Constrained hierarchical modeling of degradation data in tissue-engineered scaffold fabrication

ABSTRACT

In tissue-engineered scaffold fabrication, the degradation of scaffolds is a critical issue because it needs to match with the rate of new tissue formation in the human body. However, scaffold degradation is a very complicated process, making degradation regulation a challenging task. To provide a scientific understanding on the degradation of scaffolds, we propose a novel constrained hierarchical model (CHM) for the degradation data. The proposed model has two levels, with the first level characterizing scaffold degradation profiles and the second level characterizing the effect of process parameters on the degradation. Moreover, it can incorporate expert knowledge in the modeling through meaningful constraints, leading to insightful inference on scaffold degradation. Bayesian methods are used for parameter estimation and model comparison. In the case study, the proposed method is illustrated and compared with existing methods using data from a novel tissue-engineered scaffold fabrication process. A numerical study is conducted to examine the effect of sample size on model estimation.

Keywords:

• Bayesian inference
• Bayes factor
• constrained hierarchical model
• Gibbs sampling
• tissue-engineered scaffolds

Funding

Li Zeng gratefully acknowledges financial support from the National Science Foundation under grant CMMI-1266225. Jian Yang acknowledges support from the National Science Foundation under grant CMMI-1266116.

Appendices

Appendix A. proofs of equations (6)(6) and (7)(7)

By Equations (1) to (4): (A.1) thus the joint posterior of {c₀, … ,c_p-1, c_σ} is where π(·) is the prior. Since flat priors are used for all of the parameters:

The conditional posteriors in Equations (6)(6) and (7)(7) can be obtained from the joint posterior by conditioning on the given quantities.

Appendix B. proof of equations (8)(8) and (9)(9)

By Equation (A1)(A.1) :

Given {c₁, … ,c_p-1, c_σ}, the first term is a constant, so can be moved to the left. Let thus, (A.2)

Equation (A2)(A.2) can be viewed as a polynomial model of z with coefficient c₀ and known variance–covariance matrix V₀(z; c_σ). According to Gelman et al. (2004, p. 374), under flat priors, the conditional posterior of the coefficient of a linear model given variance–covariance matrix is multivariate normal: which gives Equation (8)(8) .

The conditional posterior of c_k given {c₁, … , c_k-1, c_k+1, … ,c_p-1, c_σ}, 1 ≤ k ≤ p − 1, can be proved in a similar way. Specifically, by Equation (A1)(A.1) : Let then (A.3)

Since

Thus, Equation (A3)(A.3) can be viewed as a polynomial model of z with coefficient c_k and known variance–covariance matrix V_k(z; c_σ). Thus, which gives Equation (9).

Appendix C. proof of theorem 1.

From the definition of the BF in Equation (10)(10) :

By condition (i):

Note that the denominator in the bracket—i.e., —is a constant since the integrand only depends on . The numerator depends on ξ since the integrand, , involves and ξ (recall θ_l = [θ_s ξ]^′). Thus, the quantity in the bracket is a function of ξ.

Let then (A.4) which means that the BF is an average of ψ(ξ) over the prior of ξ under M_l. By condition (iii): (A.5)

The joint posterior with this prior is which gives the marginal posterior of ξ:

Consequently,

Thus,

By the definition of ψ(ξ), we can get (A.6)

Plugging Equations (A5) and (A6) to Equation (A4)(A.4) gives so the theorem holds.

Additional information

Notes on contributors

Li Zeng

Li Zeng is an Assistant Professor in the Department of Industrial and Manufacturing Systems Engineering at the University of Texas at Arlington. She received her B.S. degree in Precision Instruments and M.S. degree in Optical Engineering from Tsinghua University and Ph.D. in Industrial Engineering and M.S. degree in Statistics from the University of Wisconsin–Madison. Her research interest is process monitoring and control in complex manufacturing and healthcare delivery systems. She is a member of INFORMS and IIE.

Xinwei Deng

Xinwei Deng is an Assistant Professor in the Department of Statistics at Virginia Tech. He received his Ph.D. degree in Industrial Engineering from Georgia Tech and his bachelor's degree in Mathematics from Nanjing University, China. His research interests are in statistical modeling and analysis of massive data, including high-dimensional classification, graphical model estimation, interface between experimental design and machine learning, and statistical approaches to nanotechnology. He is a member of INFORMS and ASA.

Jian Yang

Jian Yang is an Associate Professor of Biomedical Engineering at the Pennsylvania State University. He is known as the inventor for citrate-based biomaterials for tissue engineering and medical devices. He has published 71 journal articles with many shown in prestigious journals such as PNAS, Advanced Materials, and ACS Nano. He has also received eight issued patents for his inventions in citrate polymers and their applications. He was a recipient of an NSF CAREER Award (2010) and Outstanding Young Faculty Award of College of Engineering at the University of Texas Arlington (2011). He serves as an Associate Editor for Frontiers in Biomaterials and on the editorial board for a number of journals in his field.