You would not be able to assess the effect on "response" from drug A alone but you would be able to assess the incremental effect of drug A+drug B (versus no drug) when compared to drug B alone (versus no drug). (So you would have half of an assessment of an "interaction".) Any further commentary would require that you better describe your data situation. Do you have measurements taken before and after drug or placebo were administered? Were these measurements perhaps taken in replicates? What is known about the reliability of the measurements of "response"?
As far as coding a model, I don't think you would use:
response variable = drug A + drug B
Rather I think you would form three categories for a single variable: baseline: 'no drugA/no drugB', DrugA_alone: 'yes drugA/no drugB', and combA_B: 'yes drugA/yes drug B'