Appendix (Supplementary Material)

Transcription

1 Appendix (Supplementary Material) Appendix A. Real and Imaginary Parts of the complex modulus E*(f), and E*(f) for the Pyramidal Proe Tip The magnitude of the complex modulus, E*(f), and the phase angle, (f), etween the measured force and the applied sinusoidal displacement (Fig. 1d) can e expressed in terms of (the real) storage modulus, E'(f), and the (imaginary) complex modulus, E''(f), as follows: E * (f ) E (f ) je (f ) (A1) * E (f ) E (f ) E (f ) (A) E(f ) tan (f ) (A3) E (f ) For the pyramidal proe tip, E*(f), was calculated using a Taylor series expansion of the Hertzian contact model (Radmacher 1997; Mahaffy 00): * 1 F(f ) E pyramidal (f ) (A) d0 tan d(f ) where : d 0 is indentation depth at zero frequency, Poisson's ratio is assumed to e 0. (Freeman 199), the pyramidal tip opening angle α is 35 (Ng 007), F(f) is the magnitude of the force and d(f) is the magnitude of the applied displacement at frequency f during dynamic oscillatory compression.

2 Appendix B. Indentation stiffness of single chondrocytes with GF treatment during time in culture For all of the experiments of Figs -, a force-displacement (F-d) curve was otained during the initial loading phase of each cell (Fig. 1c). This F-d curve was also used to characterize the quasistatic ehavior of each cell prior to force relaxation and susequent dynamic loading. The F-d curves cells in FBS and GF culture as a function of time in culture are shown in Fig. S1 for oth micron-sized and nano-sized proe tips. F-d curves varied with culture duration and tip geometry in oth medium conditions (two-way ANOVA, p<0.05) (Fig. S1). For each proe tip, the stiffness (slope of the F-d curve) depended significantly on the GF treatment and culture duration as well (two-way ANOVA, p<0.05). Chondrocytes cultured in FBS medium (Fig S1a,c) up to day 8 exhiited much lower stiffness than day 0 chondrocytes which had no cell-associated matrix, while chondrocytes treated with GF exhiited stiffness similar to or greater than day 0 cells after day 1 Fig S1,d). These trends are consistent with those reported in a previous study (Ng 007). Appendix C. Characteristic time constants for the poroelastic ehavior of tissue-engineered cell-associated matrix As descried in the main text, we estimated the characteristic poroelastic relaxation time, p ~ [L /(Hk)] to understand the mechanisms underlying the frequency dependence of the measured dynamic stiffness of the cell-associated matrix. The geometric configuration comprised of the cell-associated matrix

3 sandwiched in-etween the impermeale spherical proe tip and the cell memrane (Fig. 1a) is qualitatively approximated y a uniaxial unconfined compression geometry; the characteristic length scale, L, over which fluid flows is then approximated y the radius of the proe tip,.5 µm. The equilirium modulus H can e approximated y the quasi-static indentation stiffness (~0.1 kpa for day 1 and 0.17 kpa for day 8 in FBS culture (Ng 007). To estimate the hydraulic permeaility, k, it was hypothesized that GAG chains are the most critical determinant of the permeaility of the newly developing matrix, ased on previous measurements and models for the permeaility of native cartilage (Maroudas 1987). A unit cell model was used to calculate k (Happel 1959; Eisenerg 1988); the unit cell is comprised of a central charged GAG rod of radius, a, and its surrounding concentric cylindrical fluid shell having outer radius, (i.e., is the average center-to-center GAG spacing within the matrix (Eisenerg 1988)): k ln 3 a a a 3 ln a 1 a a (D1) where is the fluid viscosity (10-3 Pa s). Outer radius was independently calculated to e ~0-70 nm using the Poisson-Boltzmann cell model (Buschmann 1995) along with the measured total GAG content of the cell-associated matrix (C GAG = 0~130 g/ml for days 7-8, (Ng 007)): 1 / CGAG BN M (D) CS

4 where B is the distance etween charge groups along the GAG chain (0.6nm), N is the Avogadro's numer (6.0 x 10 3 ), and M cs is the molecular weight of a dissociated chondroitin sulfate disaccharide (58g/mol) (Buschmann 1995). With the estimated value of, equation D1 yields values for the permeaility, k, of the newly developing matrix, which decreased from 6.6 x10-1 m /N s for day 7 to 1.7 x10-1 m /N s for day 8. This range of values for k is much higher than that of the mature chondron (10-17 m /N s (Alexopoulos 005)), suggesting that the newly developed cell-associated matrix is much less resistant to the fluid flow. Based on the aove estimates for L, H, and k, we calculated values for p to e in the range sec for day 1-8 cells in FBS culture. This corresponds to characteristic frequencies f p ~[1/ p ] of Hz Appendix D. Analysis and correction for the effects of hydrodynamic drag on amplitude and phase angle of measured force We performed the same dynamic oscillatory compression after the rampand-hold on a silicon sustrate as a control sample, which is elastic and approximated as a hard surface comparing to the cartilage and AFM cantilever (Han 008). Under lower frequencies (< 100 Hz), the phase difference etween the acquired z-piezo movement and the cantilever deflection are negligile; under higher frequencies ( 100 Hz), there is a significant phase difference etween the z-piezo and the cantilever deflection measured on the mica sample, possily due to the data acquisition process of a cantilever and the z-piezo movement and hydrodynamic drags at high frequencies. Hence, the phase angle

5 difference (e.g. for frequency f, f, control ) was calculated at each tested frequency and we sutracted the corresponding phase angle difference from the value otained on the silicon surface to correct the artificial phase differences from the AFM instrument and hydrodynamic effects: f,real (E1) f,measured f,control

6 Force (nn) Force (nn) Force (nn) Force (nn) Figure S1 (a).0 10% FBS(n=-5) ().0 IGF-1+OP-1 (n=5) (mean± SEM, n=,5) (n=5) 1.5 Day Day 8 Day 0 Loading Day 8 Day 1 Day 1 Day 1 Day 1 (c).0 10% FBS (n=3-5) 1.5 Day 0 Day 8 Day 1 Day 1 (d).0 IGF-1+OP-1(n=5) (n=5) 1.5 Day 8 Day 1 Day 0 Day 1 Figure S1 Force-displacement curves of individual chondrocytes and chondrocyte-matrix composites using each proe tip: (a,c) FBS culture and (,d) GF culture (IGF-1+OP-1). n = numer of cells, mean ± SEM, n=3 for day 0 and n = -5 for day 7-8