Estimating centre of mass and mass moments of inertia is an important aspect of many studies in biomechanics. Characterising these parameters accurately in three dimensions is challenging with traditional methods requiring dissection or suspension of cadavers. Here, we present a method to quantify the three-dimensional centre of mass and inertia tensor of birds of prey using calibrated computed tomography (CT) scans. The technique was validated using several independent methods, providing body segment mass estimates within approximately 1% of physical dissection measurements and moment of inertia measurements with a 0.993 R2 correlation with conventional trifilar pendulum measurements. Calibrated CT offers a relatively straightforward, non-destructive approach that yields highly detailed mass distribution data that can be used for three-dimensional dynamics modelling in biomechanics. Although demonstrated here with birds, this approach should work equally well with any animal or appendage capable of being CT scanned.
Estimates of centre of mass (CoM) and mass moment of inertia (MoI) are necessary features of many studies in animal biomechanics and locomotion. The CoM informs questions of stability and balance while MoI relates angular accelerations and moments via Newton's second law. In terrestrial locomotion, for example, the CoM is required for studies of stability at rest and in motion, while limb MoI is utilised for quantifying the energetics of the limb swing phase (Biewener and Patek, 2018). In flying animals, CoM and MoI are crucial parameters for modelling stability and flight dynamics (Durston et al., 2019). Wing MoI plays a role in understanding the energetics of flapping flight (Berg and Rayner, 1995), and is used by bats for controlling low-speed head-over-heels manoeuvres (Bergou et al., 2015). Similar examples may be found for swimming, climbing, suspension and jumping (Biewener and Patek, 2018).
Methods for CoM estimation in animal cadavers typically fall into four categories: balance, suspension, scales and digital modelling (Macaulay et al., 2017). The first of these involves use of a balance board or straight edge to find segment or whole-organism CoM. With the suspension approach, the cadaver is hung from multiple orientations such that the intersection of superimposed vertical reference lines provides the position of the CoM (Kilbourne et al., 2016; Ros et al., 2015; Thomas and Taylor, 2001). An example of the scales approach involves laying the cadaver on a beam supported by two mass balances and solving the corresponding moment equation to obtain CoM (Kilbourne, 2013; Macaulay et al., 2017). The digital modelling approach utilises image contrast from volume scans such as computed tomography (CT), magnetic resonance imaging or photogrammetric reconstruction to segment the scan into regions of uniform density based on tissue type (i.e. air, muscle, bone) (Allen et al., 2009; Brassey and Sellers, 2014; Hutchinson et al., 2007; Peyer et al., 2015). A study into the accuracy of the last three approaches concluded that suspension methods yield lower accuracy and repeatability compared with scales and digital modelling (Macaulay et al., 2017).
Methods for MoI estimation in animal cadavers also typically fall into four categories: shapes, pendulum, strips and digital modelling. The shapes method models the anatomy as a series of shapes (i.e. either primitive shapes or a scanned cadaver outline), each of uniform density, whose combined MoI can be calculated analytically (Kilbourne et al., 2016; Meriam and Kraige, 2012; Tucker, 1992). Pendulum methods relate the time period of oscillation of the cadaver with MoI via a dynamic model (Kilbourne, 2013; Ros et al., 2015). The strips method uses the recorded masses and positions of dissected cadaver segments to calculate MoI (Kirkpatrick, 1990; Thollesson and Norberg, 1991; Berg and Rayner, 1995). The digital modelling approach described above can also yield MoI estimates, although the magnitude of the error due to local variation in tissue density is unclear (Macaulay et al., 2017). The error associated with the primitive shapes and pendulum approaches is also unclear because studies often do not provide validation data. The accuracy of the strips method can be improved by increasing the number of strips, but this can also lead to increased damage caused by the dissection procedure.
Here, we describe a method for the generation of high spatial resolution three-dimensional (3D) mass distribution/inertial models for birds of prey using calibrated CT scanning. Estimates for CoM and MoI are provided alongside validation data used to quantify the accuracy of the technique. The advantages of this digital modelling approach are highlighted with novel visualisations of mass distribution in birds of prey. In contrast to the digital modelling method outlined previously, this approach describes mass variation throughout the organism at the level of individual voxels. Birds represent a challenging test case because of their feathered appendages (possibly explaining why inertial properties data for this class are relatively scarce (Mills et al., 2018)). This method should therefore apply equally well to animals with less challenging anatomical features.
MATERIALS AND METHODS
This basic calibration works reasonably well for biological materials with densities close to that of water, and has been used to estimate the mass of human organs (Lescot et al., 2008; Malbouisson et al., 2001). To improve accuracy for densities farther from water, calibrations can be obtained using objects of known density. This approach has yielded accurate mass estimates for different types of wood (Davis and Wells, 1992; Freyburger et al., 2009; Wei et al., 2011). Although calibrated CT has been used successfully for mass estimation in these instances, it should be noted that results can vary depending on the scanner type, model, calibration, settings and scan date (Levi et al., 1982). In particular, the approach assumes that the correlation between X-ray attenuation and object density is linear.
Cadavers of naturally deceased barn owls [Tyto alba (Scopoli 1769)], peregrine falcons (Falco peregrinus Tunstall 1771) and a sparrow hawk [Accipiter nisus (Linnaeus 1758)] were used in this study (Table 1). These were obtained from animal rescue centres across the south-west of the United Kingdom and frozen as soon as possible after death in a standard freezer (approximately −18°C) for approximately 1 year. Details of the cause of death and cadaver mass at the time of freezing were provided where possible. As with previous studies using bird cadavers (Kirkpatrick, 1990; Berg and Rayner, 1995), several specimens were emaciated. Only one barn owl cadaver and one peregrine falcon cadaver exhibited a healthy mass based on comparison with records for these species (Robinson, 2005).
A trifilar pendulum method was used to validate the calibrated CT estimates of MoI, so it was important to ensure the bird adopted a consistent position and pose during data collection with both techniques. The wings of each cadaver were stretched out and pinned securely (dorsal-side down) to a flat, cross-shaped board. Nylon threads were used to firmly secure the legs, head and primary flight feathers to the board to minimise movement during transportation. A sheet of expanded polystyrene foam (12.5 mm thickness) was placed between the bird and the board to act as a radiolucent cushion to ease segmentation of the CT data. During data collection over several days, the highest mass loss was from the sparrow hawk cadaver (4.3%), with the peregrine falcons and barn owl cadavers losing less than 1.7% mass (Table 1). One cadaver gained 0.5% of its initial mass, which may have been due to moisture absorption by the feathers during handling.
Each cadaver was CT scanned (LightSpeed RT16, General Electric, Boston, MA, USA) with a voltage and current of 120 kVp and 200 mA, respectively, and a spiral pitch factor of 0.9375. Each image in the stack had a resolution of 512×512 pixels and images were separated by 1.25 mm with pixel widths ranging from 0.68 mm to 0.98 mm depending on the field of view used for each cadaver (these varied depending on the size of the cadaver to maximise use of the measurement volume). Image stacks were extracted in uncompressed 16-bit TIFF format along axes roughly equivalent to the bird's dorsoventral and mediolateral axes (Fig. 1A). The peregrine falcon and barn owl cadavers bo1 and pf4 were also scanned a second time with their wings fully retracted.
Linear calibrations (Fig. S1) were generated correlating the voxel grey-values with eight tissue characterisation phantoms (Gammex 467, Sun Nuclear, Melbourne, VIC, Australia) whose densities ranged from 450 to 1820 kg m−3. Images from the 25%, 50% and 75% planes, approximately orthogonal to each phantom's longitudinal axis, were used to obtain their mean grey-values by manually drawing an ellipse around each phantom. Regions of air were also sampled in a similar manner. The phantoms were placed next to each bird cadaver, and individual calibrations were generated for each scan. The calibration curves were different between scans/cadavers, possibly due to the different fields of view used to maximise the resolution of each scan.
CoM and MoI estimation
To estimate the CoM and MoI tensors, the grey-values from the CT image stack were converted to voxel density and mass using the calibration and voxel volume. The scanned air around the bird was removed using manual thresholding followed by manual segmentation in CloudCompare v2.8.1 (https://www.cloudcompare.org/) to remove unwanted features of the scan (i.e. parts of the CT scanner). The segmentation threshold for the bird was then optimised by minimising the difference between the total cadaver mass estimate from the calibrated scan data (i.e. the sum of the voxel masses) with the known mass of the bird obtained with a mass balance (LE341001P, Sartorius, Göttingen, Germany). This optimised threshold resulted in scan data where the majority of the bird's covert feathers and flight feather shafts remained visible (Fig. 1B) and provided a general indication that the calibration was working as expected (i.e. grey-value thresholding based on total mass yielded the expected visualisation).
Appendage mass comparison
In a first approach to validate the accuracy of calibrated CT, comparisons were made between ‘virtual’ and ‘physical’ dissections of the birds’ appendages. All the cadavers, except one peregrine falcon (pf3), were dissected to separate the head and cervical spine, torso, tibiotarsae, tarsometatarsae (including talons) and wings. Each wing was cut at the elbow and wrist joints to separate the humerus, radius/ulna and manus (with associated feathers). The segments were then weighed using a mass balance (LE341001P, Sartorius) to a resolution of 0.1 g. The scan data for these cadavers was manually segmented in the same way as the dissected cadavers (Fig. 1B) and mass was estimated using the CT calibration. A comparison between the mass of the dissected cadaver and segmented scan data was then carried out. From here on, the dissection of the cadaver is referred to as ‘physical dissection’, while the segmentation of the CT scan data is referred to as ‘virtual dissection’.
Dorsoventral MoI comparison
In a second approach to validate the accuracy of calibrated CT, the MoI about the birds’ dorsoventral axis (defined as normal to the board) was measured using a trifilar pendulum (referred to as pendulum from here on). Fig. 1C shows the plan view of the pendulum, consisting of an equilateral triangular base suspended by three nylon threads (60 lb fishing line) supporting the board, bird and a gyroscope. The position of the board and gyroscope was determined so that their combined CoM was at the pendulum base CoM. The bird was placed on the board such that its approximate CoM (judged by eye) was as close as possible to this point. A correction for this misalignment between the bird's CoM and the CoM of the remaining pendulum components is described in Eqn 7.
The pendulum method was validated with machined nylon blocks whose MoI was calculated analytically to very high accuracy (±0.003%, assuming uniform material density). This showed that for objects of equivalent mass and size to a barn owl and peregrine falcon, the MoI using the pendulum was within 3.3% and 0.1% of the analytical estimates, respectively, with the larger error for the barn owl object being due to its smaller mass and dimensions.
RESULTS AND DISCUSSION
In this section, we present results comparing appendage mass breakdown and dorsoventral MoI between calibrated CT and the two independent methods described previously. Estimates of CoM and the principal components of inertia of the birds used in this study are provided alongside illustrative examples of how calibrated CT can be used to describe mass distribution in birds.
A comparison between the mass breakdown of the physical and virtual cadavers is shown in Fig. 2A, which reveals the effectiveness of calibrated CT for quantifying the overall mass distribution. The relative contributions of the appendages were well matched, with differences typically of 1% or less (versus total cadaver mass) between physical and virtual dissections. Some of the differences may be due to imprecisely matched cut locations between the physical and virtual dissections. In future work, this could be assessed by scanning the physically dissected cadaver.
It should be noted that although the assignment of mass to voxels appears to work relatively effectively, this is not necessarily the case with density. For example, voxels containing several tissue types, including air (such as feather shafts and vanes), yield a density that is a weighted average of all the materials (including air) present within the voxel. This inclusion of air in the volume measurements influences average calculated density measurement and is why other studies focusing on density measurement have removed the feathers (Larramendi et al., 2021).
A comparison is shown in Fig. 2B between the dorsoventral MoI estimated with calibrated CT and the trifilar pendulum for each cadaver. Correcting for the bird's CoM misalignment improved the correlation between the two methods (R2=0.993, N=7). Comparison was also made with wing MoI estimations obtained with strip analysis (Fig. S2; Berg and Rayner, 1995) and showed that the calibrated CT data were consistent with this previously described allometry. Overall, Fig. 2 shows that calibrated CT provides accurate quantification of the global inertial properties in these birds.
We investigated the effect of assuming a uniform density (density=cadaver mass/cadaver volume) for each voxel instead of using a calibration curve to relate grey-value to density. This resulted in up to 5 mm movement of the CoM for the barn owl and peregrine falcon (approximately 5% mean aerodynamic chord; Durston et al., 2019) and an increase of 58% to the dorsoventral MoI. This suggests that volume data alone provide a reasonable estimate for CoM depending on the accuracy required (Macaulay et al., 2017), but lead to more significant errors in MoI from overestimating the mass of the distal appendages.
Fig. 3A shows dorsoventral, mediolateral and anteroposterior projections of the 3D scan data for the healthy mass peregrine falcon, pf4. The colour maps show mass and MoI sampled and summed on a uniform grid applied to each plane for the top and bottom three views. The data quantify how mass (Fig. 3A, top) and MoI (Fig. 3A, bottom) are distributed for a given viewing direction. In the dorsoventral projection, the mass generally follows the thickness of the cadaver in the view direction. In contrast, local MoI contributions are concentrated more distally in the wing bones, head and feet, highlighting the strong local impact of the distance-squared term in MoI (see Fig. 3C). The anteroposterior view shows mass concentration dorsally, largely due to the dorsal position of the head and legs on the board, and possibly due to the dorsal movement of internal organs (cadavers were dorsal-side down and thawed during scanning).
The relative contributions of the dissected appendages to mass and MoI for cadavers bo1 and pf4 are shown in Fig. 3B and absolute data are given in Tables S1 and S2. The torso contributed 54–59% of the total mass and the wings contributed 18–19% of the total mass, consistent with previous findings (Kirkpatrick, 1990). The tail (pygostyle and rectrices) contributed no more than 4% of the total mass, consisting mostly of feathers. The bird's head was more than half the total mass of the wings and the combined mass of the legs (tibiotarsae, tarsometatarsae and digits) was approximately three-quarters the total mass of the wings.
Fig. 3B also shows significant differences between the mass and MoI contributions of the appendages. Despite contributing only approximately 20% of the total mass, the wing contributed approximately 85% of roll MoI (Ixx) and almost 50% of yaw MoI (Izz). This effect was particularly evident in the manus (hand wing), which constituted only 5–7% of the total mass, yet contributed 57–63% towards total roll MoI (Ixx) and 32–40% towards yaw MoI (Izz). Similarly, the head, legs and tail contributed up to 80% of total pitch MoI (Iyy) despite comprising no more than 30% of the total mass. In contrast, the wings contributed only 10% to pitch MoI (Iyy) as a result of their low mass and close proximity to the y-axis. About all axes, the contribution of the torso towards all MoI parameters was relatively small (<17%), with the exception of the peregrine falcon's pitch MoI (29% of Iyy). The peregrine falcon's torso contributed slightly more towards total mass (59%) than did the barn owl’s torso (54%). It also had a higher body length to wingspan ratio (0.41) compared with the barn owl (0.34).
Fig. 3C shows the contribution of the wings and body towards roll MoI (Ixx) as a function of normalised wing span (referred to body centreline), similar to plots produced from strip analysis (Berg and Rayner, 1995). Several features match strip analysis findings for a house sparrow (Passer domesticus), blackbird (Turdus merula) and house swift (Apus affinis) (Mohan et al., 1982; Berg and Rayner, 1995), including the ‘double-bell’ feature in the vicinity of the wrist joint and manus. The trough of the double-bell proximally of ±0.5 normalised span locates the intermetacarpal spatium (in the manus) while the peaks locate the wrist joint and the pila cranialis (Baumel et al., 1993). The significant asymmetry in pf4 corresponds to trauma in the right wing, where the locally increased MoI contribution may have been due to excess fluid from inflammatory response.
The principal components of inertia (Ixx, Iyy, Izz) are shown in Fig. 3D for the healthy weight barn owl (bo1) and peregrine falcon (pf4) cadavers with fully extended and retracted wings. Retracting the wings reduced roll (Ixx) and yaw (Izz) MoI by approximately 80% and 50%, respectively. In contrast, the pitch MoI and CoM were relatively unchanged between these configurations.
In this paper, we have described the generation of 3D mass distribution/inertial models for birds of prey using calibrated CT scanning, provided estimates for CoM and MoI including validation with several independent methods and described avian mass distribution using novel visualisations.
As a digital modelling approach, calibrated CT offers clear advantages over physical techniques (i.e. suspension, scales, primitive shapes and strips methods) for estimation of inertial properties in birds. Most of these traditional methods do not easily yield 3D CoM and MoI and require destructive dissection of cadavers that may be difficult to obtain. Calibrated CT is non-destructive and provides 3D inertial properties via datasets that can be manipulated as required (i.e. coordinate transformation, segmentation, thresholding, etc.).
The use of this approach clearly requires access to CT scanners, which are costly to purchase or hire. It is also important to remember that results can vary depending on the scanner, model, calibration, settings and scan date (Levi et al., 1982) and that they rely on a linear relationship between the scanner grey-value output and physical density. Validating data from this method is therefore good practice, not only as a check against these potential pitfalls but also as a check against human error in the experimental procedure. The extent of the validation process need not be as extensive as that provided here; we suggest a mass breakdown comparison with a single cadaver may suffice for future studies. Attention to detail is important; for example, providing a calibration for every scan, including air in the calibration curves, picking scanner settings to maximise contrast and carefully preparing the cadaver.
The animals used in this study, birds of prey, represent a challenging test case mainly because of the delicate nature of flight feathers. This suggests that the method ought to be applicable to a wide range of animals with similar or less challenging anatomy. We hope that this paper leads to widespread use of calibrated CT in biomechanics studies, and that the technique can be used to build a comprehensive database of mass/inertial properties across many taxa for the benefit of the whole research community. In support of this endeavour, it would be worth further testing of this approach, especially with different scanners and species from the scale of insects to larger mammals.
Thanks to RSPCA West Hatch Animal Centre, Gower Bird Hospital and The Barn Owl Trust for the provision of naturally deceased bird cadavers. We also thank Carys Davies (University of Bristol), James Shelton (University of Oxford), Dr Marianna Biggi and Prof. John Hutchinson (Royal Veterinary College) for their assistance with experimental preparation and CT scanning.
Conceptualization: N.E.D., S.P.W.; Methodology: N.E.D., Y.M., S.P.W.; Software: N.E.D., Y.M.; Validation: N.E.D., Y.M.; Formal analysis: N.E.D., Y.M., S.P.W.; Investigation: N.E.D., Y.M., S.P.W.; Resources: S.P.W.; Data curation: N.E.D.; Writing - original draft: N.E.D.; Writing - review & editing: Y.M., S.P.W.; Visualization: N.E.D., S.P.W.; Supervision: S.P.W.; Project administration: N.E.D., S.P.W.; Funding acquisition: S.P.W.
This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement no. 679355 to S.P.W.); and a Doctoral Training Grant (to N.E.D.) from the Engineering and Physical Sciences Research Council (EPSRC). Open access funding provided by University of Bristol. Deposited in PMC for immediate release.
The original images used in this study are available from the Dryad digital repository (Durston et al., 2021): https://doi.org/10.5061/dryad.xsj3tx9gp.
The authors declare no competing or financial interests.