14) andharvesting of arterial conduits in coronary artery bypass.15)
These studies show that the use of the Harmonic Scalpel
leads to less bleeding, reduction of complications andshortening of operation time compared with conventional
techniques using other mechanical surgical knives such as anelectrocautery scalpel. Ultrasonic surgical knives of several
operating frequencies are available (e.g., Johnson & Johnson, Harmonic Scalpel : 55.5 kHz; Autosuture, AutoSonix: 55.5 kHz; Olympus, SonoSurg: 23.5/47 kHz; Aloka, SONOP 5000: 23/35 kHz), however, the eﬀects of thevibration frequency on cutting have not been thoroughly
investigated. In recent work, some important parameters,that aﬀect tissue fragmentation, including the operation
frequency of the knife and stroke amplitude of the tip, havebeen studied.16) Furthermore, the temperature elevation intissue during operation has been examined,17) and theacoustic output of the Harmonic Scalpel has been designated

E-mail address: hkanai@ecei.tohoku.ac.jp

for safe use.18) Moreover, the inﬂuence of various tissue
types on the working characteristics of the Harmonic Scalpel
has been studied in order to increase its eﬃciency.19)
However, the eﬀects of the ultrasound vibration of theblade, particularly the eﬀects on the tissue caused by thevibration of the knife tip have not yet been revealed
thoroughly. In this paper, we focus on the operating
frequency of an ultrasonic surgical knife. The objective of
this study is to investigate the frequency characteristics inthe cutting of soft tissue using prototype knives operated atseveral diﬀerent frequencies (e.g., 20 and 40 kHz andhigher) to identify the optimum operation frequency of theultrasonic surgical knife. As the ﬁrst stage of this investigation, prototype ultrasonic knives were constructed usingbolt-clamped Langevin-type transducers (BLTs) at operation
frequencies of 20, 39, and 60 kHz. Then, soft tissue wascoagulated with the developed knives, and the frequency
characteristics in the cutting of soft tissue were evaluatedusing two parameters. Furthermore, these results are discussed in terms of the heat generation caused by the stress–
strain relationship and ultrasound absorption in soft tissue.
2.

Structures of Prototype Knives

Figure 1 shows the technical drawings of the prototype
knives. These knives were constructed using BLTs operated
at frequencies of 20, 39, and 60 kHz (Honda Electronics,
HEC-3020P2B, HEC-3039P4B, HEC-1560P4B, respectively). The structure of the vibration ampliﬁer was a two-stagehorn and it was made from duralumin. The stepped horn,which can yield a higher transmutation ratio compared withother shapes, was selected. The theoretical transmutation
ratios of these horns were 3.5 and 9.1. The ﬂange, which
was 2 – 4 mm in thickness, was equipped to clamp the knife
at the ultrasonic vibration node. The lengths of the knives
including transducers were 338, 176, and 113 mm corresponding to 1.5 times the ultrasonic wavelength at therespective frequencies. Moreover, the prototype knives
shown in Fig. 1 were used to evaluate tissue coagulation.
Thus, these shapes and conﬁgurations are diﬀerent fromthose of existing ultrasonic surgical knives. Furthermore,
impedance matching will be eﬀective in increasing thevibration velocity. Thus, an electric transformer was

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K. E BINA et al.

Fig. 1. Technical drawing of the prototype knife.

500

24.3 kHz

400

100
0
18

20

22

24

26

28

30

Transformer

Laser Doppler
vibrometer

Power amplifierLavibration
300
velocity[mm/s] 200
32

frequency [kHz]

(b)
600

se

44.3 kHz

38

40

42

44

46

48

50

d

Signal generatorea0Flangerh

vibration
velocity 400[mm/s] 200(c)Prototype knife

(a)

52

frequency [kHz]300

71.9 kHz

5 mm

100
0
62

θ

Oscilloscope

400

vibration
velocity 200[mm/s]

Vibration
direction

Tissue phantom
64

66

68

70

72

74

76

z

frequency [kHz]Fig. 3. Experimental setup for measurement of frequency characteristic.
Fig. 2. Frequency response of vibration velocity of the tip of the knife.

mounted between the power ampliﬁer and the knife for
impedance matching.
Figure 2 shows the frequency responses of vibration
velocities for all knives. Vibration velocities shown in Fig. 2
are peak-to-zero values. From Fig. 2, the maximum vibration velocities were obtained at 24.3, 44.3, and 71.9 kHz forthese knives. Thus, these frequencies were employed when
the knives were applied to a tissue phantom.
3.

Experimental Procedure

Figure 3 shows the experimental setup used in this study.
A developed knife was clamped at the ﬂange. In this study,
chicken muscle was used. A laser Doppler vibrometer(LDV; Ono Sokki, LV-1300) was used to measure thevibration velocity of the tip of the knife. The laser head was
ﬁxed at angle  relative to ultrasound propagation. The tilted
laser ray was incident at a point 5 mm from the tip of theknife. A retroreﬂective tape reﬂector, which reﬂects rays
along their incident direction, was attached near the tip ofthe blade. It was conﬁrmed that there is no change in thecondition of the tape reﬂector, such as detachment, after
driving the knife at a velocity of 4 m/s (peak-to-zero). The
laser ray must be perpendicular ( ¼ 180 ) to the measurement site in order to measure the correct vibration velocity.

However, it is diﬃcult to keep the laser ray perpendicular
to the tip in this setup because the tip was in contact withthe tissue phantom. Moreover, near the tip of the knife,tissue fragments and water mist on the surface give rise toundesired eﬀects on the laser ray during the operation,
decreasing the laser reﬂection intensity. Therefore, before
the setting of tissue, the LDV was calibrated by analyzing
two signals from perpendicular to the tip of the knife( ¼ 180 ) and from the angle . As a result, the vibration
velocity was quantitatively deﬁned. The angle  was set to37, 43, and 43 for the knives at the driving frequencies of
24.3, 44.3, and 71.2 kHz, respectively.
The coagulation test is described as follows. First, each
prototype knife was operated and tissue was coagulated by
the vibration at the tip of the knife. Then, the coagulated
tissue was cut at the center of the area of tip contact using
a surgeon’s knife to obtain a cross-sectional image. This
image was used to measure the coagulated depth to evaluate
the coagulation along the z direction. The coagulation ratio
was deﬁned as the area of the coagulated region divided bythat of the surface of the tip. It was used for the evaluation of
the coagulation on the surface of the tissue. These twoparameters were used in the investigation of the frequency
characteristics in the cutting of soft tissue. These procedures
were repeated eight times for each knife.

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(a)
b

a

g

f

e

d

c

j

i

h

z

2 mm

(b)

i
c

a

g

e
d

j

h

f

b

z

1 mm

1s

5s

3s

15 s
cutting plane

10 s

Fig. 4. Results of the cutting test using the 20 kHz knife. (a) Tissue surface after the cutting test. (b) Magniﬁed cross-sectional views
along broken lines in (a).

(a)
a

c

b

e

d

g

f

i

h

j

2 mm

z

(b)
c

a

g

f

e

d

b

h

j

i

z

1 mm

1s

3s

5s

15 s
cutting plane

10 s

Fig. 5. Results of the cutting test using the 40-kHz knife. (a) Tissue surface after the cutting test. (b) Magniﬁed cross-sectional views
along broken lines in (a).

(a)
a

c

b

d

e

f

g

i

h

j

1 mm
z

(b)
a

b

c

d

g

e

h

f

j

i

1 mm

1s

3s

5s

10 s

z

15 s
cutting plane

Fig. 6. Results of the cutting test using the 60-kHz knife. (a) Tissue surface after the cutting test. (b) Cross-sectional views along
broken lines in (a).

4.

Results

4.1 Results of tissue coagulation test
First, the vibration velocity was set to 2.0 m/s for all
knives. In this experiment, the prototype knives wereoperated at 24.2, 44.8, and 71.4 kHz. The peak-to-zero valueof the vibration velocity is shown in all ﬁgures in this paper.
The tip was brought into contact with the soft tissue for 1, 3,
5, 10, and 15 s with or without load.
Figure 4 shows the coagulated tissue sample for various
operating periods using the 20 kHz knife. The images of thecoagulated tissue using the 40 kHz knife and the 60 kHz
knife are shown in Figs. 5 and 6, respectively. In Figs. 4(a),5(a), and 6(a), the area of the circle is equivalent to that of
the tip of a knife. The broken lines in Figs. 4(a), 5(a), and6(a) show the positions of cutting planes of the crosssectional images shown in Figs. 4(b), 5(b), and 6(b). In
Figs. 4(a), 5(a), and 6(a), the positions of the letters (from ato j) correspond to those in Figs. 4(b), 5(b), and 6(b). In eachof Figs. 4(a), 5(a), and 6(a), the coagulation on the surface oftissue spread from the central part to the entire region of tip
contact. Moreover, the tissue was coagulated more deeply as

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Jpn. J. Appl. Phys., Vol. 46, No. 7B (2007)(a)71.4 kHz24.2 kHz

20
2

4

6

8
10
driving period [s]

12

14

16

80

44.3 kHz

60
40

24.3 kHz

20
0
0

5

24.3 kHz

40
20
0
0.5

15

1

44.8 kHz

1.5
171.4 kHz
16

20

71.7 kHz

00

Fig. 7. (a) Measured coagulation ratio. (b) Measured coagulated depth.

2

4

6

8

60

20

24.3 kHz0010vibration acceleration [×105 m/s2]

1

2

3

4

ultrasound intensity [×106 W/m2]

Fig. 8. Measured coagulation ratio versus (a) vibration amplitude, (b)vibration velocity, (c) vibration acceleration, and (d) ultrasound intensity.

(a)

(b)

coagulated depth [mm]

2.5

2.5

24.3 kHz2

44.3 kHz
1.5

71.7 kHz

1
0.5
0
0

5

10

24.3 kHz

2
1.5
1

44.3 kHz

0.5

71.7 kHz
0
0.5

15

1

(c)

2

2.5

(d)

2.5

coagulated depth [mm]1.5vibration velocity [m/s]vibration amplitude [µm]2.5

2 24.3 kHz

44.3 kHz
1.5

71.7 kHz
1
0.5
0
0

2

4

6

8

10

5

2

24.3 kHz
2
1.5
1

44.3 kHz

0.5

71.7 kHz

0
0

100

200

300

400

4

vibration acceleration [×10 m/s ]2

ultrasound intensity [×10 W/m ]Fig. 9. Measured coagulated depth versus (a) vibration amplitude, (b)vibration velocity, (c) vibration acceleration, and (d) ultrasound intensity.

1.0

100

0.9

90

0.8

70

]

0.7

80

coagulated depth [mm]%coagulated depth [mm]the contact time increased as shown in Figs. 4(b), 5(b), and6(b). These tendencies were similar for the three diﬀerent
frequencies.
Figure 7 shows the measured coagulation ratio andcoagulated depth for various driving periods as the mean
and ranges (minimum and maximum) of eight measurements. In Fig. 7(a), the coagulation ratio became larger as
the driving period increased, as evident in the images in
Figs. 4(a), 5(a), and 6(a). The coagulation ratio at a higher
frequency tends to be larger than that at a lower frequency.
That is, at a lower frequency, coagulation on the surface isslow compared with that at a higher frequency. On the other
hand, Fig. 7(b) indicates that the tissue was coagulated moredeeply at a lower frequency than at a higher frequency. At
44.8 and 71.4 kHz, the average coagulation ratio at 15 s
approached 100%, as shown in Figs. 5(a) and 6(a). These
results show that coagulation on the soft tissue approaches
saturation.
Next, the relationship between tissue coagulation and theoutput of the knife, such as amplitude of vibration velocity
and displacement, was investigated using the same systemthat illustrated in Fig. 3. In this measurement, the vibration
velocity was changed from 0.6 to 2.0 m/s (peak-to-zero).
The prototype knives were operated at 24.3, 44.3, and71.7 kHz. Figures 8 and 9 respectively show the measured
coagulation ratio and coagulated depth obtained after a 10-s
drive for various vibration amplitudes, velocities, accelerations, and ultrasound intensities as mean and ranges(minimum and maximum) of eight measurements. The
ultrasound intensity I was calculated as I ¼ ð1=2Þ  cv2 (v:vibration velocity, : density = 1,055 kg/m3 , c: soundspeed = 1,575 m/s20)). In Figs. 8 and 9, both the coagulation ratio and coagulated depth increased as the output of theknife increased. In addition, the change in the coagulation
ratio among the driving frequencies in Fig. 8(b) was smaller
than those in Figs. 8(a) and 8(c). That is, the coagulation on
the tissue surface depends on the vibration velocity.
As shown in Fig. 9(a), when the vibration amplitude issmaller than 5 mm (peak-to-zero), the tissue coagulation
along the z axis (depth) is proportional to the vibration
amplitude, and does not depend on the vibration frequency.
Figure 10 shows the coagulation ratio in Fig. 8(a) and thecoagulated depth in Fig. 9(a) for three operation frequencies,

44.3 kHz

40

0.6

60

o[

14

71.7 kHz

80

ati

12

40

coagulated depth [mm]1044.3 kHzcoagulated depth [mm]8driving period [s]

80

nr

6

100

0.5

50

tio

4

2.5

ula

2

24.3 kHz

60

2

ag

0.5

(d)

100

coagulation ratio [%]

(c)

1.5

vibration velocity [m/s]

0.4

40

co

2

coagulation ratio [%]coagulated depth [mm]24.2 kHz

2.5

0
0

10

60

vibration amplitude [µm]344.3 kHz71.7 kHz

80

0.3

30

0.2

20

0.1

10

0

24.3

44.3

71.2

coagulation ratio [%]44.8 kHz

40

100

coagulation ratio [%]

60

0
0

(b)

100

71.7 kHz80

coagulation ratio [%]coagulation ratio [%]100

K. E BINA et al.

0frequency [kHz]Fig. 10. Results of measuring the coagulation ratio and the coagulated
depth at a constant vibration amplitude (4 mm) for each operation
frequency.

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(b)

tip of the knife

tip of the knife(a)cavity bubbles
vibration
direction

2 mm

2 mm

Fig. 13. Magniﬁed view of the tip of the knife. (a) Before operation. (b)During operation.

Fig. 11. Experimental setup. (a) Tip of the knife was kept contact with thesurface. (b) Tip of the knife was separated from the tissue surface.

(a)flaked fascias

a

c

b

g

d

e

5 mm

h

f

5 mm

5 mm

5 mm

cutting plane

(b)
b

a

1 mm
l = 0 mm

e

f

d

c

1 mm

1 mm
l = 0.5 mm

l = 1.0 mm

h

g

1 mm
l = 1.5 mm

Fig. 12. Coagulated tissue in degassed saline. (a) Tissue surface. (b) Cross-sectional views of (a).

where the vibration amplitude and the driving period were
4 mm and 10 s, respectively. The coagulated depth did notchange for various operation frequencies. However, thecoagulation ratio became larger with frequency because thevibration velocity increased with frequency.
4.2 Identiﬁcation of dominant eﬀect in tissue coagulation
Eﬀects on the tissue caused by the vibration at the tip of
the knife is divided into the following two phenomena. The
ﬁrst is the mechanical eﬀect due to tissue expansion andcontraction caused by the tip vibration. The second is theacoustic eﬀect caused by the exposure to strong ultrasoundand resulting in the cavitation and ultrasound absorption.
The absorption coeﬃcient of ultrasound absorption isroughly 0.07 dB/cm at the frequency of 1 MHz in tissue.
That is, half the energy of the ultrasound is converted to heat
when 1-MHz-frequency ultrasound propagates for 5 cm.21)
In addition, the absorption coeﬃcient is proportional to thesquare of the ultrasound frequency, and it is roughly
estimated to be 2:52  104 dB/cm at 60 kHz. Thus, it isvery small within the frequency range employed in thisstudy. That is, the temperature elevation due to ultrasoundabsorption is small. Thus, under the conditions adopted inthis study, the acoustical eﬀect on tissue coagulation islimited to the cavitation phenomenon.
In order to investigate the above phenomena, the soft
tissue was coagulated in a water bath ﬁlled with degassed
saline, and the distance l from the tip of the knife to thetissue surface was changed from 0 to 1.5 mm, as shown in
Fig. 11. When the distance l is larger than 0 mm, the direct
mechanical eﬀect does not aﬀect the tissue and tissue iscoagulated by the cavitation phenomenon. The vibration
velocity was set to 2 m/s (peak-to-zero), the driving period
was 10 s, and the driving frequency was 44.3 kHz.
The solid circles in Fig. 12(a) show the positions of the tipof the knife. Each line segment in Fig. 12(a) corresponds tothat in Fig. 12(b). When the tip of the knife was brought into
contact with the tissue surface (l ¼ 0 mm), the coagulated
tissue (white part) became visible, as shown in Fig. 12.
However, when the tip of the knife was separated from thetissue (l ¼ 0:5 mm or more), tissue was not coagulated.
When l ¼ 0:5, 1, or 1.5 mm, a slight abruption of the tissue
surface was alternatively observed.
Figure 13 shows the magniﬁed view of the driving tip of
the knife in water. In Fig. 13(b), cavity bubbles appear as awhite haze that is distributed in the region within 0.9 mm
from the tip. Thus, damages on the tissue surface would becaused by shock waves due to the breakdown of cavity
bubbles. Although such acoustical eﬀect contributes somewhat to the coagulation of the tissue, it is not a major factor
in tissue coagulation in Fig. 12 (l ¼ 0 mm).
4.3

Relationship between static load and tissue
coagulation
The coagulation ratio and coagulated depth were measured
for various values (from 0 to 89 kPa) of the static pressure of
the contact of the tip with tissue. The vibration velocity anddriving period were set at 2 m/s (peak-to-zero) and 5 s, respectively. The driving frequencies of the knives were24.2, 44.8, and 71.5 kHz. Figure 14 shows the experimental
results. In Fig. 14, the coagulation ratio and the coagulated
depth became larger for all frequencies as the pressure in-4797

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coagulation ratio [%]

100
80
60

71.5 kHz44.8 kHz

40

24.2 kHz

20
0

0

10

20

30

40

50

60

70

80

90

pressure [kPa]coagulated depth [mm]2.5

Fig. 15. Experimental setup for investigating the tissue coagulation
depending on vibration direction. (a) Longitudinal direction. (b) Lateral
direction.

24.2 kHz2
44.8 kHz

1.5
1
0.5
0

0

71.5 kHz
10

20

30

40

50

60

70

80

90

pressure [kPa]

Fig. 14. Relationship between load and tissue coagulation. (a) Coagulation ratio. (b) Coagulated depth.

creased. In the experiments described above, the tip of theknife was in contact with the tissue surface without load.
When the load was applied, it compressed the tissue, and mechanical viscoelastic properties were changed by the pressure. However, the cavitation phenomenon, which is an
acoustical eﬀect, depends not on the applied pressure buton the ultrasound intensity. On the other hand, the heat generation caused by expansion and contraction of tissue in themechanical eﬀect depends on the viscoelasticity. This resultshows that the heat generation increases with pressure force.
4.4

Comparison of tissue coagulation due to the diﬀerencein vibration direction
Figure 12 shows that the mechanical eﬀect due to thevibration of the knife tip is a dominant factor in soft tissue
coagulation. The tissue coagulation due to the mechanicaleﬀect is caused by thermal denaturation due to heatgeneration. Heat generation is divided into the following
two phenomena: frictional heat at the interface between the
knife and the soft tissue and heat generation caused byexpansion and contraction of the tissue. To investigate thediﬀerence between these two types of heat generations, thetip (edge or side) was brought into contact with tissue in thetwo manners as shown in Figs. 15(a) and 15(b). Heat will be

generated by longitudinal strain in the case of Fig. 15(a) andby shear strain or friction in the case of Fig. 15(b). This
experiment was carried out using the 39 kHz knife at a
driving frequency of 44.3 kHz.
Figure 16 shows the diﬀerence in the coagulated tissues.
In Fig. 16(c), the points marked by letters (from a to d)correspond to those in Fig. 16(d). In Fig. 16(a), for thevibration velocity of 2 m/s, the tissue was seen to becoagulated by the longitudinal vibration but not by thelateral vibration. Since the frictional force is proportional to
the normal pressure, the tissue load was increased from 100
to 400 g. However, tissue was not coagulated even with theload of 400 g, as shown in Fig. 16(b). On the other hand,when using the vibration velocity of 4 m/s, as shown in
Fig. 16(c), the tissue phantom was coagulated for both casesof the vibration direction. However, as seen from the crosssectional view in Fig. 16(d), the depth of the coagulated
regions due to longitudinal and lateral vibrations were 2.67
and 0.67 mm, respectively. Heat generation by longitudinalstrain was larger than those by shear strain and frictional
heat. Therefore, tissue coagulation is mainly caused by heatgeneration due to longitudinal strain.
5.

Discussion

In this study, since coagulation did not occur when the tipof the knife was separated from tissue, tissue coagulation
was caused not by ultrasound absorption but by heatgeneration due to the deformation of tissue. Therefore, thecoagulation resulting from heat generation is considered tooccur via the deformation of tissue. The viscoelasticity oftissue is approximated using the Voigt model, as shown in
Fig. 17. The basic equation of the Voigt model is given by

(c)

(a)

a.

b.

longitudinalc.

d.

longitudinallaterallateral

5 mm

(b)

5 mm

(d)

100 g

200 g

400 g

b.

a.

lateral
5 mm

c.

d.

longitudinal
5 mm
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Fig. 16. Diﬀerence in tissue coagulation dependingon (a) vibration direction (vibration velocity = 2 m/s, load = 200 g, driving period = 10 s), (b) pressureload (lateral vibration, vibration velocity = 2 m/s,
driving period = 20 s, tissue load = 100 g (left),
200 g (center), and 400 g (right), and (c) vibration
direction (vibration velocity = 4 m/s, load = 200 g,
driving period = 10 s). (d) Cross-sectional view of(c).

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K. E BINA et al.

stress σ

σ

G

γ

η

O

20 kHzstrain τ

40 kHzFig. 17. Voigt model.

60 kHzstress σ

Fig. 19. Stress–strain diagrams at diﬀerent frequencies.

Z 2=!
1!0 0fsinð2!t  Þ þ sin g dt
20¼ 0 0 sin :

strain τO¼

Fig. 18. Illustration of stress–strain diagram of tissue.

 ¼ E þ 

d
;
dt

ð5:1Þ

where E is Young’s modulus,  is the viscosity coeﬃcient, is the stress, and  is the strain. In this study, prototype
ultrasonic knives were driven by harmonic oscillation.
Therefore, the stress  is expressed as follows: ¼ 0 sin !t:ð5:2Þ

By substituting eq. (5.2) into eq. (5.1), the strain  can be
described by0
 ¼ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
 2ﬃ sinð!t  Þ ¼ 0 sinð!t  Þ; ð5:3Þ
E 1þ !
E ¼ tan1 ! ¼ tan1 !:ð5:4Þ
E
From eq. (5.3), strain  has a phase delay  from stress , asshown in eq. (5.4). Therefore, the stress–strain relationshipgenerally shows a hysteresis property, as shown in Fig. 18.
The area within the stress–strain diagram Wh corresponds tothe evaporated energy per period.
I
Wh ¼  d
ð5:5Þ
The time derivative of eq. (5.3) is shown as follows:d¼ !0 cosð!t  Þ:
dt

ð5:6Þ

By substituting eqs. (5.2) and (5.6) into eq. (5.5), theevaporated energy Wh per period is given byI
Wh ¼  dZ 2=!
¼!0 0 sin !t cosð!t  Þ dt
0

ð5:7Þ

As shown by eq. (5.4), phase delay  is a function of !, and becomes larger as ! increases. Thus, the evaporated
energy per period is proportional to the operation frequency!. In this section, the diﬀerence in heat generation is
discussed on the basis of the hysteresis property of thestress–strain relationship.
First, the heat generation at the tissue surface is discussed.
The stress–strain relationship at diﬀerent frequencies is
illustrated in Fig. 19. According to eq. (5.7), the area of thestress–strain relationship in Fig. 19 depends on stress ,strain , and phase delay . The area becomes larger withhigher frequency because phase delay  becomes larger withhigher frequency, as shown in eq. (5.4). Moreover, thecoagulation of the tissue surface per unit time depends on thevibration frequency because the vibration frequency determines the rotation number of the stress–strain diagram perunit time. As shown in Fig. 10, the coagulated depths are
similar for various frequencies when the vibration amplitude
is constant. However, even in the case of the same vibration
velocity, the vibration amplitude at a lower frequency
becomes larger than that at a higher frequency. Thus, thecoagulated depth becomes larger with lower frequency, as
shown in Fig. 7(b).
Next, the heat generation in the case of applying pressureload is discussed. Figure 20(a) illustrates the nonlinear
stress–strain relationship of tissue. Young’s modulus is thegradient of the stress–strain curve shown in Fig. 20(a).
When pressure load is applied, the tissue is compressed andbecomes hard. That is, Young’s modulus E is increased dueto the pressure load and the stress–strain relationship shifts
towards the right-hand side, as shown in Fig. 20(b). Thus,
the heat generation due to the expansion and contraction oftissue is increased due to the increase in the area of thestress–strain relationship.
Finally, the diﬀerence in heat generation depending on thevibration mode is discussed. Viscoelasticity in longitudinal
vibration is determined by Young’s modulus E and viscosity
coeﬃcient . On the other hand, it is determined by shear
modulus G and shear viscosity coeﬃcient  in shear
vibration. Young’s modulus is equivalent to three times

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Jpn. J. Appl. Phys., Vol. 46, No. 7B (2007)

K. E BINA et al.

stress σ(a)with pressure(b)stress σ

σP

b.

σP

without pressure
O

a.
γP

strain γOγPstrain γ

2γ 0
2γ 0

Fig. 20. Illustrations of (a) change in elasticity due to the static pressure and (b) change in hysteresis property due to static pressure.

pressure load, contact of the tip with tissue, and vibration
mode was investigated. These results were discussed interms of heat generation caused by the stress–strain relationship of soft tissue. The results will be eﬀective in theconsideration of frequency characteristics in the cutting ofsoft tissue using an ultrasonic surgical knife, and also usefulfor the identiﬁcation of the optimum operation frequency.

stress σ

longitudinal vibration

shear vibrationOstrain γ

Fig. 21. Illustration of stress–strain diagrams for diﬀerent vibration
modes.

the shear modulus (E ¼ 3G); the same is true for theviscosity coeﬃcient ( ¼ 3).22) Figure 21 shows the stress–
strain relationship for these two vibration modes when thesame amplitude of strain is generated. In Fig. 21, the heatgeneration caused by shear vibration is smaller than thatcaused by longitudinal vibration, which corresponds to thediﬀerence in Fig. 16.
In this section, we qualitatively discussed heat generation
using the stress–strain relationship, because the physicalconstants or stress used for the calculation of heat generation
was unknown. A quantitative discussion of heat generation
still remains for future work.
6.

Conclusions

In this paper, we focused on the vibration frequency of an
ultrasonic surgical knife. Frequency characteristics in the
cutting of soft tissue were investigated using prototype
knives operated at 24.2, 44.8, and 71.4 kHz. The coagulation
ratio and coagulated depth were diﬀerent for each knife. At alower frequency, the region of the coagulated tissue wasdeeper compared with at a higher frequency. At the sametime, when the operation frequency was higher, coagulated
tissue was distributed in the region closer to the surface
compared with that at a lower frequency. The coagulation oftissue along the depth direction depends on the vibration
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