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.

CT scanners measure the attenuation of X-rays through a measurement volume, from multiple orientations, to build a 3D contrast image of the object of interest (Goldman, 2007). The image consists of an array of voxels (3D pixels), each with an intensity dependent on local X-ray attenuation. The attenuation is a function of the atomic composition of the scanned materials and the distance travelled. Modern CT scanners use algorithms such as filtered back projection to provide a measure of the X-ray attenuation in each voxel in Hounsfield units (HU):(1)where μvoxel and μwater are the attenuation coefficients of the voxels representing the object of interest and of distilled water, respectively. For biological material, the relationship between the mean absolute density of the tissue in a given voxel and the X-ray attenuation is fairly linear (Mull, 1984). An approximate measure of absolute density, ρ (kg m−3) (referred to just as ‘density’ from here on) for biological materials can therefore be obtained from a CT image using:(2)

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.

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.

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’.

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 MoI of the pendulum, Ip, was estimated assuming it behaved as a horizontal, single degree of freedom dynamic system with negligible thread mass (1 mm steel cables did not yield accurate results), rotating about its CoM through small angles such that:(5)where m is the mass of the pendulum, g is gravitational acceleration, R is the distance between the threads and the centre of rotation, τ is the period of oscillation and L is the mean length of the threads (Korr and Hyer, 1962). The difference between the pendulum MoI with and without the bird provided an estimate of the bird's MoI about the axis of rotation (without correction for potential misalignment between the bird's CoM and the pendulum CoM). The mass of the bird, board, base and gyro was obtained using an electronic balance (LE341001P, Sartorius) with a resolution of 0.1 g. The angular velocity of the pendulum was sampled at 70 Hz using a smartphone gyroscope (Xperia Z1 Compact, Sony, Tokyo, Japan) and software application to collect the data (Sensor Kinetics Pro, Innoventions Inc., Houston, TX, USA). The curve fit for a damped harmonic oscillator was applied to the last 60 s of approximately 180 s of angular velocity data using Matlab (MATLAB 2017a), based on the following equations (Rao, 1995):(6)where x is angular displacement, x′ is angular velocity, t is time, ζ is damping coefficient, ωd and ωn are damped and undamped natural frequencies and X and ϕ are the amplitude and phase, respectively. The curve fit coefficient corresponding to the damped natural frequency was used to obtain the period of oscillation, τ, and was based on the mean of three repeated measurements in each case. To compare MoI estimates from the pendulum and calibrated CT data, an axis normal to the board and coincident with the bird's CoM (i.e. dorsoventral) was defined. This dorsoventral axis was aligned to the scanned bird by plane fitting to the scanned board to obtain the correct orientation, followed by translation of the origin to the bird's CoM. The pendulum orientation was matched to this transformation, except for the bird's CoM, which was unknown during its placement on the pendulum. Any misalignment between the bird and pendulum CoM was quantified based on the CT scan data, and was found to range from 8 to 31 mm. The dorsoventral MoI of the bird, accounting for bird CoM misalignment, was then calculated using (du Bois et al., 2009):(7)where Idv is the dorsoventral MoI of the bird, mb and mp are the mass of the bird and the pendulum without the bird, respectively, Ip is the MoI of the pendulum without the bird, τ is the time period of oscillation with the bird and D is the CoM offset between the bird and the pendulum without the bird (the remaining variables are as described for Eqn 5).

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.

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.

Fig. 2.

Validation of the CT approach. (A) Validation through mass breakdown for the physical versus virtual dissection, averaged for each species (barn owls, n=2; peregrine falcons, n=3; sparrow hawk, n=1). (B) Independent estimates of the dorsoventral moment of inertia (Idv) from the calibrated CT method and trifilar pendulum (corrected for centre of mass misalignment). Bird ID as in Table 1.

Fig. 2.

Validation of the CT approach. (A) Validation through mass breakdown for the physical versus virtual dissection, averaged for each species (barn owls, n=2; peregrine falcons, n=3; sparrow hawk, n=1). (B) Independent estimates of the dorsoventral moment of inertia (Idv) from the calibrated CT method and trifilar pendulum (corrected for centre of mass misalignment). Bird ID as in Table 1.

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).

Fig. 3.

Visualising avian inertial properties obtained from calibrated CT. (A) Projected normalised distribution of mass (top) and moment of inertia (MoI bottom) distribution in the peregrine falcon pf4. (B) Relative mass and MoI breakdown by individual cadaver (barn owl bo1, peregrine falcon pf4) from segmentation of the calibrated CT data. Ixx, roll MoI; Iyy, pitch MoI; Izz, yaw MoI. (C) Distribution of roll MoI versus normalised span for barn owl bo1, peregrine falcon pf4 and sparrow hawk sh1. (D) Principal components of inertia for the barn owl bo1 (top) and peregrine falcon pf4 (bottom) based on CT scan data for wings fully extended and retracted.

Fig. 3.

Visualising avian inertial properties obtained from calibrated CT. (A) Projected normalised distribution of mass (top) and moment of inertia (MoI bottom) distribution in the peregrine falcon pf4. (B) Relative mass and MoI breakdown by individual cadaver (barn owl bo1, peregrine falcon pf4) from segmentation of the calibrated CT data. Ixx, roll MoI; Iyy, pitch MoI; Izz, yaw MoI. (C) Distribution of roll MoI versus normalised span for barn owl bo1, peregrine falcon pf4 and sparrow hawk sh1. (D) Principal components of inertia for the barn owl bo1 (top) and peregrine falcon pf4 (bottom) based on CT scan data for wings fully extended and retracted.

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.