# Abstract

Correct torqueing of bone screws is important for orthopaedic surgery. Surgeons mainly tighten screws *ad hoc*, risking inappropriate torqueing. An adaptive torque-limiting screwdriver may be able to measure the torque-rotation response and use parameter identification of key material properties to recommend optimal torques. This paper analyses the identifiability and sensitivity of a model of the bone screwing process. The accuracy with which values of the Young modulus (E) of the bone were identified depended on the value of E, with larger values being less accurately identified. The error in identified *E*. Experimental validation is still required for the model and identification process, but this approach is feasible and promising from a theoretical perspective.

# Zusammenfassung

Das korrekte Befestigen von Knochenschrauben ist bedeutend für die orthopädische Chirurgie. Chirurgen ziehen Schrauben hauptsächlich ad hoc an, wodurch ein suboptimaler Anpressdruck riskiert wird. Ein adaptiver Drehmomentbegrenzer könnte die Drehmoment-Dreh-Reaktion während des Einschraubvorgangs messen, die Daten zur Parameteridentifikation der wichtigsten Materialeigenschaften verwenden und diese wiederum, um optimale Drehmomente vorherzusagen. In diesem Artikel wird die Identifizierbarkeit und Sensitivität eines Modells des Knochenschraubprozesses untersucht. Die Identifizierbarkeit von E (Elastizitätsmodul des Knochens) hing vom Wert von E selbst ab, wobei größere Werte schlechter zu identifizieren waren. Die identifizierten σ_{uts}-Werte (Zugfestigkeit) lagen in den getesteten Fällen innerhalb von ungefähr 0,5 % des realen Werts, und dies unabhängig von den gleichzeitig bestimmten Werten von E. Eine experimentelle Validierung steht für das Modell und den Identifizierungsprozess noch aus, aber dieser Ansatz ist aus theoretischer Sicht machbar und plausibel.

## 1 Introduction

A major part of orthopaedic surgery is the fixation of implants using bone screws. Correct torqueing of the screws is critical to prevent implant failure due to thread stripping [1] or screw loosening [2]. Implant failures may require revision surgery, with all associated costs and risks, or may cause further tissue damage and other complications [3].

Surgeons currently tighten screws *ad hoc*, without any specific torque guidelines. The success rate of this varies between surgeons, and incorrect tightening can easily occur [4]. It has been proposed that by monitoring the screwing process, the bone material properties can be identified using a model, and then used to predict the optimal tightening torque [5]; as these bone material properties are also dependent on factors like age and disease, this method takes these into account. This first requires an identifiable [6] model of the screwing process in terms of the most important material properties, and then a method for predicting optimal torque from these properties. This method is applicable to procedures where screws are fixed in bone. In many cases, the geometric placement of screws is more important than the correct torqueing. However, correct torque is still important to prevent tissue damage or screw loosening.

Other non-model-based methods have been proposed to regulate bone screw tightening torque. Reynolds et al. measures the plateau/steady-state torque while inserting a lag screw, and stops screwing when the torque reaches a multiple of this [7]. However, Reynolds’ method might not be applicable to non-lag screws. Thomas et al. suggest monitoring the derivative of torque with respect to time, and halting the process when a spike is detected, indicating the screw has started binding [8]. This approach is effective at preventing overtightening but does little to ensure a minimum torque. Torque-limiting devices exist for some cases [9], especially in orthodontics, which uses small screws with low torque values [10], but these still require the surgeon to know and select the correct torque or limiter manually, in contrast to the proposed automatic model-based method.

Previously, a very simple model was demonstrated to be identifiable and robust for an arbitrary set of parameters [5]. This model was based on the material failure energy density of the bone, viscous friction between the screw and bone, and a threshold torque to advance the screw; this model was derived using an power balance between the input torque times speed and the material failure plus viscous friction. This simple model was significantly extended by Wilkie et al. [11], but the application of parameter identification was not investigated. This paper computationally analyses the model from Wilkie et al. [11], focusing on identifiability, recovery of the input variables in noisy data sets, and sensitivity to parameter values. The proposed model has greater physical descriptiveness than the model in Wilkie et al. [5], as specific material and geometric parameters rather than arbitrary lumped parameters are identified.

## 2 Methods

### 2.1 Model summary

The key model equations 1–11 were derived in Wilkie et al. [11] and have some basis in the model of Seneviratne et al. [12]. This model assumes that a simple screw design is self-tapping into a pre-drilled cylindrical hole of greater diameter than the minor diameter of the screw (*θ* is the thread helix angle. Equation 8 models how critical torque changes as the screw advances. Equation 9 models how the effective torsional spring constant (*k*) for a rigid screw in material of Young’s modulus *E* changes with screw depth. Equations 10 and 11 model how angular velocity and torque are related to the input effort *ϵ* over time, using critical torque (*ϕ*).

### Figure 1

### Figure 2

The value of *ϵ* is an input basis function that varies from 0–1 and represents the effort that the surgeon is exerting; this controls the torque and speed of the advancing screw by modulating a linear torque-speed relationship defined by *a* and *b*, shown in fig. 1. This was required to keep angular velocity defined in the absence of a viscous friction assumption [11]. The outputs are *ϕ* [rad], and *T* [Nm]. The model considers a critical torque (*α* defines the angular length of the thread cutting section of the screw. *μ* is the friction coefficient between the screw and bone, *E* [Pa] is the Young’s modulus of the bone.

### Figure 3

### 2.2 Parameter identification overview

The overall parameter identification and sensitivity analysis process is outlined in Figure 3.

### 2.3 Input variable recovery

As it is not possible to measure the input effort of the surgeon directly, the effort must be determined from the measured angular speed and torque. It is not possible to do this exactly, as the relationship between *T*, *ϕ*, and *ϵ* depends on the unknown parameters, *E* and *a* and *b* are known. The basic heuristic was to calculate the value of *k* at each time point using Hooke’s law, *k* is relatively low, the position is changing a lot without much change in torque, indicating that the screw is advancing into the material, so the standard torque-speed model is used: *k* value is higher than a threshold (3 N/rad was arbitrarily defined), that indicates the screw is only moving as torque changes, and therefore elastic behaviour is occurring, so the zero-speed assumption was used with the torque-speed model, giving *T*, it must be filtered before estimating *k* values; this was achieved with a 20 sample median filter, followed by a 20 Hz filter using the MATLAB “lowpass” function with a steepness of 0.8 and a stopband attenuation of 80 dB.

Parameter | Value | Units | Notes/Source |

E | 300 | MPa | Normal femoral head [14] |

ν | 0.3 | Generic estimate | |

3.5 | MPa | Normal femoral head [14] | |

μ | 0.4 | [15] | |

3.2 | mm | [16] | |

6.5 | mm | SYNTHES 216.060 [16] | |

β | 15 | ° | Estimate from photo [17] |

p | 2.75 | mm | SYNTHES 216.060 [16] |

α | 2 | rad | Estimate from photo [17] |

a | 4 | Rad.s^{−1} | Expected value |

b | 4 | Rad.s^{−1} (N.m)^{−1} | Expected value |

Standard deviation of noise in T | N.m | 0.5 % class with 10× overrating | |

Standard deviation of noise in | 0.002 | Rad.s^{−1} | e. g., MPU-9250 gyro [18] |

### 2.4 Time variant objective-function weighting

Changes in *E* have a very small effect on the *T* and *ϕ* response [11]. Hence, to ensure that *E* was not overfit to random noise, it was desirable to place more weighting on time periods that specifically exhibit elastic behaviour. This was done by changing the weight on

### 2.5 Identification methodology

Initially data is generated using the model in eqns. 1–11. To simulate the effects of sensor noise and sampling, the model was first forward-simulated with a 20-cycle trapezoidal input signal

The identification process first takes the measured variables and performs the input recovery (Section 2.3) to get the approximated input effort over time. The input variables were then analysed to determine the time periods for the time-variant weighting (Section 2.4). Then simulated annealing [13] parameter identification was used to identify *E* and *E*. The standard deviation proportion for *E* went from 0.5 to 0.05.

The parameters/constants from Wilkie et al. [11] were used. The constants *a* and *b* were selected based on expected values. Variations in *a* and *b* are very unlikely to have a significant effect if they are consistent between the initial forward simulation, input variable recovery, and forward simulation during the identification; this is because *a* is mainly for scaling, and, with rearrangement, can always be combined with *ϵ* to write the equations in terms of

### 2.6 Identifiability/sensitivity analysis

The effects of varying the two parameters (*E* and *E* and

## 3 Results

The result of the input effort recovery with default parameters is shown in Figure 4.

### Figure 4

Figure 5 shows how the distributions of the identified *E* and *E* and

### Figure 5

E [MPa] | Comp. 1 S. D. [log(MPa)] | Comp. 2 S. D. [log(MPa)] | Comp. 1 Angle [°] | Comp. 2 Angle [°] | |

3 | 1 | 0.279 | 0.00164 | −0.082 | 89.918 |

3.2 | 0.0583 | 0.00178 | −0.157 | 89.843 | |

10 | 0.0674 | 0.00236 | −0.195 | 89.805 | |

30 | 1 | 0.855 | 0.00175 | 0.020 | 90.020 |

3.2 | 0.502 | 0.00191 | 0.002 | 90.002 | |

10 | 0.428 | 0.00202 | −0.023 | 89.977 | |

300 | 1 | 1.73 | 0.00177 | 0.004 | 90.004 |

3.2 | 1.23 | 0.00198 | −0.002 | 89.998 | |

10 | 1.04 | 0.00204 | 0.007 | 90.007 |

Figures 6 and 7 show the sensitivity of the model output to changes in input parameters, with differences from the nominal values shown quantitatively in Tab. 3.

### Figure 6

### Figure 7

## 4 Discussion

Figure 4 shows that the input variable recovery works as expected. The recovered results with the given settings closely match the known data. Figure 5 shows that the spread of the identified *E* to a reasonable accuracy when it is a low value (3 MPa). However using the model in eqns. 1–11, *E* becomes hard to identify at higher, more physiological values such as 300 MPa (typical femoral head trabecular bone [14]), as there is very ambiguous estimation variability over many orders of magnitude. This identification inaccuracy is due to a higher Young’s modulus leading to less significant elastic behaviour in the data, reducing the signal-to-noise ratio to almost zero. However, the orthogonality of the principle components in Tab. 2 shows that there was no discernible correlation between the identified *E* values. Therefore, the poor identification of *E* did not affect the good identification of *E* around an expected value for bone (300 MPa) had almost no effect on the output of the simulation, suggesting that *E* is of minor significance. Whereas Fig. 6 and Tab. 3 show that even small changes in *E*. It should be noted that the default values for *E* are plausible for the application of this method, but there is a large range of biologically significant values [20].

Nominal Value | Comparison Value | ||

279.35 | 41.94 | ||

247.92 | 37.66 | ||

0.0199 | 0.0173 | ||

0.0221 | 0.0192 |

The model was demonstrated to be capable of identifying *E* from noisy data in simulation, with plausible material properties. The model is especially promising for identifying *E*. Fortunately, the low sensitivity of the model output to changes in *E* suggests that accurate identification of *E* may not be very important to the bone screw-torqueing application, and the lack of correlation between *E* and *E* is not detrimental to the identifiability of *E* [21] to infer *E* from the accurately identified *a priori* information such as age, and apparent bone mineral density (e. g., from quantitative computed tomography imaging [22]).

As the model was derived from physical principles, one may expect it to be a reasonable mathematical description of the physical phenomena. However, as mentioned in Wilkie et al. [11], there are a few assumptions used to simplify the model. Bone is a complex anisotropic, non-homogeneous, and discontinuous material, and this analysis modelled bone as an isotropic, homogeneous, and continuous material. This will result in some error in the identified parameter values. However even imperfect values may be useful if experimental testing shows that the error is sufficiently small. A noteworthy discrepancy is in the non-homogeneous structure of bone, often there is a harder outer layer and softer inner layer. This paper focused on the simple case of homogeneous bone, but the model, and specifically eqn. 8, could be modified to account for a varying

Due to noisy torque signals, filtering was required to remove noise before input variable recovery, however it also removed some desired higher frequency components of the signal. This can potentially introduce artefacts in the processed signal precisely in the regions governed by elasticity, and the parameter identification may over fit these artefacts. There are some notable fluctuations in the recovered data, some of this will be random, and due to noise, but there will be some systematic overshoots in the trapezoidal transitions due to the bandwidth limiting nature of the filtering approach. This was mitigated using median filtering, which can smooth out noise without smoothing out sharp transitions in the signal. However, the effectiveness will be dependent on the shape of the underlying signal. A possible solution might be an adaptive filtering span that reduces when sensor noise is low (by increasing cut-off frequency). Additionally, the threshold of k used for detecting elastic and non-elastic sections during input variable recovery was a constant in this paper, but there may be better heuristics for this. If the threshold is too high, it will treat non-elastic sections assuming elasticity, and if too low it will treat elastic sections with the non-elastic assumption.

The sampling rate and noise levels used were based on relatively low-cost general-purpose sensors (0.5 % of full-range torque sensor such as NCTE-2300 Series [23], and 0.002 rads^{−1} noise at 92 Hz (∼100 Hz) from an MPU-9250 gyroscope [18]), using more accurate sensors with less noise and/or higher sampling rates may make it possible to identify higher Young’s moduli with reasonable accuracy. Conversely, this paper only looks at simulated results, and experimental noise may be non-Gaussian, and may contain other corruption, such as gyroscope drift, torque hysteresis, sensor non-linearities, and temporal shifts across sensors.

The weights used for the different variables in the objective function were determined subjectively. More work is required to determine what the optimal weights are, and it may be possible to heuristically choose better values by analysing the measured data, and the tendencies of clinicians during the screwing process. These optimal values will also depend on the levels of sensor noise, as too much weight on a noisy sensor may result in overfitting to that sensors’ noise and overwhelming the small weight of the other data. The threshold used for identifying time-periods for time-variant weighting was chosen for simplicity in this analysis, and may not work well when the input is not a perfect repeating trapezoidal signal; however this is simply a form of edge detection, and there are potentially many other algorithms that could be used to make the process more robust.

Experimental verification is required to determine whether the modelling assumptions lead to unacceptable variation in the identified values; some variation is expected, but further work will determine how much variation there is, and how much is acceptable. Experimental verification is also important for the rest of the identification process, as the models of sensor noise may differ from reality, and other real-world limitations will apply, for example, measuring rotation at the handle end of a screw-driver will also measure any elastic behaviour due to torsion in the screwdriver shaft. If a significant relationship can be found between screwing speed and torque, all of the complexity introduced with the torque-speed model of the operator, and input variable recovery could potentially be eliminated. Chumakov [24] suggests that there is no significant relationship between screw speed and torque, although they used aluminium, which is not very strain rate dependant [25], while bone is [26], so more research is required.

## 5 Conclusion

This paper analysed the use of a bone screwing model for identifying material properties in simulation. Input variable recovery was successfully demonstrated, as was time-variant objective-function weighting. It was demonstrated that *E* could be identified. *E* could be identified effectively at low values such as 3 MPa, however at more realistic values such as 300 MPa, the identified parameters had a prohibitively large variance; this may improve if more accurate sensors are assumed, or if other aspects of the modelling approach are improved/tuned, particularly in the heuristic sections of input recovery and time-variant weighting. Further study may also reveal *E* to be less significant than *E* from *a priori* relationships. Experimental testing is also required to validate all parts of the identification and model and ensure that they operate correctly under real-world conditions.

**Funding statement: **Partial support by grants “CiD” and “Digitalisation in the OR” from BMBF (Project numbers 13FH5E02IA and 13FH5I05IA).

# Ethical approval

The conducted research is not related to either human or animal use.

**Conflict of interest:**Authors state no conflict of interest.

### References

1. A. Feroz Dinah, S. C. Mears, T. A. Knight, S. P. Soin, J. T. Campbell and S. M. Belkoff, “Inadvertent screw stripping during ankle fracture fixation in elderly bone,” *Geriatric orthopaedic surgery & rehabilitation*, vol. 2, no. 3, pp. 86–89. doi: 10.1177/2151458511401352, 2011. Search in Google Scholar

2. M. Evans, M. Spencer, Q. Wang, S. H. White and J. L. Cunningham, “Design and testing of external fixator bone screws,” *Journal of biomedical engineering*, vol. 12, no. 6, pp. 457–462. doi: 10.1016/0141-5425(90)90054-Q, 1990. Search in Google Scholar

3. N. J. Hallab and J. J. Jacobs, “Biologic effects of implant debris,” *Bulletin of the NYU hospital for joint diseases*, vol. 67, no. 2, p. 182, 2009. Search in Google Scholar

4. M. J. Stoesz, P. A. Gustafson, B. V. Patel, J. R. Jastifer and J. L. Chess, “Surgeon perception of cancellous screw fixation,” *Journal of orthopaedic trauma*, vol. 28, no. 1, pp. e1–e7. doi: 10.1097/BOT.0b013e31829ef63b, 28 January 2014. Search in Google Scholar

5. J. Wilkie, P. Docherty and K. Möller, “A simple screwing process model for bone material identification,” in *Proceedings on Automation in Medical Engineering*, Lübeck, 2020. doi: 10.18416/AUTOMED.2020. Search in Google Scholar

6. P. D. Docherty, J. G. Chase, T. F. Lotz and T. Desaive, “A graphical method for practical and informative identifiability analyses of physiological models: A case study of insulin kinetics and sensitivity,” *Biomedical engineering online*, vol. 10, no. 1, p. 39. doi: 10.1186/1475-925X-10-39, 2011. Search in Google Scholar

7. K. J. Reynolds, A. A. Mohtar, T. M. Cleek, M. K. Ryan and T. C. Hearn, “Automated Bone Screw Tightening to Adaptive Levels of Stripping Torque,” *Journal of orthopaedic trauma*, vol. 31, no. 6, pp. 321–325. doi: 10.1097/BOT.0000000000000824, 2017. Search in Google Scholar

8. R. L. Thomas, K. Bouazza-Marouf and G. J. S. Taylor, “Automated surgical screwdriver: automated screw,” *Proceedings of the Institution of Mechanical Engineers, Part H: Journal of Engineering in Medicine*, vol. 222, no. 5, pp. 817–827. doi: 10.1243/09544119JEIM375, 2008. Search in Google Scholar

9. C. Ioannou, M. Knight, L. Daniele, L. Flueckiger and E. S. Tan, “Effectiveness of the surgical torque limiter: a model comparing drill-and hand-based screw insertion into locking plates,” *Journal of orthopaedic surgery and research*, vol. 11, no. 1, p. 118, 2016. Search in Google Scholar

10. J. P. Standlee and A. A. Caputo, “Accuracy of an Electric Torque-Limiting,” *Int J Oral Maxillofac Implants*, vol. 14, pp. 278–281, 1999. Search in Google Scholar

11. J. Wilkie, P. Docherty and K. Möller, “Developments in Modelling Bone Screwing,” *In Press*, *Current Trends in Biomedical Engineering*, 2020. Search in Google Scholar

12. L. D. Seneviratne, F. A. Ngemoh, S. W. E. Earles and K. A. Althoefer, “Theoretical modelling of the self-tapping screw fastening process,” *Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science*, vol. 215, no. 2, pp. 135–154. doi: 10.1243/0954406011520562, 2001. Search in Google Scholar

13. S. Geman and D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” *IEEE Transactions on pattern analysis and machine intelligence*, vol. 6, pp. 721–741. doi: 10.1109/TPAMI.1984.4767596, 1984. Search in Google Scholar

14. B. Li and R. M. Aspden, “Composition and mechanical properties of cancellous bone from the femoral head of patients with osteoporosis or osteoarthritis,” *Journal of Bone and Mineral Research*, vol. 12, no. 4, pp. 641–651. doi: 10.1359/jbmr.1997.12.4.641, 1997. Search in Google Scholar

15. A. Shirazi-Adl, M. Dammak and G. Paieme, “Experimental determination of friction characteristics at the trabecular bone/porous-coated metal interface in cementless implants,” *Journal of Biomedical Material Research*, vol. 27, no. 2, pp. 167–175. doi: 10.1002/jbm.820270205, 1993. Search in Google Scholar

16. M. Tsuji, M. Crookshank, M. Olsen, E. H. Schemitsch and R. Zdero, “The biomechanical effect of artificial and human bone density on stopping and stripping torque during screw insertion,” *Journal of Mechanical Behaviour of Biomedical Materials*, vol. 22, pp. 146–156. doi: 10.1016/j.jmbbm.2013.03.006, 2013. Search in Google Scholar

17. WestCoast Medical Resources, LLC, “SYNTHES: 216.240,” [Online]. Available: https://www.westcmr.com/synthes-216-240-216240. [Accessed 01 04 2020]. Search in Google Scholar

18. InvenSense Inc., “MPU-9250 Datasheet, PS-MPU-9250A-01 v1.1,” 20 06 2016. [Online]. Available: https://invensense.tdk.com/download-pdf/mpu-9250-datasheet/. [Accessed 01 05 2020]. Search in Google Scholar

19. K. Pearson, “LIII. On lines and planes of closest fit to systems of points in space,” *The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science*, vol. 2, no. 11, pp. 559–572. doi: 10.1016/0169-7439(87)80084-9, 1901. Search in Google Scholar

20. E. F. Morgan, G. U. Unnikrisnan and A. I. Hussein, “Bone mechanical properties in healthy and diseased states,” *Annual review of biomedical engineering*, vol. 20, pp. 119–143. doi: 10.1146/annurev-bioeng-062117-121139, 2018. Search in Google Scholar

21. T. M. Keaveny, E. F. Wachtel, C. M. Ford and W. C. Hayes, “Differences between the tensile and compressive strengths of bovine tibial trabecular bone depend on modulus,” *Journal of biomechanics*, vol. 27, no. 9, pp. 1137–1146. doi: 10.1016/0021-9290(94)90054-X, 1994. Search in Google Scholar

22. H. K. Genant, K. Engelke, T. Fuerst, C. C. Glüer, S. Grampp, S. T. Harris, M. Jergas, T. Lang, Y. Lu, S. Majumdar and A. Mathur, “Noninvasive assessment of bone mineral and structure: state of the art,” *Journal of Bone and Mineral Research*, vol. 11, no. 6, pp. 707–730. doi: 10.1002/jbmr.5650110602, 1996. Search in Google Scholar

23. NCTE AG, “NCTE torque sensors Series 2300,” [Online]. Available: https://ncte.com/wp-content/uploads/2019/08/DataSheet_Series2300_EN.pdf. [Accessed 01 05 2020]. Search in Google Scholar

24. R. Chumaakov, “Optimal control of screwing speed in assembly with thread-forming screws,” *The International Journal of Advanced Manufacturing Technology*, vol. 36, no. 3-4, pp. 395–400. doi: 10.1007/s00170-006-0839-1, 2008. Search in Google Scholar

25. Y. Chen, A. Clausen, O. Hopperstad and M. Langseth, “Stress–strain behaviour of aluminium alloys at a wide range of strain rates,” *International Journal of Solids and Stuctures*, vol. 46, no. 21, pp. 3825–3835. doi: 10.1016/j.ijsolstr.2009.07.013, 2009. Search in Google Scholar

26. F. Linde, P. Nørgaard, I. Hvid, A. Odgaard and K. Søballe, “Mechanical properties of trabecular bone. Dependency on strain rate,” *Journal of Biomechanics*, vol. 24, no. 9, pp. 803–809. doi: 10.1016/0021-9290(91)90305-7, 1991. Search in Google Scholar

**Received:**2020-05-11

**Accepted:**2020-09-01

**Published Online:**2020-10-28

**Published in Print:**2020-11-26

© 2020 Wilkie et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.