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It has been recognized that using time-varying initialization functions to solve the initial value problem of fractional-order systems (FOS) is both complex and essential in defining the dynamical behavior of the states of FOSs. In this paper, we investigate the use of the initialization functions for the purpose of estimating unknown parameters of linear non-commensurate FOSs. In particular, we propose a novel "pre-initial" process that describes the dynamic characteristic of FOSs before the initial state and consists of designing an appropriate time-varying initialization function that ensures accurate convergence of the estimates of the unknown parameters. To do so, we propose an estimation technique that consists of two steps: (i) to design of practical initialization function that is output-dependent and which is employed; (ii) to solve the joint estimation problem of both parameters and fractional differentiation orders (FDOs). A convergence proof has been presented. The performance of the proposed method is illustrated through different numerical examples. Potential applications of the algorithm to joint estimation of parameters and FDOs of the fractional arterial Windkessel and neurovascular models are also presented using both synthetic and real data. The added value of the proposed "pre-initial" process to solve the studied estimation problem is shown through different simulation tests that investigate the sensitivity of estimation results using different time-varying initialization functions.