Perfusion-enhanced growth of. tissue-engineered cartilage in a. bioreactor: A finite element study

Transcription

1 BMTE 03.0 Perfuion-enhaned growth of tiue-engineered artilage in a bioreator: A finite element tudy Martijn Cox Supervior: dr. ir. Cee Oomen ir. Bram Senger Date:

2 Abtrat Experiment on tiue-engineering artilage in a tati bioreator often how an inhomogeneou matrix-prodution. The method deribed in thi artile i aimed at anwering the quetion if it i poible to enhane matrix-prodution by perfuion through the artilage-ontrut. Goal i to invetigate the poibilitie of different kind of flow-hitorie on the inreae and homogenization of extraellular matrix (- prodution) in a bioreator for tiue-engineered artilage. The method to be ued i a Finite Element Model uing biphai mehani, advetive and diffuive olute tranport, uptake, bioynthei and hanging mehanial propertie of the tiueengineered ontrut. The key idea i that matrix prodution depend on loal oxygenlevel and onentration of (an arbitrary) growth fator. The diffuion peed of oxygen i high ompared to the peed of tranport by perfuion. Oxygen-tranport i therefore thought to be little influened by perfuion. However, applying a flow to the bio-ontrut an ontrol the tranport of the bigger growth-fator moleule, where diffuion take plae very low. Reult how the importane of flow in enhaning matrix-prodution, and the poitive effet of alternating flow on reating a more homogeneou matrix. Experiment are propoed for qualitative determination of the importane of oxygen or growth fator in matrix growth. Further experiment are needed to evaluate and validate the numerial method a propoed in thi artile.

3 Introdution The reearh deribed in thi artile form part of IMBIOTOR, a European Commiion projet in whih everal European univeritie and ompanie have ombined their fore to develop an intelligent bioreator for the prodution of tiue-engineered artilage. One of the major diffiultie in tiue-engineering artilage in tati bioreator i inhomogeneou matrix-prodution (Martin et al., 999; Freed and Vunjak-Novakovi, 000; Martin et al., 000). Peripheral artilage growth exeed that of the inner layer. Peripheral irumtane for matrix-prodution are ideal, with high onentration of oxygen and growth fator, whih are aumed to be the mot important fator for matrix prodution (Obradovi et al., 000; Bailón-Plaza et al., 00; Blunk et al., 00; Pei et al., 00). Peripheral uptake and utilization of oxygen and growth fator lead to low onentration in deeper layer of the bioontrut. Thi i likely to reult in inhomogeneou matrix-prodution. Beaue higher matrix-prodution lead to lower permeability thi proe i elf-enforing. The more matrix-growth in outer region, the le permeability of the bioontrut, whih lead to even lower onentration in deeper layer. The quetion i if it i poible to inreae inner onentration of growth fator by applying a preure gradient, whih will lead to flow through the bioreator. The relatively low tranport of growth-fator moleule (e.g. I-I, T-β) will be influened by thi flow. Diffuion of oxygen i relatively fat; therefore oxygen onentration will differ only lightly from the ituation without flow. Thi artile aim at invetigating the poibilitie of different kind of flowhitorie to inreae and homogenize extraellular matrix (-prodution) in a bioreator for tiue-engineered artilage 3

4 The Finite Element Model ued in thi artile i that propoed by Senger (ubmitted). Thi model provide a numerial framework for tiue engineering, whih an be ued for modeling many different kind of apet of the proe of tiue engineering of artilage. The modeling approah deribed here relate mehanial loading to olute tranport and uptake. Thi lead to bioynthei of extra ellular matrix, whih lead to hanging mehanial propertie (tiffne and permeability) of the artilage ontrut. Growth fator onentration in deeper region are thought to be inreaed by flow due to mehanial loading. Different kind of load-hitorie are ompared to the ituation without flow. Poible negative ide effet of applying a load to the artilage tiue i the ompreion of the pore in the medium, whih may lead to a great dereae in permeability of the ontrut. In the eond problem, to better illutrate the poible effet of ompreion on the permeability an extenion i made to the model, uing a more realiti permeability law. Reult how that, under modeling aumption, flow doe inreae and homogenize matrix prodution. An experiment i propoed to determine the relative importane of oxygen and growth fator in matrix prodution. Further experiment are needed to validate and evaluate the reult of the method a deribed here. Value and deription of ontant ued in thi artile are ummarized in table. 4

5 Table : Value of ontant ued Contant Value Deription ano 0.5 Relative anaerobi matrix prodution C F 8. 0 N m -4 Iotropi frition oeffiient out M External onentration of growth fator O out mol mm External onentration of oxygen Cm Variable Contant for growth fator dependent matrix prodution Cm Variable Contant for oxygen dependent matrix prodution 0-3 CmO 0 mol mm Mihaeli-Menten ontant oxygen Cm 0. M Mihaeli-Menten ontant growth fator 8 D DO.5 0 m - Growth fator diffuion ontant m - Oxygen diffuion ontant G 4. 5 kpa Shear modulu at t = 0 0 Kmax Max. rate of oxygen onumption O 3 Kmax 0 - Max. rate of growth fator onumption M 8.0 Deformation-dependent permeability ontant f n Fluid volume fration at t = 0 n 0.30 Max. olid volume fration max dn 7 max dt 4 0 P 3 - Max. rate of matrix prodution - 0 N mm External hydrotati preure κ 4. kpa Bulk modulu at t = 0 0 5

6 Method Mathematial model Fig. how a hemati repreentation of the model. The fluid aturated Mehanial Loading Biphai Mehani Permeability & Stiffne Diplaement & veloity field figure : A hemati repreentation of the model Oxygen tranport Growth fator tranport Matrix Prodution affold i modeled by uing biphai mixture mehani. The omputed veloity field i ued to alulate the loal onentration of oxygen and growth fator due to advetion and diffuion. The volume fration of ell and affold material i aumed to be ontant and equal to zero. Therefore, the olid volume fration an be aumed to be eentially equal to the volume fration of the extra ellular matrix. Thi olid volume fration i hoen to be 0.03 at the tart of the imulation, whih orrepond to a poroity of 97% for a PGA affold (Freed and Vunjak-Novakovi, 998). Matrix prodution depend only on growth fator and oxygen onentration and i alulated uing operator plitting. More matrix lead to higher tiffne and le permeability, whih will be ued in the biphai mehani part. A detailed deription of the onervation law and ontitutive equation i found in Senger (ubmitted). Summarizing the biphai mixture i deribed by the following et of oupled equation: r r p = 0 r e σ, () r v + r p = 0 r - K, () 6

7 r r v r + v r = 0, (3) r v β r v K = f n r D r p β β (4) in whih σ e i the effetive Cauhy tre tenor for the olid matrix, p the hydrotati pore preure, while v r i the veloity of the olid phae. f n repreent f the fluid volume fration. We tart our imulation with n 0 = (Freed and r f r f r Vunjak-Novakovi, 998). The peifi diharge i defined a v n ( v v ) i the permeability tenor. =. K The relative veloity of a olute β (oxygen or growth fator) with repet to the olid matrix i given by (4)., while D denote the diffuion tenor, and β the onentration of olute β. The diffuion ontant of oxygen i et to DO = m - (Obradovi et al., 000; Haelgrove et al., 993). The growth fator diffuion ontant i equal to 8 D = 5 0 m -, whih i in the ame order of magnitude a the diffuion ontant of growth fator T-β a ued by Bailón- Plaza et al., 00. The right-hand ide of equation (4) exit of a perfuion part and a diffuion part. In oxygen tranport, the diffuion part i dominant. Growth fator tranport i mainly determined by the perfuion part. The uptake of oxygen by the ell i deribed by Mihaeli-Menten kineti: O d dt Kmax = Cm O O + O O (5) d O in whih dt repreent the oxygen uptake rate of the ell in mol mm Kmax O i the maximum rate of oxygen onumption and O Cm the Miheali- 7

8 Menten ontant for O. Kmax O and Cm O are hoen in uh a way that the oxygen onentration profile along the depth of the bioreator i approximately imilar to that of the model by Obradovi et al., 000. Kmax O equal 0 - -, Cm O equal 0 0 mol mm -3. The diappearane of growth fator i a muh more diffiult proe. Growth fator might diappear due to utilization or uptake, but alo by biodegradation. In thi ae fou will not be on the exat proe of growth fator diappearane, therefore we aume that thi proe an be deribed in a imilar way a the uptake of oxygen: d dt Kmax = Cm + (6) d denote the uptake, uage or degradation rate of growth fator. Kmax and dt Cm are hoen in uh a way that the time ontant for uptake of growth fator i in the ame order of magnitude a the time ontant for tranport of growth fator. 3 Kmax i 0 -, Cm i 0. M. Matrix prodution i aumed to depend on both oxygen and growth fator loal onentration (Obradovi et al., 000; Bailón-Plaza et al., 00; Blunk et al., 00; Pei et al., 00). In abene of oxygen, till matrix prodution (or degradation) take plae (Lee et al., 997). For large onentration of oxygen and/or growth fator, matrix prodution reahe a maximum level. Due to before-mentioned retrition, matrix prodution repreent an inreae in olid volume fration, n. Thi reult in: dn dt = dn max dt k Cm + O ( ) ano + ano n O (7) Cm + 8

9 dn in whih i the rate of matrix growth or degradation ( - ), dt dn max i the dt maximum rate of matrix prodution, et to Cm and Cm are ontant that determine the growth fator and oxygen dependent part of matrix prodution, repetively. They are varied during the imulation. The relative ontribution of anaerobi matrix prodution i repreented by ano, it i aumed to be equal to 0.5 (Lee et al., 997). A an be een in (eq. 7), a dereae in growth fator and/or oxygen may lead to matrix degradation. k finally i a ontant that limit the maximum amount of matrix in the tiue-engineered ontrut. k i defined by: k = Cm out + out n max out ( ano ) + + ano O out Cm O (8) in whih n max i the maximum relative amount of matrix, i.e. the maximum olid volume fration. out O out and denote the onentration of oxygen and growth fator in the urrounding olution. out i hoen equal to M, beaue it i an arbitrary growth fator. Obradovi et al., 000). O out i equal to 0 0 mol mm -3, (Senger, ubmitted; For out O Oout = and = equation (7) redue to: dn dt dn max = max dt [ n n ] (9) The maximum olid volume fration i hoen to be 0.30, whih orrepond with a fluid volume fration of 60% to 85% in natural artilage (Haler et al., 999). The matrix i aumed to be a ompreible hyperelati neo-hookean material. The effetive tre tenor i given by: σ e G = κ I, (0) ( ) 3 J I + F F J J 9

10 where κ i the bulk modulu and G the hear modulu. F i the deformation tenor and J = det(f ). The bulk and hear moduli are aumed to be ontant for the matrix fration. For the whole mixture they are aumed to be proportional with the olid volume fration: κ = n n 0 κ 0 () G = n n 0 G 0 () In whih κ and G are the bulk and hear moduli of the mixture. Baed on the order of magnitude of the equilibrium modulu found in onfined ompreion for three day old tiue-engineered artilage ontrut [4], and a Poion ratio of ν = 0. (Senger, ubmitted; Wong, 000), the reulting moduli at t = 0 are κ = 4. 0 kpa and G = 4.5 kpa. 0 0

11 Numerial implementation Oxygen and growth fator tranport and uptake are relatively fat proee, whih et to an equilibrium in le than an hour. To be able to imulate thee proee a numerial model need to ue mall time tep (minute). Matrix prodution, on the other hand, i a proe of day and week. It would take very many mall time tep to ee any differene in matrix prodution at all. Therefore we work with two different time ale. Firt, an equilibrium i reahed for the tranport of oxygen and growth fator on a mall time ale ( t = min). Equilibrium i aumed to be reahed when the following inequality hold for every z: β β ( z, t) ( z, t t) < 0.0 (3) in whih ( z t) β, i either oxygen or growth fator onentration at poition z and time t, t denote a mall time tep. From eq. (3) we an ee that equilibrium i reahed when relative hange in olute onentration during one mall time tep i le then % everywhere in the medium. After equilibrium i reahed a large timetep i applied ( t = day) in whih matrix i formed. Then again oxygen and growth fator tranport take plae until teady tate and o on. Thi way it i poible to imulate the fat tranport proee a well a the low matrix prodution in a reaonable imulation time.

12 Geometry The bioreator i modeled a a di with a radiu of 0 mm and a height of 3.0 mm. The di i urrounded by a fluid with high onentration of oxygen and growth fator. In thi artile only a D-problem i onidered for a mall bar in the enter of the di, a hown in figure. The urrounding fluid i aumed to be well mixed, whih lead to ontant onentration of oxygen and growth fator at both end of the bar. The problem i olved uing the SEPRAN finite element pakage (Segal, 000). A three-dimenional 7-node brik element with quadrati interpolation funtion i ued. For a more detailed deription ee Senger (ubmitted). 0 mm 0. mm 3.0 mm 3.0 mm figure : Geometry of the modeled bioreator For the firt problem, the permeability tenor K i defined a (Senger, ubmitted; Huyghe and Janen, 997; Almeida and Spilker, 997): K f - ( n ) C F =, (4) in whih C F i the frition tenor between the olid and the fluid phae. The iotropi frition oeffiient C F = 8. 0 N m -4, whih reult from before-mentioned hear and bulk moduli and the order of magnitude of permeability found in onfined ompreion for three day old tiue-engineered artilage ontrut (Martin et al., 000). All imulation are done with different ombination of Cm and Cm (eq. 7). Cm and Cm are firt mall ompared to the outer onentration of oxygen and

13 out O out growth fator ( Cm = 0. 0 and Cm = 0.0 ). Small variation in oxygen and growth fator onentration have little influene on matrix prodution. After that we onider a ituation where Cm = out. In thi ae mall dereae in growth fator level lead to fat dereae of matrix prodution. After that, the ame i done for oxygen. Reality i expeted to be omewhere between thee extreme ituation. In the firt imulation the right ide of the bar (figure ) i fixed in pae, while the left ide of the bar i loaded with an external hydrotati preure of 3 0 N mm -. The preure i hoen to keep deformation mall (<0%). There i only flow from left to right. The eond imulation ue alternating flow. In the firt large time tep, load i applied on the left ide of the bar. In the eond large time tep, load i applied on the right ide. For modeling onveniene, the right ide i till fixed in pae. Flow i moving from right to left now, until equilibrium i reahed. After that, matrix prodution i alulated and flow i alternated again. Experimental reult how that permeability dereae rapidly, even when fluid volume fration dereae only lightly (Martin et al., 000; Freed et al., 998). Therefore in the eond problem the permeability law (eq 3) i extended with a deformation-dependent part (Lai et al., 98): K f - ( n ) M = J C F, (4) in whih J i the determinant of the deformation tenor F. Literature value for M vary between 5 and 0 (Chen et al., 00; Holme et al., 985; Holme and Mow, 990). In thi ae M i et to 8. Cm and Cm are varied in the eond problem jut like in the firt. 3

14 The number of element ued varie between 40 and 80. To avoid numerial oillation in growth fator onentration at the right edge of the bar, a trong bia i neeary. In ae of alternating flow, a double bia i ued. 4

15 Reult Generally after approximately 30 to 75 minute, the olute tranport reahe an equilibrium. An example of thu obtained oxygen and growth fator onentration profile along the depth of the bio-ontrut i hown in figure 3. Solute onentration O ( x 0-7 mol l - ) ( mol l - ) z(mm) figure 3: Steady tate oxygen and growth fator onentration profile. In figure 4, matrix volume fration along the depth of the bioreator are hown in deformed tate for different ombination of flow and Cm and Cm. Eah line orrepond with a time tep of approximately 4.6 day. Matrix growth i imulated for 46 day. Perfuion from left to right reult in a relatively homogeneou matrix volume fration profile when Cm and Cm are low, exept for a harp dip around a depth of approximately mm. When growth fator onentration limit matrix prodution ( Cm i high ) thi profile i muh more kew, whih reemble the growth fator onentration profile in figure 3. With oxygen a limiting fator the matrix onentration profile i almot ymmetri, jut like the oxygen onentration in figure 3. With alternating flow, profile beome muh more ymmetri and homogeneou. When there i no flow, reult how only peripheral matrix prodution. 5

16 Cm = 0.0 out Cm = 0.0 Oout Cm = out Cm = 0.0 Oout Cm = 0. 0 Cm = Oout out flow from left to right alternating flow no flow figure 4: Solid volume fration profile along the depth of the bioreator for different ombination of Cm and Cm and flow hitorie. Eah line orrepond with a time tep of 4.6 day. A hown in figure 5 and 6, the eond problem, uing deformation-dependent permeability lead to le matrix prodution. Differene in matrix volume fration i bet een at the right edge of the bioontrut. When Cm = out the differene between imple permeability and deformation-dependent permeability are mot obviou. Matrix volume fration profile are hown in figure 5. Differene in matrix growth i aued by a differene in relative fluid veloity f v v, a hown in figure 6. The relative fluid veloity i muh lower for deformation-dependent permeability when Cm = out. 6

17 n Cm = 0.0 out Cm = 0.0 Oout n Cm = out Cm = 0.0 Oout n Cm = 0. 0 Cm = Oout out z (mm) z (mm) z (mm) figure 5: Solid volume fration after 46 day with different ombination of Cm and Cm for imple permeability (olid line), and deformation-dependent permeability (dotted line). v f - v (mm - ) n Cm = 0.0 out Cm = 0.0 Oout v f - v (mm - ) n Cm = out Cm = 0.0 Oout v f - v (mm - ) n Cm = 0. 0 Cm = Oout out z (mm) z (mm) z (mm) figure 6: Relative fluid veloity profile after 46 day with different ombination of Cm and Cm for imple permeability (olid line), and deformation-dependent permeability (dotted line). 7

18 Diuion The objetive of thi finite element tudy wa to invetigate the poibilitie of perfuion in enhaning matrix growth in tiue-engineered artilage. We hypotheized that perfuion would lead to higher onentration of growth fator moleule in deeper region of the artilage ontrut, whih would reult in more matrix growth. Reult indiate that, under modeling aumption, perfuion indeed help to timulate matrix growth. Oxygen onentration profile are almot ymmetri, whih indiate that oxygen tranport mainly take plae by diffuion. Growth fator onentration profile are more oblique; growth fator tranport i mainly aued by the preure gradient in the bioreator. Without perfuion, only peripheral matrix growth take plae, whih i in aordane to data found in literature (Martin et al., 999; Freed and Vunjak-Novakovi, 998, Martin et al., 000). Perfuion inreae matrix growth in deeper layer. When matrix prodution i limited by oxygen onentration the profile i nearly ymmetrial; growth fator a limiting fator lead to a more oblique profile. With thi information, a perfuion experiment will how the relative importane of oxygen and growth fator in matrix growth. A ymmetri matrix profile indiate oxygen a mot important fator, an oblique profile i a reult of growth fator a main limiter of matrix growth. When flow i alternated, reulting matrix volume fration profile are muh more homogeneou, epeially when matrix growth i limited by growth fator. In thi ae flow i alternated with a frequeny of 4.6 day. It might be intereting to imulate matrix growth a a funtion of the alternating flow frequeny, in order to find an optimum. 8

19 In the eond problem fou ha been on the permeability of the artilage ontrut. Reult how that growth fator onentration ignifiantly dereae when a deformation-dependent permeability law i ued. Relative fluid veloity i lower, while diappearane rate of growth fator tay the ame. When matrix growth i limited by growth fator, the differene in relative fluid veloity i mot obviou. Thi i aued by the aymmetri matrix profile, whih lead to differene in loal train along the depth of the bioreator. High loal train lead to low loal permeability and therefore lower relative fluid veloity. Experimental data how a large dereae in permeability for relatively mall hange in fluid volume fration, even without deformation, whih an not be explained by the permeability law a propoed in thi tudy (Martin et al., 000; Freed et al., 998). More experimental data are needed on loal permeability in a bioreator to be able to implement a more realiti permeability law. Virtually no quantitative experimental data an be found on matrix and olute onentration along the depth of a bioreator, exept for Obradovi et al. (000), in whih experimental data on oxygen onentration profile are obtained. Therefore in thi tudy only ome extreme ituation are hown. Experiment hould be done to obtain loal matrix volume fration at different tage in the tiueengineering proe to validate the model propoed here. In future, in vitro meaurement of oxygen and growth fator onentration profile hould be poible. When thee reult are ombined with the model thi will provide more inight in the omplex proee of tiue-engineering artilage. 9

20 Referene Almeida, E. and Spilker, R., 997. Mixed and penalty finite element model for the nonlinear behavior of biphai oft tiue in finite deformation: Part II nonlinear example. Comput. Meth. Biomeh. Biomed. Eng. (), Bailón-Plaza, A., Meulen, van der, M.C.H., 00. A Mathematial Framework to Study the Effet of Growth Fator Influene on Frature Healing. J. theor. Biol., Blunk, T., Sieminki, A.L., Gooh, K.J., Courter, D.L., Hollander, A.P., Menahem Nahir, A., Langer, R., Vunjak-Novakovi, G. and Freed, L.E., 00. Differential Effet of Growth Fatore on Tiue-Engineered Cartilage. Tiue Engineering 8, Chen, A.C., Bae, W.C., Shinagl, R.M., Sah, R.L., 00. Depth- and train-dependent mehanial and eletrohemial propertie of full-thikne bovine artiular artilage in onfined ompreion. Journal of Biomehani 34, -. Freed, L.E., Hollander, A.P., Martin, I., Barry, J.R., Langer, R., and Vunjak- Novakovi, G., 998. Chondrogenei in a Cell-Polymer-Bioreator Sytem. Experimental Cell Reearh, 40, Freed, L.E. and Vunjak-Novakovi, G., 998. Culture of organized ell ommunitie. Advaned Drug Delivery Review 33, Haelgrove, J., Shapiro, I., and Silverton, S., 993. Computer modeling of the oxygen upply and demand of ell of the avian growth artilage. Am. J. Phyiol. 65, Haler, E.M., Herzog, W., Wu, J.Z., Muller, W. and Wy, U., 999. Artiular artilage biomehani: Theoretial model, material propertie, and 0

21 bioyntheti repone. Critial review in Biomedial Engineering 7(6), Holme, M.H., Lai, W.M., Mow, V.C., 985. Singular perturbation analyi of the non-linear, flow-dependent ompreive tre relaxation behavior of artiular artilage. J. Biomeh. Eng. 07, Holme, M.H., Mow, V.C., 990. The nonlinear harateriti of oft gel and hydrated onnetive tiue in ultrafiltration. J. Biomeh. 3, Huyghe, J., and Janen, J., 997. Quadriphai mehani of welling inompreible porou media. Int. J. Eng. Si. 35(8), Lai, M.W., Mow, V.C., Roth, V., 98. Effet of Nonlinear Strain-Dependent Permeability and Rate of Compreion on the Stre Behavior of Artiular Cartilage. J. Biomeh. Eng. 03, Lee, R.B., Urban, J.P.G., 997. Evidene for a negative Pateur effet in artiular artilage. Biohem. J. 3, Martin, I., Obradovi, B., Freed, L. and Vunjan-Novakovi, G., 999. Method for quantitative analyi of glyoaminoglyan ditribution in ultured natural and engineered artilage. Annal of Biomedial Engineering 7, Obradovi, B., Meldon, J., Freed, L. and Vunjak-Novakovi, G., 000. Glyoaminoglyan depoition in engineered artilage: Experiment and mathematial model. AIChE. 46(9) Martin, I., Obradovi, B., Treppo, S., Grodzinky, A.J., Langer, R., Freed, L.E. and Vunjak-Novakovi, G., 000. Modulation of the mehanial propertie of tiue engineered artilage. Biorheology 37, 4-47.

22 Pei, M., Seidel, J., Vunjak-Novakovi, G. and Freed, L.E., 00. Growth fator for equential ellular de- and re-differentiation in tiue engineering. Biohemial and Biophyial Reearh Communiation 94, Segal, G., SEPRAN uer manual, 000. Ingenieurbureau SEPRA, Leidhendam, The Netherland. Senger, B.G., Oomen, C.W.J. and Baaijen, F.P.T., ubmitted. An integrated finite element approah to mehani, tranport and bioynthei in tiue engineering. J. Biomeh. Eng. Wong, M., Pontiiello, M., Kovanen, V., and Jurvelin, J., 000. Volumetri hange of artiular artilage during tre relaxation in unonfined ompreion. J. Biomeh. 33,