Abstract

In present paper we considered if temperature effects the growth of microcracks in a broken femur with AMBI. We studied three particularly points of fracture. We used theory of adaptive elasticity neglecting and accounting temperature and energy density theory. We showed for both cases: that after a long time femur locally at points of our interest: i) will be (quickly or normally or delayed) united. Our results are verified by clinical studies. We concluded that temperature plays no role to growth of microcracks.

1. Introduction

The purpose of this paper is to study if temperature effects the growth of microcracks in a broken femur with AMBI. For this reason we will use the theory of adaptive elasticity ^{1, 2} neglecting and accounting temperature and energy density theory ^{3, 4, 5}.

Macroscopically the bone has a volume V and a surface S. The volume V microscopically consists of microvolumes which generally are not homogenous ^{4, 5}. Expanding theory of adaptive elasticity [ ¹, p. 322] at microscopic area we assume:

i) A microvolume of bone consists of an elastic microvolume (micromatrix bone) and of microporous (microcracks)

(1)

where Δp is the volume of microcracks.

From the other hand Sih [ ⁴, p.179] showed that every microvolume contains an homogenous microvolume. Thus we suppose that an elastic microvolume ΔV given by (1) is homogenous microvolume.

ii) The mechanical properties of microvolume of bone coincides with the mechanical properties of homogenous microvolume of micromatrix bone.

iii) The fraction of microvolume of the micromatrix bone is defined as [ ², p. 322]:

(2)

where is the density of microvolume , while is the density of material (bone) and assume to be constant. From the above it follows .

v) The porosity that is the mean length of microcraks of microvolume alters with mass added /removal to /from micro matrix bone and linearly depends from the history of microstrain [ ¹, p. 322]. The above is characterized by a parameter ê ²:

(3)

where is the initial fraction of the microvolume of micromatrix bone. With other words parameter is the change of the mean value of microcracks.

2. The Problem and Its Physical Approximation

Assume that someone breaks his /her left femur due to fall or traffic accident. Suppose that we deal with an intertrochanteric fracture type Α1 and case 3 [ ⁶, p.120; ⁷, p. 60]. The fracture starts from proximal femur and results to the last third of its diaphysis. We impose into injury bone a prosthetic device AMBI as indicates in Figure 1 and suggest to patient to stay at bed immobilization three months. We want to predict the situation of femur when patient start again walking, locally at three points of fracture area. The lasts have no contact with AMBI as indicated in Figure 2 and are the followings:

Γ: at middle distance between endosteal and periosteal surface and at the end of diaphysis of femur. The origin and end of the diaphysis of femur are defined as: nearest to joint knee and to proximal femur respectively.

Δ: at endosteal surface of femur and at 5/6 of the distance between origin and end of femur’s diaphysis.

Ε: at periosteal surface of femur and at 2/3 of the distance between origin and end of femur’s diaphysis.

At t=0 the last third of the diaphysis of femur was separated into two pieces due to fracture. Consequently all points that belong to fracture area ABEZ had and the same goes for points of our interest Γ, Δ and Ε. At t >0 we impose a prosthetic device AMBI into broken femur as indicated in Figure 1 and Figure2. We want to predict ê(t) after long time period at points Γ, Δ and Ε.

3. The Mathematic Modeling and the Solution of Our Problem

Femur’s diaphysis is modeled as a hollow circular cylinder with length L, inner and outer radii a and b respectively. These radii corresponds to endosteal and periosteal surfaces of bone and assume to be constant due to internal remodeling ⁸. Since we deal with microscopic area, we use the density energy theory ^{3, 4, 5}.

The equations of the above theory in cylindrical coordinates are:

i) the stress relations between macroscopic and microscopicarea [ ⁴, p. 182]

where _, and (dV/dA) are respectively: the macroscopic displacement, the stress on microvolume, the stress at macrovolume and the change of volume with surface ^{4, 5}

ii) the macrostress - equations [ ⁵, p.182]:

iii) the microstress equations [ ⁵, p.182]:

iv) the microstrain-microdisplacement relations [ ⁵, p.179):

v) the microstress - microstrain relations:

where:

where Ε_Α, Ε_Τ and v_A, v_Τ are Young’s modulus and Poisson ratio in transverse and axial direction at macroscopic area.

Finally rate remodeling equation ² at microscopic area without /and accounting temperature are respectively:

where A_T(ê), Α_Α(ê) are rate remodeling coefficients in transverse and axial direction respectively while B(ê) is a rate remodeling coefficient depends from temperature.

The boundary conditions of our problem are:

i) at point Γ:

ii) at point Δ:

iii) at point Z:

Our problem has a unique solution [ ⁵, p.186] and assume that microdisplacements are of the form:

where A(t), B(t), C(t) are unknowns. Then (7) are written as:

Therefore microstress - microstrains relations (8) because of (12) take the forms:

Applying the boundary conditions (11)-(12)-(13) into (16) it is possible to obtain unknown functions A(t), B(t) and C(t):

i) at point Γ:

ii) at point Δ:

iii) at point E

Employing (17)-(18)-(19) into (14) it is possible to calculate microdisplacements for points Γ, Δ and Ε. Also (15) due to (17)-(18)- (19) become:

i) at point Γ:

ii) at point Δ:

iii) at point Ε:

At continuity we distinguish the following cases:

i) Internal remodeling of femur does not depends upon temperature:

Then substituting (20), (21), (22) into (10)₁ it follows:

i) at point Γ:

(23)

ii) at point Δ:

(24)

iii) at point Ε:

(25)

At continuity we impose ^{9, 10}

(26)

Therefore (23)-(24)-(25) conclude to the following form:

(27)

where:

i) at point Γ:

ii) at point Δ:

iii) at point Ε:

The solution of (27) must satisfy the following condition

(31)

since the mean length of microcracks can not be negative.

From the other hand (27) satisfies the initial condition:

(32)

because as we stated earlier points Γ, Δ and Ε belong to fracture area. We defined and distinguish the following cases ¹⁰:

1) . The solution of (27) that satisfies (32) is:

(33)

i) If , then for , then it results that . We distinguish the following subcases: i_a) If the solution has no physical sense, since contradicts to (31). i_b) If then when the patient get of the bed immobilization, femur at points Γ, Δ and Ε will be under osteopetrosis. I_c) Finally if , then when patient get of bed immobilization, femur at points Γ, Δ, Ε will be united. In addition if value of parameter is great, then value of is small. The last means that femur at points of our interest will be quickly united. If value of parameter α⁽ⁱ⁾ is small, then the value of is great. The last means that femur at points of our interest will be normally united.

ii) If , then for it follows that . The last means that femur at points Γ, Δ and Ε, may be under a delayed union (healing).

iii) If then . This is the worst case because it means that the femur at points of our interest has not a union (healing).

2) . The solution of (27) is:

(34)

i) If K = 0, then employing initial condition (32) into (34) it results that . The last contradicts to (32) and there is no solution. ii) If , then employing initial condition (32) into (34) it possible to obtain . Specially if K>0 then , that is and we coincide with case 1. iii). From the other hand if K<0 then , that is and we coincide with case i).

3) . Then the solution of (27) that satisfies (32) is:

(35)

where:

From the above it follows that:

(37)

Employing the initial condition (32) into (37) it follows:

(38)

From the above we conclude that K=1 and . The last contradicts to (37). Therefore (35) has no solution.

The solutions for finite and infinite t are in Table 1. and Table 2. respectively. Accordingly to all that results when patient get of the bed immobilization, his (hers) femur at points of our interest will be: quickly, or normally, or delay united, or it will under osteopetrosis, or will be not united.

ii) Internal remodeling depends upon temperature:

Τhen (10)₂ because of (20), (21), (22) for all points Γ, Δ and Ε is written as:

(39)

Finally substituting (26) and

(40)

into (39) we again result to (27), where:

i) at point Γ:

ii) at point Δ:

iii) at point Ε:

The solution of (27) satisfying initial condition (32) are given by (33), (34) and (35). The acceptable solutions are the same as at previous case and are given by Table 1 and Table 2.

4. Discussion and Conclusion

Our theoretical results for both cases: neglecting and accounting temperature are verified by clinical studies. Papasimos [ ⁶, p. 92] and ¹¹ studied 40 cases of broken femur with AMBI. All fractures were united (100%) in a period ranged from 45 days until six months. Particularly 24 fractures were united in 45 days (quickly union 60%) 13 of them in 3 months (normally union 32%) and rests in 6 months (delay union7.5%). The mean time of union for all cases was 3.15 months and corresponds to normal union [ ¹¹, p.93].

Also other examples of quick union to humans diaphysis and to animals methaphysis has been reffered ^{12, 13, 14}. The acceptable solutions are in Table 3. From the above we result that temperature plays no role to growth of microcracks in a broken femur with prosthetic device AMBI.

References