Xiaoxiao Han

A phenomenological model for the degradation of biodegradable polymers

This paper presents a phenomenological diffusion-reaction model for the biodegradation of biodegradable polymers. The biodegradation process is modelled using a set of simplified reaction-diffusion equations. These partial differential equations are non-dimensionalised giving two normalised parameters which control the interplay between the hydrolysis reaction and the monomer diffusion. The equations are firstly solved for simple cases of plates and pins. The numerical results are presented in the form of biodegradation maps which show the conditions where the biodegradation is controlled by auto-catalysed hydrolysis, non-catalysed hydrolysis, a combination of auto-catalysed and non-catalysed hydrolyses, or a combination of hydrolysis and monomer diffusion, respectively. The degradation maps provide a clear guide for the design of biodegradable fixation devices used in orthopaedic surgeries. Finally the diffusion-reaction equations are solved using the finite element method for strip and square meshes, showing how the model can be used to assist the design of sophisticated fixation devices.

A model for simultaneous crystallisation and biodegradation of biodegradable polymers

This paper completes the model of biodegradation for biodegradable polymers that was previously developed by Wang et al. (Wang Y, Pan J, Han X, Sinka, Ding L. A phenomenological model for the degradation of biodegradable polymers. Biomaterials 2008;29:3393-401). Crystallisation during biodegradation was not considered in the previous work which is the topic of the current paper. For many commonly used biodegradable polymers, there is a strong interplay between crystallisation and hydrolysis reaction during biodegradation - the chain cleavage caused by the hydrolysis reaction provides an extra mobility for the polymer chains to crystallise and the resulting crystalline phase becomes more resistant to further hydrolysis reaction. This paper presents a complete theory to describe this interplay. The fundamental equations in the Avramis theory for crystallisation are modified and coupled to the diffusion-reaction equations that were developed in our previous work. The mathematical equations are then applied to three biodegradable polymers for which long term degradation data are available in the literature. It is shown that the model can capture the behavior of the major biodegradable polymers very well.

Analysis of degradation data of poly(l-lactide-co-l,d-lactide) and poly(l-lactide) obtained at elevated and physiological temperatures using mathematical models.

The degradation of resorbable polymeric devices often takes months to years. Accelerated testing at elevated temperatures is an attractive but controversial technique. The purposes of this paper include: (a) to provide a summary of the mathematical models required to analyse accelerated degradation data and to indicate the pitfalls of using these models; (b) to improve the model previously developed by Han and Pan; (c) to provide a simple version of the model of Han and Pan with an analytical solution that is convenient to use; (d) to demonstrate the application of the improved model in two different poly(lactic acid) systems. It is shown that the simple analytical relations between molecular weight and degradation time widely used in the literature can lead to inadequate conclusions. In more general situations the rate equations are only part of a complete degradation model. Together with previous works in the literature, our study calls for care in using the accelerated testing technique.

An entropy spring model for the Young’s modulus change of biodegradable polymers during biodegradation

This paper presents a model for the change in Youngs modulus of biodegradable polymers due to hydrolysis cleavage of the polymer chains. The model is based on the entropy spring theory for amorphous polymers. It is assumed that isolated polymer chain cleavage and very short polymer chains do not affect the entropy change in a linear biodegradable polymer during its deformation. It is then possible to relate the Youngs modulus to the average molecular weight in a computer simulated hydrolysis process of polymer chain sessions. The experimental data obtained by Tsuji [Tsuji, H., 2002. Autocatalytic hydrolysis of amorphous-made polylactides: Effects of L-lactide content, tacticity, and enantiomeric polymer blending. Polymers 43, 1789-1796] for poly(L-lactic acid) and poly(D-lactic acid) are examined using the model. It is shown that the model can provide a common thread through Tsujis experimental data. A further numerical case study demonstrates that the Youngs modulus obtained using very thin samples, such as those obtained by Tsuji, cannot be directly used to calculate the load carried by a device made of the same polymer but of various thicknesses. This is because the Youngs modulus varies significantly in a biodegradable device due to the heterogeneous nature of the hydrolysis reaction. The governing equations for biodegradation and the relation between the Youngs modulus and average molecular weight can be combined to calculate the load transfer from a degrading device to a healing bone.

A model for biodegradation of composite materials made of polyesters and tricalcium phosphates

A saturation behaviour has been observed when incorporating tricalcium phosphate (TCP) in various polyesters to control the degradation rate. This paper presents an understanding of this behaviour using a mathematical model. The coupled process of hydrolysis reaction of the ester bonds, acid dissociation of the carboxylic end groups, dissolution of the calcium phosphates and buffering reactions by the dissolved phosphate ions is modelled together using a set of differential equations. Two non-dimensional groups of the material and chemical parameters are identified which control the degradation rate of the composites. An effectiveness map is established to show the conditions under which incorporating TCP into polyesters is effective, saturated or ineffective. Comparisons are made between the model predictions and existing experimental data in the literature. The map provides a useful tool to guide the design of polyester/TCP composites for tissue engineering and orthopaedic fixation applications.

Polymer chain scission, oligomer production and diffusion: a two-scale model for degradation of bioresorbable polyesters.

This paper presents a computer model for the biodegradation of polyesters such as poly(lactic acid), poly(glycolic acid) and their copolymers. The model can take polymer details such as molecular weight distribution, different end and random scission rates and copolymer ratio as input data. A multi-scale approach is developed: polymer chain scission and oligomer production which occur at the molecular scale are modelled using a kinetic Monte Carlo scheme, oligomer diffusion which occurs at the device scale is modelled using a diffusion equation, and the two are connected at the finite difference nodes of the diffusion equation. The two-scale model can be used to predict the temporal evolution and spatial distribution of molecular weight distribution in a device as well as the weight loss as a function of time. It is shown that the kinetic Monte Carlo scheme can accurately predict the effect of copolymer ratio on the degradation rate. Grizzi and co-workers observed in their experiments that a PLA film 0.3mm thick degrades much more slowly than one that is 2mm thick. The numerical study shows that the conceptional reaction diffusion model suggested by Grizzi et al. needs to be extended in order to explain the size effect fully.

Engineering design of artificial vascular junctions for 3D printing

Vascular vessels, including arteries, veins and capillaries, are being printed using additive manufacturing technologies, also known as 3D printing. This paper demonstrates that it is important to follow the vascular design by nature as close as possible when 3D printing artificial vascular branches. In previous work, the authors developed an algorithm of computational geometry for constructing smooth junctions for 3D printing. In this work, computational fluid dynamics (CFDs) is used to compare the wall shear stress and blood velocity field for the junctions of different designs. The CFD model can reproduce the expected wall shear stress at locations remote from the junction. For large vessels such as veins, it is shown that ensuring the smoothness of the junction and using smaller joining angles as observed in nature is very important to avoid high wall shear stress and recirculation. The issue is however less significant for capillaries. Large joining angles make no difference to the hemodynamic behavior, which is also consistent with the fact that most capillary junctions have large joining angles. The combination of the CFD analysis and the junction construction method form a complete design method for artificial vascular vessels that can be 3D printed using additive manufacturing technologies.

Design of bifurcation junctions in artificial vascular vessels additively manufactured for skin tissue engineering

Construction of an artificial vascular network ready for its additive manufacturing is an important task in tissue engineering. This paper presents a set of simple mathematical algorithms for the computer-aided design of complex three dimensional vascular networks. Firstly various existing mathematical methods from the literature are reviewed and simplified for the convenience of applications in tissue engineering. This leads to a complete and step by step method for the construction of an artificial vascular network. Secondly a systematic parametric study is presented to illustrate how the various parameters in the vascular junction model affect the key factors that have to be controlled when designing the bifurcation junctions of a vascular network. These results are presented as a set of simple design rules and a design map which serve as a convenient guide for tissue engineering researchers when constructing artificial vascular networks. Construction of an artificial vascular network ready for its additive manufacturing is an important task in tissue engineering.A set of simple mathematical algorithms for computer-aided design of complex three dimensional vascular networks are provided.A systematic parametric study is presented.A set of simple design rules and a design map which service as a convenient guide for tissue engineering researchers are generated.An example of application and its flowchart are presented.

Modelling degradation of biodegradable polymers

Abstract Computer modelling is a modern technique that helps research in medical applications to design and develop their devices. In this chapter, small molecule diffusion theories based on Fick’s law are introduced first. A simplified Fickian model is used to derive diffusion kinetic models with the aid of classical mathematics. Based on the outcomes of the kinetic models, Fickian and non-Fickian diffusion kinetics were discussed for biodegradable polymers. This is followed by computer modelling of the polymer degradation. It starts with introduction of degradation mechanism and then computer modelling. Basic computer modelling methodologies are discussed. The outcomes of the modelling with experimental results are discussed. These computing models can be used to predict biodegradation rate, molecular weight, oligomer diffusion, as well as mechanical properties of a bioresorbable polymer system.