WARNING: Actual video of hip surgery…….scroll down if you dare.
Objective
Use DFT to predict proper alloy choice for biological implants applying the following materials constraints; phase stability (bcc vs. fcc), and proper elastic stiffness (comparable to human cortical bone).
Introduction
As technical skills become more specialized in all walks of life the independent discipline of biomedical engineering has emerged from the pool of knowledge generated by mechanical engineering, materials engineering, chemistry, biology, and physics. In general, biomedical engineering is a merging of engineering design with the scientific foundations of medicine in an effort to repair a damaged or missing biological component. As a whole, biomedical engineering is focused on combining the problem solving capabilities of both fields to improve our ability to diagnose and treat medical problems. Although biomedical engineering can be applied to the any life form in the entire food chain, the current state of technology is focused on mending human deficiencies, which are obviously most crucial to our advancement as a species. One such form of expanding biomedical technology is the biomedical implant. A biomedical implant is a device that is designed and implemented to replace or repair a missing or damaged biological structure. An implant differs from a medical transplant in the sense that a transplant indicates the use of existing biological tissue as opposed to an implant which requires the use of foreign non-biological material. Biomedical implants may be functional if they are coated with a drug delivery or bioactive material. Currently, widely used biomedical implants include heart pacemakers, cochlear implants, the Jarvik artificial heart, hip prostheses, knee prostheses, and dental implants.
Fig. 1 Schematic of broken hip (left), hip implant (center), and repaired hip (right). [1]
The history of the hip implant is relatively minimal because of the enormous surgical challenges that the procedure entails, often life threatening in nature. The original design of the hip implant from a geometrical point of view has not changed much since its conception, mainly because each implant is designed specifically to fit the exact shape and size of the original bone material that has failed. The material choice has however undergone quite drastic alterations recently, including the use of alloys rather than pure metal components, as well as the use of a functional surface to further promote bone in growth and stymie infection. The major advancements in this area include the shift from dense, pure titanium components toward α-β titanium alloys (i.e. Ti 6Al-4V) and porous surface components. The advantage of the porous surface is that it more closely mimics natural bone by allowing the native bone surrounding the implant to grow into the implant, thus securing the position and strength of the implanted device in an interwoven network. On average, bone ingrowth reaches a depth of 1500 microns over the course of 3-5 years. Another interesting progression in hip implant materials is the use of a coating material that is bioactive. In this case the typical material of choice is Hydroxyapatite (Ca5(PO4)3(OH), which itself comprises 50% of the natural bone material in humans, thus making it nearly perfectly biocompatible.
The scope of this project is to use first principles to investigate materials (metal alloy) selection in terms of the mechanical properties of the implanted hip stem. In this way the expensive and arduous task of metling, casting, homogenizing, and characterizing the implant alloy can be avoided until the final experimental validation stage. One major motive for such an inquiry is based on the Food and Drug Adminstration (FDA) regulation on biomedical materials, which fall under three basic categories. Class I devices (i.e. bandages) present minimal harm for the user and are simple in design. Class II devices (i.e. x-ray machines) are usually non-invasive but do require some specialized training for their proper use. Class III devices (i.e. hip implants) are usually very invasive to implement and therefore require premarket approval and highly trained medical specialists [2]. The requirement of intensive premarket approval is an important line of protection for the consumer because a scientific review of the device ensures its safety and effectiveness.
Hip Implant 3-D Animation
The first video shows the hip implant process. Beginning with measurement of pre-surgery range of motion, followed by the size/shape fitting process in which the damaged bone is hollowed out to make room for the implant, then the mold filling, next the implant insertion and finally the range of motion gained.
[3]
Caution!!!! Not for the weak of stomach. This video shows the process of opening up the flesh around the hip, and the underlying skeletal structure at the hip implant site. I have included this to prove how invasive the procedure is, justifying the need to limit the surgery to a one time event.
[4]
The following studies take aim at reducing the experimental throughput time involved with design and implementation of new hip implant alloys. The key discrepancy currently being that titanium implants have a bulk modulus that is much higher than that of natural bone, 110 GPa and 20 GPa respectively [5]. This bulk modulus mismatch has the detrimental effect of inducing a phenomenon known as stress shielding. Stress shielding is an extremely harmful result in which natural bone surrounding the implant device undergoes a loss in mass and a loss in mechanical strength due to it no longer actively participating in the load bearing process. The resulting loss in bone mass surrounding the implant first appears as osteoporosis, but inevitably results in loosening of the implant, which can lead to infection and/or sudden failure. Any significant infection or failure requires additional surgery to repair or replace the component, making the hip implant device exponentially expensive and justifiably closely regulated.
A brief outline of the case studies to follow…..
1. Stabilization of the low bulk modulus beta-phase in titanium.
2. Alloying of titanium with bulk modulus lowering and/or beta-phase stabilizing elements.
*Experimental validation will be provided where available.
Phase Transformations in Titanium
Titanium exists as an hcp crystal (alpha-phase) under ambient temperature and pressure conditions. Above 1155K at ambient pressure the alpha-beta phase transformation occurs resulting in the bcc (beta-phase) structure. When pressure is increased from the ambient the alpha-omega phase transformation is induced, resulting in the hexagonal AlB2 structure which shows alternating triangular and honeycombed layers.
Fig. 2 Titanium P-T phase diagram. [6]
As mentioned previously in the introduction, the use of an alpha-beta two phase alloy (i.e. Ti 6Al-4V) is attractive for implant applications because the alloy shows a lowering of the bulk modulus and an improvement in creep resistance due to the presence of the beta phase in an alpha matrix. Therefore, it is the goal of Hennig and company to define phase transformation conditions in pure titanium before investigating such transformations in more complex titanium alloy systems. The basic guidelines used for determination of the phase boundaries is as follows: (1) determine the martensitic phase transformation pathway in pure titanium, (2) use the determined pathway to model the effect of impurities on pure titanium, and (3) determine the nucleation and growth dynamics of the phase transformation in a more realistic non-homogeneous system. Comparison of calculated data to experimental data is provided for certain parameters in the following case studies.
Conveniently, the martensitic phase transformation pathway (crystallographic) in pure titanium and the pathway for the alpha to omega phase transformation were determined previously by Burgers and Trinkle respectively (note the hysteresis at the alpha-omega phase boundary due to range of measured transformation onset pressures from 2.9 to 9.0 GPa) . The exploitation of this information with a carefully guided systematic screening of all possible transformation pathways allows for the generation of the pressure-temperature phase diagram shown above, Fig. 2 [6]. Hennig and co-workers utilize energy estimates generated via the elastic theory, the tight-binding method, and density-functional theory to build their diagram. An additional set of data is calculated using the Modified Embedded Atom Method (MEAM) to do the simulation, however the details of these computations will not be divulged any further.
The following details describe the density functional approach and initial input data [6]:
-Vienna Ab initio Simulation Package (VASP)
-Vanderbilt-type pseudo-potentials
-Generalized gradient approximations (GGA)
-400 eV plane-wave cut-off energy to ensure convergence to 0.3 meV/atom
-1 meV/atom accuracy via specific k-point mesh sampling for each phase structure
-Ti 3p orbital states are included as valance states along with the traditional 4s and 3d states
-When atomic forces < 20 meV/A, atomic positions are considered to be fully relaxed
For determination of specific parameters for each phase (i.e. alpha, beta, omega, fcc, A15, and hexagonal) it is necessary to modify the DFT approach slightly in order to maximize the accuracy of each specific calculation. The ground state energy and equilibrium lattice parameter of pure titanium at various temperature and pressure conditions are determined by relaxations of the supercell geometry and relaxations of the atom positions respectively. Elastic constants and equilibrium volumes are calculated accordingly, Table 1 and Table 2 respectively.
Table 1. Titanium elastic constants comparing MEAM, DFT, and experimental data. [6]
Fig. 3 Titanium unit cell equilibrium volume for alpha, beta, and omega phases. Data markers indicate DFT results while the curves represent the MEAM results. [6]
Determination of defect structures and defect formation energies involve relaxations in both single vacancy and single interstitial conditions. For the alpha to omega phase transformation pathway, these values are calculated using a 96-atom (4 x 4 x 3) supercell for the alpha phase and a 108-atom (3 x 3 x 4) supercell for the omega phase. In both cases a 2 x 2 x 2 k-point mesh sampling is used to model a 1 at% defect concentration.
Table 3. Defect energies for the alpha and omega phases, as calculated via MEAM and DFT. [6]
Phonon energies are calculated using the direct force method, while surface energies are determined by separating the titanium crystal along a high index plane for the alpha and omega phases. Phonon calculations for the alpha phase employ a 150 atom (5 x 5 x 3) supercell, while the beta phase and omega phase properties are determined using 125 atom (5 x 5 x 5) and 135 atom (3 x 3 x 5) supercells respectively. Surface energies are calculated by stacking 18-20 angstrom thick slabs of the respective phase unit cells, with 10 angstrom vacuum regions between each slab to ensure no interaction between surfaces. Comparison of the slab geometry and slab energy with their bulk counterparts reveal the surface energy of each phase studied based on 13 x 13 x 1 k-point sampling.
Fig. 4 Phonon spectra of alpha, beta, and omega titanium via MEAM (black lines), DFT (red dashed lines) and experimentally (blue circles). [6]
Table 4. Surface energies of alpha and omega titanium via MEAM and DFT. [6]
Feasibility of Titanium Alloy Biomaterials
Now that the base metal of choice has been thoroughly examined, the next stage in development of implants is to investigate possible alloy compositions that may have favorable mechanical and/or biological properties. Because the scope of this project is to determine alloying elements that lower the bulk modulus of titanium to a level that is more compatible to that of human bone, the biological effects will only be mentioned briefly at this junction. Titanium components are generally stable within bodily fluids, but as mentioned, the more attractive choice is an alpha-beta alloy such as Ti 6Al-4V because they exhibit a lower bulk modulus than that of pure titanium. One drawback of Ti 6Al-4V concerns biotoxicity, since it is proposed that aluminum and vanadium within the alloy will dissolve into the bloodstream causing harmful side effects. While the toxicity of the component is certainly a critical factor, the exact effect is less understood than the effect of a mechanical bulk modulus mismatch, thus the focus of Wong et al and Raabe et al is to simulate and validate beta-phase stabilizers in titanium alloys that effectively lower the bulk modulus of the component.
While pure titanium shows excellent strength to weight ratio, and superb fatigue resistance, it has an elastic modulus that is too high for bone stem implantation. To avoid the stress shielding effect, thermodynamic modeling based on Factsage software is employed to calculate the phase equilibria of titanium with various alloying elements (i.e. Fe, Al, Sn, and Zr). Note that this calculation is not a DFT based simulation but it does establish the beta-phase formation temperature in titanium alloy systems. Combining the information obtained from the Factsage simulation with ab initio calculations of the bulk modulus, a systematic screening of desirable alloy compositions is achieved. Differential thermal analysis (DTA) is employed to experimentally validate the phase transformation temperature results obtained via the Factstage simulation because DTA clearly shows compositional changes as a function of temperature. X-ray diffraction (XRD) studies are also undertaken for experimental validation of the phase present in the alloy systems.
Fig. 5 DTA matching to simulate phase diagram. [5]
The DFT parameters utilized for the lattice parameter and bulk modulus calculation are as follows [5]:
-VASP
-Projector augmented wave method potentials (PAW)
-Generalized gradient approximations (GGA)
-128 atom supercell (4 x 4 x 4), 127 titanium with 1 impurity atom (alloying element) at the center
-Beta-phase bcc supercell structure
-Mean field interaction
-Bulk modulus estimated by Murnaghan equation of state, equation (1)
[5]
The threshold value identified by this calculation is the bulk modulus of pure titanium (107 GPa), which establishes the upper limit that any alloying element can exhibit while still contributing to the lowering of the bulk modulus of the final implant material. From Table 5 it can be seen that in order to lower the bulk modulus of the final product the desired alloying elements are Al (65 GPa), Zr (88 GPa), or Sn (46 GPa).
Fig. 6 Supercell construction for defect energy calculation. [5]
Table 5. Bulk modulus of pure alloying elements in when in the bcc structure. [5]
Continuing on the same pathway, calculation of the bulk modulus of multi-component (i.e. more than one alloying element) systems is performed, again showing satisfactory accuracy to values obtained experimentally. Note that although Fe has a higher bulk modulus than that of titanium it has the advantage of lowering the beta-phase transformation temperature, thus Fe appears in the multi-component calculations because it offers the opportunity to decrease processing costs via lowering the required heat treatment temperature of the final implant device. Bulk modulus values for multi-component titanium alloys obtained experimentally as well as those calculated using DFT are given in Table 6. [5]
Table 6. Bulk modulus of mutli-component titanium alloys. [5]
A second investigation into the effect of alloying elements in endured by Raabe and co-workers in effort to utilize low cost, high accuracy screening of biomedical implant materials via ab initio calculations. By avoiding the arduous task of repeated trial and error experimentally, the cost associated with medical implant devices can be lowered while the technology being put into their development is continually improved.
Titanium alloys are not considered to be toxic because osteoblastic cells (mononucleate cells responsible for bone growth) do not show any negative reaction when in contact. Additionally as established in the previous case study, the huge bulk modulus mismatch between the native bone and implant material leads to mechanical stress related issues. Those issues are revealed first as osteoporotic degradation of the surround bone, followed by tissue resorption and the migration of wear debris to the implant interface, and finally the large stiffness gap between the structures leads to mico-motions at the interface resulting in loosening of the device. In order to investigate the effect of alloying element on the bulk modulus of binary titanium alloys, DFT calculations are performed in parallel with experimental testing. DFT results are particularly accurate because their assumed potentials are based on the electronic tendencies of the system as a function of composition.
The DFT details utilized by Raabe and co-workers are as follows:
-VASP
-GGA
-Plane wave cutoff energy of 170 eV
-8 x 8 x 8 Monkhorst-Pack mesh k-point sampling
-Supercells (2 x 2 x 2) of cubic or hexagonal unit cells with 16 atoms
Alloy compositions are varied stepwise by removing one atom of titanium from the 16 atom supercell, replacing it with one beta-phase stabilizing atom (i.e. Nb or Mo), for a lower limit on the alloy composition of 6.25 at% (1/16). The composition is varied by continuing to replace titanium with the proper alloying element, one atom a time, in a total of 48 possible bcc structures and 28 possible hcp structures. During simulation, the supercell geometry is minimized by iteration until forces and strains on all the atoms in system reach zero. Applying this calculated ground state energy to equation (2) gives the alloy formation energy as a function of alloy composition [7]. The temperature dependence of all calculations neglect vibrational contributions because they are so taxing computationally. Instead the ideal mixing approximation is used to calculate the configurational entropy term, which is applicable to alloys that exhibit formation energies that depend only on the atomic concentration, not the local configuration.
(2) [7]
When the formation energy is negative the alloy is considered to be stable, therefore from Fig. 7 it can be seen that Mo forms a stable alloy with titanium at lower concentrations than for Nb alloying [7]. The top two plots (grey background) are values calculated using DFT at 0K (ground state models), while the bottom two plots (yellow and green) show the formation energies calculated at 1K below the experimentally obtained alpha-beta phase transformation temperature (881C). The left two plots show the effect of Nb alloying, while the right two plots show the effect of Mo alloying with titanium. The general conclusion from these plots is that Mo would be a more suitable choice for alloying because it forms a stable, low bulk modulus, bcc alloy at lower atomic concentrations. This is important from a manufacturing point of view because less alloying material is required to make the alloys experimentally, thus the cost associated with their production is kept to a minimum.
[7]
Fig. 7 Formation energies for Nb and Mo alloying with titanium as calculated using DFT. Red triangles represent hcp (alpha) structure formation energies, while blue squares represent the formation energies of the bcc (beta) structure. [7]
Experimental validation is achieved by melting and casting of the alloys with the correct composition, after which solutionizing heat treatment is applied to ensure the desired phases. Subsequent characterization of the alloys included identifying their structure via XRD and measuring their elastic moduli via tensile testing. While the XRD results are not shown, they do not indicate the formation of any additional unwanted phases during the casting and heat treating processes, thus the DFT models appear to be correct. The elastic modulus values obtained via DFT predictions and by experimental validation are given in Fig. 8 [7]. The major discrepancy between the measured and predicted elastic properties of such titanium alloys are due to the predicted model making the assumption that the material is a bulk single crystal, when in reality the as cast samples tested by uniaxial tensile stretching are certainly polycrystalline. This is a major flaw because the modeled values are calculated based on the assumption that the single crystal is deformed along a single soft crystallographic direction (the bcc [001] direction), where in the polycrystalline samples the measured elastic modulus is an average of the deformations along all the hard and soft crystallographic directions. The DFT predicted elastic moduli are calculated by applying an elongation of the unit cell along the [001] using a linear approximation (Hooke's Law) that mimics an applied elastic stress. The total energy is recorded according to the relaxation of the two stress perpendicular to the applied load axis, which is then utilized in equation (1) to predict the resulting bulk modulus.
Fig. 8 Young's moduli obtained via DFT calculations (blue squares) and via experimental testing (green stars).
The results of the DFT predicted phase stability and bulk moduli of pure titanium and titanium alloy systems provide clear validation of the effectiveness of DFT simulation as a guide for future experimental endeavors. While computational methods continue to improve in accuracy, they continue to decrease in computation time as multiple high speed processors are applied in parallel to perform the simulation simultaneously. The result of such a high throughput material screening method has established DFT as a valuable tool for the prediction of materials properties such as phase stability and bulk modulus. In light of these breakthroughs the biomedical engineering community has sought to employ DFT models for the prediction of suitable titanium based alloys for implant device applications. The desire to process and apply titanium alloys that exhibit a large fraction of low bulk modulus beta phase have been successful in theory and subsequently proven via experimental analysis.
Works Cited
[1] Hip implant schematic. Orthopedic Surgery. North America Medical Tourism Agency. http://canada-indiamedicaltourism.com/ortho.html
[2] Medical device classifications. United States Food and Drug Administration. 2009.
[3] Hip implant 3-D modeling. General Medicine Reel. Ghost Productions. http://www.youtube.com/v/fuhcnx19I0o
[4] Hip resurfacing surgery. Birmingham Hip Resurfacing. Dr. Vijay Bose. http://www.youtube.com/v/uL3dNIuQp4E
[5] Wong, C., Bai, K., Tham, Y., Tong, K., Yong, M., Wu, P. "Feasibility study into the computer-aided design of titanium alloy for bone implant applications," SIMTech technical reports, Volume 7, Number 2, Apr-Jun 2006.
[6] Hennig, R., Lenosky, T., Trinkle, D., Rudin, S., Wilkins, J. "Classical potential describes martensitic phase transformations between the alpha, beta, and omega titanium phases," PHYSICAL REVIEW B 78, 054121, 2008.
[7] Raabe, D., Sander, B., Friak, M., Ma, D., Neugebauer, J. "Theory-guided design of beta-titanium alloys as biomaterials based on first principles calculations: theory and experiments," Max-Planck-Institut für Eisenforschung, Max-Planck-Str. 1, 40237 Düsseldorf, Germany.
