Computational design of curvilinear bone scaffolds fabricated via direct ink writing

Highlights

• A computational design method for ceramic bone scaffolds is proposed.

• We design scaffolds made of printed, alternate layers of curvilinear rods.

• Cellular solids theory is used to relate effective properties to design parameters.

• Gradient-based optimization renders optimal rods separation and curvature.

• Demonstrate that curvilinear-rod scaffolds mechanically outperform periodic ones.

Abstract

Bone scaffold porosity and stiffness play a critical role in the success of critical-size bone defect rehabilitation. In this work, we present a computational procedure to design ceramic bone scaffolds to provide adequate mechanical support and foster bone healing. The scaffolds considered in our study consist of a lattice of curved rods fabricated via direct ink writing. We develop cellular solids models of the scaffold’s effective elastic constants as functions of its geometric parameters, up to some unknown coefficients. To determine numeric values for these coefficients, we execute a computational design of experiments whereby effective elastic properties are obtained using numerical homogenization with the finite element method. In order to automate these experiments and circumvent re-meshing for every scaffold geometry, we project a representative volume element of the scaffold onto a fixed uniform mesh and assign an ersatz material for the analysis. We use these calibrated models in conjunction with finite element analysis and efficient gradient-based optimization methods to design patient-specific scaffolds (i.e., shape, location, and loading) by varying geometric parameters that can be controlled in the fabrication, namely the separation between rods in the lattice, and the printing path of the rods. At present, our methodology is restricted to 2-d idealizations of flat bones subject to in-plane loading. The optimization procedure renders element-wise values of these parameters. As this representation is not amenable to fabrication, lastly, we generate the final scaffold geometry by posing a differential equation whose solution is a function such that its level set lines at specified values correspond to the directrices of the rods in the scaffold. We present examples where we perform the maximization of stiffness of a scaffold implant with a constraint on porosity.

Introduction

Bone grafts are necessary to enhance biologic repair of defect sites which are too large to heal naturally, known as critical-size bone defects. Having surpassed four million annual procedures, there is a substantial and growing demand for bone grafts [1]. The current standard is the use of autografts, i.e. grafts from bone harvested from the patient’s own body. However, several issues arise with autografts, including donor site morbidity, limited availability, and graft quality. Alternatively, surgeons use grafts donated from cadavers (known as allografts) but these are likewise limited in availability, may incite an immunogenic response, and although sterilized, do not preclude the possibility of disease transfer [[2], [3], [4]].

The alternative to biological origin grafts is synthetic scaffolds, which vary in materials, fabrication techniques and structure. These scaffolds are designed with considerations such as biocompatibility, osteointegration, osteoconduction, porosity, permeability, stiffness, manufacturing process, angiogenesis, cellular seeding, and growth factor delivery [[5], [6]]. Each of these design considerations dictates geometric and mechanical requirements on the scaffold. Angiogenesis of a defect site, for example, demands facilitated nutrient transport, waste removal and void regions to allow vasculature in-growth. This demand requires a large pore size and interconnectivity and consequently a high porosity scaffold [[5], [6]]. In load-bearing applications, stiffness requirements are paramount. A graft that is too stiff will exhibit low strain energy density and thus reduce the mechanotransduction necessary to induce bone growth and stave off bone resorption, a phenomenon called stress shielding [[4], [7]]. A graft that is too compliant, on the other hand, will not provide load-bearing support so as to return function to the afflicted region. Porosity is also an essential design consideration to promote osteoconduction, osteointegration, angiogenesis, nutrient and waste transportation, and osteoinduction [[5], [6], [7]].

In this work, we focus on porosity and stiffness so as to enhance scaffold implants through osteoconduction and adequate levels of mechanical support. Therefore, we must select a material system that renders scaffolds with feasible magnitude ranges for these two objectives, i.e., we must choose a fabrication technique which is conducive to high porosity, and use biocompatible materials with stiffness similar to that of bone. Here, we consider direct ink writing (cf., [8]) of ceramic hydroxyapatite (HA) scaffolds. By extruding colloidal HA through a nozzle in a layer-by-layer fashion, we have a high level of control of the porosity and stiffness. The ability to use HA, a bioactive material naturally found in bone [9], makes this material system particularly attractive.

To balance these conflicting design considerations, researchers have looked to numerical optimization. Dias et al. used topology optimization to determine the optimal material distribution within a periodic representative volume element (RVE) of the scaffold to attain a target stiffness while maximizing permeability, tailored to selective laser sintering (SLS) of poly-$ϵ$-caprolactone-4%hydroxyapatite [10]. Hollister et al. employ a ‘restricted optimization’ problem whereby homogenization theory is used to link the scaffold’s unit cell micro-structural design parameters (i.e. voids defined by mutually orthogonal cylinders of diameters $d_1$, $d_2$, and $d_3$) to its effective stiffness tensor using a nonlinear least squares-fitted polynomial to minimize the difference to a target effective elasticity tensor. Lin et al. built upon this approach and broadened the design space by employing topology optimization where, again, the error between the homogenized effective and target stiffness tensors is minimized subject to a porosity constraint, although their porosities do not guarantee diffusivity [11]. To address the diffusivity issue, some, including Kang et al. have adopted diffusivity as part of the optimization objectives, whereby a multiobjective topology optimization formulation matches the effective elasticity tensor and diffusivity to target values subject to porosity constraints [12]. Sturm et al. also use topology optimization to match a target stiffness, and subsequently evaluate the bone adaptation of scaffolds with stiffness at, above, or below the stiffness of bone [13]. More recently, lattice-shaped bone scaffolds were optimized by Boccaccio et al. to maximize bone in-growth using a mechano-biology-based algorithm [14]. Chen et al. explore the effects of scaffold design on implant degradation and tissue ingrowth by maximizing the stiffness and permeability of the RVE subject to a volume fraction constraint in a weighted multi-objective topology optimization problem [15]. Focused on facilitating manufacturing, Makowski and Kuś tailor the compliance of structures made by fused deposition modeling (FDM) structures to that of bone by using an evolutionary optimization algorithm to update the dimensions of the rectangular cross-section of the beams in a scaffold [16]. Similarly, and relevant to this work, Entezari et al. consider optimization of HA bone scaffolds created by direct ink writing, where they maximize the scaffold’s compressive strength subject to a constant 60% porosity by varying rod layer overlap and rod spacing [17]. All these works produce scaffolds of constant porosity and orientation, i.e., the geometric elements that define the scaffold geometry (such as rectangular beams, cylindrical holes, etc.) are evenly separated throughout the scaffold and have the same orientation. As we demonstrate in this work, scaffolds with curvilinear, unevenly spaced geometric elements can attain drastic improvements in stiffness and porosity when compared to their periodic counterparts.

To design these curvilinear scaffolds, we start by implementing cellular solids models to establish functional relationships between effective elastic properties and geometric parameters, up to some unknown constants. These cellular solids models are subsequently curve-fit by least-squares regression to a data set of effective elastic constants. This data set is generated by an automated computational design of experiments that employs numerical homogenization. To automatically generate and analyze these numerical experiments, we must avoid the cumbersome, inefficient and difficult to automate generation of finite element meshes that conform to the different RVE geometries. To this end, we smoothly project the RVE geometry onto a density field of a fixed, regular finite element grid, and employ an ersatz material model to perform the numerical homogenization.

Once calibrated, we employ these cellular solids models of the scaffold’s effective properties to design patient-specific implants. We create a finite element model of the defect region and of the surrounding bone and impose appropriate boundary conditions. In the defect mesh, we define the elasticity tensor of each element as a function of the rod separation and orientation. Subsequently, these element-wise geometric parameters constitute the design variables in the optimization. We then employ efficient gradient-based optimization methods to design a mandibular implant for (a) maximal stiffness given a minimum porosity requirement; and (b) maximal porosity given a constraint that the scaffold’s stiffness must at least equal that of the native bone that would otherwise occupy the defect. The resulting element-wise optimal parameters are not conducive to manufacturing. To facilitate manufacturing, we therefore establish a differential equation that has as solution a function whose level set lines at specified values correspond to the rod directrices, which we then use to automatically produce a computer aided design (CAD) geometry suitable for fabrication. The outcome of our methodology is an automated process for efficiently designing and printing patient-specific bone scaffolds for stiffness and porosity.

The layout of the paper is as follows. The derivation of the uncalibrated cellular solids models is detailed in Section 2.1. Section 2.2 describes the generation and execution of the computational design of experiments to calibrate the cellular solids models. The optimization problem we consider is detailed in Section 2.3. We explain the level set tool-path generation function in Section 2.4, and we present the results of the mandible implant design examples in Section 3. Lastly we discuss conclusions of this work in Section 5.