## 1 Introduction  
Lattice-based models have seen broad application in the study of a variety of mechanisms and dynamics through simulation in environments where incorporation of spatial effects is important. For example, they have been commonly used for modeling biotransport, cell dynamics and tissue dynamics with applications in many biological and biomedical areas [1, 2, 3, 4, 5, 6]. One example is the cellular Potts model, in which a biological cell is defined over a region comprising multiple lattice sites [2, 5, 6]. This lattice-based modeling framework enables the study of coupled mechanisms in biological cell dynamics and has been applied to various biomedical problems, including epidermal biology, cancer, angiogenesis and vasculogenesis [1, 4]. Such discrete, lattice-based models commonly involve diffusion (e.g., of mobile solutes) as a key transport mechanism. Alternatively, agent-based models [7, 8, 9] often restrict movement to a lattice, with random walks commonly used as a simple or convenient approach to advance individual agent locations at each time increment. In such cases, diffusion of nutrients and cell-synthesized solutes is an important coupled mechanism. As computing power has advanced, direct simulation in continuous-domain (off-lattice) models is a useful tool for investigating systems where particles or agents interact with a background medium or with each other. However, the direct approach can become tedious or computationally intensive in situations involving several coupled mechanisms or complex geometry or in the presence of many particles or agents. The ability to systematically connect off-lattice (continuous) models to lattice-based (discrete) models, accounting for a few key underlying model features or mechanisms, has the potential to improve the overall accuracy of discrete models. Moreover, such discrete models can also be connected to estimates of theoretical diffusivities for random walks on a two-dimensional (2D) square lattice. An appropriate spatial length scale (grid size) must be determined for the discrete model, especially when diffusion is one of several coupled mechanisms in a 2D lattice-based model. In this study, we evaluate the extent to which 2D lattice-based surrogate models can preserve measures of particle diffusivity observed in direct simulation of continuous models based on 2D random walks with internal obstacles. Our overall approach is illustrated in Figure 1. Results from continuous model simulations (Figure 1A) on a 2D representative volume element (RVE), and over many realizations, are used to develop lattice-based surrogate models (Figure 1B) that reproduce key features of the continuous model. This is accomplished by exploiting internal symmetries in the continuous model representation. Probabilities in the lattice-based surrogate models are also connected to theoretical diffusivities for 2D random walks on a square lattice (Figure 1C). This final step requires calibration of the discrete length scale (grid size) in the square lattice surrogate model. The random walker is prescribed a fixed particle radius, and obstacle radii are varied in the context of two different obstacle configurations, each having distinct internal symmetries. Particle dynamics observed for the continuous model simulations are used to develop probabilistic lattice-based surrogate models for transitions across RVE subdomains. The subdomains are chosen to preserve internal symmetries for each of the two obstacle arrangements. A primary advantage of this approach is the capability of the surrogate models to bypass the need to directly represent the obstacles in the simulation domain. The well-known linear relationship between the mean squared particle displacement and time for the continuous model [10; 11; 12] is used to estimate and compare diffusivities for both the continuous and lattice-based surrogate models.