Bone Adaptation PDF

Adaptation in Bones Béla Suki Department of Biomedical Engineering Boston University Boston, MA  Stress analysis: simple tension (Wikipedia) Assume a st...

Adaptation in Bones Béla Suki Department of Biomedical Engineering Boston University Boston, MA  Stress analysis: simple tension (Wikipedia) Assume a straight rod, with uniform material and cross section, is subjected to tension by opposite forces of magnitude F along its axis. If the system is in equilibrium and not changing with time, and the weight of the bar can be neglected, then through each transversal section of the bar the top part must pull on the bottom part with the same force F. Therefore the stress throughout the bar, across any horizontal surface is s = F/A, where A is the area of the cross-section. On the other hand, if one imagines the bar being cut along its length, parallel to the axis, there will be no force (hence no stress) between the two halves across the cut. This type of stress is called (simple) normal stress or uniaxial stress. If the load is stretching the bar, it’s called tensile stress, if the load is compressing the bar, the analysis is the same except that F and s change sign, and is called compressive stress.  Stress analysis: shear (Wikipedia) Another simple type of stress occurs when an uniformly thick layer of elastic material like glue or rubber is firmly attached to two stiff bodies that are pulled in opposite directions by forces parallel to the layer; or a section of a soft metal bar that is being cut by the jaws of a scissors-like tool. Let F be the magnitude of those forces, and M be the midplane of that layer. Just as in the normal stress case, the part of the layer on one side of M must pull the other part with the same force F. Assuming that the direction of the forces is known, the stress across M can be expressed by the single number t = F/A, where F is the magnitude of those forces and A is the area of the layer. However, unlike normal stress, this simple shear stress is directed parallel to the cross-section considered, rather than perpendicular to it. For any plane S that is perpendicular to the layer, the net internal force across S, and hence the stress, will be zero because of equilibrium.  Stress analysis: complex stresses The ith component of the traction stress is: where the stress tensor is given by: The traction force is the traction stress times the area of a surface and is a real physical quantity. The stress is a mathematical concept, the force divided by the area of an oriented surface. The stress tensor is independent of the coordinate system. However, the components of both the force and the oriented area depend on the chosen coordinate system and so do the components of the stress tensor.  Stress transformations The vector T can be expressed in both the (x,y) and the (x’,y’) coordinate system. We need to find the relations. From elementary trigonometry, one finds the following relation:  T x '   cos  sin    T x  T '     which can be written in vector form as T’ = RT and the inverse is  y    sin  cos   T y   T x   cos   sin    T x '  T      cos   T y ' which can be written as T = R T’. But notice that R =R also and R is -1 -1 T  y   sin  the matrix representing the transformation of rotation by an angle . From the previous slide, we have Tn=sn where s is the stress tensor, n is the unit normal in a particular direction and Tn is the traction stress in that direction. How is a tensor transformed when the coordinate system is rotated?  Stress transformations continued From the previous slide, we have T=sn and, in general, for any vector v, v’=Rv and v=RTv’. We can now use these transformations to find out how a tensor is transformed. We start with T=sn and express both sides in the rotated coordinate system: RTT’ = sRTn’ Multiply both sides with R: RRTT’ = RsRTn’ Since RT=R-1, we find that T’ = RsRTn’ But this is also the traction in the rotated system and unit normal in the rotated system and hence s ’ = RsRT This is the general rule to transform a second order tensor when the coordinate system is rotated. Next, we ask the question: can we rotate the system such that the stress becomes normal stress? In other words, let’s find a direction  such that the stress vector T is parallel to one of the axes. T = sn = spn where sp is a scalar. We can thus write this as (s  spI)n = 0 where I is the identity matrix. A theorem in algebra guarantees that this has a nontrivial solution only if the determinant of the matrix [(s  spI)] is zero:  Principal Stresses These I parameters are known as the invariants of the stress state; they do not change with transformation of the coordinate system and can be used to characterize the overall nature of the stress. For instance I1, which is the trace of the stress matrix, is known to be a measure of the tendency of the stress state to induce hydrostatic dilation or compression. Thus, the stress state is one of pure shear if its trace vanishes. Since the characteristic equation is cubic in sp, it will have three roots, and it can be shown that due to the symmetry of the stress tensor, all three roots must be real. These roots are called the principal stresses: sp1, sp2, sp3.  Stresses in the new coordinate system Let us now express the new stress values in terms of the old and the angle of rotation in 2D! From 2 slides ago, we have s’=RsRT. In terms of the components of R, we obtain: After some algebraic manipulations: s xx  s yy s xx  s yy s ' xx   cos 2  s xy sin 2 2 2 s xx  s yy s xx  s yy s ' yy   cos 2  s xy sin 2 2 2 s xx  s yy s ' xy   sin 2  s xy cos 2 2 Check that the trace is independent of rotation: s ’xx+s ’yy = sxx+syy. Thus, indeed, the trace is an invariant of the stress state and it represents pressure.  Principal Stresses once more Since the principle stress state is one where we should have only normal stresses, we can seek the angle for which the shear stress disappears: s xx  s yy 0 sin 2  s xy cos 2 From which we obtain: 2 But one gets the same expression if we seek the maximum of the normal stress as a function . Thus, when the coordinate system is rotated such that the shear stress disappears, the normal stress is maximum or minimum. The principal stresses are therefore the maximum and minimum stresses in orthogonal directions and they can change from place to place. In other words, the principal state of stress imposes pure extension or compression on the material but not shear.  Wolff’s Law The basis of Wolff's trajectorial theory is illustrated above. On the left is a midfrontal section of the proximal femur, showing trabecular architecture ; in the middle the schematic representation is drawn by Meyer (1867); and on the right the stress trajectories are visualized in a model analyzed by Culmann, using graphical statics. Stress trajectories are curves representing the orientations of the maximal and minimal principal stresses in the material under load. The maximal and minimal stress trajectories always intersect perpendicularly. (Adapted from Wolff, 1986 (translation).) Wolff's law is a theory developed by the German anatomist and surgeon Julius Wolff (1836–1902) in the 19th century that states that bone in a healthy person or animal will adapt to the loads under which it is placed based on some mathematical optimization principle.  Wolff’s Law in practice If loading on a particular bone increases, the bone will remodel itself over time to become stronger to resist that sort of loading. The internal architecture of the trabeculae undergoes adaptive changes, followed by secondary changes to the external cortical portion of the bone, perhaps becoming thicker as a result. The inverse is true as well: if the loading on a bone decreases, the bone will become weaker due to turnover, it is less metabolically costly to maintain and there is no stimulus for continued remodeling that is required to maintain bone mass. But how does the bone do that? What is the mathematical rule and what is the objective function to optimize? Where is the “mechanical sensor”? What feedback processes operate in the bone microscopically?  Examples of bone adaptation The racquet-holding arm bones of tennis players become much stronger than those of the other arm. Their bodies have strengthened the bones in their racquet-holding arm since it is routinely placed under higher than normal stresses. Surfers who knee-paddle frequently will develop bone bumps, also known as exostoses, on the tibial eminence and the dorsal part of the navicular tarsal bone from the pressure of the surfboard's surface. These are often called surf knots. Astronauts who spend a long time in space will often return to Earth with weaker bones, since gravity has been greatly diminished and therefore has exerted little force on their bodies. Weightlifters often display increases in bone density in response to their training. Martial artists who strike objects with increasing intensity (e.g., repeated elbow strikes), display increases in bone density in the striking area. Evidence for bone adaptation Limb’s composition in professional tennis players and sedentary subjects (control group). Values are means ± SEM (vertical bars). *P < 0.05. (BMC: bone mineral content) Calbet et al. Calcif Tissue Int (1998)  Gravity seriously affects bones Iwamoto et al. Keio J Med, 2005. Low gravity High gravity  Bone modeling and remodeling Bone “modeling” refers to the adaptation of external form of the bone either by adding or removing bone material at the external surface. The function of bone modeling is to adapt to external loads: roughly, bone is added where the strain is high and removed where the strain is low. This process is not fully understood. Bone “remodeling” refers to the production of new, secondary bone within the old bone by formation of secondary osteons within long cavities eaten out by osteoclasts. Remodeling has many functions attributed to including taking out microcracks, preventing too highly mineralized bone, or responding to local strains. Indeed, mineral content of the new bone is always less than that of the old bone because the mineral ions have to diffuse into the collagenous matrix. In trabelular bone, the remodeling occurs along the trabecular surface and it is a faster process than in cortical bone. Thus the Young’s modulus of trabecular bone is usually lower than that of cortical bone.  FEA to test Wolff’s hypothesis Results of the mFE analysis of the proximal canine femur, showing the distribution of strain energy density (SED) for a vertical force on the femoral head, in a midfrontal section. The intensity of the SED increases from white up to red colors. The trabeculae are not equally stressed. The square in the head represents a cube of trabecular bone for which SED, stress and strain values were sampled (see text). (Reproduced from van Rietbergen et al. 1999) Wolf’s law of trabecular function in optimal stress transfer can now be checked using FEA. If the architecture pairs optimal resistance against mechanical failure with minimal weight, then all trabeculae must be used for stress transfer in some part of the hip loading cycle of daily living. The trabecular architecture of the specimen, 27 mm high, was evaluated in a mCT scanner and reconstructed with a cubic voxel size of 35 mm. From the reconstruction, a FEA model was made, consisting of 7±3 million cubic elements 70 mm in size. All elements, cortical and trabecular, were assigned an elastic modulus of 15 GPa. Stress transfer was calculated for 3 different forces working on the femoral head, 1 in vertical, 1 in lateral and 1 in posterior direction. The figure shows the stress patterns obtained for the vertical force (representing the stance phase of gait). What is shown is the strain-energy density (SED), which is the elastic energy stored in the material, per unit volume. It is obvious that the trabeculae are not equally stressed. In a cubic volume of interest (VOI), the average SED was 0.00102, s.d. 0.00148 MJ/m3. Similar results were found for the stress and strain values in the VOI (e.g. average 279 mstrain, s.d. 212). From the 3 load cases, results for a total of 91 forces in various directions could be obtained by linear superposition. It was found that no single force on the femoral head produced equal stressing of the trabecular tissue.  What does this mean? The study discussed on the previous slide has limitations in accuracy due to the assumed loading conditions and limited mesh refinement (van Rietbergen et al. 1999). However, the results still provide strong evidence against suggestions, made repeatedly in the literature in the 19th and 20th centuries that bone architecture would be a mechanically optimal structure for a particular `average' or ` typical ' force system. If the trabecular structure of the proximal femur is mechanically optimal, then it must be for a large range of forces experienced during a longer period of time. But although the results from the previous slide do not refute that hypothesis, they do not make it plausible, either. So, despite the advanced tools of finite element stress analysis based on real mCT-based structure, the old question as to whether bone architecture answers requirements for maximal strength and minimal weight, still remains unanswered. Huiskes, R. J. Anat. (2000)  Roux’s paradigm Wilhelm Roux (1881) suggested that formation and functional adaptation of trabecular architecture in bone is regulated locally by cells, governed by mechanical stimuli, in a self-organizational process. It was not stated in those terms then, but this is what he meant. Although this sounds quite reasonable in our time, in which regulatory processes and control theory are well known, it hardly moved the scientific community in bone biology. Biochemists and cell biologists have considered biological processes in bone, but the mechanical aspect was somehow lost. It was Frost (Calcified Tissue International, 1987) who stepped in Roux's footprints in the most direct sense. In his conceptual `mechanostat' theory, local strains are assumed to regulate bone mass. Frost pointed out that this control process is purely biological in its components, but governed by mechanical loads. His theories have contributed greatly to the awareness of mechanical factors in the regulation of bone modeling, remodeling and repair in orthopedics and bone biology. Using large-scale computer simulation, these kinds of proposals can now be tested in an integral quantitative sense, an example of which is discussed next.  Evolution and structure From an evolutionary point of view, bones are not required to be designed according to a mechanical optimization rule. They must be sufficiently light, adaptable to environmental factors and repairable; they must be stiff, and their strength must be adequate for daily usage without fractures. While the authority of Wolff's Law made biomechanicians concentrate on bone design, the process of formation was largely neglected. Yet it is here that the real question is found. Bone architecture is an answer to the above requirements for evolutionary endurance. It is only because nature happened to come up with a suitable cell-based regulatory system which realizes these requirements that we have bones at all. Structurally, they just are what they are. But of course, adequacy for their evolutionary survival implies that the architecture holds a sensible balance between weight and mechanical fitness, meaning that it must answer to the principles of elastomechanics. Nature has found its design by trial and error, over time, by creating a metabolic process responsive to environmental mechanical factors, which inherently dictate the design requirements. By wondering about what mathematical rules bone architecture might be the answer to, we do not learn anything useful at all. The key to information is in the metabolic process of bone production and maintenance. Huiskes, R. J. Anat. (2000) Evolution of bony structures  Effects of mechanical forces on maintenance and adaptation of form in trabecular bone Huiskes et al. NATURE, 405: 706, 2000 Osteocytes may have a prominent function in the transduction of mechanical signals (Fig. 1). Osteocytes derive from osteoblasts which are entrapped in their own matrix and survive there, mutually connected by a network of canaliculi. This network also connects with the lining cells, which also derive from osteoblasts, that cover the trabecular surface. Hence, osteoblasts, osteocytes and lining cells form a large network. It is possible that increased strain in the local mineralized matrix signals the osteocyte to transmit stimuli to the surface, where bone is formed until the strains are normalized. This paradigm provides an explanation for the mechanical control (growth and adaptation) of bone modeling. Bone remodeling (or bone maintenance) is the formation of cavities by osteoclasts, and the subsequent filling of these cavities by osteoblasts (Fig. 2). Figure 1: Osteocytes are probably mechanosensors which send strain-related signals to lining cells located at the bone surface through the canalicular cell network. These stimuli are thought to recruit osteoblasts, which add bone to the surface when strains increase.  Strain-driven modeling and remodeling It is assumed that the coupling factor between osteoblasts and osteoclasts is mechanical. While higher external forces increase the strains in the bone at large, the resorption cavities produced in bone remodeling have a similar strain-enhancing effect locally (Fig. 2). Thus, modeling and remodeling could both be described as being governed by strain perturbations, whether they are generated external load or internally by resorption cavities. Using finite-element methods of stress analysis in a regulatory computer-simulation model, they evaluated these local strain perturbations and determined whether this unified mechanical paradigm can indeed explain the morphological phenomena in bone. Figure 2: The trabecular surface is covered by lining cells (LC) and the osteocytes (Ocy) are inside the mineralized tissue. When an osteoclast (Oc) is recruited to resorb bone, a cavity is made which weakens the trabecula and causes a local elevation of strain. After the osteoclast has gone, osteoblasts (Ob) are thought to be recruited by osteocytes to form bone. During the bone formation process, some of the osteoblasts are entrapped in the bone matrix (e-Ob), where they differentiate to new osteocytes (n-Ocy). After repair, the remaining osteoblasts become lining cells, covering the new bone surface.  Regulatory process (1) The mechanical variable that triggers feed-back from the external forces to bone metabolism is a typical strain-energy density (SED) rate in bone, as produced by a recent loading history. SED can be linked to trabecular architecture as an optimal structure. Both the frequency and the amplitude of the external loads have a role, and a limited amount of loading cycles per day suffice for the maintenance of bone mass. (2) Osteocytes react to the loading in their local environments by producing a biochemical messenger in proportion to the typical SED rate. (3) This biochemical messenger causes signals to be dissipated through the osteocytic network towards the bone surface, where they create an osteoblast recruitment stimulus. The strength of this signal, which is produced by all osteocytes in the environment, stimulates osteoblast recruitment and bone formation as long as it exceeds a threshold value. The nature of such a signal is unknown, although both electric current and ion transport have been suggested. In this model, we link the bone formation stimulus at the surface directly to the SED rate per osteocyte, subject to osteocyte mechanosensitivity and attenuation by distance. (4) To portray resorption in the model, the probability p of osteoclast activation per surface site at any time is considered to be regulated either by the presence of microcracks within the bone matrix (hypothesis I) or by disuse (hypothesis II). Hypothesis I is based on the idea that osteoclasts are activated where microcracks occur in the bone. The dynamic forces of daily living are known to produce microcracks. As the trabecular bone structure is a mechanically optimal one, implying that all material is frequently stressed in daily life, and the strength of the mineralized tissue is non- homogeneously distributed, it is likely that microcracks can occur anywhere, at any time, for a normally functioning individual; in other words, the distribution of microcracks is spatially random. Hence, for hypothesis I the probability of resorption, p, would be equal for all surface sites, and independent of mechanical strain; whereas for hypothesis II the local osteoclast activation probability, p, would be high in disused areas and low in areas of high strain. This activation process could originate from a signal of mechanical disuse in the bone matrix, sent by osteocytes.  The computational algorithm Figure 3: The proposed regulatory process. Both enhanced external load intensity (amplitude and frequency) and resorption cavities provoke bone formation. Reductions in loading reduce bone mainly through unbalanced resorption. Quantitative modeling Quantitative model continued  Simulation results Figure 4: a, Iterative steps out of the simulation process, starting from a regular grid (spatially random resorption (hypothesis I): bone is black, marrow is white; iteration numbers are indicated). A homeostatic configuration is obtained, in which bone remodeling continues without changing the architecture at large. Trabecular orientation lines up with the external load. b, Starting from another initial configuration, the eventual homeostatic architecture is similar. c, After rotating the external loads, the architecture reorients until adapted to the new loading directions, a distinct characteristic of trabecular architecture, in accordance with mechanical optimality. A 20% reduction in external loading reduced trabecular thickness to a loss of bone mass by 15.8%, which is similar to that in individuals subjected to disuse. Conversely, a 20% increase in loads produced a 17.5% increased bone mass, which is of the order of effects found from high-impact gymnastics. When loading intensity was set back to original values, the original homeostatic configuration gradually re-emerged. d, The only effect of strain-regulated resorption (hypothesis II) was in the kinetics of the process. The homeostatic architecture developed much faster which makes sense, as the osteoclasts help the osteoblasts to shape the form in the latter case, whereas they do not in the former case.  Interpretation The total mass in the homeostatic architecture, at any time, is the result of a balance between resorption and formation. If the amount of bone resorbed per volume per day (roc) remains constant, homeostatic bone mass is determined by the external loading magnitude and frequency, the osteocyte mechanosensitivity (mi), the osteocyte numbers (n), the stimulus distance attenuation function (fi) and the osteoblast recruitment threshold (ktr). On the other side of the balance, homeostatic bone mass is determined only by the total amount of bone resorbed per time, which depends on osteoclast-activation frequency and amount of bone resorbed per osteoclast. When the osteoclast-activation frequency (roc) was increased by 50%, as can occur after menopause in women, trabecular thickness was reduced, until a new balance was found at a 26% lower bone mass. The time curve obtained was similar to the one reported from epidemiological studies of bone mineral density development in women after menopause.  Conclusions from the study They conclude that mechanical feed-back, the existence of which is undeniable, can be a potent and stable regulator of the complex biochemical metabolic machinery towards lasting optimality of form. The regulatory paradigm that forms the basis of the model is relatively simple and amazingly stable. This again confirms the principles governing the emergence of complex consistent forms in nature under the influence of environmental stimuli, speculated upon long ago, but only now understood and made operational through the application of large-scale computer simulation—the ‘‘third method of science’’.  Tissue engineering of bone Definition: Tissue engineering is the use of a combination of cells, engineering and materials methods, and suitable biochemical and physico-chemical factors to improve or replace biological functions. Tissue engineering bone is a difficult process due to the multiscale hierarchical organization of bone tissue. One hopes that if a suitable scaffold can be made in which bone cells are happy, then the rest will be done by bone cells. This involves matching the chemical composition so that cells can adhere to the construct, the stiffness that is important for bone cells to receive proper mechanical stimuli, the structure so that cells do not loose phenotype and can communicate with each other. An example to bone tissue engineering Most tissue engineering approaches start with isolating cells, building a scaffold, seeding the cells and incubating them in the scaffold under various conditions. If the cells survive and if the construct seems to have similar architecture and mechanical properties as the native tissue, then implant it and see what happens. This study did the opposite: they built in the construct in the beginning and hypothesized that this in vivo incubation would result in a better “nature” engineered tissue.  Methods Both distal femurs of 6 female Wistar rats were subjected to surgery. In each leg, a hole of 3 mm in diameter and height was made using a metal drill bit on the lateral side of the distal femur underneath the growth plate. Afterwards, 80% porous bioresorbable poly(L-lactic acid) (PLA) and two ceramic powders, hydroxyapatite (HA), and b-tricalcium phosphate (b-TCP)of the same size were implanted inside the holes. No cells or growth factors were added in the scaffold. The scaffolds were perfused with PBS prior to implantation to remove the air bubbles. The loading started 2 weeks after the surgery. The animals were kept under anesthesia during the loading session. The right knee joint of all animals was loaded in the anterio-posterior direction by loading the tibia axially. The left leg was kept as control. Compressive load of 10 N at 4 Hz for 5 min was applied by a custom-made machine. The magnitude of loading was chosen in a way that animals don't show any sign of pain using their loaded joint after recovery. The animals were loaded every other day resulting in a total of 5 loading sessions being spread out over 10 days. For the remainder of the study, the bone-scaffold constructs were left without a loading intervention. MicroCT was taken at several time points to determine bone fraction, mineral fraction and from their changes in time, bone formation rate and resorption rate were obtained. Finite element modeling with the usual assumptions (linear, homogeneous, single E value) was used to compute the in vivo stiffness of a piece of the implanted construct.  Results Young's modulus of bone-scaffold constructs. The data are shown as average and 95% confidence (a) 3D reconstruction of bone in a scaffold at 15 weeks (control interval of the average. The control group is in solid group), (b) Minimum principle strain distribution inside the same blue line, and the loaded group is in dashed red bone-scaffold construct. The low values of strain represented in line. At 35 weeks, the Young's modulus of the red, mostly follows the bone structure. The strain was markedly loaded group was 60% higher than the control lower in the bone compared to the pore space and it increased group (p