Internal Bone Remodeling Induced By The Distance-Running And The Unknown Remodeling Rate Coefficients.
adaptive elasticity, distance - running, tibial stress fracture, unknown remodeling rate coefficients.
M Tsili. Internal Bone Remodeling Induced By The Distance-Running And The Unknown Remodeling Rate Coefficients.. The Internet Journal of Bioengineering. 2008 Volume 4 Number 2.
We used the theory of adaptive elasticity (Hegedus and Cowin, 1976) and we qualitatively studied the internal remodeling of the tibia, induced by the distance- running. We showed that after a long time, an athlete’s tibia initially will be weaken and if the runner will not stop or decrease his (her) activity, eventually will be rupture. The result predicts “Tibial Stress Fracture” an overuse injury of the tibia (Kaplan et., al., 1997; Monaco et., al., 1997; Amendola et., al., 1999; Bouche, 1999; Walker, 1999; Jones, 2002; Romani, 2002; Mc Ginnis, 2005) due to suddenly increase in intensity and/ or du-ration of training activity, participation to new one activity, exersizing to hard surfaces and poor footwear . After that we proved that only 12 possible combinations, for the sign of the five unknown coefficients co, c1, c2, αΤ and αΑ are mathematically predicted.
Living bone is continually undergoing processes of growth, reinforcement and resorption , termed collectively “remodeling”. There are two kinds of bone remodeling: surface and internal (Frost, 1964). The internal bone remodeling ca-pacity has been investigated by many authors (Martin, 1972; Cowin and Hege-dus, 1976; Hegedus and Cowin, 1976; Cowin and Nachlinger, 1978; Cowin and Van-Buskirk, 1978; Carter et., al., 1987; Whalen, et., al., 1988; Carter et., al.,1989;
Tsili, 2000, Qin and Ye, 2004; Tsili, 2009a).
The purpose of this work is to study the internal remodeling of the tibia, induced by the distance-running. For that reason we will use the theory of a-daptive elasticity (Hegedus and Cowin, 1976).
Biomechanical analysis of the distance -running:
We assume that a person with bones which are not under osteoporosis or osteopetrosis (Hegedus and Cowin, 1976) starts distance- running activity and he (she) continues to be exercised , for a long time period. Initially the athlete was
walking with a constant velocity v o . Consequently the 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 pha-se during walking, given by:
G o =G w – W f (2.1)
where G w , W f are respectively: the vertical component of ground reaction force
at late stance phase during normal walking and the weight of foot. Andriacchi
et., al., (1977) and Rohrle et., al., (1984) used force-plate studies and evaluated the magnitude of the running load for certain magnitudes of the running velo-city. Selecting the above data and using a linear regression analysis, it is pos-sible to find: (see also Whalen et., al., 1988)
G w = 0.213v o + 0.913 in units of B.W. (2.2)
Also accordingly to Harless (1860) :
W f =0.019B.W. in units of B.W. (2.3)
Then (2.1) because of (2.2) and (2.3) becomes:
G o = 0.213v o + 0.894 in units of B.W. (2.4)
Accordingly to Whalen et., al., (1988) the walking activity corresponds to a ve-locity whose magnitude lies in the range [0.5m/sec, 2m/sec], starting with slow and ending with fast walking. Then from (2.4) it implies that:
1.0005B.W. < G o < 1.32B.W. (2.5)
Also initially the tibia had a uniform volume fraction ξ ο , 0< ξ ο <1 and a uni-form relative volume fraction e o = 0.
At t = 0 the athlete starts running activity as it seems in Fig. 1,taken from
Clisouras (1984). Accordingly to Clisouras (1984) , runners are classificated into
three categories: sprinters, middle - distance and distance - runners. The last cate-gory contains: the cross - country and track runners who participate to 5000m,
10000m and marathon race of men / women, as well new soldiers or students
of military Academies, who follow their basic training. At late stance phase du-
ring running the foot of the sprinter, middle distance - running and distance – running, contacts the ground: with the toes (forefoot strikers), the second third of the foot (middlefoot strikers) and the heel (rearfoot strikers ) respectively (Ca- vanagh and LaFortune, 1980; Clisouras,1984). The abovementioned styles of running seem in Figs. 2a, 2b and 2c (see also Tsili, 2009b).Also accordingly to Cli- souras speaking for distance - running we mean that the athlete / new soldier / student of military Academy runs with a 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) such that:
G z = G zf –W f (2.6)
where G zf is the vertical component of ground reaction force at late stance
phase for the foot during running.
Alexander and Jayes (1980); Bates et., al., (1983); Cavagna (1964); Cavagna and La-Fortune (1980); Fukanaga et., al.,(1980); Winter (1983) used force-plate studies and computed the magnitude of the run-ning load, for certain magnitudes of running velocity. Selecting the above data and using a linear regression analysis, it is possible to obtain (see also Whalen, et., al., 1988):
G zf = 0.46v + 0.55 in units of B.W (2.7)
while v is the running velocity. Then (2.6) because of (2.3), (2.7) becomes:
G z = 0.46v + 0.531 in units of B.W. (2.8)
We will study the dynamic problem of a hollow circular cylinder, consising of a material which behavior is described by the theory of adaptive elasticity (Hegedus and Cowin, 1976). The cylinder has an inner and outer radii: a and b respectively and a length L. These radii are constants, since in present work we deal only with the internal remodeling ( Frost, 1964; Hegedus and Cowin, 1976; Cowin and Van-Buskirk, 1978; Cowin and Van-Buskirk, 1979).
where B is athlete’s body weight and assumed to be constant during the training period.
Accordingly to Cowin and Nachlinger (1978), our problem has a unique so-lution. Assume the followings:
u r = A(t)r +B(t) /r u θ =0 and u z = C(t)z (3.9) 1-2-3
where A(t), B(t) , C(t) are unknown functions. Then (3.2) becomes:
Consequently the stress - strain relations (3.4) are now written as:
Applying the boundary conditions, it is possible to find that:
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:
A(e) = c o +c 1 e + c 2 e 2 A T (e) = α T + eα T A A (e) = α A + eα A λ 1 (e) = Λ 1 + eΛ 1
λ 2 (e) = Λ 2 + eΛ 2 μ 1 ( e) = Μ 1 + eΜ 1 and μ 2 ( e) = Μ 2 +eΜ 2 (3.15)1-2-3-4-5-6- 7
where c o , c 1 , c 2 , α Τ, α Α , Λ 1 , Λ 2 , Μ 1 , Μ 2 are constant coefficients. Particularly Λ 1 ,
Λ 2 , Μ 1 and Μ 2 are known (Cowin and Van- Buskirk, 1978), while the rests are
Thus we deal with an equation of the form:
ė = α( e 2 −2βe + γ) (3.17)
The mathematical analysis, the exploration of the solutions of (3.17) that satisfy the initial condition e(0) =0 and the final results, are in my earlier work (see Tsili, 2000, pp.237-238 ) and they will be repeated here. We only remind that :
Our model predicts
Tibial stress fracture occur among persons with normal bones who are undergoing physical activity to which they are unaccustomed (Devas,1969; Devas 1970;Belkin,1980;Jones et., al., 2002).Increased or different activity results in an altered relationship of bone growth and repair (Wolff law). When remodeling predominates over repair, the cortex transiently weakens and if stress continues, eventually ruptures (Walker, 1999) . The resulting stress may run the spectrum from a microfracture to rupture of bony cortices with a fracture line (Mark-ley, 1987; Walker, 1999).
In the early stages of stress fracture, bone scans are negative. As the stress fracture begins to mature, typical findings show i): subperiosteal resorption and . occasionally a small fracture line for the posteromedial area of tibia and ii): “the dreaded black line” of the horizontal fracture, for the anterior area of this bone (Monaco et., al., 1997 ; Walker , 1999) as it seems in Fig. 3. Therefore the
acceptable solutions of our problem are in Table 1. Accordingly to this table after a long time, the tibia of the runner initially will be weaken. At continuity if the athlete will not temporally interrupt or decrease his (her) running activity, the tibia eventually will be ruptured.
As we stated earlier, in the remodeling rate equation (3.16) there are 5 unknown coefficients c o , c 1 , c 2 , α Α , α Τ . Consequently there are 3 5 = 243 possible cases, concerning their signs. Employing the initial condition e(0) = 0 into (3.16), it is possible to obtain:
If c o =0, then from (5.1) it results that Λ 1 α Τ – (Λ 2 + Μ 2 )α Α = 0. Consequently the rate remodeling equation (3.16) concludes to the form:
ė = c 2 e 2 +c 1 e (5.2)
and its meaning is that: bone remodeling process is not depended upon the applied stresses, or with other words the mechanical loads do not affect the bone remodeling process. The last contradicts to the law of Wolff (1884; 1892) and therefore c o ≠0.
We distinguish two cases, for the type of the solutions of Table 1.
i) If the first or the third solutions of Table 1 hold, then from (3.18)1 we ob-tain that c 2 >0. Also from (3.19)1-2 it is easy to find that γ < 0 .Then (3.18)3 because of (3.18) 1 and (5.1) results to:
[Λ 1 α Τ (Λ 2 +Μ 2 )α Α ](G f –G o )B / π(b 2 −a 2 )F <0 (5.3)
Accordingly to Cowin and Van_-Buskirk (1978, p. 274):
Λ 1 = 40GPα Λ 2 = 40GPα Μ 1 = 7GPα and Λ 2 = 4GPα (5.4) 1-2-3-4
Consequently from (3.18) 4 and (5.3) we conclude that:
F>0 and [Λ 1 α Τ – (Λ 2 +Μ 2 )α Α ]R < 0 (5.5) 1-2
Accordingly to Whalen et., al., (1988) the walking and running activities cor- respond to velocities whose magnitudes are of the order of : v o = 1.25m/sec and v =4.5m/sec respectively. Then from (2.4) and (2.8) it is possible to conclude that G o and G f are of the order of 1.1B.W. and 2.601 B.W. respectively.
Then from (5.6) we obtain that Q>1. 501 and R>0, while (5.5) 2 by the help of (5.6) 1 becomes:
Λ 1 α Τ – (Λ 2 +Μ 2 )α Α <0. (5.7)
Then from (5.1) and (5.5) 1 we find that c o > 0. Αlso (5.7) because of (5.4) 1-2-4 concludes:
α Τ <1.1α Α (5.8)
The parameters α Τ and α Α might both be positive or negative, or it might α Τ <0 and α A >0. The case α Α =0 is excluded, because from (3.15) 3 it follows that the material function A A (e) vanishes. The last contradicts to the fact that the bone is a transversely isotropic material (Reilly and Burstein, 1975).Table 2. contains all possible combinations, for the sign of the unknown remodeling rate coefficients.
ii) If the second or forth solutions of Table 1. hold, then from (3.18)1 it im-plies that c 2 > 0. From (3.19) 2 it is possible to obtain that β <0 and γ >0. The
first because of (3.18) 2 gives that c 1 > 0, while the second by the help of
(3.18) 1 , (3.18) 3 and (5.1) concludes to:
[Λ 1 α Τ −(Λ 2 +Μ 2 ) α Α ](G f −G o )B / π (b 2 –a 2 )F >0 (5.9)
The last by the help of (5.5) 1 leads to :
[Λ 1 α Τ −(Λ 2 +Μ 2 )α Α ] R >0 (5.10)
where R is again defined from (5.6) 1 . Then (5.10) gives us that:
Λ 1 α Τ − (Λ 2 +Μ 2 )α Α >0 (5.11)
Τhen from (5.1) it implies that c o <0. Also (5.11) because of (5.4) 1-2-4 con-
α T >1.1 α Α (5.12)
The parameters α Α and α Τ might be both positive or negative or it might be α T >0 and α A < 0. Table 3. contains all possible combinations for the sign of the unknown coefficients. Accordingly to the results of Tables 2. and 3., totally 12 cases for the sign of c o , c 1 , c 2 , α Α , α Τ are theoretically predicted. Thus the i-nitial number of 243 mathematically possible cases, is dramatically restricted.