**Irene Buj-Corral \*, Ali Bagheri and Oriol Petit-Rojo**

Department of Mechanical Engineering, Universitat Politècnica de Catalunya, Av. Diagonal, 647, 08028 Barcelona, Spain; ali.bagheri@upc.edu (A.B.); oriol.petit@hotmail.com (O.P.-R.)

**\*** Correspondence: irene.buj@upc.edu; Tel.: +34-93-4054015

Received: 27 June 2018; Accepted: 22 August 2018; Published: 25 August 2018

**Abstract:** 3D printed scaffolds can be used, for example, in medical applications for simulating body tissues or for manufacturing prostheses. However, it is difficult to print porous structures of specific porosity and pore size values with fused deposition modelling (FDM) technology. The present paper provides a methodology to design porous structures to be printed. First, a model is defined with some theoretical parallel planes, which are bounded within a geometrical figure, for example a disk. Each plane has randomly distributed points on it. Then, the points are joined with lines. Finally, the lines are given a certain volume and the structure is obtained. The porosity of the structure depends on three geometrical variables: the distance between parallel layers, the number of columns on each layer and the radius of the columns. In order to obtain mathematical models to relate the variables with three responses, the porosity, the mean of pore diameter and the variance of pore diameter of the structures, design of experiments with three-level factorial analysis was used. Finally, multiobjective optimization was carried out by means of the desirability function method. In order to favour fixation of the structures by osseointegration, porosity range between 0.5 and 0.75, mean of pore size between 0.1 and 0.3 mm, and variance of pore size between 0.000 and 0.010 mm<sup>2</sup> were selected. Results showed that the optimal solution consists of a structure with a height between layers of 0.72 mm, 3.65 points per mm2 and a radius of 0.15 mm. It was observed that, given fixed height and radius values, the three responses decrease with the number of points per surface unit. The increase of the radius of the columns implies the decrease of the porosity and of the mean of pore size. The decrease of the height between layers leads to a sharper decrease of both the porosity and the mean of pore size. In order to compare calculated and experimental values, scaffolds were printed in polylactic acid (PLA) with FDM technology. Porosity and pore size were measured with X-ray tomography. Average value of measured porosity was 0.594, while calculated porosity was 0.537. Average value of measured mean of pore size was 0.372 mm, while calculated value was 0.434 mm. Average value of variance of pore size was 0.048 mm2, higher than the calculated one of 0.008 mm2. In addition, both round and elongated pores were observed in the printed structures. The current methodology allows designing structures with different requirements for porosity and pore size. In addition, it can be applied to other responses. It will be very useful in medical applications such as the simulation of body tissues or the manufacture of prostheses.

**Keywords:** fused deposition modeling; 3D printing; scaffolds; porosity; pore size; multiobjective optimization
