SENSITIVITY ANALYSIS FOR MODEL COMPARISON AND SELECTION IN TISSUE ENGINEERING

Paim, Ágata; Cardozo, Nilo S. M.; Pranke, Patricia; Tessaro, Isabel C.

doi:10.1590/0104-6632.20190361s20170268

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Brazilian Journal of Chemical Engineering

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Process Systems Engineering • Braz. J. Chem. Eng. 36 (1) • Jan-Mar 2019 • https://doi.org/10.1590/0104-6632.20190361s20170268 copy

SENSITIVITY ANALYSIS FOR MODEL COMPARISON AND SELECTION IN TISSUE ENGINEERING

Authorship SCIMAGO INSTITUTIONS RANKINGS

ABSTRACT

Computational modeling has been proven to be very useful in tissue engineering over the past years. Because the model is a simplification of the experimental system, the processes accounted for in the model should be analyzed carefully. However, new and complex models are usually proposed without a clear comparison with the basic ones. In this study, the contribution of oxygen to Contois growth kinetics and porosity variation with time due to polymer degradation was evaluated through a sensitivity analysis. The effect of initial glucose concentration, porosity and thickness of the scaffold on the cell volume fraction and substrate concentration was analyzed for three models. Even with the inclusion of oxygen concentration in the model, the output variables are more affected by the initial cell number, while the model with variable porosity is quite robust to variations in the input variables.

Key words:
Computational modeling; Tissue engineering; Scaffold porosity; Mass transport; Cell growth

Model	Mechanisms	Equations
I	Glucose limitation of cell growth (Contois)	2
I	Glucose transport (Michaelis-Menten)	8
II	Glucose and oxygen limitation of cell growth (double Contois)	3
II	Glucose and oxygen transport (Michaelis-Menten)	8, 9
III	Glucose and oxygen limitation of cell growth (double Contois)	3
	Glucose and oxygen transport (Michaelis-Menten)	8, 9
	Time dependence of the scaffold porosity	13

Parameter	Value	Unit	Reference
D_g,β	1×10^-8	m²∙s^-1	Chung et al. (2006)Chung, C.A., Yang, C.W., and Chen, C.W., Analysis of Cell Growth and Diffusion in a Scaffold for Cartilage Tissue Engineering, Wiley Period, 94, 1138-1146 (2006). https://doi.org/10.1002/bit.20944 https://doi.org/10.1002/bit.20944...
D_g,σ	1×10^-7	m2∙s^-1	Chung et al. (2006)Chung, C.A., Yang, C.W., and Chen, C.W., Analysis of Cell Growth and Diffusion in a Scaffold for Cartilage Tissue Engineering, Wiley Period, 94, 1138-1146 (2006). https://doi.org/10.1002/bit.20944 https://doi.org/10.1002/bit.20944...
D_o	3.093×10^-7	m²∙s^-1	Lin et al. (2011)Lin, T.-H., Lin, C.-H., and Chung, C.A., Comutational Study of Oxygen and Glucose Transport in Engineered Cartilage Constructs, J. Mech., 27, 337-346 (2011). https://doi.org/10.1017/jmech.2011.36 https://doi.org/10.1017/jmech.2011.36...
Τ	1.93	---	Yan et al. (2012)Yan, X., Bergstrom, D.J., and Chen, X.B., Modeling of cell cultures in perfusion bioreactors, IEEE Trans. Biomed. Eng., 59, 2568-2575 (2012). https://doi.org/10.1109/TBME.2012.2206077 https://doi.org/10.1109/TBME.2012.220607...
μ_max	1.6-4.5×10^-6	s^-1	Chung et al. (2006)Chung, C.A., Yang, C.W., and Chen, C.W., Analysis of Cell Growth and Diffusion in a Scaffold for Cartilage Tissue Engineering, Wiley Period, 94, 1138-1146 (2006). https://doi.org/10.1002/bit.20944 https://doi.org/10.1002/bit.20944...
ρc_ell	0.182×10⁶	cell∙m^-3	Chung et al. (2006)Chung, C.A., Yang, C.W., and Chen, C.W., Analysis of Cell Growth and Diffusion in a Scaffold for Cartilage Tissue Engineering, Wiley Period, 94, 1138-1146 (2006). https://doi.org/10.1002/bit.20944 https://doi.org/10.1002/bit.20944...
R_om	1.86×10^-18	mol∙s^-1∙cell^-1	Coletti et al. (2006)Coletti, F., Macchietto, S., and Elvassore, N., Mathematical Modeling of Three-Dimensional Cell Cultures in Perfusion Bioreactors, Ind. Eng. Chem. Res., 45, 8158-8169 (2006). https://doi.org/10.1021/ie051144v https://doi.org/10.1021/ie051144v...
R_gm	8×10^-3	kg∙s^-1∙m^-3	Lin et al. (2011)Lin, T.-H., Lin, C.-H., and Chung, C.A., Comutational Study of Oxygen and Glucose Transport in Engineered Cartilage Constructs, J. Mech., 27, 337-346 (2011). https://doi.org/10.1017/jmech.2011.36 https://doi.org/10.1017/jmech.2011.36...
K_om	6×10^-3	mol∙m^-3	Lin et al. (2011)Lin, T.-H., Lin, C.-H., and Chung, C.A., Comutational Study of Oxygen and Glucose Transport in Engineered Cartilage Constructs, J. Mech., 27, 337-346 (2011). https://doi.org/10.1017/jmech.2011.36 https://doi.org/10.1017/jmech.2011.36...
K_gm	6.3×10^-2	kg∙m^-3	Lin et al. (2011)Lin, T.-H., Lin, C.-H., and Chung, C.A., Comutational Study of Oxygen and Glucose Transport in Engineered Cartilage Constructs, J. Mech., 27, 337-346 (2011). https://doi.org/10.1017/jmech.2011.36 https://doi.org/10.1017/jmech.2011.36...
K_c	0.154	---	Chung et al. (2006)Chung, C.A., Yang, C.W., and Chen, C.W., Analysis of Cell Growth and Diffusion in a Scaffold for Cartilage Tissue Engineering, Wiley Period, 94, 1138-1146 (2006). https://doi.org/10.1002/bit.20944 https://doi.org/10.1002/bit.20944...
Ω	2.1×10⁶	s	Yan et al. (2012)Yan, X., Bergstrom, D.J., and Chen, X.B., Modeling of cell cultures in perfusion bioreactors, IEEE Trans. Biomed. Eng., 59, 2568-2575 (2012). https://doi.org/10.1109/TBME.2012.2206077 https://doi.org/10.1109/TBME.2012.220607...
R_d	3.3×10^-7	s^-1	Chung et al. (2006)Chung, C.A., Yang, C.W., and Chen, C.W., Analysis of Cell Growth and Diffusion in a Scaffold for Cartilage Tissue Engineering, Wiley Period, 94, 1138-1146 (2006). https://doi.org/10.1002/bit.20944 https://doi.org/10.1002/bit.20944...
R_m	3×10^-2	s^-1	Chung et al. (2006)Chung, C.A., Yang, C.W., and Chen, C.W., Analysis of Cell Growth and Diffusion in a Scaffold for Cartilage Tissue Engineering, Wiley Period, 94, 1138-1146 (2006). https://doi.org/10.1002/bit.20944 https://doi.org/10.1002/bit.20944...
V_cell	5.49×10^-16	m³∙cell^-1	Chung et al. (2006)Chung, C.A., Yang, C.W., and Chen, C.W., Analysis of Cell Growth and Diffusion in a Scaffold for Cartilage Tissue Engineering, Wiley Period, 94, 1138-1146 (2006). https://doi.org/10.1002/bit.20944 https://doi.org/10.1002/bit.20944...
K_eq	0.1	---	Chung et al. (2006)Chung, C.A., Yang, C.W., and Chen, C.W., Analysis of Cell Growth and Diffusion in a Scaffold for Cartilage Tissue Engineering, Wiley Period, 94, 1138-1146 (2006). https://doi.org/10.1002/bit.20944 https://doi.org/10.1002/bit.20944...
D_eff,cell	1.7×10^-12	m²∙s^-1	Chung et al. (2006)Chung, C.A., Yang, C.W., and Chen, C.W., Analysis of Cell Growth and Diffusion in a Scaffold for Cartilage Tissue Engineering, Wiley Period, 94, 1138-1146 (2006). https://doi.org/10.1002/bit.20944 https://doi.org/10.1002/bit.20944...

Case	ε_σ/ε_σ,max (%)	ε/ε_max (%)	H/H_max (%)	C_g,0,β/C_g,0,β,max (%)
A	40	100	100	100
B	10	100	100	100
C	100	100	100	100
D	40	100	100	22.22
E	40	80	100	100
F	40	60	100	100
G	20	100	28.95	100
H	20	100	38.16	100
I	20	100	55.26	100
J	20	80	28.95	100
K	20	80	38.16	100
L	20	80	55.26	100
M	20	60	28.95	100
N	20	60	38.16	100
O	20	60	55.26	100

Case	Maximum relative error (%)	Normalized time	Minimum relative error (%)	Normalized time
A	34.76	0.33	3.90	1.00
B	78.33	0.33	41.08	1.00
C	109.04	0.07	21.55	0.77
D	64.57	0.77	19.66	0.07
E	34.83	0.33	3.32	1.00
F	34.96	0.33	2.35	1.00
G	47.59	0.07	5.12	0.77
H	51.69	1.00	8.80	0.53
I	35.32	0.33	14.00	1.00
J	47.57	0.07	5.34	0.77
K	51.01	1.00	8.60	0.53
L	35.36	0.33	14.11	1.00
M	47.55	0.07	5.73	0.77
N	49.83	1.00	8.25	0.53
O	35.42	0.33	13.98	1.00

Case	Maximum relative error (%)	Normalized time	Minimum relative error (%)	Normalized time
A	37.08	0.33	6.47	1.00
B	78.66	0.33	45.06	1.00
C	106.66	0.07	9.42	0.77
D	68.51	0.77	20.76	0.07
E	37.15	0.33	6.90	1.00
F	37.27	0.33	7.63	1.00
G	43.79	0.07	13.64	1.00
H	24.51	0.07	6.75	0.53
I	38.48	0.33	12.96	0.07
J	43.78	0.07	13.82	1.00
K	24.49	0.07	6.88	0.53
L	38.52	0.33	13.05	0.07
M	43.76	0.07	4.52	0.53
N	24.47	0.07	7.11	0.53
O	38.57	0.33	12.93	0.07

Case	Maximum relative error (%)	Normalized time	Minimum relative error (%)	Normalized time
E	37.13	0.33	6.64	1.00
F	37.21	0.33	6.86	1.00
J	43.79	0.07	13.71	1.00
K	24.49	0.07	6.83	0.53
L	39.83	0.33	12.67	0.07
M	43.76	0.07	13.79	1.00
N	24.47	0.07	6.93	0.53
O	38.54	0.33	12.94	0.07

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[1] Corresponding author: Ágata Paim - E-mail: agata@enq.ufrgs.br