Introduction

Living bone is continually undergoing process of growth, reinforcement and resorption, termed collectively “remodeling”. Accordingly to Frost (1964) there are two kinds of bone remodeling: surface and internal. Many theories of internal bone remodeling have been suggested (Martin, 1972; Cowin and Hegedus 1976; Hegedus and Cowin, 1976; Cowin and Nachlinger, 1978; Cowin and Van -Buskirk,1 978; Carter et., al.,1987; Carter et., al., 1989).The purpose of this paper is to consider the internal remodeling of the tibia, induced by the volley-ball. For that reason, we will base upon the theory of Hegedus and Cowin (1976) which is known as “theory of adaptive elasticity”.

During a volley-ball game, it is possible to distinguish that the following hold:

i) The players are running in order to successfully rebut the ball, before it co-mes to contact with the ground.

ii) The volleyball spiker is vertically jumping as high as possible, in order to hit the ball. Also the players of the opposite team, are simultaneously vertically jumping as high as possible, in order to rebut the hit of the volleyball spi-ker.

iii) Finally the player who serves the ball, is sometimes jumping as high as possible, but he (she) is not landing to his (her) initial location. This jump can be modeled as an oblique shoot in the plane Oxz (x, z are the horizontal and vertical axons respectively). Particularly the center of the mass of the player is launched from the origin, with a velocity u_o. An angle α is formed between the direction of the vector of velocity u_o and the horizontal axon, as it seems in Fig. 1. The maximums high h_M and the horizontal displacement S_M are gi-ven by respectively:

Figure 1

where g is the acceleration of the gravity.

Figure 7

The server is mainly interested to jump as high as possible than as long as possible, in order to succesfully send the ball towards the region of the op-posite team. Also it is forbidden the server to leave his (her) area during the service, because of the international rules of volleyball playing. The last means that his (her) horizontal displacement is always under restriction. Therefore it holds h_M >S_M from which we result to:

Figure 3

Since sinα >sin76º = 0.97 ≈1,the jump of the server can approximatelly be con- sidered as a vertical.Therefore during the volleyball game, the tibia of the ath- lete is under an axial impact load, due as to running as to vertical jumps.

Biomechanical analysis of the volleyball

We assume that a person with healthy bones, that is with bones which are not under osteoporosis or osteopetrosis ( Hegedus and Cowin,1976) participates to a volley-ball game or training and he (she) continues to be exersized with the same way, for a long time period. Initially the athlete was not dealed with the volley-ball playing, but he (she) was following a normal lifestyle, by wal-king with a constant velocity v_o. Consequently his (her) tibia was in a steady state at which no remodeling occurred, subjected only to a constant compres-sive load G_o, due to the vertical component of ground reaction force, at late

stance phase during walking. Selecting the experimental data from Andriacchi et., al., (1977); Rohrle et., al., (1984), neglecting the weight of the foot because is small (Harless,1860) and using a linear regression analysis, it is possible to find:

Figure 4

The walking activity corresponds to a velocity v_o, whose magnitude lies be-tween 0.5m/sec and 2m/sec (Whalen et., al., 1988). Then because of eq. (2.1), it is possible to conclude that the magnitude of the initial load G_o, _{lies bet-ween 1.01B.W. and 1.329B.W. In addition we suppose that initialy the tibia had a uniform reference volume fraction ξο, such that 0< ξο<1 and a uniform relative volume fraction eo= 0.}

At t = 0 the athlete starts his (her) volleyball activity. We assumed that the player is x times running and is also ψ times vertically jumping. Then the ti-bia is under an average load G_z such that:

Figure 5

where Gz _i, i = 1,2, ..., x and Gz_j, j = x +1, x+2, ..., ψ are respectively: the vertical component of ground reaction force at late stance, during the i time of running and the vertical component of ground reaction force at landing phase, during the j time of jumping. Taking the experimental data from Alexander and Jayes (1980); Bates et.,al., (1983), Cavagna (1964); Cavanagh and LaFortune (1980); Fu-kanaga et., al., (1980); Winter et., al., (1983) and using a linear regression analy-sis, it is possible to obtain:

Figure 6

where v_i is the running velocity for the i time of running and assumed to be constant. The running activity corresponds to a velocity v, whose magnitude is at least 2.5m/sec (Whalen et., al., 1988). Then because of (2.3), it is possible to obtain that the magnitude of the running load Gz_i , is _{at least 1.7 B.W. Also it holds: 2B.W.< Gzj < 4B.W (Gross and Nelsson ,1987; Nigg ,1985; Valiant and Ca-vanagh , 1983). Therefore accordingly to the abovementioned and by the help of eq. (2.2) we conclude that Gz > Go.}

A hollow circular cylinder, subjected only to an axial impact load

In this section we will study the dynamic problem of a hollow circular cy- linder, consisting of a material whose behavior is described by the theory of adaptive elasticity (Hegedus and Cowin, 1976).The cylinder has an inner and an outer radii: a and b respectively and a lenght L. These radii are constants, sin-ce in present work we deal only with the internal remodeling which does not results changes to bone’s shape (Frost, 1964; Cowin and Van-Buskirk, 1978; Co-win and Van-Buskirk, 1979). The equations of the theory of Hegedus and Co-win (1976) in cylindrical coordinates,are the followings: (see also Tsili, 2000, pp. 235 -236). The remodeling rate equation:

Figure 8

The stress equations:

Figure 9

where ρ is the density of the tibia.

Also the stress-strain relations for a transversely isotropic material are:

Figure 10

where:

Figure 11

In addition the boundary conditions are:

Figure 12

where B is the weight of the athlete in units of B.W. and assumed to be con-stant during the training period.

Accordingly to Cowin and Nachlinger (1978), our problem accepts a unique solution. Assume the followings:

Figure 13

where A(t), B(t) and C(t) are unknown functions. Then (3.2) becomes:

Figure 14

Substituting the assumed strain field into equations (3.4), it is possible to find:

Figure 15

Applying the boundary conditions, we find that A(t), B(t) and C(t) are given by:

Figure 16

The displacements, the strains and the stresses, can be calculated by replacing (3.12)-(3.13) into (3.9), (3.10) and (3.11) respectively. Finally when the obtained strains are employed, the remodeling rate equation (3.1) concludes to the form:

Figure 17

Since we don’t know the exact values of the material functions, we will use approximate forms of them, for small values of e. Accordingly to Hegedus and Cowin (1976); Cowin and Van-Buskirk (1978), the proper approximations are:

Figure 18

where c_o, c₁, c₂, α_Τ, α_Α, Λ₁, Λ₂, Μ₁, Μ₂ are constant coefficients. Therefore equa-tion (3.14) after the above approximations, concludes to:

Figure 19

that is we deal with an equation of the form:

Figure 20

Τable 1: The asymptotic solution of (3.17) and its physical sense, for all subcases of the case: − ξ< e < e

The mathematical analysis and the exploration of the solutions of (3.17) that satisfies the initial condition e(0) =0, are in my earlier work ( Tsili, 2000, pp. 237-238) and it will not be repeated here. Only the final results, that are in Tables 1., 2., 3. and have been taken from my paper (Tsili, 2000) are registered.

Figure 21

Table 2: The asymptotic solution of (3.17) and its physical sense, for all subases of the case: −ξ

Figure 22

Table 3: The asymptotic behavior of the solution of (3.17) and its physical sense, for all subcases of the case e < −ξ

Figure 23

Table 4: All the acceptable solutions of our problem

Discussion – Results

Our model predicts the clinical results of several researchers who have e-valuated the bone mineral density (BMD) and the muscle bone strength indice (MBSI) of the tibia of the volley-ball players ( Fehling et., al.,1995; Calbet et., al., 1999; Rittweger et., al., 2000; Ito et., al., 2001). In their studies as the BMD as the MBSI of the tibia of the volleyball players (and generally of the athle-tes whose bones are subjected to high impact loads) are significant higher, than the corresponding findings of the normally active control subjects.

In addition, there are some clinical findings concerning the assessment of the BMD or the hole body of the volley-ball players (Alfredson et., al., 1997; Hara et., al., 2001; Nikander et., al., 2005; Nikander et., al., 2006; Nichols et., al., 2007). Accordingly to the abovementioned citations, the BMD of the hole bo-dy of the volleyball players (and generally of the athletes whose sports deal with high impact loads),are significant higher compared with the corresponding findings of the normally active control subjects.Therefore the acceptable solu-tions of our problem, are in table 4. Accordingly to the results of this table, after a long time, the tibia of the athlete will be strengthened, that is it will be less porous and more stiff.

{image:23}