Fisica

SIAM J. APPL. MATH. Vol. 65, No. 6, pp. 2128–2140

c 2005 Society for Industrial and Applied Mathematics

TRANSPORT OF NUTRIENTS IN BONES∗
GUILLERMO H. GOLDSZTEIN† Abstract. Lacunar-canalicular systems are networks of pores (lacunae) interconnected by thin channels (canaliculi) that are embedded in bones. The efficient transport of nutrients within lacunarcanalicular systems is necessary tokeep bones healthy. Several theories have been proposed to identify the physical phenomena responsible for this efficient transport. In this paper, we develop and study a mathematical model motivated by one of those theories. Key words. bones, porous media, nutrient transport, solute transport, effective diffusion, mathematical modeling AMS subject classifications. 76S05, 76R50, 92B05 DOI.10.1137/040616632

1. Introduction. As illustrated in Figure 1.1, we consider long bones of our extremities such as the femur. Bones are porous media with complex microgeometry. The particular components of bones that we discuss here are osteons. These are cylindrical structures of about 120 μm radii that extend along the long axis of the bone (see Figure 1.1). An osteonal canal is located at the center ofosteons. This canal contains blood vessels, a nerve, and bone fluid (see Figure 1.1). Pores, called lacunae, are distributed within the osteon. Thin channels, called canaliculi, and the lacunae form a connected system known as the lacunar-canalicular system, which is filled with fluid and is connected to the osteonal canal. A cartoon of the osteon microgeometry is given in Figure 1.1. More details onthe structure of bones are given in [4, 12, 13, 2, 14] among numerous other articles.

Fig. 1.1. At left is the cartoon of a bone. A horizontal cross section (view from the top) is shown in the middle figure. The right-most image is a cartoon of the microgeometry of a section of an osteon (also viewed from the top); the white region is the solid part of the bone.

Bones consume nutrients andproduce waste products. It is believed that nutrients are transported from the osteonal canal into the rest of the osteon through the lacunarcanalicular system (see [17] and references therein). Waste products, on the other hand, are produced within the osteon and need to be transported to the osteonal
∗ Received by the editors October 8, 2004; accepted for publication (in revised form) March 23,2005; published electronically August 9, 2005. This research was supported by the NSF. http://www.siam.org/journals/siap/65-6/61663.html † School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332 (ggold@math.gatech. edu).

2128

TRANSPORT OF NUTRIENTS IN BONES

2129

Fig. 1.2. Cartoon of a vertical longitudinal section (parallel to the direction of the bone) of part of anosteon. The bone is supporting a load in the left-hand image. The load is removed in the right-hand image. The arrows at the bottom denote the direction of the flow.

Fig. 1.3. One-dimensional lacunar-canalicular system.

canal so that they can be disposed of. Thus, efficient transport within the lacunarcanalicular system is necessary to maintain a healthy bone. The solid part of the bone is anelastic material. Thus, activities that apply and remove loads to bones, such as walking, produce small deformations of the bone. As a consequence, the incompressible fluid that fills the lacunar-canalicular system is squeezed, producing fluid flow. This is illustrated in Figure 1.2. As shown in the left-hand image, when a bone is subjected to a load, fluid is squeezed out of the lacunar-canalicularsystem and into the osteonal canal. Once the load is removed, an equal volume of fluid flows back from the osteonal canal into the lacunar-canalicular system (see the right-hand image in Figure 1.2). After each cycle of a periodic load, there is no net volume of fluid transported from the osteonal canal into the lacunarcanalicular system simply because the volume of the bone does not change after each...