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Multicolor cell spatio-temporal image data have become important to investigate organ development and regeneration, malignant growth or immune responses by tracking different cell types both in vivo and in vitro. Statistical modeling of image data from common longitudinal cell experiments poses significant challenges due to the presence of complex spatio-temporal interactions between different cell types and difficulties related to measurement of single cell trajectories. Current analysis methods focus mainly on univariate cases, often not considering the spatio-temporal effects affecting cell growth between different cell populations. In this paper, we propose a conditional spatial autoregressive model to describe multivariate count cell data on the lattice, and develop inference tools. The proposed methodology is computationally tractable and enables researchers to estimate a complete statistical model of multicolor cell growth. Our methodology is applied on real experimental data where we investigate how interactions between cells affect their growth. We include two case studies; the first evaluates interactions between cancer cells and fibroblasts, which are normally present in the tumor microenvironment, whilst the second evaluates interactions between cloned cancer cells when grown as different combinations.