HYDRODYNAMIC PRESSURE IN HUMAN HIP JOINT

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1 Journal of KONE Powertrain and Transport, Vol.4, No. 007 HYDODYNAMIC PEUE IN HUMAN HIP JOINT Krzysztof Wierzcholski Gdynia Maritime University Morska 83, 8-5 Gdynia, Poland tel.: , 34766, fax: Abstract This paper presents modelling and simulations for synovial fluid flow occurring in gap between two co-operating bone surfaces in human hip joint. The present consideration gives an analysis of solutions of systems of non-linear, partial, differential equations for synovial fluid flow in human joint gap. In the hip joint the spherical rotation bone and the pelvis bone create a spherical gap. In this gap between two co-operating bones synovial fluid flows, see Dowson [], Maquet [4], Mow [5], [6], [7], [8], [9], [0], Wierzcholski [], [3], Pytko et.al. []. The flow of this fluid is caused by the motion of the bone head. The theoretical considerations of the synovial non-newtonian fluid flow in thin joint gap, taking into account boundary layer simplifications, have practical applications in theory of lubrication in medicine, see Dowson et.al. [], [], Gruca [3]. The consideration given in the present section enables to find synovial fluid flow parameters and carrying capacity forces in human hip joint gap between two co-operating cartilage surfaces. Human hip joint and gap height for deformable cartilage and pressure distribution in normal and pathological spherical hip joint gap for hydrodynamic lubrication caused by rotation, cases of pressure distribution in pathological spherical hip joint gap for hydrodynamic lubrication caused by rotation, lubrication surface, area=0.38 cm, are presented in the paper. Keywords: normal, pathological hip, capacity, compressive stresses CINIENIE HYDODYNAMICZNE W BIODOWYM TAWIE CZOWIEKA treszczenie Praca przedstawia obliczenia numeryczne rozkadu cinienia hydrodynamicznego na sferycznej gowie kostnej stawu biodrowego wykonujcego ruch obrotowy. Uwzgldniona zostaa deformacja powierzchni chrzstki stawowej oraz brane s pod uwag wasnoci nie Newtonowskie cieczy synowialnej poprzez zmiany lepkoci dynamicznej wywoane zmianami prdkoci deformacji. Ponadto uwzgldnia si zarówno przypadek zdrowego jak i chorego stawu biodrowego. W tym stawie midzy dwoma wspópracujc koci przepywa pyn synowialny, patrz Dowson [], Maquet [4], kocie [5], [6], [7], [8], [9], [0], Wierzcholski [], [3], Pytko et.al. []. Przepyw tego pynu jest spowodowany przez rucch gówki koci. Teoretyczne rozwaania synowialnego nie-newtonowskiego przepyw pynu w cienkiej w cienkiej szczelinie stawu biorc pod uwag uproszczenia warstwy granicznej, maj zastosowania praktyczne teorii smarowania w medycynie, patrz Dowson et.al. [], [], Gruca [3]. ozwaanie podane w obecna artykule umoliwi znalezienie parametrów strumienia synowialnego pynui i zdolnoci do przenoszenia si w szczelinie stawu biodrowego czowieka midzy dwoma wspópracujcymi powierzchniami chrzstki. taw biodrowy czowieka oraz wysoko szczeliny dla odksztacalnej chrzstki oraz rozkad cinienia w szczelinie zdrowego i chorego stawu biodrowego przy hydrodynamicznym smarowaniu, przypadki rozkadów cinienia w chorej, sferycznej szczelinie biodra dla hydrodynamicznego,smarowania wywoanego ruchem obrotowym, przypadki rozkadów cinienia w zdrowej, sferycznej szczelinie biodra dla hydrodynamicznego smarowania wywoanego ruchem obrotowym sa przedstawione w artykule. owa kluczowe: Zdrowe i chore biodra,nono, naprenia ciskajce

2 K. Wierzcholski.Introduction We take into account equations of conservation of momentum and continuity equation for the non-stationary flow of synovial incompressible fluid between two rotational roughness cartilage surfaces in the spherical co-ordinates. Imposing the proper boundary conditions on the basic equations the pressure p we obtain from the following eynolds equation: p p 3 3 sin 6 sin sin, () in region: 0, /8/.We denote: -dynamic viscosity of synovial fluid, -angular velocity of bone head, -radius of bone head.. Deformable gap height The minimum gap height, see Dowson [], for spherical hip joint can be obtained from the formula []:, 0,6 0,4 5 min C p, E v E v E E C, () where: E, E,, the elastic module and Poisson ratios for bone head and cartilage, C load, and the quantities,, are as defined previously. elation for share rate min can be written in the following form:.,, 3 min 3 A E C o o o o p (3) By combining equations (), (3) we obtain the system of two equations for determination of two unknown quantities, namely the dynamic viscosity p of synovial fluid and the minimum value min of gap height, where elastic deformations of cartilage are taken into account. If we assume the following data: =.60 m, E=0 5 Pa, =30 m/s, =0.0 Pas, /C=30 4 m/n, o / 000, A=.88 s, C=544.6 N, then from Eqs. (), (3) we obtain: min = m=0.88 m and p = Pas. If we take in computations the following quantities: A=.88 s, o =00.00 Pas, =0.0 Pas, =0.00 m, C=544 N, 0.50 s 0.00 s, 0 5 PaE0 7 Pa, then we obtain the minimum value of gap height in the interval: 0.9 m min 9.90 m. 3. The gap height and primary calculations The gap height has the following form: 546

3 Hydrodynamic Pressure in Human Hip Joint = x cossin/+ y sinsin/ z cos/ +[( x cossin/+ y sinsin/ z cos/) +(+ min )(+D+ min )] 0.5. (4) Fig.. Human hip joint and gap height for deformable cartilage and pressure distribution in normal and pathological spherical hip joint gap for hydrodynamic lubrication caused by rotation ys.. taw biodrowy czowieka oraz wysoko szczeliny dla odksztacalnej chrzstki oraz rozkad cinienia w szczelinie zdrowego i chorego stawu biodrowego przy hydrodynamicznym smarowaniu The centre point of bone head (see Fig. ) can be written in the following form: O (x x, y y, z+ z ), while D is the distance between the centre of bone head and the acetabulum (sleeve) centre.we solve equation () for the region (,) resting on bone head and. 547

4 K. Wierzcholski indicated in Fig.. We assume atmospheric pressure on a boundary of the region (, 3 ). The centre point of bone head (see Fig. ) can be written in the following form: O (x x, y y, z+ z ), while D is the distance between the centre of bone head and the acetabulum (sleeve) centre.we solve equation () for the region (,) resting on bone head and indicated in Fig. We assume atmospheric pressure on a boundary of the region (, 3 )For this region we calculate also the capacity values. In numerical calculations we assume the following values for joint gap: x =5 m, y =5 m, z =+5 m; the radius of bone head = m. Now we calculate pressure distributions in normal and pathological hip joint. For the angular velocity of bone head =0. s, and the average value of synovial fluid dynamic viscosity =30.00 Pas for normal joint and =0.0 Pas for pathological joint, we obtain the smallest gap height min =3.3 m for normal joint and min =3.3 m for pathological joint. The hydrodynamic pressure p has its maximum value equal to N/m 8.30 at for normal joint and N/m for pathological joint, and the capacity C tot =63 N for normal joint and the capacity C tot =07 N for pathological joint, (C tot -total capacity,-gap height) see Fig.. For the angular velocity of bone head =.0 s and the average value of synovial fluid dynamic viscosity o =.00 Pas for normal joint and o =0.05 Pas for pathological joint, we obtain the smallest gap height min =3.3 m for normal joint and min =3.3 m for pathological joint. The hydrodynamic pressure p has its maximum value equal to N/m.79 at for normal joint and Pa for pathological joint, and the capacity C tot =775 N for normal joint and the capacity C tot =68 N for pathological joint. Lubrication surface area has the value cos/80.38 cm. Numerical calculations are performed by means of Mathcad Professional program with the help of the Method of Finite Differences. This method satisfies the requirement of stability of numerical solutions for the partial differential equation (). 4. Numerical calculations Now we calculate pressure distribution in pathological joint (see Fig..). For the angular velocity of bone head: =0.0 s, =7.5 s, =5.0 s, =.5 s, =.0 s, =0.75 s and the average value of synovial fluid dynamic viscosity: o = Pas, o = Pas o =0.074 Pas, o =0.00 Pas, o =0.59 Pas, o =0.359 Pas, we obtain the following maximum values of hydrodynamic pressure: p max =.383 at, p max =.3 at, p max =.4 at, p max =.64 at, p max =.08 at, p max =.067 at, and the carrying capacity: C tot =5.607 N, C tot =.97 N, C tot =5.94 N, C tot =0.77 N, C tot =5.44 N, C tot =4.393 N, respectively. Now we calculate pressure distribution in normal joint (see Fig. 3). For the angular velocity of bone head: =0.0 s ; =7.5 s, =5.0 s, =.5 s, =.0 s, =0.75 s and the average value of synovial fluid dynamic viscosity: o =6.83 Pas, o =9.6 Pas, o =3.08 Pas, o =3.63 Pas, o =4.987 Pas, o =50.8 Pas, we obtain the following maximum hydrodynamic pressure values: p max = Pa45.63 at, p max = Pa 46,50 at, p max = Pa at, p max = Pa 39.0 at, p max = Pa 9.6 at, p max = Pa 5.6 at, and the carrying capacity: C tot =935.3 N, C tot =99.5 N, C tot =88. N, C tot =505.6 N, C tot =85.0 N, C tot =69.4 N, respectively. Lubrication surface area has the value cos/80.38 cm. 548

5 Hydrodynamic Pressure in Human Hip Joint Fig.. Cases of pressure distribution in pathological spherical hip joint gap for hydrodynamic lubrication caused by rotation, lubrication surface, area=0.38 cm ys.. Przypadki rozkadów cinienia w chorej, sferycznej szczelinie biodra dla hydrodynamicznego smarowania wywoanego ruchem obrotowym 549

6 K. Wierzcholski Fig. 3. Cases of pressure distribution in normal spherical hip joint gap for hydrodynamic lubrication caused by rotation, lubrication surface area =0.38 cm ys. 3. Przypadki rozkadów cinienia w zdrowej, sferycznej szczelinie biodra dla hydrodynamicznego smarowania wywoanego ruchem obrotowym 550

7 Hydrodynamic Pressure in Human Hip Joint 5. Final comments For the above given capacities in normal joint we obtain the following compressive stresses: s = N / 0.38 cm =.440N/mm =.440 MN/m, s =.468 MN/m, s =.383 MN/m, s =.9 MN/m, s = MN/m, s = MN/m. In pathological joint, the compressive stresses are as follows: s =5.6N/0.38cm =0.03 N/mm =0.03 MN/m, s =0.004 MN/m,. These stresses are smaller than the compressive strength MN/m of normal human cartilage. Compressive strength of pathological human cartilage has the value of about MPa [], [3]. The present paper shows the method of determination of approximate solutions of the partial non-linear differential equations of non-newtonian, asymmetrical synovial fluid flow in the thin gap occurring in human joint in curvilinear, orthogonal co-ordinates. The presented method enables to obtain solutions in the form of Taylor series with increasing powers of the small parameter A obtained in experimental way for synovial fluid. In the particular case of the symmetrical flow we can, by virtue of the presented theory, find analytical solutions in a simple form. The percentage corrections of pressure caused by the non- Newtonian properties of synovial fluid are examined numerically. For large shear rates within the range: 00 s 000 s, the viscosity of synovial fluid is small and has values: 0 Pas Pas. In this case we obtain small pressure changes from % to 4%. For small shear rates within the range: 0 s 0 s, when viscosity of synovial liquid is large, i.e. within the range: 0 Pas00 Pas, from Eq. (5) we obtain pressure changes from 7% to 5%. - Note : If Young Modulus of cartilage and bone decreases then the least value of joint gap height increases. - Note : If Young Modulus of cartilage decreases then viscosity of synovial fluid increases. Acknowledgement The present research is financially supported within the frame of the TOK FP6, MTKD-CT eferences [] Dowson, D., Van Mow, C., atcliffe, A., Woo,. L. Y., Biomechanics of Diarthrodial Joint. pringer-verlag, Chap.9, New York-Berlin, 990. [] Dowson, D., Advances in Medical Tribology, Mech. Eng. Publ. Limited Bookcraft UK Limited Bath., 998. [3] Gruca, A., Chirurgia Ortopedyczna, (in Polish), PZWL, Warszawa, 993. [4] Maquet, P. G. J., Biomechanics of the knee, pringer-verlag, Berlin-Heidelberg- New York, 984. [5] Mow, V. C., atcliffe A., Woo., Biomechanics of Diarthrodial Joints, pringer-verlag, Berlin-Heidelberg-New York,990. [6] Mow, V. C., Guilak, F., Cell Mechanics and Cellular Engineering, pringer-verlag, Berlin- Heidelberg-New York, 994. [7] Mow, V. C., Foster. J., Tribology. (Chapter 5) In: The Adult Hip, edited by JJ Callaghan, osenberg AG, ubash HE; Lippincott-aven Publishers, Philadelphia, 998. [8] Mow, V. C., oslowsky L. J., Friction, lubrication and wear of diarthrodial joints. In: Mow V. C., Hayes W. C., Eds. Basic orthopedic biomechanics, aven Press; pp New York, 99. [9] Mow, V. C., Holmes, M. H., Lai, W. M., Fluid transport and mechanical properties of articular cartilage, Journal of Biomechanics, Vol.7, pp ,

8 K. Wierzcholski [0] Mow, V. C., Wilson, C. Hayes, W. C., Biomechanics, Lippincott-Wiliams, Wilkins Publishers, Philadelphia-Baltimore-New York-London, 998. [] Pytko,., Wierzcholski, K., Analytical Biobearing Calculation for Experimental Dependencies Between hear ate and ynovial Fluid Viscosity, Proceedings of International Tribology Conference in Yokohama, 3, pp , Yokohama 995. [] Wierzcholski, K., The method of solutions for hydrodynamic lubrication by synovial fluid flow in human joint gap, Control and Cybernetics, Vol. 3 No., pp.9-6,warszawa 00. [3] Wierzcholski, K., ynovial fluid squeeze film flow in curvilinear biobearing human gap, Polish ociety of Medical Informatics, Computers in Medicine 5, Volume II pp ód