- Email: [email protected]
The effect of charge density on the velocity and attenuation of ultrasound waves in human cancellous bone
Journal of Biomechanics xxx (2018) xxx–xxx
Contents lists available at ScienceDirect
Journal of Biomechanics journal homepage: www.elsevier.com/locate/jbiomech www.JBiomech.com
The effect of charge density on the velocity and attenuation of ultrasound waves in human cancellous bone Young June Yoon ⇑ Center for Integrated General Education, Republic of Korea College of Engineering, Hanyang University, Republic of Korea
a r t i c l e
i n f o
Article history: Accepted 31 July 2018 Available online xxxx Keywords: Cancellous bone Poroelasticity Ultrasound Charge density
a b s t r a c t Cancellous bone is a highly porous material, and two types of waves, fast and slow, are observed when ultrasound is used for detecting bone diseases. There are several possible stimuli for bone remodelling processes, including bone fluid flow, streaming potential, and piezoelectricity. Poroelasticity has been widely used for elucidating the bone fluid flow phenomenon, but the combination of poroelasticity with charge density has not been introduced. Theoretically, general poroelasticity with a varying charge density is employed for determining the relationship between wave velocity and attenuation with charge density. Fast wave velocity and attenuation are affected by porosity as well as charge density; however, for a slow wave, both slow wave velocity and attenuation are not as sensitive to the effect of charge density as they are for a fast wave. Thus, employing human femoral data, we conclude that charged ions gather on trabecular struts, and the fast wave, which moves along the trabecular struts, is significantly affected by charge density. Ó 2018 Elsevier Ltd. All rights reserved.
1. Introduction Cancellous bone is a highly porous material, so general poroelasticity is appropriate to detect bone diseases when ultrasound is used. Two types of waves are observed in cancellous bone (Cardoso et al., 2008; Fellah et al., 2004; Haire and Langton, 1999; Hughes et al., 2003; Lee et al., 2003; Sebaa et al., 2006; Williams et al., 1996; Williams, 1992; Yoon et al., 2012), and general poroelasticity was first formulated by Biot (Biot, 1956). Hosokawa (Hosokawa and Otani, 1998) employed the Biot theory for cancellous bone because the two waves are observed in cancellous bone, and they found a close fit with experimental data. Williams (Williams, 1992) also introduced the Biot theory, or the so-called poroelasticity theory, to cancellous bone. All equations used in these studies have an isotropic assumption for bone materials and are solved numerically. Thus, simple solutions for isotropic generalized equations are given in the literature. Yoon and his colleagues (Yoon et al., 2012) showed that the shape of trabeculae significantly affects the velocity of wave propagation using the Biot theory.
⇑ Address: Center for Integrated General Education and College of Engineering, Hanyang University, 222 Wangsimni-ro, Seongdong-gu, Seoul 04763, Republic of Korea. E-mail address: [email protected]
Bone cells are exposed to mechanical forces and electrokinetic forces (Hung et al., 1996; Schneider et al., 2004). Hung et al. (Hung et al., 1996) applied fluid flow to bone cells, exposing mechanical forces as well as electrokinetic forces. Electrokinetic forces are created by an applied convective current that produces mobile ion transport. As a result, the authors found that fluidinduced shear stress is a primary stimulus rather than electrokinetic forces. However, Schneider et al. (Schneider et al., 2004) found that positively charged hydrogel scaffolds result in greater cell attachment compared to negative and neutral charge densities. The cell attachment is altered by manipulating a fixed charge density (Alsberg et al., 2001; English et al., 1998). Mesenchymal stem cells are differentiated to osteoblasts during osteogenesis. Mineralization is influenced by extracellular matrix proteins and by substrates (Schneider et al., 2001). It is believed that the flow of charged ions generates the convective current, and the mechanical deformation of bone generates both piezoelectricity and the electrokinetic effect (or charge density). However, piezoelectricity may be determined by the electrokinetic double layer that remains in dry bone because charged ions remain on the surface of bone after it is dried (Pollack et al., 1984b). Osteoblastic cells respond to charged hydroxyapatite particles (Liang et al., 2011) The positively charged hydroxyapatite nanoparticles have better interaction with osteoblastic cell membranes because the cell membrane is negatively charged and the size of
https://doi.org/10.1016/j.jbiomech.2018.07.048 0021-9290/Ó 2018 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Yoon, Y.J. The effect of charge density on the velocity and attenuation of ultrasound waves in human cancellous bone. J. Biomech. (2018), https://doi.org/10.1016/j.jbiomech.2018.07.048
2
Y.J. Yoon / Journal of Biomechanics xxx (2018) xxx–xxx
hydroxyapatite mineral crystals increases. Also at the early stage of mineralization, charged hydroxyapatites affect the mineralization as well as cell adhesion (Bodhak et al., 2009). The wave propagation of ultrasound is similar to the mechanical deformation because the governing equation of wave propagation is solved for the displacement of solid and fluid, respectively. Thus we believe that the wave propagation of ultrasound has the similar effect to mechanical deformation and the charge density affects both mineralization and cell adhesion. Poroelasticity is widely used for elucidating the bone fluid flow stimulating bone cells, and these bone cells communicate with each other (Cowin, 1999). The electrokinetic effect is another parameter in the bone remodeling process. These two parameters – bone fluid flow and the electrokinetic effect - are coupled to each other. The strain-generated potentials induce charged ions, called the charge density, in bone fluid flow through the canalicular network, and excessive positive ions move along the bone fluid flow (Pollack et al., 1984a; Salzstein and Pollack, 1987; Salzstein et al., 1987). Thus, the general bone poroelasticity equation is combined with charge density in the electrokinetic effect of bone. 2. Poroelasticity theory The governing equations for the poroelasticity theory are given by (Biot, 1956; Neev and Yeatts, 1989), but Neev and Yeatts (Neev and Yeatts, 1989) include the charge densities qs and qf , which are those of solid and fluid, respectively, into the governing equations of poroelasticity. These two quantities, qs and qf , can be assumed to be identical, the details of which can be found in Neev and Yeatts (Neev and Yeatts, 1989). The governing equations of poroelasticity, including electrokinetic effects and charge densities qs and qf , are as follows:
q11 u€ þ q12 U€ Nui;jj ðA þ NÞuj;ji QUj;ji þ b u_ i U_ i þ qs U;i ¼ 0
ui ¼ u0i exp½iðki xi xt Þ,
U i ¼ U 0i exp½iðki xi xtÞ,
U ¼ U exp½iðki xi xt Þ,
and
0
Eqs. (1), (2), and (3) are expressed by
h h i i 2 2 Pk x2 q11 þ ixb u01 þ Qk x2 q12 ixb U 01 iqf k U0 ¼ 0; ð7Þ h h i i 2 2 Qk x2 q12 ixb u01 þ Rk x2 q22 þ ixb U 01 þ iqf k U0 ¼ 0; ð8Þ iqf Qkj u01 þ iqf Rkj U 01 þ ½ð1 ixebÞU0 ¼ 0:
ð9Þ
In order to produce non-trivial solutions, the determinant of Eqs. (7), (8), and (9) should be zero, i.e., we can find the wave velocity, Re½v ¼ xk . The last term of Eq. (9) is reformulated from Eq. (3) for the case when the electric potential induces charged ion movement though the fluid only. Note that, in this paper, we assume that the charged ions are moving along the fluid only so that the permittivity for this calculation is used for the constant value of a fluid (or water). From the definition of the wave number, k is a complex number, k ¼ Re½k þ ia, where a is the attenuation constant. Because the wave number k is expressed as (Cardoso and Cowin, 2012)
k¼
x
v
¼
x
Re½v þ iIm½v
¼
xðRe½v iIm½v Þ jv j2
¼ Re½k þ ia;
ð10Þ
the attenuation a is obtained by
a¼
xIm½v jv j2
:
ð11Þ
In this calculation, we set the wave number k to be 1 so that the frequency is identical to the wave velocity because we cannot change the wave velocity but can manually change the frequency.
ð1Þ
3. Numerical application to human cancellous bone
q22 U€ i þ q12 u€ i RU j;j þ Quj;j ;i b u_ i U_ i þ qs U;i ¼ 0
ð2Þ
n h i o _ þ ð/G=eÞ U ¼0 qf RU j;j þ Quj;j eb U
ð3Þ
;ii
Here, N, R, Q, and A are poroelastic parameters, u and U are displacements of the fluid and solid constituents, respectively, and the variable bis defined by /2 l=K, where l is the fluid viscosity, K is the permeability of bone fluid, e is the permittivity, and / is the porosity. G is a constant showing the linear relationship between the electric current and electrical potential due to the applied electrical potential, i.e., J i ¼ /GU;i , where J i is the current density. After we define P A þ 2N, the poroelastic parameters, or the socalled Biot parameters P, Q, and R, are given by (Williams, 1992; Yoon et al., 2012) 2
P¼
/ðK s =K f 1ÞK b þ / K s þ ð1 2/ÞðK s K b Þ 4G ; þ 3 1 / K b =K s þ /K s =K f
ð4Þ
Q¼
ð1 / K b =K s Þ/K s ; 1 / K b =K s þ /K s =K f
ð5Þ
and
K s /2 ; R¼ 1 / K b =K s þ /K s =K f
To obtain the bulk moduli described in the poroelastic parameters P, Q, and R as illustrated in Eqs. (4), (5), and (6), the following relation is employed:
Ki ¼
Ei ; 3ð1 2mi Þ
ð12Þ
where the subscript i can be replaced by b, s, and f for bone, solid bone material, and fluid (or water), respectively. The elastic modulus and Poisson’s ratio of 141 human cancellous bones are theoretically estimated for the porosity / (Yang et al., 1999; Yoon et al., 2012),
Eb ¼ 822:367Et /1:95 ; Gb ¼ 345:540Et /1:99 and mb ¼ 0:198/0:16 : ð13Þ The unit of the elastic and shear moduli in Eq. (13) is GPa. To obtain the unknown variable, Et , we set Eb ¼ Es , as the porosity / is zero. The variable Et is obtained as 0.023, as the elastic modulus of solid bone material is 18.9 GPa. Then, we can estimate the shear modulus of solid bone material as 7.94 GPa, and the poroelastic parameters P, Q, and R are numerically calculated with the bulk modulus of fluid (or water) to be 2.3 GPa. The parameter b intro1/3
ð6Þ
K i is the bulk modulus of i, where i is substituted by solid (s), fluid (f ), and bone (b). By assuming that the displacements of the fluid and solid, u and U, and the potential U are harmonic, i.e.,
duced in Eqs. (1), (2), and (3) is /2 l=K, where K ¼ lSv 26 3þ1/ and ð2 6 Þ
Sv ¼ 26:23/ 81:73/2 þ 121:80/3 92:71/4 þ 26:55/5 . The unit for b is kg=ðm3 sÞ (Daish et al., 2017). The permittivity e for the fluid, which we assume to be same as that of water, is calculated by
the
formula
water Þe1 ðT water Þ eðx; T water Þ ¼ e1 ðT water Þ þ e0 ðT1i , xrðT water Þ
Please cite this article in press as: Yoon, Y.J. The effect of charge density on the velocity and attenuation of ultrasound waves in human cancellous bone. J. Biomech. (2018), https://doi.org/10.1016/j.jbiomech.2018.07.048
Y.J. Yoon / Journal of Biomechanics xxx (2018) xxx–xxx
e0 ðT water Þ ¼ 87:9 0:404T water þ 9:59 104 T water 2 1:33 10 T water 3 , e1 ðT water Þ ¼ e0 ðT water Þ 80:7exp½4:42 103 T water , where 6
and rðT water Þ ¼ 1:37 1013 exp½651=ðT water ð CÞ þ 133Þ. The unit for the temperature is Kelvin, except for the last equation, which is denoted by Celsius. (Mount and Comas, 2014). In this formulation, it is assumed that the charged ions are moving through the fluid so that the permittivity for the fluid (or water) is selected for the calculation, but numerically the permittivity is not sensitive to either the wave velocity or attenuation. The unit for permittivity is Farad/m. We also find the poroelastic densities q11 , q22 , and q12 , which are defined by
q11 þ q12 ¼ ð1 /Þqs ;
ð14Þ
q22 þ q12 ¼ /qf ;
ð15Þ
and
q12 ¼ ð1 tÞ/qf :
ð16Þ
Here, qs and qf are the densities of the solid and fluid, respectively, and a is the tortuosity (Williams, 1992). In this paper, we follow the expression of Williams (Williams, 1992), and the tortuosity t is
t ¼ 1 0:25ð1 1=/Þ:
ð17Þ
4. Results and discussion Fig. 1 shows the variation of fast wave velocity against porosity and charge density, and Fig. 2 shows the variation of fast wave attenuation against porosity and charge density. As can be seen in Figs. 1 and 2, the charge density and porosity affect the fast wave velocity and attenuation. However, the slow wave velocity and attenuation are not as sensitive to the charge density as are the fast wave velocity and attenuation (Figs. 3 and 4). We already know
Fig. 1. The fast wave speed plotted against porosity and charge density.
3
that the fast wave moves along the trabecular struts and the slow wave moves along the fluid in pores. Then, we can simply conclude that the excessive charged ions are attached to the trabecular struts and increase the fast wave velocity and attenuation. The charged ions are gathered on the trabecular struts, which recruit osteoblasts to attach. Experimentally, the hydrogel, which is positively charged, recruits more osteoblasts to the surface (Schneider et al., 2004), where a hydrogel is commonly used as the scaffold material for bone tissue engineering (Zhu and Marchant, 2011). Similar research found that positively charged coated titanium shows better osteoblastic focal adhesion formation than vinculin and paxillin and active cytoskeleton development (Zhu and Marchant, 2011). However, osteopontin is an acidic phosphorylated glycoprotein and is an inhibitor of hydroxyapatite formation, as it prevents the mineral crystals from growing (Hunter et al., 1996). It appears that it is associated with a negative charge density (Pampena et al., 2004). The sensitivity to a negative charge density promotes osteopontin to inhibit hydroxyapatite mineral crystallite growth (Pampena et al. 2004). At this point, it is not clear if the charge density calculated in this study is positive or negative. However, the strain-generated potentials induce excessive positive ions in the bone fluid flow, so we assume the generated ions due to the wave are positive. This is because the strain-generated potentials induce charged ions, called the charge density, in bone fluid flow through the canalicular network, and excessive positive ions move along the bone fluid flow. It is known that the fast wave moves along the trabeculae. If we calculate the bulk wave of the cancellous bone, it is identical to the fast wave. This means that the fast wave moves when the solid matrix is locked with the fluid phase, and the slow wave is the relative motion between the solid matrix and the fluid phase. Thus, it is believed that some charged ions attach to the trabeculae are positive and recruit osteoblasts to attach to the trabecular surface due to the ultrasound stimulation, which is similar to mechanical
Fig. 3. Slow wave speed plotted against porosity and charge density.
Fig. 2. Attenuation of fast wave speed plotted against porosity and charge density.
Please cite this article in press as: Yoon, Y.J. The effect of charge density on the velocity and attenuation of ultrasound waves in human cancellous bone. J. Biomech. (2018), https://doi.org/10.1016/j.jbiomech.2018.07.048
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Y.J. Yoon / Journal of Biomechanics xxx (2018) xxx–xxx
Fig. 4. Attenuation of slow wave speed plotted against porosity and charge density.
loading. However, this is only a hypothesis because no experiments regarding this matter have been performed. We need further experiments to examine the charged ion effect on trabecular struts, but it would be a challenge to set up the experiment. In a future study, we will combine the piezoelectric effect, which is another bone remodelling stimulus, into the governing equations shown in this paper to simplify the experimental setup of future experiments. Conflict of interest The author declare no conflict of interest. Acknowledgement This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (grant number 2015R1D1A1A 09057878). References Alsberg, E., Anderson, K.W., Albeiruti, A., Franceschi, R.T., Mooney, D.J., 2001. Cellinteractive alginate hydrogels for bone tissue engineering. J. Dent. Res. 80, 2025–2029. Biot, M.A., 1956. Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency range. the. J. Acoust. Soc. Am. 28, 168–178. Bodhak, S., Bose, S., Bandyopadhyay, A., 2009. Role of surface charge and wettability on early stage mineralization and bone cell–materials interactions of polarized hydroxyapatite. Acta Biomater. 5, 2178–2188. Cardoso, L., Cowin, S.C., 2012. Role of structural anisotropy of biological tissues in poroelastic wave propagation. Mech. Mater. 44, 174–188. Cardoso, L., Meunier, A., Oddou, C., 2008. In vitro acoustic wave propagation in human and bovine cancellous bone as predicted by Biot’s theory. J. Mech. Med. Biol. 8, 183–201. Cowin, S.C., 1999. Bone poroelasticity. J. Biomech. 32, 217–238. Daish, C., Blanchard, R., Gulati, K., Losic, D., Findlay, D., Harvie, D., Pivonka, P., 2017. Estimation of anisotropic permeability in trabecular bone based on microCT imaging and pore-scale fluid dynamics simulations. Bone Rep. 6, 129–139. English, A.E., Tanaka, T., Edelman, E.R., 1998. Polymer and solution ion shielding in polyampholytic hydrogels. Polymer 39, 5893–5897. Fellah, Z.E.A., Chapelon, J.Y., Berger, S., Lauriks, W., Depollier, C., 2004. Ultrasonic wave propagation in human cancellous bone: application of Biot theory. J. Acoust. Soc. Am. 116, 61–73. Haire, T., Langton, C., 1999. Biot theory: a review of its application to ultrasound propagation through cancellous bone. Bone 24, 291–295.
Hosokawa, A., Otani, T., 1998. Acoustic anisotropy in bovine cancellous bone. J. Acoust. Soc. Am. 103, 2718–2722. Hughes, E.R., Leighton, T.G., Petley, G.W., White, P.R., Chivers, R.C., 2003. Estimation of critical and viscous frequencies for Biot theory in cancellous bone. Ultrasonics 41, 365–368. Hung, C.T., Allen, F.D., Pollack, S.R., Brighton, C.T., 1996. What is the role of the convective current density in the real-time calcium response of cultured bone cells to fluid flow? J. Biomech. 29, 1403–1409. Hunter, G.K., Hauschka, P.V., Poole, A.R., Rosenberg, L.C., Goldberg, H.A., 1996. Nucleation and inhibition of hydroxyapatite formation by mineralized tissue proteins. Biochem. J. 317, 59–64. Lee, K.I., Roh, H.-S., Yoon, S.W., 2003. Acoustic wave propagation in bovine cancellous bone: application of the Modified Biot-Attenborough model. J. Acoust. Soc. Am. 114, 2284–2293. Liang, C., Joseph, M.M., James, C.M.L., Hao, L., 2011. The role of surface charge on the uptake and biocompatibility of hydroxyapatite nanoparticles with osteoblast cells. Nanotechnology 22, 105708. Mount, G.J., Comas, X., 2014. Estimating porosity and solid dielectric permittivity in the Miami Limestone using high-frequency ground penetrating radar (GPR) measurements at the laboratory scale. Water Resour. Res. 50, 7590–7605. Neev, J., Yeatts, F., 1989. Electrokinetic effects in fluid-saturated poroelastic media. Phys. Rev. B 40, 9135. Pampena, D.A., Robertson, K.A., Litvinova, O., Lajoie, G., Goldberg, H.A., Hunter, G.K., 2004. Inhibition of hydroxyapatite formation by osteopontin phosphopeptides. Biochem. J. 378, 1083–1087. Pollack, S., Salzstein, R., Pienkowski, D., 1984. The electric double layer in bone and its influence on stress-generated potentials. Calcif. Tissue Int. 36, S77–S81. Salzstein, R.A., Pollack, S.R., 1987. Electromechanical potentials in cortical bone—II. Experimental analysis. J. Biomech. 20, 271–280. Salzstein, R.A., Pollack, S.R., Mak, A.F., Petrov, N., 1987. Electromechanical potentials in cortical bone—I. A continuum approach. J. Biomech. 20, 261–270. Schneider, G.B., English, A., Abraham, M., Zaharias, R., Stanford, C., Keller, J., 2004. The effect of hydrogel charge density on cell attachment. Biomaterials 25, 3023–3028. Schneider, G.B., Zaharias, R., Stanford, C., 2001. Osteoblast integrin adhesion and signaling regulate mineralization. J. Dent. Res. 80, 1540–1544. Sebaa, N., Fellah, Z.E.A., Fellah, M., Ogam, E., Wirgin, A., Mitri, F., Depollier, C., Lauriks, W., 2006. Ultrasonic characterization of human cancellous bone using the Biot theory: inverse problem. J. Acoust. Soc. Am. 120, 1816–1824. Williams, J., Grimm, M., Wehrli, F., Foster, K., Chung, H., 1996. Prediction of frequency and pore size dependent attenuation of ultrasound in trabecular bone using Biot’s theory. In: Mechanics of Poroelastic Media. Springer, pp. 263– 271. Williams, J.L., 1992. Ultrasonic wave propagation in cancellous and cortical bone: prediction of some experimental results by Biot’s theory. J. Acoust. Soc. Am. 91, 1106–1112. Yang, G., Kabel, J., Van Rietbergen, B., Odgaard, A., Huiskes, R., Cown, S.C., 1999. The anisotropic Hooke’s law for cancellous bone and wood. J. Elast. 53, 125–146. Yoon, Y.J., Chung, J.-P., Bae, C.-S., Han, S.-Y., 2012. The speed of sound through trabecular bone predicted by Biot theory. J. Biomech. 45, 716–718. Zhu, J., Marchant, R.E., 2011. Design properties of hydrogel tissue-engineering scaffolds. Expert Rev. Med. Devices 8, 607–626.
Please cite this article in press as: Yoon, Y.J. The effect of charge density on the velocity and attenuation of ultrasound waves in human cancellous bone. J. Biomech. (2018), https://doi.org/10.1016/j.jbiomech.2018.07.048
Contents lists available at ScienceDirect
Journal of Biomechanics journal homepage: www.elsevier.com/locate/jbiomech www.JBiomech.com
The effect of charge density on the velocity and attenuation of ultrasound waves in human cancellous bone Young June Yoon ⇑ Center for Integrated General Education, Republic of Korea College of Engineering, Hanyang University, Republic of Korea
a r t i c l e
i n f o
Article history: Accepted 31 July 2018 Available online xxxx Keywords: Cancellous bone Poroelasticity Ultrasound Charge density
a b s t r a c t Cancellous bone is a highly porous material, and two types of waves, fast and slow, are observed when ultrasound is used for detecting bone diseases. There are several possible stimuli for bone remodelling processes, including bone fluid flow, streaming potential, and piezoelectricity. Poroelasticity has been widely used for elucidating the bone fluid flow phenomenon, but the combination of poroelasticity with charge density has not been introduced. Theoretically, general poroelasticity with a varying charge density is employed for determining the relationship between wave velocity and attenuation with charge density. Fast wave velocity and attenuation are affected by porosity as well as charge density; however, for a slow wave, both slow wave velocity and attenuation are not as sensitive to the effect of charge density as they are for a fast wave. Thus, employing human femoral data, we conclude that charged ions gather on trabecular struts, and the fast wave, which moves along the trabecular struts, is significantly affected by charge density. Ó 2018 Elsevier Ltd. All rights reserved.
1. Introduction Cancellous bone is a highly porous material, so general poroelasticity is appropriate to detect bone diseases when ultrasound is used. Two types of waves are observed in cancellous bone (Cardoso et al., 2008; Fellah et al., 2004; Haire and Langton, 1999; Hughes et al., 2003; Lee et al., 2003; Sebaa et al., 2006; Williams et al., 1996; Williams, 1992; Yoon et al., 2012), and general poroelasticity was first formulated by Biot (Biot, 1956). Hosokawa (Hosokawa and Otani, 1998) employed the Biot theory for cancellous bone because the two waves are observed in cancellous bone, and they found a close fit with experimental data. Williams (Williams, 1992) also introduced the Biot theory, or the so-called poroelasticity theory, to cancellous bone. All equations used in these studies have an isotropic assumption for bone materials and are solved numerically. Thus, simple solutions for isotropic generalized equations are given in the literature. Yoon and his colleagues (Yoon et al., 2012) showed that the shape of trabeculae significantly affects the velocity of wave propagation using the Biot theory.
⇑ Address: Center for Integrated General Education and College of Engineering, Hanyang University, 222 Wangsimni-ro, Seongdong-gu, Seoul 04763, Republic of Korea. E-mail address: [email protected]
Bone cells are exposed to mechanical forces and electrokinetic forces (Hung et al., 1996; Schneider et al., 2004). Hung et al. (Hung et al., 1996) applied fluid flow to bone cells, exposing mechanical forces as well as electrokinetic forces. Electrokinetic forces are created by an applied convective current that produces mobile ion transport. As a result, the authors found that fluidinduced shear stress is a primary stimulus rather than electrokinetic forces. However, Schneider et al. (Schneider et al., 2004) found that positively charged hydrogel scaffolds result in greater cell attachment compared to negative and neutral charge densities. The cell attachment is altered by manipulating a fixed charge density (Alsberg et al., 2001; English et al., 1998). Mesenchymal stem cells are differentiated to osteoblasts during osteogenesis. Mineralization is influenced by extracellular matrix proteins and by substrates (Schneider et al., 2001). It is believed that the flow of charged ions generates the convective current, and the mechanical deformation of bone generates both piezoelectricity and the electrokinetic effect (or charge density). However, piezoelectricity may be determined by the electrokinetic double layer that remains in dry bone because charged ions remain on the surface of bone after it is dried (Pollack et al., 1984b). Osteoblastic cells respond to charged hydroxyapatite particles (Liang et al., 2011) The positively charged hydroxyapatite nanoparticles have better interaction with osteoblastic cell membranes because the cell membrane is negatively charged and the size of
https://doi.org/10.1016/j.jbiomech.2018.07.048 0021-9290/Ó 2018 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Yoon, Y.J. The effect of charge density on the velocity and attenuation of ultrasound waves in human cancellous bone. J. Biomech. (2018), https://doi.org/10.1016/j.jbiomech.2018.07.048
2
Y.J. Yoon / Journal of Biomechanics xxx (2018) xxx–xxx
hydroxyapatite mineral crystals increases. Also at the early stage of mineralization, charged hydroxyapatites affect the mineralization as well as cell adhesion (Bodhak et al., 2009). The wave propagation of ultrasound is similar to the mechanical deformation because the governing equation of wave propagation is solved for the displacement of solid and fluid, respectively. Thus we believe that the wave propagation of ultrasound has the similar effect to mechanical deformation and the charge density affects both mineralization and cell adhesion. Poroelasticity is widely used for elucidating the bone fluid flow stimulating bone cells, and these bone cells communicate with each other (Cowin, 1999). The electrokinetic effect is another parameter in the bone remodeling process. These two parameters – bone fluid flow and the electrokinetic effect - are coupled to each other. The strain-generated potentials induce charged ions, called the charge density, in bone fluid flow through the canalicular network, and excessive positive ions move along the bone fluid flow (Pollack et al., 1984a; Salzstein and Pollack, 1987; Salzstein et al., 1987). Thus, the general bone poroelasticity equation is combined with charge density in the electrokinetic effect of bone. 2. Poroelasticity theory The governing equations for the poroelasticity theory are given by (Biot, 1956; Neev and Yeatts, 1989), but Neev and Yeatts (Neev and Yeatts, 1989) include the charge densities qs and qf , which are those of solid and fluid, respectively, into the governing equations of poroelasticity. These two quantities, qs and qf , can be assumed to be identical, the details of which can be found in Neev and Yeatts (Neev and Yeatts, 1989). The governing equations of poroelasticity, including electrokinetic effects and charge densities qs and qf , are as follows:
q11 u€ þ q12 U€ Nui;jj ðA þ NÞuj;ji QUj;ji þ b u_ i U_ i þ qs U;i ¼ 0
ui ¼ u0i exp½iðki xi xt Þ,
U i ¼ U 0i exp½iðki xi xtÞ,
U ¼ U exp½iðki xi xt Þ,
and
0
Eqs. (1), (2), and (3) are expressed by
h h i i 2 2 Pk x2 q11 þ ixb u01 þ Qk x2 q12 ixb U 01 iqf k U0 ¼ 0; ð7Þ h h i i 2 2 Qk x2 q12 ixb u01 þ Rk x2 q22 þ ixb U 01 þ iqf k U0 ¼ 0; ð8Þ iqf Qkj u01 þ iqf Rkj U 01 þ ½ð1 ixebÞU0 ¼ 0:
ð9Þ
In order to produce non-trivial solutions, the determinant of Eqs. (7), (8), and (9) should be zero, i.e., we can find the wave velocity, Re½v ¼ xk . The last term of Eq. (9) is reformulated from Eq. (3) for the case when the electric potential induces charged ion movement though the fluid only. Note that, in this paper, we assume that the charged ions are moving along the fluid only so that the permittivity for this calculation is used for the constant value of a fluid (or water). From the definition of the wave number, k is a complex number, k ¼ Re½k þ ia, where a is the attenuation constant. Because the wave number k is expressed as (Cardoso and Cowin, 2012)
k¼
x
v
¼
x
Re½v þ iIm½v
¼
xðRe½v iIm½v Þ jv j2
¼ Re½k þ ia;
ð10Þ
the attenuation a is obtained by
a¼
xIm½v jv j2
:
ð11Þ
In this calculation, we set the wave number k to be 1 so that the frequency is identical to the wave velocity because we cannot change the wave velocity but can manually change the frequency.
ð1Þ
3. Numerical application to human cancellous bone
q22 U€ i þ q12 u€ i RU j;j þ Quj;j ;i b u_ i U_ i þ qs U;i ¼ 0
ð2Þ
n h i o _ þ ð/G=eÞ U ¼0 qf RU j;j þ Quj;j eb U
ð3Þ
;ii
Here, N, R, Q, and A are poroelastic parameters, u and U are displacements of the fluid and solid constituents, respectively, and the variable bis defined by /2 l=K, where l is the fluid viscosity, K is the permeability of bone fluid, e is the permittivity, and / is the porosity. G is a constant showing the linear relationship between the electric current and electrical potential due to the applied electrical potential, i.e., J i ¼ /GU;i , where J i is the current density. After we define P A þ 2N, the poroelastic parameters, or the socalled Biot parameters P, Q, and R, are given by (Williams, 1992; Yoon et al., 2012) 2
P¼
/ðK s =K f 1ÞK b þ / K s þ ð1 2/ÞðK s K b Þ 4G ; þ 3 1 / K b =K s þ /K s =K f
ð4Þ
Q¼
ð1 / K b =K s Þ/K s ; 1 / K b =K s þ /K s =K f
ð5Þ
and
K s /2 ; R¼ 1 / K b =K s þ /K s =K f
To obtain the bulk moduli described in the poroelastic parameters P, Q, and R as illustrated in Eqs. (4), (5), and (6), the following relation is employed:
Ki ¼
Ei ; 3ð1 2mi Þ
ð12Þ
where the subscript i can be replaced by b, s, and f for bone, solid bone material, and fluid (or water), respectively. The elastic modulus and Poisson’s ratio of 141 human cancellous bones are theoretically estimated for the porosity / (Yang et al., 1999; Yoon et al., 2012),
Eb ¼ 822:367Et /1:95 ; Gb ¼ 345:540Et /1:99 and mb ¼ 0:198/0:16 : ð13Þ The unit of the elastic and shear moduli in Eq. (13) is GPa. To obtain the unknown variable, Et , we set Eb ¼ Es , as the porosity / is zero. The variable Et is obtained as 0.023, as the elastic modulus of solid bone material is 18.9 GPa. Then, we can estimate the shear modulus of solid bone material as 7.94 GPa, and the poroelastic parameters P, Q, and R are numerically calculated with the bulk modulus of fluid (or water) to be 2.3 GPa. The parameter b intro1/3
ð6Þ
K i is the bulk modulus of i, where i is substituted by solid (s), fluid (f ), and bone (b). By assuming that the displacements of the fluid and solid, u and U, and the potential U are harmonic, i.e.,
duced in Eqs. (1), (2), and (3) is /2 l=K, where K ¼ lSv 26 3þ1/ and ð2 6 Þ
Sv ¼ 26:23/ 81:73/2 þ 121:80/3 92:71/4 þ 26:55/5 . The unit for b is kg=ðm3 sÞ (Daish et al., 2017). The permittivity e for the fluid, which we assume to be same as that of water, is calculated by
the
formula
water Þe1 ðT water Þ eðx; T water Þ ¼ e1 ðT water Þ þ e0 ðT1i , xrðT water Þ
Please cite this article in press as: Yoon, Y.J. The effect of charge density on the velocity and attenuation of ultrasound waves in human cancellous bone. J. Biomech. (2018), https://doi.org/10.1016/j.jbiomech.2018.07.048
Y.J. Yoon / Journal of Biomechanics xxx (2018) xxx–xxx
e0 ðT water Þ ¼ 87:9 0:404T water þ 9:59 104 T water 2 1:33 10 T water 3 , e1 ðT water Þ ¼ e0 ðT water Þ 80:7exp½4:42 103 T water , where 6
and rðT water Þ ¼ 1:37 1013 exp½651=ðT water ð CÞ þ 133Þ. The unit for the temperature is Kelvin, except for the last equation, which is denoted by Celsius. (Mount and Comas, 2014). In this formulation, it is assumed that the charged ions are moving through the fluid so that the permittivity for the fluid (or water) is selected for the calculation, but numerically the permittivity is not sensitive to either the wave velocity or attenuation. The unit for permittivity is Farad/m. We also find the poroelastic densities q11 , q22 , and q12 , which are defined by
q11 þ q12 ¼ ð1 /Þqs ;
ð14Þ
q22 þ q12 ¼ /qf ;
ð15Þ
and
q12 ¼ ð1 tÞ/qf :
ð16Þ
Here, qs and qf are the densities of the solid and fluid, respectively, and a is the tortuosity (Williams, 1992). In this paper, we follow the expression of Williams (Williams, 1992), and the tortuosity t is
t ¼ 1 0:25ð1 1=/Þ:
ð17Þ
4. Results and discussion Fig. 1 shows the variation of fast wave velocity against porosity and charge density, and Fig. 2 shows the variation of fast wave attenuation against porosity and charge density. As can be seen in Figs. 1 and 2, the charge density and porosity affect the fast wave velocity and attenuation. However, the slow wave velocity and attenuation are not as sensitive to the charge density as are the fast wave velocity and attenuation (Figs. 3 and 4). We already know
Fig. 1. The fast wave speed plotted against porosity and charge density.
3
that the fast wave moves along the trabecular struts and the slow wave moves along the fluid in pores. Then, we can simply conclude that the excessive charged ions are attached to the trabecular struts and increase the fast wave velocity and attenuation. The charged ions are gathered on the trabecular struts, which recruit osteoblasts to attach. Experimentally, the hydrogel, which is positively charged, recruits more osteoblasts to the surface (Schneider et al., 2004), where a hydrogel is commonly used as the scaffold material for bone tissue engineering (Zhu and Marchant, 2011). Similar research found that positively charged coated titanium shows better osteoblastic focal adhesion formation than vinculin and paxillin and active cytoskeleton development (Zhu and Marchant, 2011). However, osteopontin is an acidic phosphorylated glycoprotein and is an inhibitor of hydroxyapatite formation, as it prevents the mineral crystals from growing (Hunter et al., 1996). It appears that it is associated with a negative charge density (Pampena et al., 2004). The sensitivity to a negative charge density promotes osteopontin to inhibit hydroxyapatite mineral crystallite growth (Pampena et al. 2004). At this point, it is not clear if the charge density calculated in this study is positive or negative. However, the strain-generated potentials induce excessive positive ions in the bone fluid flow, so we assume the generated ions due to the wave are positive. This is because the strain-generated potentials induce charged ions, called the charge density, in bone fluid flow through the canalicular network, and excessive positive ions move along the bone fluid flow. It is known that the fast wave moves along the trabeculae. If we calculate the bulk wave of the cancellous bone, it is identical to the fast wave. This means that the fast wave moves when the solid matrix is locked with the fluid phase, and the slow wave is the relative motion between the solid matrix and the fluid phase. Thus, it is believed that some charged ions attach to the trabeculae are positive and recruit osteoblasts to attach to the trabecular surface due to the ultrasound stimulation, which is similar to mechanical
Fig. 3. Slow wave speed plotted against porosity and charge density.
Fig. 2. Attenuation of fast wave speed plotted against porosity and charge density.
Please cite this article in press as: Yoon, Y.J. The effect of charge density on the velocity and attenuation of ultrasound waves in human cancellous bone. J. Biomech. (2018), https://doi.org/10.1016/j.jbiomech.2018.07.048
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Y.J. Yoon / Journal of Biomechanics xxx (2018) xxx–xxx
Fig. 4. Attenuation of slow wave speed plotted against porosity and charge density.
loading. However, this is only a hypothesis because no experiments regarding this matter have been performed. We need further experiments to examine the charged ion effect on trabecular struts, but it would be a challenge to set up the experiment. In a future study, we will combine the piezoelectric effect, which is another bone remodelling stimulus, into the governing equations shown in this paper to simplify the experimental setup of future experiments. Conflict of interest The author declare no conflict of interest. Acknowledgement This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (grant number 2015R1D1A1A 09057878). References Alsberg, E., Anderson, K.W., Albeiruti, A., Franceschi, R.T., Mooney, D.J., 2001. Cellinteractive alginate hydrogels for bone tissue engineering. J. Dent. Res. 80, 2025–2029. Biot, M.A., 1956. Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency range. the. J. Acoust. Soc. Am. 28, 168–178. Bodhak, S., Bose, S., Bandyopadhyay, A., 2009. Role of surface charge and wettability on early stage mineralization and bone cell–materials interactions of polarized hydroxyapatite. Acta Biomater. 5, 2178–2188. Cardoso, L., Cowin, S.C., 2012. Role of structural anisotropy of biological tissues in poroelastic wave propagation. Mech. Mater. 44, 174–188. Cardoso, L., Meunier, A., Oddou, C., 2008. In vitro acoustic wave propagation in human and bovine cancellous bone as predicted by Biot’s theory. J. Mech. Med. Biol. 8, 183–201. Cowin, S.C., 1999. Bone poroelasticity. J. Biomech. 32, 217–238. Daish, C., Blanchard, R., Gulati, K., Losic, D., Findlay, D., Harvie, D., Pivonka, P., 2017. Estimation of anisotropic permeability in trabecular bone based on microCT imaging and pore-scale fluid dynamics simulations. Bone Rep. 6, 129–139. English, A.E., Tanaka, T., Edelman, E.R., 1998. Polymer and solution ion shielding in polyampholytic hydrogels. Polymer 39, 5893–5897. Fellah, Z.E.A., Chapelon, J.Y., Berger, S., Lauriks, W., Depollier, C., 2004. Ultrasonic wave propagation in human cancellous bone: application of Biot theory. J. Acoust. Soc. Am. 116, 61–73. Haire, T., Langton, C., 1999. Biot theory: a review of its application to ultrasound propagation through cancellous bone. Bone 24, 291–295.
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Please cite this article in press as: Yoon, Y.J. The effect of charge density on the velocity and attenuation of ultrasound waves in human cancellous bone. J. Biomech. (2018), https://doi.org/10.1016/j.jbiomech.2018.07.048