Achievable rate for MIMO channel with estimated channel matrix
I found the following relation which answers my question

$$R=\log\det \left(\boldsymbol{C}_{MMSE}^{-1}\right)$$

where $\boldsymbol{C}_{MMSE}$ is the covariance matrix of the error in the data $\boldsymbol{e}=\boldsymbol{x}-\hat{\boldsymbol{x}}$

I found the answer in the paper "An Iteratively Weighted MMSE Approach to Distributed Sum-Utility Maximization for a MIMO Interfering Broadcast Channel" but it is said to be well known.

Therefore, we need to assume some method for estimation the data using the estimated channel, for example, match-filter or MMSE estimator, and then calculate its error (as a function of the real and estimated channels).