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Does Temperature Effects the Growth of Cracks in a Tibia due to Distance – running?

M. Tsili , D. Zacharopoulos
Biomedical Science and Engineering. 2017, 5(1), 5-8. DOI: 10.12691/bse-5-1-2
Published online: April 19, 2017

Abstract

In present paper we investigated if temperature plays any role to the growth of cracks in a tibia due to distance-running. We used modified theory of adaptive elasticity taking into account the temperature. We compared our results with that of the corresponding problem neglecting temperature and we concluded that temperature effects the bone disease.

1. Introduction

It is well known that bone fracture due to many factors: as age 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, microstructure 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, bone density 2, 3, 8, 9, 10, 14, 17, 24, 25 and loading mode 14, 23, 24, 25, 26, 27, 28. From the other hand very few known studies investigated the effect of temperature in bone disease 23, 25.

The purpose of this paper is to study if temperature plays role to growth of cracks in a tibia due to distance-running. For that reason we will base upon the theory of adaptive elasticity 29 assuming that rate remodeling equation depends also from temperature.

2. The Problem and Its Physical Approximation

i) The internal remodeling of tibia due to distance running neglecting temperature:

In earliest paper we studied the internal remodeling of tibia due to distance-running neglecting temperature 30. We modeled tibia as a hollow circular cylinder with constant inner and outer radii a and b respectively and we assumed that it was under a constant axial load Gf. We follow a process analytically described in 30 and we used the rate remodeling equation 29:

(1)

where e(t) is the change of the volume fraction of the bone, from its reference state, that is the change of the mean length of its cracks, while A(e), AT(e), ΑΑ(e) are rate remodeling coefficients. At continuity we imposed 30:

and we concluded to:

(3)

where:

In (4)3 B is the weight of athlete assumed to be constant during the period of training.

The solutions of (1) satisfying initial condition 29:

(5)

were:

Since e(t) is defined as the mean length of the cracks of the bone, it must be and . Therefore the acceptable solutions for are in Table 1.

ii) The internal remodeling of tibia due to distance running accounting temperature.

We use the modified rate remodeling equation:

(7)

where B(e) is an unknown rate coefficient depends from temperature of bone. We follow exactly the same process as in 30 and in addition with (2) we impose:

(8)

where is an unknown rate remodeling equation depending upon temperature. We again conclude to

(9)

where:

The solutions of present problem are:

and the acceptable solutions are again in Table 1. with the only difference that instead of (6)1-2 we replace (11)1-2. The atrophia arised initially in tibia at continuity will be worse and after a long time period will progressively result to “stress fracture” 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57.

3. Comparison of the Solutions of Earliest and Present Problem

If then from (4)2 and (10)2 we obtain respectively:

Then the solution (6)2 has no physical sense since . Also from (6)1 it is possible to conclude that . We impose

Therefore (4)3 and (10)3 due to the above can be written as respectively:

Since 30, from (4)1 and Table 1. it follows that: .

We distinguish the following cases:

i) If , then from (12)2 it is possible to obtain that . Since from (11)2 it implies that . Therefore (14)2 gives . Consequently and it follows that: . Therefore: and due to (14)1-2 it is possible to obtain . Finally the combination of (6)1,(10)1, (12)2 and (14)1-2 gives that:

(15)

that is the porosity of tibia accounting temperature is greater than that neglecting temperature. With other words the mean length of cracks at our case seem to be greater than the corresponding mean length of cracks of the case neglecting temperature 30.

ii) If , then from (12)2 it results that . Then the solution (10)2 has no physical sense since . Also from (10)1 it is possible to obtain that . Therefore from (14)2 it results that . Consequently and it follows that . Therefore and due to (14)1-2 it is possible to conclude that .

Since it follows that . The last implies that: and consequently: . Taking into account (4)1, (10)1 it is possible to conclude:

(16)

that is the porosity of tibia neglecting temperature is greater than that accounting it. With other words the mean length of cracks of the case neglecting temperature 30 seem to be greater than the case of present problem.

4. Discussion and Conclusion

Our theoretical results come to accordance with previous studies 25, 58, 59. Particularly the dependence between fatigue fracture of bone and temperature has been investigated by Carter and Hayes 58. Also Carter et., al., 25 dealed with fatigue tests to failure of bone specimens at four temperature levels (21-45°C). The test results demonstrated highly significant correlation between fatigue life and temperature. In addition Yan et., al., 59 investigated the depence of temperature on the fracture toughness of compact bone and resulted that there is a revengelly analogous relation. Finally Murcia et., al., 60 investigated if temperature effects the fracture resistance in cyprinus fishes. The results showed that there was a significant reduction in tear resistance with decreasing temperature and the lowest resistance to fracture was obtained at -150°C.

Therefore we conclude that temperature plays role to the growth of cracks in a tibia due to distance running.

References

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In article      View Article
 
[2]  Thompson, D. (1980). “Age changes in bone mineralization, cortical thickness, and haversian canal area.”Calcif Tissue, Int. 31, 5-11.
In article      View Article
 
[3]  Grynpas, M. D.and Holmyard, D. (1988). “Changes inquality of bone mineral on aging and in disease”. Scan Microsc. 2, 1045-1054.
In article      PubMed
 
[4]  Hui, S. L., Slemenda, C. W. and Johnston, C. C. (1988). Age and bone mass as predictors of fracture in a prospective stu- dy. J. Clin. Invest. 81, 1804-1809.
In article      View Article  PubMed
 
[5]  Kiebzak G. M. (1991). “Age-related bone changes”. Exp. Ge- rontol. 26, 171-187.
In article      View Article
 
[6]  Simmons, E. D., Pritzker, K. P. and Grynpas, M. D. (1991). “Age - related changes in the human femoral cortex.” J. Orthop. Res.9, 155-167
In article      View Article  PubMed
 
[7]  Melvin JW.. “Fracture mechanics of bone.” J. Biomech. Eng. 1993. Nov; 115 (4B): 549-554.
In article      View Article
 
[8]  Currey, J. D., Brear, K. and Zioupos, P. (1996). “The effects of aging and changes in mineral content in degrading the toughness of human femora”. J. Biomech. 29, 257-260.
In article      View Article
 
[9]  Francis, R. M. (1996). “Low bone mineral content is com-mon but osteoporotic fractures are rare in elderly rural Gam- bian women.” J. Bone Miner. Res.11, 1019-1025.
In article      PubMed
 
[10]  Aspray, T. J., Prentice, A., Cole, T. J., Sawo, Y., Reeve, J. and Francis, R. M. (1996). “Low bone mineral content is com- mon but osteoporotic fractures are rare in elderly rural Gam- bian women.” J. Bone Miner. Res.11, 1019-1025.
In article      View Article  PubMed
 
[11]  Yeni, Y. N. and Norman, T. L. (2000). “Fracture toughness of human femoral neck: Effect of microstructure, composition and age.” Bone 26, 499-504.
In article      View Article
 
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In article      View Article
 
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In article      View Article  PubMed
 
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In article      View Article
 
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In article      View Article
 
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In article      View Article
 
[17]  Yeni, Y. N., Brown, C.U. and Norman, T.L. (1998). “Inf- luence of bone composition and apparent density on fracture toughness of the human femur and tibia”. Bone 22, 79-84.
In article      View Article
 
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In article      View Article
 
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In article      View Article
 
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In article      View Article
 
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In article      View Article
 
[22]  Seeman, E. (1999). “The structural basis of bone fragility in men.” Bone 25, 143-147
In article      View Article
 
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In article      View Article
 
[24]  Ford C.M. and Keaveny, T.M. (1996). “The dependence of shear failure properties of trabecular bone on apparent density and trabecular orientation.” J. Biomech.29, 1309-1317.
In article      View Article
 
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In article      View Article
 
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In article      View Article
 
[27]  Feng, Z., Rho, J., Han, S. and Ziv, I. (2000). “Orientation and loading condition dependence of fracture toughness in cortical bone. Mater. Sci. Engng CC11, 41-46.
In article      View Article
 
[28]  Feng X. and M. McDonald J. (2011). “Disorders of Bone Remodeling,” Annu Rev Pathol.; 6: 121-145.
In article      View Article  PubMed
 
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In article      View Article
 
[30]  Τsili M. (2008b). “Internal bone remodeling induced by the distance - running and the unkown remodeling coefficients.” in: www.ispub.com/journal- of- internet journal of bioengineering, Vo- lume 4. number 2,
In article      
 
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In article      
 
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In article      
 
[33]  Amendola A., Clatworthy M., and Μagness S. (1999). “Overuse injuries of the lower extremity” Chapt, 35., From the book OKY Orthopedic Knowledge Update Sports Medicine (Ed. by the Arenth) American Academy of Orthopaedic Sur- geons, Rosemont, Illinois.
In article      
 
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Published with license by Science and Education Publishing, Copyright © 2017 M. Tsili and D. Zacharopoulos

Creative CommonsThis work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

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M. Tsili, D. Zacharopoulos. Does Temperature Effects the Growth of Cracks in a Tibia due to Distance – running?. Biomedical Science and Engineering. Vol. 5, No. 1, 2017, pp 5-8. http://pubs.sciepub.com/bse/5/1/2
MLA Style
Tsili, M., and D. Zacharopoulos. "Does Temperature Effects the Growth of Cracks in a Tibia due to Distance – running?." Biomedical Science and Engineering 5.1 (2017): 5-8.
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Tsili, M. , & Zacharopoulos, D. (2017). Does Temperature Effects the Growth of Cracks in a Tibia due to Distance – running?. Biomedical Science and Engineering, 5(1), 5-8.
Chicago Style
Tsili, M., and D. Zacharopoulos. "Does Temperature Effects the Growth of Cracks in a Tibia due to Distance – running?." Biomedical Science and Engineering 5, no. 1 (2017): 5-8.
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[1]  Burstein, A., Reilly, D. and Martens, M. (1976). “Aging of bone tissue mechanical properties. J. Bone Joint Surg”. A58, 82-86.
In article      View Article
 
[2]  Thompson, D. (1980). “Age changes in bone mineralization, cortical thickness, and haversian canal area.”Calcif Tissue, Int. 31, 5-11.
In article      View Article
 
[3]  Grynpas, M. D.and Holmyard, D. (1988). “Changes inquality of bone mineral on aging and in disease”. Scan Microsc. 2, 1045-1054.
In article      PubMed
 
[4]  Hui, S. L., Slemenda, C. W. and Johnston, C. C. (1988). Age and bone mass as predictors of fracture in a prospective stu- dy. J. Clin. Invest. 81, 1804-1809.
In article      View Article  PubMed
 
[5]  Kiebzak G. M. (1991). “Age-related bone changes”. Exp. Ge- rontol. 26, 171-187.
In article      View Article
 
[6]  Simmons, E. D., Pritzker, K. P. and Grynpas, M. D. (1991). “Age - related changes in the human femoral cortex.” J. Orthop. Res.9, 155-167
In article      View Article  PubMed
 
[7]  Melvin JW.. “Fracture mechanics of bone.” J. Biomech. Eng. 1993. Nov; 115 (4B): 549-554.
In article      View Article
 
[8]  Currey, J. D., Brear, K. and Zioupos, P. (1996). “The effects of aging and changes in mineral content in degrading the toughness of human femora”. J. Biomech. 29, 257-260.
In article      View Article
 
[9]  Francis, R. M. (1996). “Low bone mineral content is com-mon but osteoporotic fractures are rare in elderly rural Gam- bian women.” J. Bone Miner. Res.11, 1019-1025.
In article      PubMed
 
[10]  Aspray, T. J., Prentice, A., Cole, T. J., Sawo, Y., Reeve, J. and Francis, R. M. (1996). “Low bone mineral content is com- mon but osteoporotic fractures are rare in elderly rural Gam- bian women.” J. Bone Miner. Res.11, 1019-1025.
In article      View Article  PubMed
 
[11]  Yeni, Y. N. and Norman, T. L. (2000). “Fracture toughness of human femoral neck: Effect of microstructure, composition and age.” Bone 26, 499-504.
In article      View Article
 
[12]  Wang, X., Shen, X., Li, X. and Agrawal C. M. (2002). “Age- related changes in the collagen network and toughness of bone.” Bone 31, 1-7.
In article      View Article
 
[13]  Akkus, O., Adar, F. and Schaffler, M. B. (2004). “Age-rela- ted changes in physicochemical properties of mineral crystals are related to impaired mechanical function of cortical bone.” Bone 34, 443-453.
In article      View Article  PubMed
 
[14]  Ritchie. R, Kinney H., Kruzic R., and Nalla R. (2005). “A fracture mechanics and mechanistic approach to the failure of cortical bone.” Fatigue Fract. Engng Mater Struct. 28, 345-37
In article      View Article
 
[15]  Behiri, J. C. and Bonfield, W. (1989). “Orientation dependence of the fracture mechanics of cortical bone.” J. Biomech,. 22, 863-872.
In article      View Article
 
[16]  Yeni, Y. N., Brown, C. U., Wang, Z. and Norman, T. L. (1997). “The influence of bone morphology on fracture tough- ness of the human femur and tibia.” Bone 21, 453-459.
In article      View Article
 
[17]  Yeni, Y. N., Brown, C.U. and Norman, T.L. (1998). “Inf- luence of bone composition and apparent density on fracture toughness of the human femur and tibia”. Bone 22, 79-84.
In article      View Article
 
[18]  Feng, Z., Rho, J., Han, S. and Ziv, I. (2000). “Orientation and loading condition dependence of fracture toughness in cortical bone.” Mater. Sci. Engng CC11, 41-46.
In article      View Article
 
[19]  Brown, C. U., Yeni, Y. N. and Norman, TL.(2000). “Fractu- re toughness is dependent on bone location-A study of the femoral neck, femoral shaft and the tibial shaft.” J. Biomed. Mater. Res.49, 380-389.
In article      View Article
 
[20]  Phelps, J. B., Hubbard, G. B., Wang, X. and Agrawal, C. M.(2000). “Microstructural heterogeneity and the fracture toughness of bone. J. Biomed.” Mater. Res.51, 735-471.
In article      View Article
 
[21]  Yeni, Y. N. and Norman, T. L.(2000). “Fracture toughness of human femoral neck: Effect of microstructure, composition and age.” Bone 26, 499-504.
In article      View Article
 
[22]  Seeman, E. (1999). “The structural basis of bone fragility in men.” Bone 25, 143-147
In article      View Article
 
[23]  Rimnac, C. M., Petko, A. A., Santners, T. J. and Wright, T. M (1993). “The effect of temperature, stress and microstructure on the creep of compact bovine bone”. J. Biomech.26, 219-228.
In article      View Article
 
[24]  Ford C.M. and Keaveny, T.M. (1996). “The dependence of shear failure properties of trabecular bone on apparent density and trabecular orientation.” J. Biomech.29, 1309-1317.
In article      View Article
 
[25]  Carter, D. R. and Hayes, W. C. (1976). “Fatigue life of com- pact bone-I. Effects of stress amplitude, temperature and density.” J. Biomech.9, 27-30, Biomech. 26, 219-228.
In article      View Article
 
[26]  Norman, T.L., Nivargikar, S. V. and Burr, D. B. (1996). “Resistance to crack growth in human cortical bone is greater in shear than in tension”. J. Biomech.29, 1023-1031.
In article      View Article
 
[27]  Feng, Z., Rho, J., Han, S. and Ziv, I. (2000). “Orientation and loading condition dependence of fracture toughness in cortical bone. Mater. Sci. Engng CC11, 41-46.
In article      View Article
 
[28]  Feng X. and M. McDonald J. (2011). “Disorders of Bone Remodeling,” Annu Rev Pathol.; 6: 121-145.
In article      View Article  PubMed
 
[29]  Hegedus D. and Cowin S. (1976). “Bone remodeling II: Theory of adaptive elasticity.” J. Elastic. 6, pp, 337-352.
In article      View Article
 
[30]  Τsili M. (2008b). “Internal bone remodeling induced by the distance - running and the unkown remodeling coefficients.” in: www.ispub.com/journal- of- internet journal of bioengineering, Vo- lume 4. number 2,
In article      
 
[31]  Kaplan M., William C. and James A. (1977). “Injuries to the leg and ankle. Chapter 14. From the book: On field eva- luation and treatment of common athletic injuries: (Ed. By Andrews, W. Chaney. and J. Whiteside), Mosby” Year Book, St. Louis Mis- souri”.
In article      
 
[32]  Monaco R., Halpern B., LeeRice E. and J Catalano M (1997). “Lower leg injuries” Chapter 13 of the book “Imaging in musco- skeletal and sports medicine” (Edit., by B. Halpern, S. Herring, Altchek and R.Herog) Blackwell Science.
In article      
 
[33]  Amendola A., Clatworthy M., and Μagness S. (1999). “Overuse injuries of the lower extremity” Chapt, 35., From the book OKY Orthopedic Knowledge Update Sports Medicine (Ed. by the Arenth) American Academy of Orthopaedic Sur- geons, Rosemont, Illinois.
In article      
 
[34]  Boucher R. (1999). “Exercise - induced leg -pain: Chapter 16., of the book “Sports medicine of the lower extremity” (Ed. By St., Subotnick 2 edition).” J. Biomech., 20., pp. 785-794.
In article      
 
[35]  Walker W. (1999). “Lower pain “Chapter 16. From the book “Handbook of sport medicine” (Edited by Lillegard, J. Butc- her and K. Rucker, sec, Edition, Butterworth - Heinemann).”
In article      
 
[36]  Romani W., Gieck J., Perrin D.et., al., (2002). “Mechanisms and management of stress fractures in physically active persons” J. Athl., Train Jul - Sep.,37, pp. 306-314.
In article      View Article
 
[37]  Jones B.,Thacker S., Gilchrist J. et., al., (2002). “Preven- tion of the lower extremity stress fracture in athletes and soldiers: A systematic review.” Epidem., Rev., 24., pp. 228-247
In article      View Article
 
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