It depends on the drug you are using. If it is chemotherapy, you can use a MTS assay or Xcelligence on cells (to measure apoptosis). If your carrier has immunomodulatory activity, you can do a variety of assays with flowcytometry checking activation markers on APC and T cells, T cell proliferation etc. You can also check for cytokines secretion using ELISA.
If you want to check for the anti-proliferative effect of your nanoparticle / drug carrier on cancer cells, you could easily employ MTS/MTT and Live/Dead assay. In the case of inflammatory responses to your carriers, you should go for specific ELISA tests, for example to detect pro-inflammatory cytokines (TNF-alpha, IL-1beta) or anti-inflammatory ones (TGF-beta, IL-6, IL-1ra).
As mentioned above the definitely the ability of your nanoparticle to kill or inhibit cancer growth needs to the quantified with proliferation and or apoptosis assays. You can assay for anti-inflammatory markers in the cancer cells using RT-PCR, ELISA, western blots etc