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 $q_1$?

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.