The advanced computational design of tissue scaffolds was carried out utilising topology optimisation techniques, and studies had looked into the manipulation of microstructure based on multiple design criteria [13, 132, 164, 168]. Multi-objective topological optimisation helps address a multitude of design considerations, conflicts and constraints simultaneously [12, 169]. Optimisation algorithms such as bi-directional evolutionary structural optimisation (B/ESO), level-set [170, 171], and inverse homogenisation [164] have been applied and all-rounded design solutions have been obtained. However, a design method for time-dependent criteria has yet been established. Nevertheless, it has been suggested using a dynamically evolving culturing condition to match tissue maturity and maximise the efficiency of tissue regeneration process [172].
To evaluate and select the most suitable methods for scaffold design, various topology optimisation methods and results obtained from past researches were analysed. In this section, the merits and issues associated with computational techniques are discussed in the
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context of structural modelling, along with their potential and limitations in tackling the key challenges in tissue engineering.
2.2.2.1 Modelling and optimisation methods
There exist a number of fully-developed topology optimisation methods, such as the Evolutionary Structural Optimisation and the Bi-directional variant, the Solid Isotropic Material Penalisation method, and the Level Set method. These techniques provide the design capability essential to generate and manipulate complex structural.
Evolutionary Structural Optimisation (ESO) technique and its improved version, Bi- directional Evolutionary Structural Optimisation (BESO), are two well-established finite element methods for topological design [173-175]. They are based on the idea of gradual structural evolution by eliminating less contributing elements and reinforcing the more critical ones (only in BESO) in a fixed finite element domain. The end result is a design with the highest average criteria across all individual constituent elements, hence the maximisation of design objective such as stiffness. The BESO approach allows the addition of efficient material, or the restoration of erroneously removed elements and provides a more flexible evolution path to the global optimum [176]. Recently studies have also demonstrated the capability of BESO of generating complex micro- and macro-structures [177-179]. While these methods enjoy computation robustness and versatility, the fact that they are using fixed, often square and cubic, mesh makes them strongly resolution dependent.
The Solid Isotropic Material with Penalisation (SIMP) approach is a density or volume fraction based representation originally used in topological optimisation of macrostructures [175, 180] but later implemented in the design of porous materials [162]. Often combined with the method of moving asymptote [181], it uses partially solid elements with penalised material properties. The concept of ersatz material, i.e. conceptual material that only exists in
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computational environment, is closely associated with SIMP only with an ultra-low density to mimic void. The implementation of sensitivity filtering is critical to the rigorousness of this density approach through the elimination of numerical artefacts [182]. Ersatz material and sensitivity filtering are also applied to numerical methods other than BESO when tackling similar issues. A major drawback is that volume fraction representation results in a grey scale model with a blurred boundary that is difficult to track. This approach makes better sense in solid evolution than fluidic design.
It has been well known that conventional topology optimisation methods frequently encounter two issues: (1) one being the ambiguous, blurred boundary as a result of voxelisation or filtering, and dependence on resolution; (2) the common use of smeared Heaviside and δ functions that is responsible for level set function deterioration [183]. To tackle the second issue, numerous re-initialisation algorithms have been developed and implemented, but more than often the process is accompanied by adverse numerical side effects and inconsistency [184]. Meanwhile, the advances in unstructured mesh generation [185-189] and the development of new level-set based adaptive meshing methods [190] makes unstructured mesh a promising alternative to voxelised models and offers a possible optimisation pathway to a more accurate solution [171, 191].
To solve dynamic fluid boundary problems, the level set method stands out as a more favourable approach because of its surface tracking capability. Level set provides an implicit means for defining a stationary or dynamic boundary evolving in space, and is particularly useful for interface tracking. Level set method is well-established and has been incorporated into many numerical techniques such as image segmentation [192], fluid dynamics [170, 193, 194], and shape optimisation [183, 190, 191] due to its versatility in tracking boundaries of random or complex bodies. A fictitious energy technique has been recently introduced to
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allow formation of holes and the capability of topological optimisation [195]. The level set based topology optimisation was first developed by Sethian [196] and a more standard method was established by Osher et al. [197], Wang et al. [198], and Allaire et al. [183]. This technique clearly defines boundaries that divide a design domain into separate regions. For general solid design, the concept of ersatz material has also been incorporated into the level- set method to improve computational robustness [199, 200].
Many researches on the topic of level-set optimisation emphasize building more robust model with relatively low computational expense by avoiding re-meshing, but fail to take full advantage of such prospective boundary tracking technique. One of the most common approaches involves combining signed distance function and smeared Heaviside function when defining a fixed-grid level-set function and evaluate sensitivity as an evolution criterion [198]. However, this method suffers from a number of issues: it experiences certain degree of numerical diffusion which is a necessary process in sensitivity filtering; the simulation is not performed on the same design boundary (voxelised) drawn by the level-set function; also, the boundary sharpness is strictly limited by grid resolution used in finite element analysis. Implementing the re-meshing step is becoming inevitable in level-set based optimisation if a better boundary definition beyond the current tracking capacity is to be found. Recently, there are developments on level-set optimisation with implementation of unstructured grid [171, 191] and new adaptive meshing technique found on level-set function [190]. These studies are making steps toward a possible breakthrough of a highly versatile mesh generation suitable for implicit modelling in an evolution purpose. In computational fluid dynamics (CFD), meshing is critical to the capture of topological effect, thus it holds the key to the accurate optimal solution in flow optimisation problems. Phase-field method is similar to level-set method but will not be discussed in this study [201].
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In summary, existing optimisation methods have the potential but not the full capacity to define the optimal structure of tissue scaffolds. Some modifications must be made to their surface tracking technique to resolve the modelling issues.
2.2.2.2 Topology optimisation
Topology optimisation has recently found applications in the field of scaffold tissue engineering. The key benefit of such application is that the structural design process can be carried out away from laboratories, thus it helps reduce experimental cost. Additionally, topology optimisation can help assess and verify the optimality of different structural characteristics, and create a clearer picture of the ideal tissue scaffolds.
Past studies have shown support to the hypothesis that bone tissue regeneration in porous scaffold responds to local mechanical strain [202] or mechanical stimuli such as pulsatile pressure [203]. This implies that the basic properties of tissue scaffolds will affect tissue regeneration. For this reason, properties such as stiffness and bulk modulus can be subject to design optimisation to indirectly influence cell development. A number of studies has applied topology optimisation on microstructure using the effective stiffness [164, 204, 205] and bulk modulus [10, 206, 207] as a design criterion.
Fluid flow behaviour in a porous tissue scaffold is a complex mechanical problem and is difficult to analyse on a microscopic level. Fortunately, computational fluid dynamics (CFD) and the homogenisation technique together make it possible to study the fluid-structure interaction. Topology optimisation of tissue scaffolds based on conductivity/diffusivity criterion has already been looked into [10, 12]. Results of permeability optimisation are seen as a potential solution to general issues associated with in-scaffold cell activity [84, 134]. Recent modelling studies have also investigated fluid transport phenomena in tissue scaffolds and suggested that optimised fluid domains somewhat resemble the Schwarz’s Primitive
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surface, which led to the speculation that Schwarz’s Primitive surface construct is optimised for permeability and conductivity [170, 208, 209]. There is also a growing interest in characterising various triply minimal surfaces’ fluid-dynamics properties [66, 133, 134, 209, 210]. Examples of triply minimal surfaces are Schwarz’s Primitive surface and Schwann’s Gyroid surface. However, apart from crude resemblance, researchers have yet provided rigorous proof that the optimised surface is exactly the same as the Schwarz’s Primitive surface, or counter-proof that the Schwarz’s Primitive construct can be improved further.