The most popular in vitro assay currently used to characterize cell traction forces exerted on extracellular matrix (ECM) fibers is the fibroblast-populated collagen lattice (FPCL) assay. The compaction of a disk of cell-populated collagen gel, in terms of rate or extent of diameter reduction, is typically reported as the measure of cell traction. This measure, however, depends on assay properties incidental to the intrinsic traction, such as the initial cell concentration, the initial collagen concentration, and the geometry of the gel. Thus, there is a clear need to identify and measure an objective index of cell traction. Here, we propose as such an index a traction parameter (reflective of the cell-fiber mechanical interaction) defined in a continuum theory in which the interactive processes of cell motility and ECM deformation are modeled by expressions for cell and ECM conservation coupled to the mechanical force balance for the cell-ECM composite. The equations are formulated and solved for our adaptation of the FPCL assay in which cells are initially dispersed in a collagen gel microsphere, conferring several experimental and theoretical advantages over the popular disk geometry. The solution of the nonlinear system of partial differential equations (parameterized on the traction parameter) is then compared to compaction data for the fibroblast-populated collagen microspheres (FPCM). We show that the model predictions are consistent with the data when the initial cell concentration and the initial FPCM diameter are varied. In Part 2, we show how these results, along with the determination of the growth parameters of the cells and the viscoelastic parameters of the gel, have allowed us to estimate the magnitude of the traction parameter, which is a direct measure of the traction exerted by fibroblasts in a physiologically relevant collagen gel.