Introduction

Many years ago Carter et., al., (1987) proposed a theory for inter-nal bone remodeling. The basic idea of this theory was that the ap-parent bone density ρ, is expressed in terms of daily loading histo-ry. The last mathematically is approximated by the following equa-tion:

Figure 1

where ρ, n_i, σ_i are respectively the bone apparent density, the number of loading circles and the peak effective stress for the i ^th activity of the day, c is the average number of daily activities, while K, m are coefficients that have not been experimentally determined. However Whalen et., al.,( 1988) initially specified that 2<m <12 for the physi-cal activity and at continuity they accounted the running studies of Williams et., al., (1984); Dalen and Olsson (1974) and concluded that 2< m< 7. Also Carter et., al., (1989) determined the range of the un- known parameter K,for the case of normal activity and for the as- sumed values of m =1 and m=4.

The purpose of this work is to theoretically predict the range of the unknown parameter K, for the case of distance-running. For that reason we will use the model of Carter et., al., (1987), in order to express the density of the distal-end of the femur of a distance -run- ner, in terms of his (her) loading history. During these past years, the loads of his (her) bones due: as to training as to daily normal acti-vities that are common for all people, like normal -walking. However the loads generated by the walking ( Andriacchi et., al., 1977; Rorhle et., al.,1984) are small,compared with the loads generated by the dis- tance- running (Alexander and Jayes, 1980; Bates et., al., 1983; Cavag-na, 1964;Cavanagh and La-Fortune,1980; Fukanaga et., al., 1980; Win-ter,1983) and therefore contribute insignificantly to the regulation of bone density.For that reason,we assume that athlete’s loading history due exclusively to his (her) training- practice.

The model

We assume that we deal with a distance- runner (with an athlete who participates to 5000, or 10000, or Marathon race men/ women)

with N years exersizing background. During these N years, he (she) was training by following an annual exersizing program. Accordin-gly to Clisouras (1984), the annual exersizing of an athlete is divi-ded: to the preparation, to the gaming and to the relaxation periods. Suppose that the athlete was exersizing, accordingly to the training schedules of Tables 1., 2. and 3., taken from Clisouras (1984), as fol-lows: Initially the runner was following the instructions of Table 1. and he (she) was repeating this kind of exersize 6 times, that is the training program was performed for a period of 24 weeks.At con- tinuity he (she) was following the instructions of Table 2. and was repeating this king of training 6 times, that is the total duration of this program was also 24 weeks.Finally he (she) was completing his (her) annual training - schedule, by following the instructions of Tab-le 3.The total duration of the last program, was 4 weeks.In the be- ginning of a new athletic year,the athlete was exersizing as we des-cribed earlier,following the instructions of Tables 1., 2. and 3. With other words, the runner was training with the same manner for N years.We assume that during these years, his (her) body weight was constant.

Accordingly to Carter et., al., (1987) the density of the distal-end of the femur of the athlete, for the first day of exersizing ρ₁ is gi-ven by:

Figure 2

where ρ_ο is the initial density of the distal- end of his (her) femur and is independent of the exersizing and σ_i, n_i, β_i are respectively: the peak cyclic stress, the number of loading circles (per minute) and the total duration of running activity (in minutes) for the first day of exersizing.

By the same way, the density of the distal- end of the femur of the athlete for the second day ρ₂, is given by:

Figure 3

where σ₂, n₂ and β₂ are respectively: the peak effective stress, the number of loading circles (per minute) and the total duration of run- ning activity (in minutes) for the second day of training. Then (2.2) because of (2.1) becomes:

Figure 4

Table 1:The monthly training program for the athlete, during the preparation period. The runner permormed this exersizing schedule 6 times. From Clisouras (1984).

Figure 5

Table 2: The monthly training program of the athlete, during the gaming period. The runner performed this exersizing schedule totally 6 times. From Clisouras (1984).

Figure 6

Table 3: The weekly training program for the relaxation period of the athlete. The runner followed the instructions of this exersizing schedule, for a time period of 4 weeks, that is he (she) performed it 4 times

Figure 7

Consequently the density of the distal- end of the femur of the ath-lete for the first year of training background ρ̅ is given by:

Figure 8

Since the runner had N years training background, the final density of his (her) distal–end of the femur is:

Figure 9

The units of the parameter K can be defined accounting that ρ, ρ_ο are measured in units of mg.cm ² , while β_i, σ_i and n_i are measured in units of minutes, N /cm ² and circles/ minute respectively. Therefore from (2.5), it follows that K is measured in units of mg.cm ³ /N ^1/2 .

The physical approximation of our problem

The peak effective stress of the distal- end of the femur of the athlete during the distance -running, due to the groung reaction force at late stance phase, denoted by G_d.Τhe last is not an axial load,be- cause during the abovementioned phase, an angle α is formed by the femoral neck of the runner and by the vertical direction, as it seems in Figs.1. taken from Clisouras (1984) and 2.This angle satis-fies 0 <α<π/2.The running load G_d can be analysed into two com- ponents: the vertical Gdv and the horizontal Gdh (see Fig. 2), where:

Figure 10

Figure 11

Figure 1: An athlete is distance –running. The photo has been taken from the book of Clisouras (1984).

Figure 12

Figure 2: At late stance phase during the distance running, an angle α is formed by the femoral neck of the athlete and by the vertical direction

From (3.1)_1-2 it is possible to obtain:

Figure 13

The magnitude of the running load G_d is:

Figure 14

Replacing (3.2) into (3.3), it is possible to find:

Figure 15

The vertical component Gdv is given by:

Figure 16

where G_fi, W_f, W_t, are respectively_: the vertical component of ground reaction force at late stance phase during running for the foot, the weights of tibia and foot. Alexander and Jayes (1980); Bates et.,al., (1983); Cavagna (1964); Cavanagh and La- Fortune (1980); Fukanaga et., al., (1980); Winter (1983), calculated the magnitude of bone’s load

for certain values of the magnitude of the running velocity. Selec-ting the experimental data from the abovementioned cites and using a linear regression analysis, is possible to obtain:

Figure 17

where v_i is the running velocity of the athlete for the i day. Accor-dingly to Harless (1860):

Figure 18

Then (3.5) because of (3.6) and (3.7) is written as:

Figure 19

Τhe peak effective stress σ_i for the i day of training, is given by:

Figure 20

where: i =1., 2.,…,365 and B is the body weight of the runner in u- nits of N. Finally S is the cross − section area of the distal – end of the femur given by:

Figure 21

where a, b are the radii that correspond to endosteal and periosteal surfaces of the (distal-end) femur and are constants, since in present work we deal only with the internal remodeling (Frost, 1964; Cowin and Van - Buskirk, 1978; Cowin and Van- Buskirk, 1979).

Substituting (3.4), (3.8) and (3.10) into (3.9), it follows:

Figure 22

Also Fukanaga et., al.,(1980); Ito et., al., (1983) calculated the magni-tude of the number of loading circles,for certain values of the mag- nitude of the running velocity. Selecting these data from the above-mentioned cites and using a second- order regression analysis, it is possible to find:

Figure 23

The study of Whalen et., al., (1988, p. 830) contains a table that de-fines the magnitudes of the velocities which correspond to different physical activities: slow walking, walking, fast walking, jogging and distance- running.Accordingly to this table, the distance-running activi-ty corresponds to a velocity v₂, whose magnitude is at least of the order of:

Figure 24

that is v_i ≥ v₂. Then (3.11), (3.12) because of (3.13), take the forms respectively:

Figure 25

n(v_i) = 0.66v_i ² – 1.524v_i + 84.159 ≥ 0.66v₂ ² −1.524v₂ +84.159 =n(v₂) (3.14)_1-2

Since 0 <cosα <1, from (3.14)₁ we obtain:

Figure 26

Consequently (2.5) because of (3.14)_1-2 and (3.15) takes the form:

Accordingly to (distance) running studies of Dalen and Olsson (1974); Williams (1984) and Whalen et., al., (1988), the range of the unknown parameter m is :

Figure 27

Therefore from (3.16) we conclude:

Figure 28

and it is possible to determine that the range of K is the fol-lowing :

Figure 29

Evaluation of the range of the parameter K

In order to specify the range of K, some data will be used. The values of the initial and final density of the distal-end of the femur of the athlete, as well his (her) mean mass :

Figure 30

are taken from Nilsson and Westlin (1971). Consequently the mean weight of the runner, is:

{image:31}

Accordingly to Clisouras (1984), the ideal age for starting an ath-letic carrier in distance- running,is between 16 - 20 years old. Suppose that the athlete started his (her) track - carrier in the age of 16. Also in the work of Nilsson and Westlin (1971),the mean age of the ath- letes who participated in their measurements, was 22. Therefore we conclude that:

{image:32}

Also the values:

{image:33}

are taken from Cowin and Van- Buskirk (1978). Substituting (3.13), (4.2) and (4.4)_1-2 into (3.15) and (3.14)₂ , it is possible to obtain res-pectively that :

{image:34}

From Table 1, it is possible to obtain that:

{image:35}

Also from Table 2. it is possible to obtain:

Finally from Table 3., it is possible to conclude that:

{image:36}

Substituting (4.1)_1-2, (4.3), (4.5)_1-2 and (4.6) through (4.25) into (3.19), it is possible to find that:

0 < K < 0.3206

Discussion - Conclusions

Accordindly to all that we stated earlier, it is possible to conclude the following: The value of the unknown parameter K is not unique for all individuals, because (even we knew the accurate value of the stress exponent m) it depends from the following factors:

i) The location of the bone in the skeleton. The last means that du-ring the same activity, the magnitude of the load differs from bone to bone.For example during the distance –running,the load of the fe- mur is given by (3.4) and (3.8),while the load of the foot is given by (3.6).

ii)The kind of the physical activity. Even we want to calculate the value of K at the same bone (for example for the femur), the mag- nitudes of the loads generated by different activities, are not the sa-me. For example during walking, the magnitude of the load of the foot denoted by G_o, is given by:

{image:37}

where v_o is the walking velocity. The last belongs to the range [0.5m/sec, 2m/sec] starting with slow and ending with fast walking (Whalen et., al., 1988, p. 830). Eqn. (5.1) has been found taking the experimental data from Andriacchi et., al., (1977) and Rohrle et., al., (1984) who evaluated the magnitude of walking load for certain

magnitudes of walking velocity; and using a linear regression analy-sis (Whalen et., al., 1988) . In contrast with the abovementioned, the magnitude of foot’s load during the distance - running is given again by (3.6), where v ≥v₂= 4.5m/sec.

iii) Finally even we want to calculate the value of K at the same bone and for the same activity, as the magnitude of the load of the bone as the number of the loading circles,differ from one person to another. For example during the distance-running, the modest athlete runs with the minimum velocity v₂ = 4.5m/sec registered in Whalen et., al.,(1988, p. 830) work, while the world champion Kenenisa Beke- le runs with a velocity v = 6.6m/sec corresponding to the world re-cord performance in 5000 men race.(Iaaf-Yearbook 2008).Accordingly to the aboventioned, the magnitude of the load of the foot and the number of the loading circles for the modest athlete, could be eva- luated by replacing the value v₂ = 4.5m/sec into (3.6) and by (4.5)₂ respectively. From the other hand, the corresponding values for the world champion, could be calculated by replacing v=6.6m/sec into (3.6) and (3.12) respectively.

In addition it clearly seems that it is too difficult to evaluated the accurate value of K for each individual, because a lot of infor- mations are required and practictually is impossible to be obtained. Even it was possible to concentrate the required informations, in or-der to find the value of K a lot of evaluations and time spending would be necessary. In contrast with the aboventioned, the method introduced in present study, has the advantage that it can be ap-plied to an extensive team of people who deal with the same phy- sical activity ( for example walkers, distance –runners, sprinters, bicy-cletists), since athletes belonging to the same group has similar trai-ning histories with small divergencies concerning i) the magnitudes of waking / running/ biking velocity and ii) the duration of the daily training. However the velocity magnitudes lie in a certain range and it is adequate to know the minimum value. From the other hand di-vergencies of the duration of the daily training β_i, do not significan tly alter the extracted range of K, since in (3.19) we deal with the 14 ^th root of the total sum and not with the square or third root of it.