Is it possible to partial trace the $\chi$-matrix of $4$ qubits $q_0,q_1,q_2,q_3$ to obtain a description of what happens to $q_1$?

Question

Considering a $\chi$-matrix of a circuit with, say, 4 qubits, is it possible to trace out 3 of them from $\chi$ - for example qubits $q_0$, $q_2$ and $q_3$ - thus gaining the process matrix describing what happens to $q_1$? If yes, since it seems to me different from partial trace for a density matrix, could you please give me the definition?

The $\chi$-matrix of an $n$-qubit quantum channel $\mathcal{E}$ is a matrix $\chi$ such that the evolution of a density matrix $\rho$ is given by \begin{gather} \mathcal{E}(\rho) = \sum_{i,j}\chi_{i,j}P_i \rho P_j \end{gather}

where $\{P_0, P_1,\dots ,P_{d^{2}−1}\}$ is the $n$-qubit Pauli basis containing $d^{2} = 4^{n}$ elements.

An estimation method to compute the $\chi$-matrix is available at 8.4.2 of Nielsen & Chuang's Quantum Computation and Quantum Information.