The term primary stability refers to the inducible displacement between an implant and the surrounding bone, under physiological loading of the implant in the early postoperative stage, when osseointagration has not yet occurred. Sufficient primary stability is a prerequisite for the long term success of cementless total hip replacements (THRs). Various authors suggest that osseointegration becomes unlikely at micromotions larger than 150 μ m [1, 2]. Instead, a fibrous interface tissue will be formed, which does not give adequate support to the implant. This will compromise the endurance of the implant fixation and may lead to aseptic loosening, which is the primary cause of failure in cementless THR [1, 3].
The primary stability of cementless hip implants has been investigated extensively, in vitro as well as numerically. Finite element (FE) studies have contributed to the research on primary stability in several ways. Some studies have investigated the influence of certain factors on the primary stability, e.g. bone quality , loading conditions , amount of press-fit  and the presence of gaps at the bone-implant interface . Other FE studies have evaluated the primary stability of new prosthetic designs [8, 9]. It has also been suggested to use finite element modelling (FEM) in preoperative planning of THRs  to quantify the expected primary stability. However, the introduction of FE methods into the preoperative environment would require specialised software and high performance computing hardware to keep the runtime of the simulation within acceptable limits. This would considerably raise the cost of the procedure. Fast assessment of the primary stability is even more important when custom-made stems are designed intraoperatively, based on the geometry of the reamed cavity. Furthermore, a protocol would be needed to automatically generate accurate patient-specific models, to account for the inter-subject variability . The development of such a protocol is far from evident, and it will also result in higher costs and longer runtimes.
In vitro studies usually consider the micromotion at the interface between the prosthesis and the bone under physiological loading conditions [12–17]. However, methods of measurement, points of measurement, loading conditions, and the designs tested have varied among different studies. This has limited the comparability of these studies. A wide range of inducible displacements was found for comparable loading conditions: for instance, when loading conditions simulating stair climbing were applied, micromotions were found in the range of 10–50 μ m , 10–280 μ m , 10–380 μ m  and 240–1540 μ m . Considering this large experimental variability in measuring micromotion, several authors have suggested that a theoretical approach could help in determining the potential stability of different stem designs. It would be very useful to be able to make statements about the primary stability of a hip stem without the need for measurements.
Torsional loading (e.g. stair climbing) has been shown to cause the largest displacements at the bone-implant interface [13, 18]. Therefore, large torsional loads, e.g. stair climbing, must be avoided in the first postoperative months. New implant designs should aim for a good resistance against axial rotation, to ensure sufficient primary stability.
Several in vitro studies have investigated the influence of the stem geometry on the primary stability, by comparing the magnitudes of motion between different stem types [13, 17, 19, 20]. These studies pointed out that the geometry of the stem significantly affects the primary stability and can be important in the prevention of excessive micromotion. Very few attempts have been made so far to define a parameter able to quantify the potential stability that can be achieved with a specific stem design. Ruben et al. proposed an optimisation strategy to design new hip stems, based on two objective stability functions . The first one is a function of tangential displacement at the bone-stem interface, the second one is a function of normal contact stresses. A mapping of the relative displacements and the normal contact stresses at the bone-stem interface is obtained using FEM. To the authors' knowledge, currently no stability characteristics have been proposed without the need for FEM.
A parameter characterising the potential primary stability of a hip stem could also be of great value in pre-operative planning of THRs. It could provide the surgeon with objective information to help him choose the best stem type in patient-specific cases. The traditional way of planning a THR is to superpose transparent templates of prostheses onto a radiograph of the hip joint, to determine the most suitable stem size and type . However, this procedure does not provide the surgeon with much information about the quality of the surrounding bone and a radiograph provides only limited geometrical information. A study by Viceconti et al. has shown that, by using a preoperative planning system, the implanted stem geometry more often corresponds to the planned stem geometry than when templates are used . Furthermore, the difference in planning result is smaller among different surgeons. Currently, there is no consensus about the best criterion to predict the long term success of a THR. Therefore, the current preoperative planning systems rely on very divergent criteria as a measure of the expected success [24–26]: HipOp, a planning system developed by Viceconti et al., provides the user with two analysis modules to assess the bone quality around the implant ; a planning system developed by Duda et al.  on the other hand estimates the joint contact force, based on a musculoskeletal model; and Benedetti et al. presented two computer-based tools to be used in preoperative planning of THRs . Both tools are based on gait analysis: one tool aims at restoring correct joint motion, while the other one considers the lever arms of the abductor muscles and leg-length discrepancy. Although it has been shown that good primary stability is essential to achieve long term functionality of cementless implants [1, 2, 27], no quantitative relationship has yet been established between the primary stability and long term results. However, a parameter quantifying the primary stability could be a good predictor for the long term results, and might thus be useful as a criterion for the expected success of THRs in a preoperative planning system .
At the department of orthopaedic surgery of the Leuven university hospitals, an intraoperatively custom-made prosthesis (IMP) system is used, based on the theory that a THR stem with optimal fit and fill of the intramedullary canal will resist the daily loads on the hip better than standard stems . However, a large variety of stem geometries is obtained with this technique, and sufficient primary stability of the stems is not guaranteed.
This study proposes an algebraic formula, which allows a fast estimation of the primary stability of a given implant design under torsional loading. The analytical formula is based on a straightforward mechanical model and the principle of virtual work. The suitability of the analytical model as a measure for the primary stability was investigated and confirmed for five custom-made stem designs using FEM.