Tyrannosaurus rex
Creators: name: Farlow, James O.
name: Smith, Matt B.
name: Robinson, John M.
Publisher: name: Zenodo
Year: 1995
Subjects: Biodiversity
Taxonomy
Animalia
Chordata
Reptilia
Dinosauria
Tyrannosauridae
Tyrannosaurus
Tyrannosaurus rex

description: INTRODUCTION Large carnivorous dinosaurs like <i>Tyrannosaurus</i> were the biggest land-living predators in earth history (Far- low, 1993). Because body mass influences, or is at least correlated with, many features of the biology of ani­ mals (McMahon and Bonner, 1983; Peters, 1983; Cal­ der, 1984; Paul, 1988; Alexander, 1989; Dunham et al., 1989; Damuth and MacFadden, 1990; Spotila et al., 1991; Damuth, 1993), estimates ofthe body masses of the largest terrestrial predators are of some theo­ retical interest. One controversial aspect (of many!) of the biology of large theropods is the top speeds attained by these reptiles. Interpretations of the sprint speed of <i>Tyrannosaurus</i> range from as much as 18-20 meters/second (m/s) (Bakker, 1986; Paul, 1988) to as little as 7 m/s (Thulbom, 1982). In the present paper we report an estimate of the live mass of <i>Tyrannosaurus rex</i> based on a model res­ toration of a fairly complete skeleton (Museum of the Rockies [MOR] 555), along with an estimate of an “indicator of athletic ability” (Alexander, 1989:55), or more simply the “strength indicator,” of the femur of this specimen. We then consider the implications of our results for interpretations of the cursorial potential of this very large theropod. Finally, we discuss other matters relevant to interpreting the sprint speed of <i>Tyrannosaurus',</i> among these, particular attention is given to simple models that consider the risk of injury that this huge predator would face were it to fall while running. METHODS An account of the discovery and collection of MOR 555 was presented by Homer and Lessem (1993). This specimen is a representative of what is said to be the gracile morph of <i>Tyrannosaurus rex</i> (Horner and Les­ sem, 1993; cf. Carpenter, 1990; Molnar, 1991; Larson and Frey, 1992). One of us (<i>M</i>.B.S.) sculpted a model <i>(</i>1/20 scale) life restoration of the dinosaur, based on MOR 555 (Fig. 1); selected estimates of dimensions of the “prototype”, based on Smith’s model, are sum­ marized in Table 1. We determined the volume of this model by measuring the difference in the model’s weight in air and when submerged (cf. Alexander, 1985, 1989). Cavities in the model interior were filled with Duco cement, which was allowed to harden before measurements were made. Volume determinations were done at room temperature; measured water temperatures during the runs ranged from 18-21 °C. Three such model volume determinations were made. We checked the accuracy of our experimental tech­ nique by estimating the model’s volume in another way: by collecting water displaced from a container when the model was completely submerged. The mo­ del’s volume was estimated either by weighing the displaced water, or measuring its volume directly. Model volumes estimated by this water overflow method were a little higher, and much less precise (in the statistical sense), than those measured by determining the difference of the model’s weight in air and in water; this presumably reflects greater experimental error in measuring model mass by water overflow than by the method used by Alexander. We then calculated the volume of the full-sized, pro­ totype dinosaur by multiplying the model’s volume by the cube of the inverse of the scale of the model (i.e., by 203 = 8,000). Estimating the live mass of the pro­ totype from its volume depends on the specific gravity (SPGR) of the living dinosaur. This was probably comparable to that of water (Alexander, 1989), or a little less (Paul, 1988), depending on (among other things) the degree of pneumaticity of the animal’s bones. We made two estimates of the live mass of MOR 555, one (SPGR assumption 1) reflecting the assumption that the creature’s specific gravity was 1.00 (Alexander, 1989), and the second (SPGR assumption 2) that the dinosaur’s specific gravity was 0.85 (Paul, 1988). The mass estimate was then used to estimate Alexander’s (1989) strength indicator, a quantity that describes a long bone’s capacity for resisting bending stresses associated with vigorous activity. We measured the length, midshaft anteroposterior and mediolateral diameters, and midshaft circumference of the right femur of MOR 555 (Table 2). Unfortunately, a CT-scan of the midshaft of this bone (Fig. 2) indicated that it had been slightly crushed. Our estimate of the section modulus of this bone, and our estimate of the strength indicator of <i>Tyrannosaurus,</i> could be made only by using certain assumptions. We assumed that the before-crushing cross-sectional shape of the femoral midshaft was nearly circular (cf. Osborn, 1906:fig. 10), and estimated its external diameter as the average of the anteroposterior and mediolateral diameters of the bone as now preserved (Table 2; average =153 mm). We made several measurements of the femur’s cortical thickness from the CT-scan, and used the average of these (36 mm) for further calculations. We assumed that the pre-crushing marrow cavity was circular in shape, and estimated the diameter of the marrow cavity by subtracting twice the value of the average cortical thickness from our estimate of the external midshaft diameter. Because we assumed circular cross-sectional shapes for the entire femoral midshaft and for the marrow cavity of MOR 555, we calculated the section modulus for each of these as 0.78 x radius3 (Alexander, 1989). The section modulus of the marrow cavity was then subtracted from that of a solid bone (no marrow cavity) of external dimensions comparable to the femoral midshaft of MOR 555 (see Farlow, 1990:table 1) to obtain an overall section modulus estimate for the femoral midshaft. Like Alexander (1985, 1989) we assumed that the bending force acting on the femur acts halfway along the length of the femur. We therefore calculated the strength indicator for MOR 555 as (section modulus of femur midshaft) <i>÷</i> [(animal mass)(1/2 femur length) x (gravitational constant)] and multiplied the result by 109 newtons/giganewton to express the strength indicator in units of m2/giganewton. RESULTS As restored by Smith, <i>Tyrannosaurus rex</i> is very slimly built for its size. The dinosaur is quite deepbodied, and its torso extraordinarily narrow in mediolateral dimensions (Table 1; Fig. 1). Although reminiscent of Paul’s (1988:341) skeletal restoration of <i>T. rex</i> (based on American Museum of Natural History 5027 and Carnegie Museum of Natural History 9380), Smith’s restoration has a transversely narrower body trunk, and a relatively longer tail whose proximal end is narrower from top to bottom. Three body mass experimental determinations yielded estimates ranging 6,319-6,329 kg, with an average of 6,325 kg (SPGR assumption 1), or 5,371­ 5,380 kg, with an average of 5,376 kg (SPGR assumption 2). The section modulus obtained for the femur of MOR 555 is 0.00030 m3.4For* * *an* estimated body mass of 6,300 kg, the strength indicator for <i>Tyrannosaurus rex</i> is 7.5 m2/giganewton; for an estimated mass of 5,400 kg, the strength indicator is about 9 m2/giganewton. The assumption of circular shapes for the non-crushed femur midshaft exterior and marrow cavity does not greatly affect the values of the strength indicator. If we assume that the measured values ofthe anteroposterior and mediolateral midshaft diameters given in Table 2 are close to the real values, and that (as suggested by some of our CT-scan images) the marrow cavity was elliptical in shape (with an anteroposterior diameter of about 46 mm and a mediolateral diameter of about 92 mm), then the strength indicator for a 6,300-kg <i>Tyrannosaurus is</i> about 8 m2/giganewton (using the formula for the section modulus of an elliptical section from Alexander, 1983:table 1). The mass estimate was then used to estimate Alexander’s (1989) strength indicator, a quantity that describes a long bone’s capacity for resisting bending stresses associated with vigorous activity. We measured the length, midshaft anteroposterior and mediolateral diameters, and midshaft circumference of the right femur of MOR 555 (Table 2). Unfortunately, a CT-scan of the midshaft of this bone (Fig. 2) indicated that it had been slightly crushed. Our estimate of the section modulus of this bone, and our estimate of the strength indicator of <i>Tyrannosaurus,</i> could be made only by using certain assumptions. We assumed that the before-crushing cross-sectional shape of the femoral midshaft was nearly circular (cf. Osborn, 1906:fig. 10), and estimated its external diameter as the average of the anteroposterior and mediolateral diameters of the bone as now preserved (Table 2; average =153 mm). We made several measurements of the femur’s cortical thickness from the CT-scan, and used the average of these (36 mm) for further calculations. We assumed that the pre-crushing marrow cavity was circular in shape, and estimated the diameter of the marrow cavity by subtracting twice the value of the average cortical thickness from our estimate of the external midshaft diameter. Because we assumed circular cross-sectional shapes for the entire femoral midshaft and for the marrow cavity of MOR 555, we calculated the section modulus for each of these as 0.78 x radius3 (Alexander, 1989). The section modulus of the marrow cavity was then subtracted from that of a solid bone (no marrow cavity) of external dimensions comparable to the femoral midshaft of MOR 555 (see Farlow, 1990:table 1) to obtain an overall section modulus estimate for the femoral midshaft. Like Alexander (1985, 1989) we assumed that the bending force acting on the femur acts halfway along the length of the femur. We therefore calculated the strength indicator for MOR 555 asTotal length of the animal, snout to tailtip, measured along the curve of the dinosaur’s back—1,196 cmHead length— 162 cmNeck length —120 cmBody length from in front of the shoulder to the back of the hip—320 cmTail length—594 cmHeight of the top of the head above the ground—460 cmHeight of the lower edge of the lower jaw above the ground—346 cmHeight of the top of the hip above the ground—356 cmHeight of the lowest spot on the belly (at distal end of pubis) above the ground—146 cmMaximum dorso-ventral body depth—208 cmMaximum width across hips—96 cmMaximum chest width—98 cm (section modulus of femur midshaft) <i>÷</i> [(animal mass)(1/2 femur length) x (gravitational constant)] and multiplied the result by 109 newtons/giganewton to express the strength indicator in units of m2/giganewton. RESULTS As restored by Smith, <i>Tyrannosaurus rex</i> is very slimly built for its size. The dinosaur is quite deepbodied, and its torso extraordinarily narrow in mediolateral dimensions (Table 1; Fig. 1). Although reminiscent of Paul’s (1988:341) skeletal restoration of <i>T. rex</i> (based on American Museum of Natural History 5027 and Carnegie Museum of Natural History 9380), Smith’s restoration has a transversely narrower body trunk, and a relatively longer tail whose proximal end is narrower from top to bottom. Three body mass experimental determinations yielded estimates ranging 6,319-6,329 kg, with an average of 6,325 kg (SPGR assumption 1), or 5,371­ 5,380 kg, with an average of 5,376 kg (SPGR assumption 2). The section modulus obtained for the femur of MOR 555 is 0.00030 m3.4For* * *an* estimated body mass of 6,300 kg, the strength indicator for <i>Tyrannosaurus rex</i> is 7.5 m2/giganewton; for an estimated mass of 5,400 kg, the strength indicator is about 9 m2/giganewton. The assumption of circular shapes for the non-crushed femur midshaft exterior and marrow cavity does not greatly affect the values of the strength indicator. If we assume that the measured values ofthe anteroposterior and mediolateral midshaft diameters given in Table 2 are close to the real values, and that (as suggested by some of our CT-scan images) the marrow cavity was elliptical in shape (with an anteroposterior diameter of about 46 mm and a mediolateral diameter of about 92 mm), then the strength indicator for a 6,300-kg <i>Tyrannosaurus is</i> about 8 m2/giganewton (using the formula for the section modulus of an elliptical section from Alexander, 1983:table 1). DISCUSSION Body Mass of <i>Tyrannosaurus <i>rex</i> </i> Our estimate of the live mass of MOR 555 is roughly in the middle of previously published body mass estimates of <i>Tyrannosaurus rex</i> (Table 3). Some of the previously published estimates (e.g., Colbert, 1962; Alexander, 1985; Paul, 1988), like ours, have been based on measurements of the volumes of scale models. The accuracy of such determinations obviously depends on the quality of the model restoration. Potentially the greatest source of error in our estimate of the live mass of MOR 555 stems from inaccuracies in modeling the live animal (e.g., in reconstructing its musculature) that might result in departures from the 1/ <i>20</i> scale assumed for the linear dimensions of the model. These will be greatly magnified when the reciprocal of the scale is cubed to determine the volume of the prototype; only a little error in the model results in a great error in the volume of the fullsized animal. For example, if the overall scale of the model is actually closer to V18 than <i>V20,</i> the body mass estimate of the prototype drops to about 4,600 kg (SPGR assumption 1) or 3,900 kg (SPGR assumption 2). There is obviously no way of checking the extent to which this kind of error exists in our mass estimate for the living dinosaur, and we will assume that the scale of our model of the live animal really <i>is to­</i> Other estimates (Anderson et al., 1985; Campbell and Marcus, 1992) of the mass of <i>Tyrannosaurus</i> have been based on regressions of body mass on femoral midshaft circumferences of living mammals or birds. It is noteworthy that previously published body mass estimates for <i>T. rex</i> based on regression analyses tend to be lighter than those based on model volume determinations. The same result is obtained for MOR 555. If the mass of this dinosaur is estimated from the midshaft circumference of its femur (Table 2 —but recall that the femur is crushed, casting some doubt on our measurement of its circumference), the value obtained from the equation of Anderson et al. (1985) is 4,030 kg, and that obtained from the equation of Campbell and Marcus (1992) is 3,000 kg. Although we are uncertain why the equation of An- derson et al. (1985) yields a smaller mass estimate than the one obtained from our model (unless this reflects the small number of [perhaps questionable] data on which the regression equation is based), a plausible reason for the even lighter mass estimate obtained by Campbell and Marcus is readily apparent. The latter authors used a regression of live mass against femoral circumference in birds to estimate the mass of <i>Tyrannosaurus.</i> However, the ratio of femoral cross-section diameter to femoral length is much greater in birds than in non-avian theropods (Gatesy, 1991), presumably due to differences in the dynamics of locomotion in typical theropods as opposed to birds (Gatesy, 1990, 1991). For a given body mass, a bird probably has a much shorter and stouter femur than did a non-avian theropod, and this in turn may mean that using the avian relationship between femoral circumference and body mass will considerably underestimate the live mass of a non-avian theropod. On the assumption that our mass estimate of about 6,000 kg for MOR 555 is reasonably accurate, it is worth noting again that this specimen is a representative of what is said to be the gracile morph of <i>Tyrannosaurus rex</i> (Homer and Lessem, 1993). We would not be surprised if the robust morph were considerably heavier, with a body mass of 7,000 or even 8,000 kg. Furthermore, as noted by Paul (1988), the presently known specimens of <i>Tyrannosaurus</i> might well be average-sized adults, and not the very biggest individuals. <b>Strength Indicator</b> of <i>Tyrannosaurus <i>rex</i> </i> Alexander (1985, 1989) estimated the strength indicator of <i>Tyrannosaurus</i> to be about 9.0 m2/giganewton. Given that the model he used in his mass determination was not particularly good (Paul, 1990), that our mass estimate is based on a more complete specimen than any known at the time the model used by Alexander was constructed, and that Alexander’s model might not have been based entirely on the same specimen for which he obtained data on the femoral section modulus, we think that our data for estimating the strength indicator of <i>Tyrannosaurus rex</i> are better than Alexander’s. Even so, our estimate of the strength indicator of the animal (7.5-9.0 m2/giganewton) is remarkably close to Alexander’s. Alexander (1985, 1989) and Alexander and Pond (1992) calculated the strength indicator for a number of living animals, and found good agreement between values calculated from the femur and the known cursorial performance of those animals. Our estimate of the strength indicator of <i>Tyrannosaurus rex</i> is a little higher (as noted by Alexander) than that of an African elephant (7.0 m2/giganewton) and, again as noted by Alexander, much less than that of a human being (15 m2/giganewton), let alone runners like the ostrich (44 m2/giganewton—although differences in the way the ostrich carries its femur from the way <i>T. rex</i> presumably did may cause the ostrich’s very high strength indicator value to be inappropriate for comparison with that of a non-avian theropod), the white rhinoceros (see below), and the African buffalo (22 m2/giganewton). Although Farlow (1990) suspected that Alexander’s strength indicator might underestimate the cursorial potential of theropods, it seems unlikely, given our results, that the strength indicator underestimates the running ability of <i>Tyrannosaurus</i> to the point that this dinosaur could have attained speeds comparable to those ofthe fastest modem cursorial birds and mammals. A note of caution must be stated, however. As noted by Farlow (1990), the value of the strength indicator is greatly influenced by the body mass estimate. We believe our estimate of the live mass of MOR 555 to be the best presently available for <i>T. rex,</i> but it could nevertheless be inaccurate, as already discussed. This would in turn affect the accuracy of our estimate of the strength indicator. For example, if the true live mass of MOR 555 were about 4,000 kg, as suggested by the equation of Anderson et al. (1985), the strength indicator for the dinosaur becomes 12 m2/giganewton. Although much higher than the value obtained when using a mass estimate from Smith’s model, the magnitude of the increase is not enough greatly to affect our interpretation of the running ability of <i>Tyrannosaurus.</i> Even if our estimate of the strength indicator of MOR 555 is accurate, however, this nonetheless represents the value for a single individual of <i>Tyrannosaurus rex,</i> and it may be risky to generalize about the cursorial potential of the species on the basis of what amounts to a single data point (or two, given that Alexander’s estimate of the index was based on another individual of the species). Furthermore, our interpretations about the cursorial potential of <i>Tyrannosaurus</i> apply only to adult animals. It is not inconceivable that younger, smaller
description: Published as part of <i>James O. Farlow, Matt B. Smith &amp; John M. Robinson, 1995, Body Mass, Bone " Strength Indicator, " and Cursorial Potential of Tyrannosaurus rex, pp. 713-725 in Journal of Vertebrate Paleontology 15 (4)</i> on pages 713-723, DOI: 10.1080/02724634.1995.10011257, http://zenodo.org/record/3732361
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