estimating appropriate body density to be used with volumetric models for extinct cetaceans
Glarou et al. 2023 provided estimates of body density (figuring in lung volume) based on four lines of evidence:
- Using the mean density based on estimated volumes of three whales in Lockyer 1991 that had both length, mass and girth measured, allowing for estimation of their volume and hence their densities. 877.9 kg/m3
Length<-c(4.6,4.22,11.51)
Girth<-c(2.6,2.52,6.4)
Mass<-c(1.45,1.05,15.26)
Volume<-exp(-3.311+1.27*log(Length)+1.78*log(Girth))#estimated volume from TL and Girth
res<-data.frame(Length, Girth, Mass, Volume)
res$Density<-res$Mass/res$Volume
knitr::kable(res)
| Length | Girth | Mass | Volume | Density |
|---|---|---|---|---|
| 4.60 | 2.60 | 1.45 | 1.388053 | 1.0446289 |
| 4.22 | 2.52 | 1.05 | 1.176768 | 0.8922746 |
| 11.51 | 6.40 | 15.26 | 22.111438 | 0.6901406 |
(lengths in m, masses in metric tons, volumes in m³, densities in kg/l or t/m³)
- Based on hydrodynamic modelling results of Miller et al. 2004, using accelerometer measurements during diving and surfacing and inferred buoyancy properties. 1,002.8 kg/m3
- Based on the body composition and the respective densities of the main tissues. 886.7 kg/m3
- Based on a regression of density against volume from the three sperm whales in Lockyer 1991, resulting in the following linear model: Density=985.56-13.09*Volume. At the sizes of large adults, this produces unusually low density, some below 0.7:
Volume<-seq(10,100,10)#range of volumes in 10 m³ intervals
985.56-13.09*Volume
## [1] 854.66 723.76 592.86 461.96 331.06 200.16 69.26 -61.64 -192.54
## [10] -323.44
For application to Livyatan, we first need to produce a volumetric estimate that is comparable to the GDI estimates from Glarou et al., based on the silhouette of my size estimate based on Brygmophyseter:
Estimated volume of Livyatan based on pixel-precise mathematical slicing (GDI with elliptical cross-sections and individual segments that are 1px long, see here→) and estimated density based on model 4:
livyatan<-gdi("livd.txt","livl.txt")/1000
The resulting volume for Livyatan, at 53.9 m³, is slightly higher than the one based on my 3d model (50.8 m³)→. This is likely mostly explained by the model not having elliptical cross-sections throughout, but rather sub-elliptical ones in the tail, in order to account for the “keeled” morphology seen in Physeter. In fact I am getting a volume consistent with the GDI when constructing a 3D-model with cross-sections approximating an ellipse instead.
Indeed I suspect that deviations from a density of 1.0 kg/l may in fact be related to this factor, as they base on GDI-derived volumes that use elliptical cross-sections and are thus likely a bit inflated. Note that I think this has little impact on the accuracy of their mass estimates, as the inflation in volume would be counterbalanced by the decreased estimated density based on that volume. –––––––––———————————————————–
As for density, here is my own, similar approach (but based on a larger sample than just the three whales), herein termed model 5, which uses the formula from Glarou et al. for estimating body volume from length (assuming elliptical cross-sections);
L<-c(10:20)#range of lengths corresponding to adult sperm whales
V<-exp(-4.048+2.861*log(L))#estimated volume from TL
M<-0.0218*L^2.74#predict mass from TL
plot(M~V)
abline(lm(M~V))
summary(lm(M~V))
##
## Call:
## lm(formula = M ~ V)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.37550 -0.14726 0.08315 0.19873 0.25843
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.494696 0.152933 9.774 4.33e-06 ***
## V 0.856812 0.002953 290.182 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2474 on 9 degrees of freedom
## Multiple R-squared: 0.9999, Adjusted R-squared: 0.9999
## F-statistic: 8.421e+04 on 1 and 9 DF, p-value: < 2.2e-16
predict(lm(M~V), data.frame(V=livyatan))#mass prediction based on volume
## 1
## 47.71165
data.frame(L,V,M)->res2
res2$Density<-res2$M/res2$V#estimate densities over size range
lm.density<-lm(Density~V, data=res2)
summary(lm.density)
##
## Call:
## lm(formula = Density ~ V, data = res2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.007185 -0.005554 -0.001649 0.004288 0.012830
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 9.438e-01 4.431e-03 212.99 < 2e-16 ***
## V -9.090e-04 8.555e-05 -10.62 2.16e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.007168 on 9 degrees of freedom
## Multiple R-squared: 0.9262, Adjusted R-squared: 0.918
## F-statistic: 112.9 on 1 and 9 DF, p-value: 2.156e-06
predict(lm.density, data.frame(V=res2$V))->res2$density_lm_est
here are the resultant estimates of mass, volume and density over a range of body lengths corresponding to adult Physeter, using model 5:
| L | V | M | Density | density_lm_est |
|---|---|---|---|---|
| 10 | 12.67582 | 11.97999 | 0.9451061 | 0.9322764 |
| 11 | 16.64947 | 15.55509 | 0.9342692 | 0.9286643 |
| 12 | 21.35568 | 19.74300 | 0.9244845 | 0.9243863 |
| 13 | 26.85146 | 24.58449 | 0.9155739 | 0.9193906 |
| 14 | 33.19313 | 30.11947 | 0.9074006 | 0.9136260 |
| 15 | 40.43645 | 36.38702 | 0.8998570 | 0.9070418 |
| 16 | 48.63660 | 43.42554 | 0.8928573 | 0.8995878 |
| 17 | 57.84826 | 51.27274 | 0.8863316 | 0.8912143 |
| 18 | 68.12564 | 59.96573 | 0.8802227 | 0.8818721 |
| 19 | 79.52249 | 69.54107 | 0.8744830 | 0.8715123 |
| 20 | 92.09218 | 80.03477 | 0.8690723 | 0.8600863 |
As we can see, using the resultant linear model of density on volume, we get an estimated density of 0.895 for a volume of 53.9406055 m³. The model formula is approximately as follows: 0.9438+-9.09 × 10-4*Volume (m³).
Review of all methods
While a sound principle in theory, the predicted density from model 4 value does not appear plausible or even possible at all, given that it results in a density of less than 0.3 kg/l, and that even the least dense tissues in the body of a sperm whale, such as blubber, have densities far higher than that (≥0.94 kg/l for blubber according to Lockyer 1991), as do fats in general (which, at around 0.9, would make up the least dense non-gaseous substances in a sperm whale body):
985.56-13.09*livyatan
## [1] 279.4775
The resultant estimate at the body density of L. melvillei, 279 kg/m^3, is completely implausible and would imply a whale that would be unable to dive because two thirds of its body would, somehow, have to be made up of gas with a negligible density.
Getting such offputting results is not entirely surprising, since, as acknowledged by the authors, the model lacks statistical significance on account of being based on only three specimens. Based on this and because it produces increasingly implausible (right up to straight-up impossible) density values at higher body sizes (even going into the negative figures), I am inclined to dismiss the results from this model. Body densities of 0.7 or less (predicted over most of the size range of adult Physeter using this formula) are unheard of among animals for which reliable measurements exist, and are disputed even for highly pneumatic taxa, such as Sauropods (see Larramendi et al. 2021). There is clearly no mechanism by which an aquatic mammal, which has no such skeletal pneumaticity and whose least dense tissues are still considerably denser than this estimated overall value, could actually have such low overall body density, let alone half that. Hence, results from this approach are ignored for the remaining analysis.
Models 1 and 3, as well as my own model 5 produce density estimates of around 0.88-0.89 kg/l. While still a bit on the low end compared to what I would expect, and lower than the density of blubber, this appears more plausible as a lower-bound, and is also close to own estimate from model 5, which results in a formula giving (imo) much more reasonable density predictions. As already mentioned, I think this can at least partly be explained by the higher estimated volumes (compared to the real whale’s volume) due to the elliptical assumption inherent in the GDI. A modification of model one, exluding the specimen with an estimated density of less than 0.7 kg/l (which is likely impossible) and only using the two juvenile specimens that were weighed in one piece (therefore not requiring correction for fluid loss) gives a density of 0.968.
A density at or close to 1.0 (as estimated by Miller et al. 2004 and the model 2 in Glarou et al. based thereon) is probably still the most reasonable estimate of the real whale’s density, as acknowledged by Glarou et al. This is supported by this being the only estimate that is relatively close to neutral buoyancy in sea water, as well as the only estimate higher than the density of whale blubber. Hence, for simplicity I will stick with 1.0 kg/l for my “best estimate” for Livyatan based on the 3D model (which includes body cross-sections that are sub-elliptical, but will additionally test the effects of the values of 0.968 and 0.895 kg/l to provide alternative estimates.
Based on these, my estimate for the body mass of Livyatan ends up at 48.3 to 52.2 t. This nicely brackets my original estimate of 50.8 t based on the 3D model, and suggests that the discrepancy between the ~1.0 density (also resulting from Miller et al. 2004 when correcting for lung volume as per Glarou et al. 2023) and the >0.9 density based on comparing estimated volumes to estimated masses might more or less go away when accounting for the volume difference between a GDI and a realistically-shaped whale.
References
Glarou, M., Gero, S., Frantzis, A., Brotons, J.M., Vivier, F., Alexiadou, P., Cerdà, M., Pirotta, E. and Christiansen, F. 2023. Estimating body mass of sperm whales from aerial photographs. Marine Mammal Science 39 (1): 251–273. Lockyer, C. 1991. Body composition of the sperm whale, Physeter catodon, with special reference to the possible functions of fat depots. Rit Fiskideilda 12 (2): 1–24. Miller, P.J.O., Johnson, M.P., Tyack, P.L. and Terray, E.A. 2004. Swimming gaits, passive drag and buoyancy of diving sperm whales Physeter macrocephalus. Journal of Experimental Biology 207 (11): 1953–1967. Larramendi, A., Paul, G.S. and Hsu, S. 2021. A review and reappraisal of the specific gravities of present and past multicellular organisms, with an emphasis on tetrapods. The Anatomical Record 304 (9): 1833–1888. Larramendi, A., Paul, G.S. and Hsu, S. 2021. A review and reappraisal of the specific gravities of present and past multicellular organisms, with an emphasis on tetrapods. The Anatomical Record 304 (9): 1833–1888.