# Surface bone remodeling induced by the Distance-running and medial tibial stress syndrome (shin splints)

M Tsili

###### Keywords

decreased bone marrow density, distance- running, medial tibial stress syndrome, periosteal in-flammation, surface remodeling, tibia

###### Citation

M Tsili. *Surface bone remodeling induced by the Distance-running and medial tibial stress syndrome (shin splints)*. The Internet Journal of Bioengineering. 2008 Volume 4 Number 1.

###### Abstract

We based on the theory of Cowin and Firoozbakhsh,(1981) and qua- litatively studied the surface remodeling of tibia,induced by the distance -run- ning. We showed that its periosteal and endosteal surfaces will move outwards and inwards respectively. The result predicts

### 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.,

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:

Accordingly to Whalen et., al.,(1988) the walking activity corresponds to a ve- _{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

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 5

##### Figure 6

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:

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:}

### 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):

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:

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 :

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:

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:

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

Defining the parameters:

and accounting that:

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

From the above it follows that:

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

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

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

### 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:

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

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:

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

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

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

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:

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:

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:

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)→a_{2} 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 a_{2} ≤0. This solution has no physical sense. If 1 < R <q, then L<0 that results a_{2}>0. Also it is possible to conclude that a_{2} >a_{o } and b_{1}>b_{o}.

a_{ii}) If C <0, the solution has no sense, because M>0 which implies that a_{1} <0.

b) q < −1.

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

a_{1} <a_{o} and 0 <b_{1} <b_{o}.

b_{ii}) If C <0, then for t→+∞ it implies that a(t)→ a_{2 } and b(t) → b_{2} . 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 a_{2} ≤0. This solution has no meaning._{2}>0. Moreover if −b_{o}/a_{o} <q<−1 then a_{2} >a_{o} and 0 <b_{2} <b_{o} , while if q<−b_{o}/a_{o} <−1 then 0<a_{2} <a_{o} and b_{2} > b_{o}.

c) 0< q < 1.

c_{i}) If C >0, for t→+∞ it is possible to conclude that a(t)→ a_{2} and b(t) → b_{1}. 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 a_{2}>0. Also we obtain that 0< a_{2}<a_{o} and b_{1}> b_{o}. If 0 <q =R < 1, then L≤0 that results a_{2}≤0. This solution has no sense.

c_{ii}) If C<0, then M>0 that leads to b_{2} <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)→ a_{1 } and b(t) →b_{1}. If −1<q <−a_{o}/ b_{o} ≤0 then from (2.19) it results that K ≤0.Consequently M ≤ 0 which gives a_{1} ≤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 a_{1}≤0, while if −1 <−R <q <0, then M>0 that leads to a_{1} >a_{o}. Also it holds b_{1}>b_{o}.

d_{ii}) If C <0, then for t→+∞ it is possible to obtain that a(t) →a_{2} 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 b_{2} ≤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 b_{1} <0. This case is also rejected. All the solutions are found in Table 1.

### Discussion

Our model predicts

2. is:

{image:33}

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