In this chapter, starting from fractional Brownian motion of unique parameter H (0 < H < 1), we propose two variants which extend the classical fBm. First, the piece-wise fractional Brownian motion model of parameters $H_o$, $H_i$ and $\gamma$ presents two spectral regimes: it behaves like fBm of parameter $H_o$ for low frequencies $|f| < \gamma$ and like fBm of parameter $H_i$ for high frequencies $|f| > \gamma$. Second, the nth-order fractional Brownian motion enables the H parameter to be within the range $]n-1, n[$, where $n$ is any strictly positive integer. The properties of these two processes are investigated. Special interest is given to their increments, which extend fractional Gaussian noises. Synthesis of realizations of such processes is discussed. Finally, an application of such processes for the characterization of trabecular bone radiographs is presented. The overall objective of this application is to provide methods helping in the early diagnosis of osteoporosis for elderly people.