ESTIMATION OF THE HYDRAULIC CANALICULAR RADIUS IN CORTICAL BONE

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1 JOURNAL OF THEORETICAL AND APPLIED MECHANICS, Sofia, 2, vol. 3, No. 4, pp (Received January 2) ESTIMATION OF THE HYDRAULIC CANALICULAR RADIUS IN CORTICAL BONE N. PETROV Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl. 4, 3 Sofia, Bulgaria, petrov333@gmail.com ABSTRACT. The fact that osteocites are not more than. to.2 mm from capillary strongly suggests for canalicular via by hich osteocites are nourished and rid themselves of aste products. As possible mechanisms appear to be diffusion and stress generated fluid flo. Hoever it is not clear, hat are the iciency limits of these to ays to transport the species. Because of the presence of cell processes ithin the canaliculus, it is unknon, hat part of the canalicular cross section is occupied and hat is free for fluid flo or spices diffusion. The objective of the present study is to anser the question: hat is the order of the hydraulic canalicular radius? In the present study its value is estimated about 9 nm. In the present study for first time, in the literature, the hydraulic radius of the canaliculus is directly estimated by processing data of relaxation type experiments

2 2 NOMENCLATURE V, V o - moment and the reference, load free lacunar volume; P, P o - moment and initial lacunar fluid pressure; R - hydraulic (ective) radius of canaliculus; L - canaliculus length (center-to-center distance beteen to lacunae); n - number of canaliculus connecting to lacunae; M - number of lacunae in one lacunar-canalicular set (Fig. ); S - uniaxial stress value; µ - shear modulus of bone matrix; - bulk modulus of bone matrix; - bulk modulus of fluid; - ective bulk modulus; η - viscosity of the fluid; τˆ - characteristic lacunar-canalicular system time.. Introduction The fluid flo in bone, from point of vie of nutrients supply, has been studied by Jendruco [], elly [2], Dillaman [3], Montgomery et al. [4], Dillaman et al. [5], McCarthy and Yang [6], eanini et al. [7]. A simple mathematical model for fluid flo in an osteon, consisting of a single lacuna and canaliculus, has been offered by Pollack, Petrov et al. [8]. Many of the previous published observations of stress generated potentials in osteons ere explained for first time. ufahl and Saha [9] extended this mathematical model on the base of the anatomical model, offered by Piekarsky and Munro [], consisting of canaliculi and lacunae bounded in series as in Fig..

3 3 Fig.. Haversian canal-lacunae- canaliculi netork model of Murno and Piekarski [] The question of the ect of stress concentration around the Haversian canal on the symmetry of the fluid flo and streaming potentials has been addressed by Petrov et al. [] using finite element model to solve Biot [2] equations from the theory of consolidation. Later the model has been generalized by Zeng et al. [3]. The case of fluid flo exchange beteen the lacunar- canalicular system and microporosity has been studied by Mak et al. [4]. Opposite Weinbaum et al. [5] entirely reject the idea for such a fluid exchange. The objective of the present study is to give an anser the question: What is the order of the size of the hydraulic canalicular radius? As the canaliculus is plying role of nutrients and aste products transport via, the physiological importance of this problem is ithout doubt.

4 4 2. Physical and mathematical model Because of the complex microanatomic structure of the osteon, it is reasonable, to make the folloing simplifying assumptions, as in papers of Pollack et al. [8], Petrov et al. [], ufahl and Saha [9], Zeng et al. [3], Weinbaum et al. [5], Wang et al. [6]:. Bone matrix is isotropic and elastic; 2. Pores are spherical, fluid filled inclusions; 3. The fluid pressure in the Haversian canal is arterial and is taken to be zero; 4. Dilatation of the pores can be computed as a function of the uniaxial stress; 5. The fluid fluxes at the cement line of osteon are taken to be equal to zero. We have in mind that the fluid pressure has a local extremum at the boundary beteen to osteons, because of symmetry. 6. All distances beteen lacunae in series and all canalicular radii are equal. 7. The fluid exchange beteen lacunar- canalicular system and microporosity is neglected. Taking into account the above simplifications, e present the fluid flo in lacunar-canalicular system, in case of instantaneous loading, ith ufahl and Saha model [9]: d dt ˆ τ () { P} + [ S]{ P} =, here { } N P is a vector ith components lacunar fluid pressures P, N =,2,..., M ( M is the number of the lacunae in one lacunar-canalicular set),

5 5 (2) [ ] S , is M M matrix. The characteristic time τˆ is function of the microanatomical parameters and physical moduli (3) 8η LV ˆ τ, = 4 nπ R here V is the reference, load free volume of the lacuna, L is the canaliculus length (center-to-center distance beteen to lacunae), R is the ective (hydraulic) canalicular radius, η is the viscosity of the fluid, is bulk modulus of the bone fluid and is ective bulk modulus defined by the folloing non steady relation beteen the lacunar volume V and lacunar pressure P : dv V dp (4) =, t >. dt dt Because of the presence of processes ithin canaliculus and consequently unknon part of the canalicular cross section is block for fluid flo, opposite to ufahl and Saha [9], e use in equation (3) the ective radius anatomical radius of matrix canaliculus interface. R instead of the

6 6 3. Determination of ective bulk modulus ufahl and Saha [9] interpreted as an experimentally determined constant. In fact, hoever e have no available experimental data or experimental method of its determining. In the present paper an analytic method of explicit determining, for the case of instantaneous loading, is offered. This leads to reduction of the number of the unknon constants and gives possibility of estimating the ective canalicular radius on the base of processing the data from relaxation type experiments. Next the method is represented. Because bone is consider as elastic, the step load results in an instantaneous dilatation of a fluid filled lacunae (Hashin, [7]): (5) V (+) V = V 4µ + 3 4µ + 3 S 3, here V ( + ) V,, µ, S are the instantaneous dilatation of the lacuna, shear and bulk moduli of the bone matrix, and the macroscopic uniaxial stress respectively. The instantaneous pressure P in the lacuna, resulting from dilatation, is V (+) V 4µ + 3 (6) P = = S. V 3 4µ + 3 Folloing Pollack, Petrov et al. [8] for the volume balance of lacuna e have:

7 7 dp (7) V = J, t >, dt here J is the sum of all fluxes drained from the inclusion by the system of channels. Integrating the to sides of equation (7) over all time and taking into account that at t, in the case of instantaneous loading, the dilatation reaches steady state and the pressure of the fluid ithin the inclusion goes to zero, therefor e obtain (8) V P= J dt= V ( ) V. Because the lacunae are open for drainage into Haversian canal, their steady state dilatation is characterized ith zero fluid pressure and mimic empty pores. This case can be easily obtained from (5), formally substituting =, (9) V ( ) V V = 4µ + 3 4µ S 3. Substituting equation (6) and (9) into (8) e obtain the folloing explicit expression of the ective bulk modulus, as function of the fluid bulk modulus and shear modulus of bone matrix µ () 4µ = 8µ + 3.

8 8 Inserting () into (3) for the characteristic time τˆ e obtain () ˆ τ η µ 2 LV =. 4 nπ R µ obtain 4. Estimation of the ective canalicular radius Solving equation () ith respect to the ective canalicular radius e (2) R = 4 2η LV 4µ + 3 nπτˆ µ. For identification of τˆ, to mean experimental, relaxation type curves ere used, first for set of samples invitro and second for set of samples invivo (Otter et al. [8]). The evaluation of the parameter τˆ as realized ith inverse algorithm, based on least-square method. exp For a priori selected experimental points P r l, l =,2,..., m and the calc respective values calculated by the theoretical model P (τˆ ), l =,2,.., m, the quadratic deviation is defined as l (3) m exp r calc dev P l Pl R = exp, r calc l= P P 2 here m is the number of selected experimental points, and P exp r and calc P the initial values of the experimental and theoretically calculated pressures, respectively. are

9 9 Fig. 2. Normalized experimental data of pressure response at step loading and least square approximations ith the model of ufahl and Saha The value of τˆ is determined by minimization of dev R ithin an apriori, accepted interval. The theoretical values of the pressure in (3) ere obtained by numeric integration of equation () ith Crank-Nickolson differential scheme. The experimental values and their approximation ith the model of ufahl and Saha are represented in Fig. 2.

10 To determine R, beside τˆ e need of the values of the microanatomical parameters and physical moduli, taking part in formulas (2). For this aim e use reasonable, model anatomical parameters and material moduli as shon in Table. The sources for this data are elly [2], Moris et al. [9], Pollack et al. [8], Atkinson and Hallsorth [2]. The values of the identified ith the above minimization procedure ective canalicular radii R, characteristic times τˆ and the corresponding quadratic deviations dev R are represented in Table 2. Table 2. Model parameters, identified ith the least square method and the respective minimal deviation Data from ˆ τ (sec) R (nm) Otter et al., (992), invitro R dev min Otter et al., (992) invivo Discussion and conclusions A problem arises from the fact that the theoretical model is formulated for single osteon, but for the parameter identification the experimental data for samples, composed of many osteons are used. The reason is the absence of experimental data in the literature, measured on the level of single osteon, (in case

11 of instantaneous loading), but there are experiments on stress generated streaming potentials ith hole samples (collection of osteons). The admissibility of such use of experimental data comes from the similarity of the relaxation process in hole samples and in single osteons, if the dominant part of the fluid drainage into Haversian Canals. The experimental and theoretical studies of Otters et al. [2] and MacGinitie et al. [22] support this statement. They have measured experimentally streaming potentials and obtained that the relaxation times do not depend on sample thickness. The conclusion statement is that there is evidence to indicate that fluid does not flo to the bone surface, but rather flos into Haversian Canals". Cusp-like funnel shape osteonial streaming potentials ere experimentally measured by Innoncone et al. [23] at cyclic uniform loading of bone samples. The obtained extremes at the cement lines could be considered as indication that there is not fluid exchange beteen osteons. Recently, this supposition as theoretically demonstrated by Wang et al. [6] at cyclic non-uniform loading of model sample composed of 6 osteons. It is folloing from the above studies that the relaxation process is realized dominantly on osteonial level and consequently the relaxation process in hole samples and in single osteons should be analogous. It gives possibility the identification of τˆ to be realized on the base of processing the experimental data obtained for bone samples, composed of many osteons. The results, obtained by identification procedure (Table 2) are 7.5 nm for invitro and 2.4 nm for invivo samples. So e may do the statement that the ective canalicular radius about 9 nm. Because of the 4 th degree of the root, in equation (2), the predicted values for the ective radius are not sensitive ith respect to the values of the anatomical parameters and physical moduli. For example, if the combination of the anatomical parameters and elastic moduli - η LV 4µ + n 3 µ in (2), is varied in limits of ±5 % the ective radius is

12 2 varied in the relatively narro interval: R (3., 29.7) nm, e.g. remain at the same order of size. In the present study for first time, in the literature, the hydraulic radius of the canaliculus is directly estimated by processing data from relaxation type experiments. ACNOWLEDGMENTS: The author of the present paper has pleasant obligation to thank Professor Solomon Pollack for his valuable suggestions and supervision of the study, and Professor Stephen Coin for the copies of papers he has leaven. REFERENCES [] JENDRUCO, R. J., W. A. HYMAN, P. H. NEWELL, B.. CHARABORTY. Theoretical Evidence for the Generation of High Pressure in Bone Cells. J. Biomechanics, 9 (976), [2] ELLY, P. J. Pathays of Transport in Bone, Handbook of Physiology, The Cardiovascular System, Sect. 2, 3 (Ed. J. T. Shepherd, F. M. About), American Physiology Society, Bathesda, MD, 983. [3] DILLAMAN, R. M. Movement of Ferritin in the to-day Old Chick Femur. Anat. Res., 29 (984), [4] MONTGOMERY, R. J., B. D. SUTER, J. T. BRON, S. R. SMITH, P. J. ELLY. Interstitial Fluid Flo in Cortical Bone. Microvasc. Res., 35 (988), [5] DILLAMAN, R. M., R. D. ROER, D. M. GAY. Fluid Movement in Bone: Theoretical and Empirical. J. Biomechanics, 24 (99),

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