Hierarchical Hill Model And Its Application In Three-drug Combination Experiment

With the development of medical science,drug combination experiments be-come more and more important.Because drug combination can weaken the resis-tance to drugs owing to multiple targets,it is widely used in disease treatment.Drug combination can also enhanced efficiency owing to synergistic drug interac-tions.Moreover,mathematical models based on drug combination experiments can provide greatly aid in the generation of new combination therapies,particularly dur-ing the treatment of specific patients.In this paper,we will give a new hierarchical Hill model with latent variables to build response surface model which fitted the drug experimental data.The hierarchical Hill model with latent variables inherit-ed Hill model's global applicability and self-explanatory feature of its parameters.Meanwhile it can overcome the disadvantage of the original Hill model that it can not handle the case of multiple drugs combination.Aim at hierarchical Hill model with latent variables,we introduced Monte Carlo EM algorithm(MCEM)to get the Maximum likelihood estimation of parameters.Under the assumption of error normal,we gave the explicit solution of M step in EM algorithm.For the complex distribution of E step,we use the sampling important resampling(SIR)to solve it.The efficiency of the algorithm is used to get an effective maximum likelihood estimation.In order to test the new model and the solution method for new model,we give the corresponding simulation and provide a new application of the model to a drug combination experiment with three chemicals.We also compared the result to the method's result provided in the reference(Ning and Xu et al.(2014)).The result shows that our methods got a better fitting result and less MSE.Due to the explicit solution in our model,the computational complexity is much less than the original nonlinear Hierarchical Hill model.So the method is much easier to promote when the species of drug combination added.