Introduction

Living bone is continually undergoing processes of growth, reinforcement and resorption termed collectively “remodeling”. Accordingly to Frost (1964) the- re are two kinds of bone remodeling : internal and surface. Many theories of surface bone remodeling have been proposed (Gjelsvik,1973a and 1973b; Cowin and Van-Buskik,1979; Cowin and Firoozbakhsh,1981; Hart et., al.,1982; Hart et., al., 1984; Cowin et.,al., 1985).

The purpose of this work is to qualitative study the surface remodeling of tibia, induced by the distance - running. For that reason, we will use the propo-sed theory of Cowin –Firoozbakhsh (1981).

Biomechanical analysis of the distance-running

A person starts distance- running and suppose that he (she) continues to be training with the same way, for a long time period. Initially the athlete was fol- lowing a normal lifestyle, by walking with constant velocity v_o. Consequently his (her) tibia was in a state at which no remodeling occurred, subjected only to a constant compressive load G_o, due to the vertical component of ground reaction force, at late stance phase during walking. Accounting the 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 obtain:

Figure 4

Accordingly to Whalen et., al.,(1988) the walking activity corresponds to a ve- locity v_o whose magnitude belongs to the following range: [0.5m/sec, 2m/sec] Then, from (2.1) it implies that 1.01B.W≤G_o ≤ 1.329B.W.

At t =0 the athlete starts distance - running, as it seems in Fig. 1.,taken from Clisouras (1984). Accordingly to Clisouras (1984), the runners are classifi-cated into three categories: sprinters, middle- distance and distance - runners. The last category contains cross-country and runners who participate to 5000, 10000 and Marathon races of men ( women). At late stance phase during running, the

Figure 2

Figure 1: An athlete is distance- running. From (Clisouras, 1984)

foot of the sprinter, middle- distance runner and distance- runner contacts the ground with the toes (forefoot strikers), the second third of the foot (middlefoot strikers) and the heel (rearfoot strikers) respectively (Clisouras, 1984:Cavagna and La- Fortune, 1980). The abovementioned styles of running seem in Figs. 2a, 2b and 2c.

Figure 3

Figure 2a: The style of running at late stance phase for a sprinter. From Clisouras (1984).

Figure 5

Figure 2b: The style of running at late stance phase, for a middle-distance runner.From Clisouras (1984).

Figure 6

Figure 2c: The style of running at late stance phase, for a distance-runner. From Clisouras (1984)

Figure 7

Also accordingly to Clisouras (1984), speaking for distance - running, we mean that the athlete runs a long distance, with constant velocity. Then his (her) tibia is under a constant axial load G_z, due to the vertical component of ground reaction force, at late stance phase, during running (see Fig. 2c) Selecting the ex-perimental data from Alexander and Jayes (1980); Bates et., al.,(1983); Cavagna (1964); Cavanagh and La-Fortune (1980); Fukanaga et., al., (1980); Winter (1983) and using a linear regression analysis, we obtain:

Figure 8

where v _{is the running velocity and assumed to be constant. Since the running activity corresponds to a velocity at least of the order of 2.5m/sec (Whalen et., al., 1988, p.830), from (2.3) it is possible to conclude that Gz>1.7B.W.Therefore always holds that:}

Figure 9

The physical approximation of our problem

The tibia is modeled as a hollow circular cylinder with an inner radius a and an outer radius b, corresponding to endosteal and periosteal surfaces respe ctively. These radii are not constant, as at case of internal remodeling (Cowin and Van-Buskirk, 1978 and 1979), but they are moving, accordingly to the fol-lowing equations (Cowin and Firoozbakhsh, 1981):

Figure 10

Initially ( for t <0), the tibia was under a stress Tzz_o and the cross- section area of its diaphyseal surface was E_o where:

Figure 11

where B is the body weight of the athlete and is assumed t o be constant during the training period, while a_o and b_o are the initial inner and outer ra-dii respectively. These radii where constants ,since initially the tibia was in a steady state. Therefore from (2.4)-(2.5) it is possible to obtain :

Figure 12

Figure 13

where Gz is given by (2.2). Then (2.4) and (2.5) because of (2.6)1, (2.7), (2.8) and (2.9) are respectively written as:

Figure 14

In order to study the surface remodeling of the tibia, we will solve the sys-tem of differential equations (2.10)-(2.11) that satisfy the initial conditions:

Figure 15

and after that we will evaluate a_∞ and b_{∞ ,} where:

Figure 16

Defining the parameters:

Figure 17

and accounting that:

Figure 18

eqns (2.10)-(2.11) conclude respectively to:

Figure 19

From the above it follows that:

Figure 20

which because of (2.12)-(2.13) gives:

Figure 21

Therefore (2.21)-(2.22) because of (2.24) take their final forms respectively:

Figure 22

In order the solutions to have physical meaning, it must a(t) and b(t) be evol-ved towards possitive values b(t) > a(t) > 0, because a non positive value of b(t) means that after a long time, there will be not bone mass. Also a zero value of a_∞ have been reported by Charnay and Tschantz (1972). In their experiments, he canine ulna was under an hyperphysiological load, due to the surgical removal of the central portion of the radius. However we reject his case, because in present study we deal with an athlete, whose bones are un-der physiological loads, due to his (her) training activity and not to surgical treatment as in Charnay and Tscantz study (1972).

The solution of the problem

We dinstinguish the following cases:

1)If q = +∞, then from (2.16) and (2.20) it implies that C= 0.Consequently the system of (2.25)- (2.26) concludes to:

Figure 23

and because of (2.12)-(2.13) immediately results to:

Figure 24

This solution although has a physical sense is rejected,because it states that af-ter a long time, both endosteal and periosteal surfaces will not move, that is the tibia will be in the same steady state at which no remodeling occurs. The abovementioned contradicts to the law of Wollf (1884; 1892).

2) If q = –∞, from (2.16), (2.20) it implies that C =0 and concides with the pre-vious case.

3) If q =1, the solutions of the system of (2.25)-(2.26) satisfying (2.12)-(2.13) a-re:

Figure 25

4) If q =−1, the solutions of (2.25)-(2.26) that satisfy (2.12)- (2.13) are:

Figure 26

5) If q=0, the solutions of (2.25)-(2.26) satisfying (2.12)- (2.13) are:

Figure 27

6) If q≠+∞, q≠−∞, q≠1, q≠‒1 and q≠0, we define:

Figure 28

and distinguish the following cases:

i) Δ <0. From (2.38)2 it results that |q| >1 and the solutions of (2.25)- (2.26) satis-fying (2.14)-(2.15) are:

Figure 29

This case gives no solutions because: if C<0, then for t→+∞, it results that b(t)→+∞, while if C>0, then for t→+∞ it follows that b(t)→−∞.

ii) Δ =0. Τhen the solutions of (2.25)- (2.26) satisfying (2.12 )- (2.13) are:

Figure 30

This case has no solutions because: if C >0 then for t→+∞, it results b(t)→−∞, while if C<0 we obtain that b(t)→+∞.

iii) Δ >0. Then the solutions of (2.25)- (2.26) satisfying (2.12)- (2.13) are:

Figure 31

From (3.38)2 it results that the term 1−q2 could be positive or negative and we distinguish the following subcases:

a) q >1.

a_i) If C>0, then for t→+∞ from (2.43) we obtain that a(t)→a₂ and b(t)→b_1. We define the parameters R , L and M as follows:

{image:31}

and account the followings: If 1<q ≤ R, then L≥0 and consequently a₂ ≤0. This solution has no physical sense. If 1 < R <q, then L<0 that results a₂>0. Also it is possible to conclude that a₂ >a_o and b₁>b_o.

a_ii) If C <0, the solution has no sense, because M>0 which implies that a₁ <0.

b) q < −1.

b_i) If C>0 then for t→+∞, it implies a(t)→a₁ >0 and b(t)→b₁ >0. We distin-guish the following subcases: If −b_o/ a_o <q < −1, then a₁ >a_o and b₁ >b_{o .} If q = − b_o /a_o then Δ =0 which gives no solution. Finally if q <− b_o/a_o <− 1, then

a₁ <a_o and 0 <b₁ <b_o.

b_ii) If C <0, then for t→+∞ it implies that a(t)→ a₂ and b(t) → b₂ . In present case because of (2.19) and (2.38)2 we obtain that R <−1.If q < R ≤ −1,then L≥0 that leads to a₂ ≤0. This solution has no meaning. If R < q <−1,then L<0, that gives a₂>0. Moreover if −b_o/a_o <q<−1 then a₂ >a_o and 0 <b₂ <b_o , while if q<−b_o/a_o <−1 then 0<a₂ <a_o and b₂ > b_o.

c) 0< q < 1.

c_i) If C >0, for t→+∞ it is possible to conclude that a(t)→ a₂ and b(t) → b₁. In present case because of (2.19) and (2.38)₂ we obtain that 0< R <1. Therefore:

{image:32}

If 0 < q < R <1, then L >0 that results a₂>0. Also we obtain that 0< a₂<a_o and b₁> b_o. If 0 <q =R < 1, then L≤0 that results a₂≤0. This solution has no sense.

c_ii) If C<0, then M>0 that leads to b₂ <0. This solutionn has no physical sense.

d)−1 < q <0.

d_i) If C >0, then for t→+∞ it is possible to obtain that a(t)→ a₁ and b(t) →b₁. If −1<q <−a_o/ b_o ≤0 then from (2.19) it results that K ≤0.Consequently M ≤ 0 which gives a₁ ≤0, while if −1<−a_o/b_o <q <0 then K>0. Also from (2.46) we obtain that 0< R < 1. Therefore if: −1 < q ≤ −R <0, then M≤0 that leads to a₁≤0, while if −1 <−R <q <0, then M>0 that leads to a₁ >a_o. Also it holds b₁>b_o.

d_ii) If C <0, then for t→+∞ it is possible to obtain that a(t) →a₂ and b(t) → b_2.

If −1 < q < −(a0/b0)≤ 0, then from (2.19) it results that K≤0. Consequently M ≥0 that outcomes b₂ ≤0. This solution has no sense. If − 1<−(a0/b0)< q < 0, then K >0. From (2.19) and (2.38)₂ we obtain M>0 that results to b₁ <0. This case is also rejected. All the solutions are found in Table 1.

Discussion

Our model predicts “Medial Tibial Stress Syndrome” or “Shin Splints” a dis-crete clinical situation of the overuse injury of tibia, due to high axial impact forces generated by suddenly increase in athletic activity (duration or/and inten- sity), participation to a new one, inappropriate footwear and training to hard sur-faces (O’ Donoghue,1970; Clisouras, 1984; Cook et., al., 1985; Kaplan, et., al.,1997; Monaco,1997; Amendola et., al., 1999; Walker, 1999; Bouche,1999; Bennet, 2001; Magnusson et., al., 2003; Romansky and Karlson, 2003; Hester, 2006; American Academy of Orthopaedic Surgeons, 2007).This situation is characterized by periosteal inflammation of tibia (Monaco, 1997; Beck, 1998; American Academy of Orthopaedic Surgeons, 2007) and marrow bone density (Couture and Karlson, 2002; Magnusson et., al., 2003; Hester, 2006 ). The last means that the periosteal surfaces moves outwards, while the endosteal surface moves inwards, resulting to bone's hypertrophy (Clisouras, 1984; Kaplan et., al., 1997; Monaco, 1997; A-mendola et., al.,1999; Walker, 1999; Bouche,1999).The “MTTS” occurs very com-monly in runners who are in the beginning of their athletic carrier and have very few years of training background (Clisouras,1984).Therefore the acceptable solutions, are in Table 2. and all them are stable ( Petrovsky, 1966). The new cross-section area of the diaphyseal surface of the tibia for all cases of Table

2. is:

{image:33}

{image:34}