The drying of particulate suspensions is important to many industries such as paints, ceramics, minerals processing, and pharmaceuticals. Cakes or films first consolidate due to capillary pressure and, at a critical concentration, stop consolidating and begin to desaturate. Desaturation occurs once the compressive strength of the particulate network is greater than the maximum capillary pressure at the air-liquid interface. This work combines existing descriptions of the compressive strength and the maximum capillary pressure to give the dependencies of volume fraction, particle size, interparticle bond strength, surface tension, and contact angle on the breakthrough pressure and critical concentration. Understanding the interplay of these system parameters explains the point of desaturation in filtration and drying processes, allowing optimization of these processes, including mitigation of cracking. Air-driven filtration results are presented for the direct measurement of breakthrough pressure of coagulated calcium carbonate.