## 4 Analysis  
The change of amplitudes of the streaming potential pulse is consistent with that from other studies. Our finding is that, during the period of constant pressure, the measured streaming potential increases approximately linearly with time instead of remaining constant, as defined in Equation (1). The slopes of the streaming potential versus time of the four specimens ranged from 0.040 mV/s to 0.111 mV/s. The minimum and maximum increments of the streaming potentials measured during the 65 s were 2.6 and 7.22 respectively, which were even higher than the amplitudes of the pulses. This phenomenon may be due to the special structure of cortical bone. Bone is a hierarchically structured composite of hydroxyapatite (HA) mineral reinforced by the collagen fibrils. Figure 4 is a schematic diagram showing the microstructure of cortical bone. There are three typical porous hierarchical levels in the cortical bone [27]. They are the collagen-HA crystal composite with micropores between the crystallites about 10 nm in order of magnitude; the layered lamellae with lacunae-canaliculi pores (with a typical diameter of 0.1 \(\upmu\)m and length about 50 \(\upmu\)m); and the arrangement of lamellae into cylindrical structures (osteons) with Volkmann's canals and Haversian canals (with a typical diameter of 20-50 \(\upmu\)m and length of 2.5-4 mm) [28; 29]. All the microcanals and micropores are filled with bone fluid. Bone cells or osteocytes locate in lacunas. When the fluid flows through the microcanals in bone, the streaming potentials are induced. Because a passage through which the fluid flows may consist of tortuous microcanals, even those with different diameters, the irregular configuration can lead to nonuniform distribution of the electric  
Figure 4: A schematic diagram of a bone osteon. charges on the inner surfaces of the microcanals, which may affect the streaming potentials in bone. To explore the mechanisms for the specific characteristic observed, it is necessary to analyze streaming potentials theoretically or numerically in bone microcanals. The classical streaming potential Formula (1) is based on a straight capillary with electric charge density on its inner surface being uniform [16; 21; 30]. [PERSON] et al. [31] obtained a solution of streaming potentials in a microcanal composed of two sections with different electric charge densities on the inner surfaces. The solution strategy they employed is to solve the two sections separately with each section corresponding to a different electric charge density. It is noted that a model of a microcanal with a continuous distribution of electric charge density is relatively closer to the actual situation in bone. For ease of analysis and without loss of generality, the model selected is a round microcanal with equal diameter (Figure 5). The charge densities on the inner surface of the middle section and the two side sections are \(\sigma_{1}\) and \(\sigma_{0}\) (_let_\(\sigma_{1}>\sigma_{0}\)), respectively. The condition of the electric charge distribution on the inner surface of the microcanal is also expressed on bone solid. At the junction interface of any two sections, there is a sudden change in charge density, as shown in Figure 5 (black line). In reality, the sudden changes cannot happen, because the bone matrix or solid has electric conductivity which leads the charges to shift to eliminate the sudden change on the two sides of the junction interface. That is, the charge density around the interface must change smoothly and continuously from \(\sigma_{0}\) to \(\sigma_{1}\). Thus, the distribution of the charge density should be described in terms of a function of \(x\). A simulation function \(\sigma(x)\) was constructed as  
\[\sigma(x)=\left\{ \begin{array}{c}\sigma_{0}\,,\;x\in[0,l_{0}]\cup[l_{0}+l_{1 },L]\\ \left\{\left[\left(x-l_{0}-\frac{l_{1}}{2}\right)A\right]^{2}+B\right\}\!\! \left[1-\cos\!\left(\frac{x-l_{0}-l_{1}}{l_{1}}2\pi\right)\right]\!C+\sigma_{ 0}\,,\;x\in[l_{0},l_{0}+l_{1}]\end{array} \right.