Aaron Scheiner 1 , Thomas Shomer 2 , T.J. Sego 3 , Andres Tovar 4 1 Department of Mathematics, Rutgers University; 2 Department of Mathematics and Statistics, Valparaiso University; 3 Department of Intelligent Systems Engineering, Indiana University; 4 Department of Mechanical and Energy Engineering, IUPUI Multi-material Topology Optimization using a Cellular Potts Model INTRODUCTION Topology optimization (TO) is a process to optimally distribute material within a prescribed volume (design domain) in order to maximize the performance of the resulting structure [1,2]. TO is recognized as the most effective numerical method to generate novel and high-performance architectural layouts. Multi-material topology optimization (MMTO) is a finite element analysis (FEA)-based method to optimally distribute a plurality of materials within a prescribed volume without any constraint in the final shape [3]. While successful, most variations of MMTO are extensions of classical topology optimization and rely on traditional, gradient-based optimization. This limits their application, particularly when gradient coefficients are unknown, to simpler problems that involve linear FEA models. The hybrid Cellular Potts Model (CPM) simulates cell sorting, movement, and many additional cellular phenomena based on adhesion energies between the cells, the target sizes of the cells, and intrinsic random motility. It runs according to a Hamiltonian function, which can be modified in its complexity to incorporate a plenitude of cellular phenomena. MOTIVATION AND OBJECTIVE Motivation : Gradient-based MMTO is limited to linear or mildly nonlinear FEA problems in which the sensitivity coefficients can be analytically derived. Hence, inspired by the processes of cellular dynamics, we sought to create a gradient-free MMTO method. Objective : The objective of this research is to construct a novel, bio- inspired MMTO method based on a hybrid Monte Carlo method. The hybrid Monte Carlo method is preceded by a hybrid CPM. The proposed MMTO method would be capable of being applied to nonlinear FEA models. METHODS Figure 1: Generalized CPM evolution of cells depicting a necrotic core due to a lack of oxygen [7]. Source: IUPUI. The construction of the method entailed a bottom-up approach, beginning with the development of a fully-functional, generalized CPM [6]. The proposed MMTO method implements the Boltzmann acceptance function described in Eq. (1). This equation governs a local transition rule of material distribution as a function of the change of the Hamiltonian energy expression in Eq. (2) central to every CPM. The input parameters account for an arbitrary number of cell types when modeling spatial constraints and adhesion. In addition, the method incorporates the specification of details concerning the transition rule. The generalized CPM has the ability to simulate the dynamics of multicellular environments (Figure 1). In addition, a MMTO algorithm has been also utilized to generate multi-material designs (Figure 2). Figure 2: MMTO evolution of design domain with void (white) and two materials (blue and red) [3]. In our work, the CPM was coupled with FEA in the current hybrid structure. Following this, the strain energy field was incorporated into the model. The model is now being programmed to employ equations from MMTO during the development of new transition rules that produce emergent optimization. Most recently, chemotaxis was coded into the method. MAIN MODEL: EQUATIONS Eq. (1) where  and  represent neighboring lattice sites,  $ =  $ is the cell type of the cell  $ ,  is the boundary coefficient that determines the adhesion between two cell types,  $) represents the Kronecker delta for two neighboring cells,  $ = ( $ ) equals the volume of the cell  $ ,  $ =  $ outputs the target volume of the cell  $ , and  equates to the Lagrange multiplier that controls the power of the volume constraint [5]. While this Hamiltonian demonstrates the generalized CPM’s necessities, it can be modified to include additional cellular phenomena, as was done in our model. This Boltzmann acceptance function describes the probability of a successful copy attempt [4]. This function is given by where T refers to the Intrinsic random motility and ∆ is the change in the Hamiltonian  . In this context, The Hamiltonian is defined by  Δ =4 1 if Δ < 0 exp − ⁄ Δ  if Δ ≥ 0 Eq. (2) =A $.)   $ − ) 1− $) +A $  $ − $ D c RESULTS The generalized hybrid CPM was successfully implemented. The evolution of the associated strain energy field was also simulated using FEA. Results demonstrated the ability to sort cells and reach a homeostatic stage (Figure 3). In this work, chemotaxis was successfully incorporated in the generalized hybrid CPM. The chemotaxis demonstrated was implemented as a function of the strain energy density field (Figure 4). Figure 3: Evolution of the generalized hybrid CPM with two cell types for 3200 Monte Carlo steps. Top: cell sorting. Bottom: associated strain energy field. Figure 4: Evolution of the hybrid, chemotaxis-incorporated CPM with two cell types for 10000 Monte Carlo steps. Top: cell sorting. Bottom: associated strain energy field. SUMMARY Accomplishments : Several key accomplishments have been achieved in order to establish a novel MMTO derived from a generalized CPM. Firstly , a hybrid Monte Carlo method developed from a generalized CPM was implemented in this research. Secondly , FEA was coupled with the hybrid Monte Carlo method to evaluate overall stiffness and to predict stress and strain fields. Thirdly , chemotaxis was implemented to incorporate the anticipated total stiffness as well as the predicted stress and strain fields into cell motility. Future Work: Ongoing research includes the development of new transition rules and cellular phenomena in order to better relate the structure of the CPM to field variables of solid mechanics. It is expected that the resulting method will be a true alternative to traditional, gradient-based MMTO methods and will be capable of addressing problems with nonlinear FEA models. Comparison to other MMTO methods (mostly gradient-based) will demonstrate the capabilities of the proposed approach. This project is supported by the REU Sites program of the National Science Foundation (NSF DMS- 1852146) and by the IUPUI Department of Mathematical Sciences. ACKNOWLEDGEMENTS REFERENCES [1] Bendsøe & Sigmund, Springer, 2003 [2] Liu & Tovar, Struct. Multidisc. O., 2014 [3] Tavakoli & Mohseni, Struct. Multidisc. O., 2014 [4] Boas et al., Lousi and Nardi (ed.) Springer, 2018 [5] Oers et al., PLOS, 2014 [6] T. Hirashima, Mathworks, 2017 [7] Sego & Tovar, PLOS Comput. Biol. (MUR), 2019 MAIN MODEL: ALGORITHM