A Resonance Frequency Technique to Determine Elastic Modulus of Hydroxyapatite ANIL M. TORGALKAR, Johnson & Johnson Research Center, North Brunswick, New Jersey 08903
Summary A resonance frequency technique was applied to determine the elastic modulus of hydroxyapatite. A Free-Free Vibration Transducer was designed to determine elastic modulus in a longitudinal direction. A Fixed-Free Vibration Transducer was also designed to study elastic modulus of materials where specimens longer than 3 cm in length were not available. Six lots of hydroxyapatite were prepared utilizing the same process. The elastic modulus of hydroxyapatite varied between 3.94 X 1Olo (dyn/cm2) and 6.30 X 1O’O (dyn/cm2) in a longitudinal direction. For the same six lots, it varied between 1.95 X 101O (dyn/cmz) and 3.20 X 1O1O (dyn/cm*) in a cross direction. The elastic modulus values of cortical bone from dog tibias, fibulas, and femurs were also determined.
INTRODUCTION The method of obtaining elastic moduli of solid materials from their mechanical resonance frequencies originated almost 40 years ag0.l Investigators2 have improved and refined the basic method so that resonance frequencies can be determined with a high degree of accuracy. This improvement in experimental technique has been accompanied by a corresponding refinement in theoretical relations3 for calculation of the various elastic moduli from the resonance frequencies. The mechanical properties of bone have been studied extensively during the last two decades by destructive test methods. A summary of these results is available in the A nondestructive test method of determining elastic modulus in standardized specimens of human bone by transmitted ultrasound has been de~cribed.~-~ The concept is based on the measurement of the velocity of ultrasound transmitted longitudinally through the bone. The velocity, in turn, is a function of the modulus of elasticity and density of bone. Journal of Biomedical Materials Research, Vol. 13,907-920 (1979) 0021-9304/79/0013-0907$01.00 01979 John Wiley & Sons, Inc.
908
TORGALKAR
There is very little published information, however, concerning the application of similar techniques for determination of the elastic properties of biomaterials. This paper describes a nondestructive test method based on the principle of resonance frequency measurement. It also describes the designs of the transducers built to determine the elastic modulus of hydroxyapatite and similar materials. The method employed here is direct and simple. It also overcomes some of the difficulties associated with destructive test methods.
SPECIMEN PREPARATION Hydroxyapatite Specimen Hydroxyapatite blocks were prepared according to a procedure discussed in the 1iterature.lO These blocks were 9 to 12 in. in length, 2 in. wide and 2 in. thick. To determine the elastic modulus in a longitudinal direction, rectangular bar specimens were cut along the length of the block. A slitting saw was used to cut the specimens. The rectangular bar specimens were approximately 4l/2 in. long, l/2 in. wide and 1'8 in, thick. These specimens were then polished using a 240-grit polishing paper. Specimens about 3 cm long, 0.5 cm wide and 0.2 cm thick were also cut from the blocks of hydroxyapatite. These specimens were cut in a longitudinal direction and cross direction. The specimens were polished using a 240-grit polishing paper.
Bone Specimen Cortical bone specimen from tibias, femurs, and fibulas of a seven and a half year old dog were prepared. The bones were cut along the long axis. The length of the specimens varied between 3 cm and 6 cm. The specimens were polished on the sand paper and stored in saline solution.
EXPERIMENTAL METHOD An acoustic ~pectrometerll-'~ was used to measure the resonance frequencies of hydroxyapatite specimens. Frequency data obtained were used to calculate the complex elastic modulus of hydroxyapatite a t room temperature (24°C).
ELASTIC MODULUS OF HYDROXYAPATITE
909
Fig. 1. Top and side view of Free-Free Vibration Transducer: A,B-Vernier screws; C,D-electromagnetic coils; E,F-specimen support blocks; G,H-steel wires; I-specimen; J-vertical support block; K-platform; L,M-transducer cables; N,P-metal blocks supporting electromagnetic coils; R,S-ferromagnetic material pieces attached to the specimen; T,U-metal rods.
Apparatus-Free-Free Vibration Transducer* The Free-Free Vibration Transducer was designed to determine the resonance frequency of rectangular hydroxyapatite specimens. A diagrammatic sketch of the transducer is shown in Figure 1. A specimen setup is shown in a side view of the transducer. No specimen has been shown, however, in the top view in order to show the details of the design. The transducer was constructed of stainless steel. The transducer coils C and D were screwed in the blocks N and P, respectively. It enables the transducers to move inward or outward in order to position them precisely under the specimen. This design also permits study of specimens from 8-16 cm in length and no more than 0.7 cm in width or thickness for flexural mode of resonance. The support and coil distances from one to the other are independently
* Nemetre Co., Edison, N.J., U.S.A.
910
TORGALKAR
adjustable by the Vernier screws A and B along the length of the specimen. These adjustments can be made simultaneously. The specimen is supported on the steel wires G and H glued over blocks E and F, so that it can resonate freely. It was used to determine the elastic modulus of specimens that were cut in a longitudinal direction.
Fixed-Free Vibration Transducer This apparatus was designed to determine the elastic modulus of specimens that were cut in a cross direction. This was necessary because the hydroxyapatite blocks were only a few centimeters wide. It was not possible, therefore, to cut specimens in the cross direction longer than a few centimeters. The Fixed-Free Vibration Transducer was capable of measuring the elastic modulus of specimens that were 2-3 cm in length, 0.4-0.5 cm in width and 0.2 cm in thickness. A diagrammatic sketch of the Fixed-Free Vibration Transducer is shown in Figure 2. A piece of ferromagnetic material is glued to one end of the specimen, and it is then attached to a specimen holder
Fig. 2. Fixed-Free Vibration Transducer, side view: A-driver; B-pickup; C-specimen holder; D-specimen; E-ferromagnetic metal piece; F,G-movable metal blocks; H-stand; I-knob; J-Vernier screw; K-platform; L,M-transducer cables.
ELASTIC MODULUS OF HYDROXYAPATITE
911
as shown in Figure 2. The transducer is then connected to the acoustic spectrometer. A driver metal block, G, is then moved downward and a pickup block, F, is moved upward so that a metal piece attached to the specimen is close to an electromagnetic pickup coil. The various frequencies are then passed through the specimen. The frequency of vibration of the specimen at the free end is picked up by an electromagnetic coil and fed to a frequency counter of the acoustic spectrometer. The resonance frequency of the specimen was determined by passing various frequencies through the specimen. The procedure followed to determine resonance frequency of the specimen is discussed in the next section.
Acoustic Spectrometer An acoustic spectrometer was used to determine the resonance frequency of the specimens. A diagrammatic sketch of an acoustic spectrometer and Free-Free Vibration Transducer is shown in Figure 3. Two electromagnetic coils were used. One of these coils is used as a driver, while the other is used as a pickup. The purpose was to excite and detect the resonance. The specimen was set for testing in such a way that it was able to resonate freely. This was achieved by placing it on two thin wires as shown in Figure 3. The points of support of the specimen were calculated from the length of the specimen. It was supported to obtain fundamental resonance frequency (see Fig. 4). The output of the audio oscillator is amplified and fed to the driver, the mechanical energy of which in turn is transmitted to the specimen.
I' b I L
I
T
Fig. 3. A diagrammatic sketch of an acoustic spectrometer: 1-specimen; 2ferromagnetic metal piece; 3-electromagnetic coil, driver; 4-Electromagnetic coil, pickup; 5-support blocks; 6-Lissajou pattern on cathode ray oscilloscope (C.R.O.); V.P.-vertical plate; H.P.-horizontal plate.
TORGALKAR
912
Fig. 4. Specimen setup for first mode of flexural resonance: specimen support points from the ends of the specimen = L (cm) X 0.224, where E is elastic modulus (dyn/cm2); p is density (gm/cm2);L is length of the specimen (cm); t is thickness of A is support point; B is ferromagnetic the specimen (cm); f is resonance frequency (Hz); metallic piece; and W is width of the specimen (cm).
At the same time that the driver is energized, the oscillator output also goes to the horizontal plates of the oscilloscope. When the oscillator is not close to the resonance frequency of the specimen, a horizontal line is seen on the oscilloscope. (This is the electron beam moving back and forth at the frequency of the oscillator.) As the oscillator frequency is scanned, it eventually reaches one of the mechanical resonance frequencies of the specimen. The predominant characteristic of the specimen in regard to resonance is a large increase in the amplitude of its vibrations. The increased amplitude cletected by the pickup in contact with the specimen (but not quite touching) is then amplified and fed to the vertical plates of the oscilloscope. If the same frequency is received by horizontal and vertical plates of the oscilloscope a t the same time, a Lissajou pattern (as shown in Fig. 5 ) occurs. An elliptical loop A is observed on the oscilloscope when the specimen is vibrating at its fundamental resonance frequency. The Lissajou pattern has its maximum vertical amplitude at the resonance frequency. This freC
A
B
Fig. 5. Lissajou patterns observed a t various resonance frequencies: A-Lissajou pattern observed when the specimen is vibrating at its fundamental resonance; R,C-Lissajou pattern observed when the specimen is vibrating slightly above its fundamental resonance; D,E-Lissajou pattern observed when the specimen is not vibrating.
ELASTIC MODULUS OF HYDROXYAPATITE
913
quency is counted accurately by the frequency counter which also is connected to the oscillator. The experiments are carried out in a temperature- and humidity-controlled room a t 24OC.
CALCULATIONS Elastic Modulus Determination The elastic modulus was determined using the following equations: For Free-Free Vibration Transducer:
E (dyn/cm2) = 0.94642 X p
x
( f Px
For Fixed-Free Vibration Transducer:
E (dyn/cm2) = 38.34 X p
x
(f)2
x (L)*
(t)2
where p = density (g/cm3); f = frequency, H,; L = length of the specimen (cm); t = thickness of the specimen (cm); E = elastic modulus (dyn/cm2).
Density Determination The density of the specimen was determined from the weight of the specimen and its volume. density (g/cm3) =
(weight in g) (volume in cm3)
Determination of the Specimen Support Points for Flexural Resonance Measure the length of the specimen. Then, to determine the specimen support points for the fundamental resonance frequency, multiply the length of the specimen in cm by 0.224. The specimen should be supported at this distance from its ends. To obtain the first overtone frequency, multiply the length of the specimen by 0.132 to determine the specimen support points. (Theoretical node points tend to shift toward the ends of the specimen when the specimen is
TORGALKAR
914
nonmagnetic, and small ferromagnetic pieces are glued to each end of the specimen.)
RESULTS About six lots of hydroxyapatite were prepared using the same process. At least five specimens of each lot were prepared for testing. The average densities calculated for each lot are given in Table I. These densities were used to calculate the elastic moduli of the specimens. The average elastic modulus values obtained for each lot are given in Table 11. These values were obtained for hydroxyapatite in the longitudinal direction. The elastic modulus values obtained in a longitudinal and a cross direction using the Fixed-Free Vibration Transducer are presented in Table 111. The longitudinal modulus values obtained for hydroxyapatite using the Fixed-Free Vibration TABLE I The Average Densities of Various Lots of Hydroxyapatitea
a
Lot No.
Avg. Density (g/cm3)
1 2 3 4 5 6 7
0.977 0.965 1.064 1.13 1.01 1.05 1.03
Results based on five specimens. TABLE I1 Average Elastic Modulus of Hydroxyapatite in the Longitudinal Directiona Lot No.
Avg. Elastic Modulus (dyn/cm2) X 5.12 3.94 4.40 6.23 4.81 6.30
a The Free-Free Vibration Transducer was used to determine modulus values of hydroxyapatite specimens.
ELASTIC MODULUS OF HYDROXYAPATITE
915
TABLE 111 Average Elastic Modulus of Hydroxyapatite in Longitudinal and Cross Directionsa Elastic Modulus (dyn/cm2) X 10-lo Longitudinal Direction Cross Direction
Lot No.
4.69 3.59 4.24 5.82 4.82 5.86
2.28 1.95 2.49 2.89 3.01 3.20
a These values were determined using the Fixed-Free Vibration Transducer. A t least five specimens were tested.
Transducer were slightly lower than similar modulus values obtained using the Free-Free Vibration Transducer. The average elastic modulus values obtained for fresh cortical bone from dog femurs, tibias and fibulas are given in Table IV. The elastic modulus values determined for various dental composite materials, enamel and dentin, are given in Table V. Some of this data were previously reported in the 1 i t e r a t ~ r e . lThe ~ elastic modulus values determined for various biomaterials and composite material$.by researchers using both destructive and nondestructive test methods are given in Table VI.
DISCUSSIONS It was observed from Table I1 that lot No. 2 had the lowest average elastic modulus, while lot No. 6 had the highest elastic modulus. All these lots were prepared by the same process. The variation in the TABLE IV Average Elastic Modulus of Fresh Cortical Bone from Dog Femurs, Tibias and Fibulas Cortical Bone
Elastic Modulusa (dynes/cm2) X lo-"
Fibula Tibia Femur
1.175 f 0.15 1.46 f 0.20 1.29 f 0.10 ~
~~
~~~
These values were determined using the Fixed-Free Vibration Transducer. At least five specimens were tested. a
916
TORGALKAR
TABLE V Average Elastic Modulus of Some Other Materials Determined Using Free-Free and Fixed-Free Vibration Transducers Free-Free Vibration Transducer (ref. 21) Elastic Modulus Materiala (dyn/cm2) X A 1.95 B 1.57 C 1.62 D 3.72 Fixed-Free Vibration Transducer Enamel 6.8 Dentin 2.3 a A-Adapticm Dental Restorative, Johnson & Johnson, New Brunswick, N.J.; B-Concise, 3M Company, St. Paul, Minn.; C-Blendant, Kerr Mfg. Co., Romulus, Mich.; D-Aristoloy (Amalgam), Englehard Minerals and Chemical Corporation, Carteret, N.J.
elastic modulus values observed for different lots of hydroxyapatite is understandable since its mechanical properties depend upon various factors, such as porosity,14 pore size,15 the average grain size,16J7 and calcium/phosphorus ratio. 18~19 The elastic modulus values of hydroxyapatite determined by the Fixed-Free Vibration Transducer are given in Table 111. These values were slightly lower than similar modulus values obtained using the Free-Free Vibration Transducer. This variation observed in the modulus values obtained by two different transducers might be related to the specimen size, porosity, and pore size distribution.20 The elastic modulus values of hydroxyapatite determined in cross direction are also listed in Table I11 for the various lots. It was found that the elastic modulus in the longitudinal direction was about twice as high as its values in the cross direction for hydroxyapatite. It was not possible to determine, however, the elastic modulus of hydroxyapatite in cross direction using the Free-Free Vibration Transducer because only short-length specimens (2-3 cm long) were available. It was observed from Table IV that fresh tibia had the highest modulus values, while fibula had the lowest. These specimens were cut along the longitudinal axis of the bone. The elastic modulus values of three different commercial dental composite materials and one commercial dental amalgam were determined using the Free-Free Vibration Transducer. Rectangular bar specimens of these materials were prepared according to manu-
ELASTIC MODULUS OF HYDROXYAPATITE
917
TABLE VI Elastic Modulus of Various Materials Determined By Other Researchers Using Different Destructive and Nondestructive Test Methods Test Method (1)Ultrasonic interference
technique
(2) Pulse technique
(3)Destructive test method
Material Hydroxyapatite (ref. 21) (mineral) Hydroxyapatite (ref. 21) (synthetic) Dentin (ref. 22) Enamel (ref. 22) Phalanx (ref. 23) (animal bone) Axial (fresh) Axial (dry) Femur (ref. 23) (animal bone) Axial (dry) Dental Amalgam (ref. 24) Radiopaque composite (ref. 25) (Powder liquid ratio 1.1) Human Cortical Bone (ref. 26) Femur Tibia Fibula
Elastic Modulus (dyn/cm2) X lo-" 11.40 11.70 2.10 7.40
2.2 3.05 2.6 6.28 1.72
2.46 3.50 3.20
facturer's instructions. The elastic modulus values of dental composite materials varied between 1.57 X loll dyn/cm2 for material B and 1.95 X lo1' dyn/cm2 for material A (see Table V). An elastic modulus of 3.72 X lo1' dyn/cm2was obtained for dental amalgam. All these specimens were 2 weeks old and stored at 37OC in water. An elastic modulus of 6.8 X 10" dyn/cm2 was obtained for enamel and 2.3 X lo1' dyn/cm2 for dentin using the Fixed-Free Vibration Transducer. The elastic modulus values of dentin, enamel, and composite materials determined using the Fixed-Free Vibration Transducer and Free-Free Vibration Transducer were comparable to the modulus values for the same materials determined by other methods (see Table VI). The modulus values of hydroxyapatite determined using both Fixed-Free Vibration Transducer and Free-Free Vibration Transducer were, however, much lower than their modulus values determined using ultrasonic interference or pulse technique. A possible
918
TORGALKAR
explanation can be given as follows: In a resonance technique using either Fixed-Free Vibration Transducer or Free-Free Vibration Transducer, the specimens are prepared without altering the original physical structure of the material. Therefore, the elastic modulus values determined are the true mechanical property of the material. Determination of the mechanical properties using ultrasonic interference technique necessitates breaking of true physical structure of hydroxyapatite.21 The difference in the modulus values obtained by resonance technique and other nondestructive or destructive test methods will be minimum if the physical structure of the testing material is less porous, highly dense, and homogeneous.
CONCLUSIONS
A resonance frequency technique can be used successfully to determine the mechanical properties of hydroxyapatite and similar materials. The technique is nondestructive. It is possible, therefore, to repeat the experiments for selected specimens to confirm the results. This technique eliminates instrumental errors introduced by conventional destructive test methods. Since the technique is nondestructive, it is possible to observe changes in the mechanical properties of the material under various conditions without destroying the original physical structure of the specimen. Specimen preparation is simple, and the testing procedure is straightforward and quick. The method is also extremely useful for testing brittle materials having low flexibility, where conventional destructive test methods can be applied only with great difficulty. The elastic modulus of hydroxyapatite varied between 3.94 X 1O1O (dyn/cm2) and 6.30 X 1O1O (dyn/cm2) in the longitudinal direction. For the same six lots, elastic modulus varied between 1.95 X 1O'O (dyn/cm2) and 3.20 X 1O1O(dyn/cm2) in the cross direction. A fresh cortical bone from dog tibias, fibulas, and femurs had elastic modulus values of (1.46 f 0.02) X loll dyn/cm2, (1.175 f 0.15) X loll dyn/cm2, and (1.29 f 0.1) X lo1' dyn/cm2, respectively. The author would like to acknowledge the assistance of Mr. Merwin Krienitz of Central Analytical Laboratories, Johnson & Johnson Products, for preparing the specimens using the procedure described in the paper.
ELASTIC MODULUS OF HYDROXYAPATITE
919
References 1. J. M. Ide, “Some Dynamic Methods of Determination of Young Modulus,”Reu. Sci. Instrum., 6, 296-298 (1935). 2. F. Forster, “New Method for Determination of Modulus of Elasticity and Damping,” 2. Metallkde., 29,109-115 (1937). 3. S. Spinner and W. E. Tefft, “A Method for Determining Mechanical Resonance
Frequencies and for Calculating Elastic Moduli From These Frequencies,” Proc. ASTM, 61,1221-1238 (1961). 4. F. G. Evans, “Mechanical Properties of Bone,” American Lecture Series, Charles C Thomas, Springfield, Illinois, U.S.A., 1973. 5. B. S. Mather, “Correlation Between Strength and Other Properties of Long Bones,” J . Trauma, 7, 633-638 (1967). 6. E. D. Sedlin, “A Rheological Method for Cortical Bone,” Acta. Orthop. Scand., Suppl., 83,l-77 (1965). 7. W. Abendschein and G. W. Hyatt, “Ultrasonics and Selected Physical Properties of Bone,” Clin. Orthop. Relat. Res., 69,294-301 (1970). 8. L. P. Floriani et al., Surg. Forum, 18,468-470 (1967). 9. B. Martin and J. H. McElhaney, “The Acoustic Properties of Human Skull Bone,” J. Biomed. Mater. Res., 5, 325-333 (1971). 10. D. M. Roy, U.S. Patent 3,929,971, Dec. 30, 1975. 11. J. V. Fitzgerald, “Acoustic Spectrometry of Adhesives,” Adhes. Age, 6, 36-38 (1963). 12. F. J. Matusik, “Structure Composition and Bond Strength,” Ceram. Age, 80,4650 (1964). 13. A. M. Torgalkar, “Comparison of Rheological Properties of Four Dental Restorative Materials,” J.Dent. Res., 52(3), 476-482 (1973). 14. R. W. Rao and R. A. Boehm, “A Study of Sintered Apatites,”J. Den. Res., 53(6), 1351-1354 (1974). 15. J. G. J. Peelen, B. V. Rejda, and J. P. W. Vermeiden, “Sintered Hydroxylapatite as a Bioceramic,” Philips Tech. Reu., 9/10(37), 234-235 (1977). 16. M. Jarcho, C. H. Bolen, M. B. Thomas, J. Bobik, J. F. Kay, and R. A. Doremus, “Hydroxylapatite Synthesis and Characterization in Dense Polycrystalline Form,” J.Mater. Sci., 11,2027-2035 (1976). 17. M. Jarcho, J. R. O’Connor, and D. A. Paris, “Ceramic Hydroxy-apatite as a Plaque Growth and Drug Screening Substrate,” J.Dent. Res., 56(2), 151-156 (1977). 18. E. A. Monroe, Votaya, D. B. Ward, and J. C. McMullen, “New Calcium Phosphate Ceramic Material for Bone and Tooth Implants,” J. Dent. Res., 50, 860-861 (1971). 19. M. Jarcho, “Hydroxylapatite Ceramic,” Dutch Patent, DN 6304A. 20. A. M. Torgalkar, “Tensile Strength Distribution Study of a Dental Restorative Material,” J. Dent. Res., 52(3), 483-486 (1973). 21. D. E. Grenoble, “The Elastic Properties of Hard Tissues and Apatites,” J. Biomed. Res., 6(3), 221-233 (1972). 22. R. S. Gilmore, R. P. Pollack, and J. L. Katz, “Elastic Properties of Bovine Dentin and Enamel,” Arch. Oral Biol., 15,787-796 (1970). 23. S. B. Lang, “Elastic Coefficients of Animal Bones,” Science, 165, 287 (July 1969).
920
TORGALKAR
24. G . Dickson and P. L. Oglesby, “Elastic Constants of Dental Amalgam,” J. Dent.
Res., 46, 1475 (Nov.-Dec. 1967). 25. J. A. Barton, Jr., C. L. Burns, H. H. Chandler, and R. L. Bowen, “An Experimental Radiopaque Composite Material,” J. Dent. Res., 52(4), 731-739 (1973). 26. F. Gaynor Evans, “Mechanical Properties of Bone,” Charles C. Thomas, Springfield, Illinois, 1973, p. 164, Table XLI.
Received April 6,1979 Revised June 15,1979
Summary A resonance frequency technique was applied to determine the elastic modulus of hydroxyapatite. A Free-Free Vibration Transducer was designed to determine elastic modulus in a longitudinal direction. A Fixed-Free Vibration Transducer was also designed to study elastic modulus of materials where specimens longer than 3 cm in length were not available. Six lots of hydroxyapatite were prepared utilizing the same process. The elastic modulus of hydroxyapatite varied between 3.94 X 1Olo (dyn/cm2) and 6.30 X 1O’O (dyn/cm2) in a longitudinal direction. For the same six lots, it varied between 1.95 X 101O (dyn/cmz) and 3.20 X 1O1O (dyn/cm*) in a cross direction. The elastic modulus values of cortical bone from dog tibias, fibulas, and femurs were also determined.
INTRODUCTION The method of obtaining elastic moduli of solid materials from their mechanical resonance frequencies originated almost 40 years ag0.l Investigators2 have improved and refined the basic method so that resonance frequencies can be determined with a high degree of accuracy. This improvement in experimental technique has been accompanied by a corresponding refinement in theoretical relations3 for calculation of the various elastic moduli from the resonance frequencies. The mechanical properties of bone have been studied extensively during the last two decades by destructive test methods. A summary of these results is available in the A nondestructive test method of determining elastic modulus in standardized specimens of human bone by transmitted ultrasound has been de~cribed.~-~ The concept is based on the measurement of the velocity of ultrasound transmitted longitudinally through the bone. The velocity, in turn, is a function of the modulus of elasticity and density of bone. Journal of Biomedical Materials Research, Vol. 13,907-920 (1979) 0021-9304/79/0013-0907$01.00 01979 John Wiley & Sons, Inc.
908
TORGALKAR
There is very little published information, however, concerning the application of similar techniques for determination of the elastic properties of biomaterials. This paper describes a nondestructive test method based on the principle of resonance frequency measurement. It also describes the designs of the transducers built to determine the elastic modulus of hydroxyapatite and similar materials. The method employed here is direct and simple. It also overcomes some of the difficulties associated with destructive test methods.
SPECIMEN PREPARATION Hydroxyapatite Specimen Hydroxyapatite blocks were prepared according to a procedure discussed in the 1iterature.lO These blocks were 9 to 12 in. in length, 2 in. wide and 2 in. thick. To determine the elastic modulus in a longitudinal direction, rectangular bar specimens were cut along the length of the block. A slitting saw was used to cut the specimens. The rectangular bar specimens were approximately 4l/2 in. long, l/2 in. wide and 1'8 in, thick. These specimens were then polished using a 240-grit polishing paper. Specimens about 3 cm long, 0.5 cm wide and 0.2 cm thick were also cut from the blocks of hydroxyapatite. These specimens were cut in a longitudinal direction and cross direction. The specimens were polished using a 240-grit polishing paper.
Bone Specimen Cortical bone specimen from tibias, femurs, and fibulas of a seven and a half year old dog were prepared. The bones were cut along the long axis. The length of the specimens varied between 3 cm and 6 cm. The specimens were polished on the sand paper and stored in saline solution.
EXPERIMENTAL METHOD An acoustic ~pectrometerll-'~ was used to measure the resonance frequencies of hydroxyapatite specimens. Frequency data obtained were used to calculate the complex elastic modulus of hydroxyapatite a t room temperature (24°C).
ELASTIC MODULUS OF HYDROXYAPATITE
909
Fig. 1. Top and side view of Free-Free Vibration Transducer: A,B-Vernier screws; C,D-electromagnetic coils; E,F-specimen support blocks; G,H-steel wires; I-specimen; J-vertical support block; K-platform; L,M-transducer cables; N,P-metal blocks supporting electromagnetic coils; R,S-ferromagnetic material pieces attached to the specimen; T,U-metal rods.
Apparatus-Free-Free Vibration Transducer* The Free-Free Vibration Transducer was designed to determine the resonance frequency of rectangular hydroxyapatite specimens. A diagrammatic sketch of the transducer is shown in Figure 1. A specimen setup is shown in a side view of the transducer. No specimen has been shown, however, in the top view in order to show the details of the design. The transducer was constructed of stainless steel. The transducer coils C and D were screwed in the blocks N and P, respectively. It enables the transducers to move inward or outward in order to position them precisely under the specimen. This design also permits study of specimens from 8-16 cm in length and no more than 0.7 cm in width or thickness for flexural mode of resonance. The support and coil distances from one to the other are independently
* Nemetre Co., Edison, N.J., U.S.A.
910
TORGALKAR
adjustable by the Vernier screws A and B along the length of the specimen. These adjustments can be made simultaneously. The specimen is supported on the steel wires G and H glued over blocks E and F, so that it can resonate freely. It was used to determine the elastic modulus of specimens that were cut in a longitudinal direction.
Fixed-Free Vibration Transducer This apparatus was designed to determine the elastic modulus of specimens that were cut in a cross direction. This was necessary because the hydroxyapatite blocks were only a few centimeters wide. It was not possible, therefore, to cut specimens in the cross direction longer than a few centimeters. The Fixed-Free Vibration Transducer was capable of measuring the elastic modulus of specimens that were 2-3 cm in length, 0.4-0.5 cm in width and 0.2 cm in thickness. A diagrammatic sketch of the Fixed-Free Vibration Transducer is shown in Figure 2. A piece of ferromagnetic material is glued to one end of the specimen, and it is then attached to a specimen holder
Fig. 2. Fixed-Free Vibration Transducer, side view: A-driver; B-pickup; C-specimen holder; D-specimen; E-ferromagnetic metal piece; F,G-movable metal blocks; H-stand; I-knob; J-Vernier screw; K-platform; L,M-transducer cables.
ELASTIC MODULUS OF HYDROXYAPATITE
911
as shown in Figure 2. The transducer is then connected to the acoustic spectrometer. A driver metal block, G, is then moved downward and a pickup block, F, is moved upward so that a metal piece attached to the specimen is close to an electromagnetic pickup coil. The various frequencies are then passed through the specimen. The frequency of vibration of the specimen at the free end is picked up by an electromagnetic coil and fed to a frequency counter of the acoustic spectrometer. The resonance frequency of the specimen was determined by passing various frequencies through the specimen. The procedure followed to determine resonance frequency of the specimen is discussed in the next section.
Acoustic Spectrometer An acoustic spectrometer was used to determine the resonance frequency of the specimens. A diagrammatic sketch of an acoustic spectrometer and Free-Free Vibration Transducer is shown in Figure 3. Two electromagnetic coils were used. One of these coils is used as a driver, while the other is used as a pickup. The purpose was to excite and detect the resonance. The specimen was set for testing in such a way that it was able to resonate freely. This was achieved by placing it on two thin wires as shown in Figure 3. The points of support of the specimen were calculated from the length of the specimen. It was supported to obtain fundamental resonance frequency (see Fig. 4). The output of the audio oscillator is amplified and fed to the driver, the mechanical energy of which in turn is transmitted to the specimen.
I' b I L
I
T
Fig. 3. A diagrammatic sketch of an acoustic spectrometer: 1-specimen; 2ferromagnetic metal piece; 3-electromagnetic coil, driver; 4-Electromagnetic coil, pickup; 5-support blocks; 6-Lissajou pattern on cathode ray oscilloscope (C.R.O.); V.P.-vertical plate; H.P.-horizontal plate.
TORGALKAR
912
Fig. 4. Specimen setup for first mode of flexural resonance: specimen support points from the ends of the specimen = L (cm) X 0.224, where E is elastic modulus (dyn/cm2); p is density (gm/cm2);L is length of the specimen (cm); t is thickness of A is support point; B is ferromagnetic the specimen (cm); f is resonance frequency (Hz); metallic piece; and W is width of the specimen (cm).
At the same time that the driver is energized, the oscillator output also goes to the horizontal plates of the oscilloscope. When the oscillator is not close to the resonance frequency of the specimen, a horizontal line is seen on the oscilloscope. (This is the electron beam moving back and forth at the frequency of the oscillator.) As the oscillator frequency is scanned, it eventually reaches one of the mechanical resonance frequencies of the specimen. The predominant characteristic of the specimen in regard to resonance is a large increase in the amplitude of its vibrations. The increased amplitude cletected by the pickup in contact with the specimen (but not quite touching) is then amplified and fed to the vertical plates of the oscilloscope. If the same frequency is received by horizontal and vertical plates of the oscilloscope a t the same time, a Lissajou pattern (as shown in Fig. 5 ) occurs. An elliptical loop A is observed on the oscilloscope when the specimen is vibrating at its fundamental resonance frequency. The Lissajou pattern has its maximum vertical amplitude at the resonance frequency. This freC
A
B
Fig. 5. Lissajou patterns observed a t various resonance frequencies: A-Lissajou pattern observed when the specimen is vibrating at its fundamental resonance; R,C-Lissajou pattern observed when the specimen is vibrating slightly above its fundamental resonance; D,E-Lissajou pattern observed when the specimen is not vibrating.
ELASTIC MODULUS OF HYDROXYAPATITE
913
quency is counted accurately by the frequency counter which also is connected to the oscillator. The experiments are carried out in a temperature- and humidity-controlled room a t 24OC.
CALCULATIONS Elastic Modulus Determination The elastic modulus was determined using the following equations: For Free-Free Vibration Transducer:
E (dyn/cm2) = 0.94642 X p
x
( f Px
For Fixed-Free Vibration Transducer:
E (dyn/cm2) = 38.34 X p
x
(f)2
x (L)*
(t)2
where p = density (g/cm3); f = frequency, H,; L = length of the specimen (cm); t = thickness of the specimen (cm); E = elastic modulus (dyn/cm2).
Density Determination The density of the specimen was determined from the weight of the specimen and its volume. density (g/cm3) =
(weight in g) (volume in cm3)
Determination of the Specimen Support Points for Flexural Resonance Measure the length of the specimen. Then, to determine the specimen support points for the fundamental resonance frequency, multiply the length of the specimen in cm by 0.224. The specimen should be supported at this distance from its ends. To obtain the first overtone frequency, multiply the length of the specimen by 0.132 to determine the specimen support points. (Theoretical node points tend to shift toward the ends of the specimen when the specimen is
TORGALKAR
914
nonmagnetic, and small ferromagnetic pieces are glued to each end of the specimen.)
RESULTS About six lots of hydroxyapatite were prepared using the same process. At least five specimens of each lot were prepared for testing. The average densities calculated for each lot are given in Table I. These densities were used to calculate the elastic moduli of the specimens. The average elastic modulus values obtained for each lot are given in Table 11. These values were obtained for hydroxyapatite in the longitudinal direction. The elastic modulus values obtained in a longitudinal and a cross direction using the Fixed-Free Vibration Transducer are presented in Table 111. The longitudinal modulus values obtained for hydroxyapatite using the Fixed-Free Vibration TABLE I The Average Densities of Various Lots of Hydroxyapatitea
a
Lot No.
Avg. Density (g/cm3)
1 2 3 4 5 6 7
0.977 0.965 1.064 1.13 1.01 1.05 1.03
Results based on five specimens. TABLE I1 Average Elastic Modulus of Hydroxyapatite in the Longitudinal Directiona Lot No.
Avg. Elastic Modulus (dyn/cm2) X 5.12 3.94 4.40 6.23 4.81 6.30
a The Free-Free Vibration Transducer was used to determine modulus values of hydroxyapatite specimens.
ELASTIC MODULUS OF HYDROXYAPATITE
915
TABLE 111 Average Elastic Modulus of Hydroxyapatite in Longitudinal and Cross Directionsa Elastic Modulus (dyn/cm2) X 10-lo Longitudinal Direction Cross Direction
Lot No.
4.69 3.59 4.24 5.82 4.82 5.86
2.28 1.95 2.49 2.89 3.01 3.20
a These values were determined using the Fixed-Free Vibration Transducer. A t least five specimens were tested.
Transducer were slightly lower than similar modulus values obtained using the Free-Free Vibration Transducer. The average elastic modulus values obtained for fresh cortical bone from dog femurs, tibias and fibulas are given in Table IV. The elastic modulus values determined for various dental composite materials, enamel and dentin, are given in Table V. Some of this data were previously reported in the 1 i t e r a t ~ r e . lThe ~ elastic modulus values determined for various biomaterials and composite material$.by researchers using both destructive and nondestructive test methods are given in Table VI.
DISCUSSIONS It was observed from Table I1 that lot No. 2 had the lowest average elastic modulus, while lot No. 6 had the highest elastic modulus. All these lots were prepared by the same process. The variation in the TABLE IV Average Elastic Modulus of Fresh Cortical Bone from Dog Femurs, Tibias and Fibulas Cortical Bone
Elastic Modulusa (dynes/cm2) X lo-"
Fibula Tibia Femur
1.175 f 0.15 1.46 f 0.20 1.29 f 0.10 ~
~~
~~~
These values were determined using the Fixed-Free Vibration Transducer. At least five specimens were tested. a
916
TORGALKAR
TABLE V Average Elastic Modulus of Some Other Materials Determined Using Free-Free and Fixed-Free Vibration Transducers Free-Free Vibration Transducer (ref. 21) Elastic Modulus Materiala (dyn/cm2) X A 1.95 B 1.57 C 1.62 D 3.72 Fixed-Free Vibration Transducer Enamel 6.8 Dentin 2.3 a A-Adapticm Dental Restorative, Johnson & Johnson, New Brunswick, N.J.; B-Concise, 3M Company, St. Paul, Minn.; C-Blendant, Kerr Mfg. Co., Romulus, Mich.; D-Aristoloy (Amalgam), Englehard Minerals and Chemical Corporation, Carteret, N.J.
elastic modulus values observed for different lots of hydroxyapatite is understandable since its mechanical properties depend upon various factors, such as porosity,14 pore size,15 the average grain size,16J7 and calcium/phosphorus ratio. 18~19 The elastic modulus values of hydroxyapatite determined by the Fixed-Free Vibration Transducer are given in Table 111. These values were slightly lower than similar modulus values obtained using the Free-Free Vibration Transducer. This variation observed in the modulus values obtained by two different transducers might be related to the specimen size, porosity, and pore size distribution.20 The elastic modulus values of hydroxyapatite determined in cross direction are also listed in Table I11 for the various lots. It was found that the elastic modulus in the longitudinal direction was about twice as high as its values in the cross direction for hydroxyapatite. It was not possible to determine, however, the elastic modulus of hydroxyapatite in cross direction using the Free-Free Vibration Transducer because only short-length specimens (2-3 cm long) were available. It was observed from Table IV that fresh tibia had the highest modulus values, while fibula had the lowest. These specimens were cut along the longitudinal axis of the bone. The elastic modulus values of three different commercial dental composite materials and one commercial dental amalgam were determined using the Free-Free Vibration Transducer. Rectangular bar specimens of these materials were prepared according to manu-
ELASTIC MODULUS OF HYDROXYAPATITE
917
TABLE VI Elastic Modulus of Various Materials Determined By Other Researchers Using Different Destructive and Nondestructive Test Methods Test Method (1)Ultrasonic interference
technique
(2) Pulse technique
(3)Destructive test method
Material Hydroxyapatite (ref. 21) (mineral) Hydroxyapatite (ref. 21) (synthetic) Dentin (ref. 22) Enamel (ref. 22) Phalanx (ref. 23) (animal bone) Axial (fresh) Axial (dry) Femur (ref. 23) (animal bone) Axial (dry) Dental Amalgam (ref. 24) Radiopaque composite (ref. 25) (Powder liquid ratio 1.1) Human Cortical Bone (ref. 26) Femur Tibia Fibula
Elastic Modulus (dyn/cm2) X lo-" 11.40 11.70 2.10 7.40
2.2 3.05 2.6 6.28 1.72
2.46 3.50 3.20
facturer's instructions. The elastic modulus values of dental composite materials varied between 1.57 X loll dyn/cm2 for material B and 1.95 X lo1' dyn/cm2 for material A (see Table V). An elastic modulus of 3.72 X lo1' dyn/cm2was obtained for dental amalgam. All these specimens were 2 weeks old and stored at 37OC in water. An elastic modulus of 6.8 X 10" dyn/cm2 was obtained for enamel and 2.3 X lo1' dyn/cm2 for dentin using the Fixed-Free Vibration Transducer. The elastic modulus values of dentin, enamel, and composite materials determined using the Fixed-Free Vibration Transducer and Free-Free Vibration Transducer were comparable to the modulus values for the same materials determined by other methods (see Table VI). The modulus values of hydroxyapatite determined using both Fixed-Free Vibration Transducer and Free-Free Vibration Transducer were, however, much lower than their modulus values determined using ultrasonic interference or pulse technique. A possible
918
TORGALKAR
explanation can be given as follows: In a resonance technique using either Fixed-Free Vibration Transducer or Free-Free Vibration Transducer, the specimens are prepared without altering the original physical structure of the material. Therefore, the elastic modulus values determined are the true mechanical property of the material. Determination of the mechanical properties using ultrasonic interference technique necessitates breaking of true physical structure of hydroxyapatite.21 The difference in the modulus values obtained by resonance technique and other nondestructive or destructive test methods will be minimum if the physical structure of the testing material is less porous, highly dense, and homogeneous.
CONCLUSIONS
A resonance frequency technique can be used successfully to determine the mechanical properties of hydroxyapatite and similar materials. The technique is nondestructive. It is possible, therefore, to repeat the experiments for selected specimens to confirm the results. This technique eliminates instrumental errors introduced by conventional destructive test methods. Since the technique is nondestructive, it is possible to observe changes in the mechanical properties of the material under various conditions without destroying the original physical structure of the specimen. Specimen preparation is simple, and the testing procedure is straightforward and quick. The method is also extremely useful for testing brittle materials having low flexibility, where conventional destructive test methods can be applied only with great difficulty. The elastic modulus of hydroxyapatite varied between 3.94 X 1O1O (dyn/cm2) and 6.30 X 1O1O (dyn/cm2) in the longitudinal direction. For the same six lots, elastic modulus varied between 1.95 X 1O'O (dyn/cm2) and 3.20 X 1O1O(dyn/cm2) in the cross direction. A fresh cortical bone from dog tibias, fibulas, and femurs had elastic modulus values of (1.46 f 0.02) X loll dyn/cm2, (1.175 f 0.15) X loll dyn/cm2, and (1.29 f 0.1) X lo1' dyn/cm2, respectively. The author would like to acknowledge the assistance of Mr. Merwin Krienitz of Central Analytical Laboratories, Johnson & Johnson Products, for preparing the specimens using the procedure described in the paper.
ELASTIC MODULUS OF HYDROXYAPATITE
919
References 1. J. M. Ide, “Some Dynamic Methods of Determination of Young Modulus,”Reu. Sci. Instrum., 6, 296-298 (1935). 2. F. Forster, “New Method for Determination of Modulus of Elasticity and Damping,” 2. Metallkde., 29,109-115 (1937). 3. S. Spinner and W. E. Tefft, “A Method for Determining Mechanical Resonance
Frequencies and for Calculating Elastic Moduli From These Frequencies,” Proc. ASTM, 61,1221-1238 (1961). 4. F. G. Evans, “Mechanical Properties of Bone,” American Lecture Series, Charles C Thomas, Springfield, Illinois, U.S.A., 1973. 5. B. S. Mather, “Correlation Between Strength and Other Properties of Long Bones,” J . Trauma, 7, 633-638 (1967). 6. E. D. Sedlin, “A Rheological Method for Cortical Bone,” Acta. Orthop. Scand., Suppl., 83,l-77 (1965). 7. W. Abendschein and G. W. Hyatt, “Ultrasonics and Selected Physical Properties of Bone,” Clin. Orthop. Relat. Res., 69,294-301 (1970). 8. L. P. Floriani et al., Surg. Forum, 18,468-470 (1967). 9. B. Martin and J. H. McElhaney, “The Acoustic Properties of Human Skull Bone,” J. Biomed. Mater. Res., 5, 325-333 (1971). 10. D. M. Roy, U.S. Patent 3,929,971, Dec. 30, 1975. 11. J. V. Fitzgerald, “Acoustic Spectrometry of Adhesives,” Adhes. Age, 6, 36-38 (1963). 12. F. J. Matusik, “Structure Composition and Bond Strength,” Ceram. Age, 80,4650 (1964). 13. A. M. Torgalkar, “Comparison of Rheological Properties of Four Dental Restorative Materials,” J.Dent. Res., 52(3), 476-482 (1973). 14. R. W. Rao and R. A. Boehm, “A Study of Sintered Apatites,”J. Den. Res., 53(6), 1351-1354 (1974). 15. J. G. J. Peelen, B. V. Rejda, and J. P. W. Vermeiden, “Sintered Hydroxylapatite as a Bioceramic,” Philips Tech. Reu., 9/10(37), 234-235 (1977). 16. M. Jarcho, C. H. Bolen, M. B. Thomas, J. Bobik, J. F. Kay, and R. A. Doremus, “Hydroxylapatite Synthesis and Characterization in Dense Polycrystalline Form,” J.Mater. Sci., 11,2027-2035 (1976). 17. M. Jarcho, J. R. O’Connor, and D. A. Paris, “Ceramic Hydroxy-apatite as a Plaque Growth and Drug Screening Substrate,” J.Dent. Res., 56(2), 151-156 (1977). 18. E. A. Monroe, Votaya, D. B. Ward, and J. C. McMullen, “New Calcium Phosphate Ceramic Material for Bone and Tooth Implants,” J. Dent. Res., 50, 860-861 (1971). 19. M. Jarcho, “Hydroxylapatite Ceramic,” Dutch Patent, DN 6304A. 20. A. M. Torgalkar, “Tensile Strength Distribution Study of a Dental Restorative Material,” J. Dent. Res., 52(3), 483-486 (1973). 21. D. E. Grenoble, “The Elastic Properties of Hard Tissues and Apatites,” J. Biomed. Res., 6(3), 221-233 (1972). 22. R. S. Gilmore, R. P. Pollack, and J. L. Katz, “Elastic Properties of Bovine Dentin and Enamel,” Arch. Oral Biol., 15,787-796 (1970). 23. S. B. Lang, “Elastic Coefficients of Animal Bones,” Science, 165, 287 (July 1969).
920
TORGALKAR
24. G . Dickson and P. L. Oglesby, “Elastic Constants of Dental Amalgam,” J. Dent.
Res., 46, 1475 (Nov.-Dec. 1967). 25. J. A. Barton, Jr., C. L. Burns, H. H. Chandler, and R. L. Bowen, “An Experimental Radiopaque Composite Material,” J. Dent. Res., 52(4), 731-739 (1973). 26. F. Gaynor Evans, “Mechanical Properties of Bone,” Charles C. Thomas, Springfield, Illinois, 1973, p. 164, Table XLI.
Received April 6,1979 Revised June 15,1979
