Duration of urination does not change with body size
Patricia J. Yanga, Jonathan Phama, Jerome Chooa, and David L. Hua,b,1
Schools of aMechanical Engineering and bBiology, Georgia Institute of Technology, Atlanta, GA 30332
Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved May 14, 2014 (received for review February 6, 2014)
Many urological studies rely on models of animals, such as rats and
pigs, but their relation to the human urinary system is poorly
understood. Here, we elucidate the hydrodynamics of urination
across five orders of magnitude in body mass. Using high-speed
videography and flow-rate measurement obtained at Zoo Atlanta,
we discover that all mammals above 3 kg in weight empty their
bladders over nearly constant duration of 21 ± 13 s. This feat is
possible, because larger animals have longer urethras and thus,
higher gravitational force and higher flow speed. Smaller mammals are challenged during urination by high viscous and capillary
forces that limit their urine to single drops. Our findings reveal
that the urethra is a flow-enhancing device, enabling the urinary
system to be scaled up by a factor of 3,600 in volume without
compromising its function. This study may help to diagnose urinary problems in animals as well as inspire the design of scalable
hydrodynamic systems based on those in nature.
urology
| allometry | scaling | Bernoulli’s principle
M
edical and veterinary urology often relies on simple, noninvasive methods to characterize the health of the urinary
system (1, 2). One of the most easily measured characteristics of
the urinary system is its flow rate (3), changes in which may be
used to diagnose urinary problems. The expanding prostates of
aging males may constrict the urethra, decreasing urine flow
rate (4). Obesity may increase abdominal pressure, causing
incontinence (5). Studies of these ailments and others often
involve animal subjects of a range of sizes (6). A study of
urination in zero gravity involved a rat suspended on two legs for
long periods of time (7), whereas other studies involve mice (8),
dogs (1), and pigs (9). Despite the wide range of animals used in
urological studies, the consequences of body size on urination
remain poorly understood.
The bladder serves a number of functions, as reviewed by
Bentley (10). In desert animals, the bladder stores water to be
retrieved at a time of need. In mammals, the bladder acts as
a waterproof reservoir to be emptied at a time of convenience.
This control of urine enables animals to keep their homes sanitary
and themselves inconspicuous to predators. Stored urine may even
be used in defense, which one knows from handling rodents
and pets.
Several misconceptions in urology have important repercussions in the interpretation of healthy bladder function. For instance, several investigators state that urinary flow is driven
entirely by bladder pressure. Consequently, their modeling of the
bladder neglects gravitational forces (11–13). Others, such as
Martin and Hillman (14), contend that urinary flow is driven by
a combination of both gravity and bladder pressure. In this study,
we elucidate the hydrodynamics of urination across animal size,
showing the effects of gravity increase with increasing body size.
generate jets. Instead, they urinate using a series of drops, which
is shown by the 0.03-kg lesser dog-faced fruit bat and the 0.3-kg
rat in Fig. 2 A–C, respectively.
Fig. 1H shows the urination time for 32 animals across six
orders of magnitude of body mass from 0.03 to 8,000 kg. Despite this wide range in mass, urination time remains constant,
T = 21 ± 13 s (n = 32), for all animals heavier than 3 kg. This
invariance is noteworthy, considering that an elephant’s bladder, at 18 L, is nearly 3,600 times larger in volume than a cat’s
bladder at 5 mL. Using the method of least squares, we fit the
data to a clear scaling law shown by the dashed line: T ∼ M0.13
(Fig. 1H).
For small animals, urination is a high-speed event of 0.01- to
2-s duration and therefore, quite different from the behavior of
the large animals observed that urinate for 21 s. Fig. 1H shows
urination time across 11 small animals, including one bat, five
rats, and five mice. Their body masses ranged from 0.03 to 0.3 kg.
The large error bar for the rats is caused by bladder fullness
varying across individuals. Fig. 2D shows the time course of the
urine drop’s radius measured by image analysis of high-speed
video of a rat. To rationalize the striking differences between
large and small animals, we turn to mathematical modeling of
the urinary system.
Modeling Assumptions. Urination may be simply described mathematically. Fig. 1E shows a schematic of the urinary system,
consisting of a bladder of volume V and the urethra, which is
assumed to be a straight vertical pipe of length L and diameter
D. We assume that the urethra has such a thin wall that its internal and external diameters are equal. Urination begins when
the smooth muscles of the bladder pressurize the urine to Pbladder,
measured relative to atmospheric pressure. After an initial transient of duration that depends on the system size, a steady flow of
speed u is generated.
Previous medical and veterinary studies, particularly cystometrography and ultrasonography, report substantial data on the
anatomy, pressure, and flow rate of the urinary system. Fig. 3
shows urethral length (8, 15–25) and diameter (15, 24–34), flow
rate (35–42), bladder capacity (25, 43–49), and bladder pressure
(1, 35, 39, 40, 43, 46, 50) for over 100 individuals across 13 species.
Significance
Animals eject fluids for waste elimination, communication, and
defense from predators. These diverse systems all rely on the
fundamental principles of fluid mechanics, which we use to
predict urination duration across a wide range of mammals. In
this study, we report a mathematical model that clarifies misconceptions in urology and unifies the results from 41 independent urological and anatomical studies. The theoretical
framework presented may be extended to study fluid ejection
from animals, a universal phenomenon that has received little
attention.
Results
In Vivo Experiments. We filmed the urination of 16 animals and
obtained 28 videos of urination from YouTube, listed in SI Appendix. Movies S1–S4 show that urination style is strongly dependent on animal size. Here, we define an animal as large if it is
heavier than 3 kg and an animal as small if it is lighter than 1 kg.
Large animals, from dogs to elephants, produce jets and sheets
of urine, which are shown in Fig. 1 A–D. Small animals, including
rodents, bats, and juveniles of many mammalian species, cannot
11932–11937 | PNAS | August 19, 2014 | vol. 111 | no. 33
Author contributions: P.J.Y. and D.L.H. designed research; J.P. and J.C. performed research; P.J.Y. and D.L.H. analyzed data; and P.J.Y. and D.L.H. wrote the paper.
The authors declare no conflict of interest.
This article is a PNAS Direct Submission.
1
To whom correspondence should be addressed. Email: hu@me.gatech.edu.
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.
1073/pnas.1402289111/-/DCSupplemental.
www.pnas.org/cgi/doi/10.1073/pnas.1402289111
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Fig. 1. Jetting urination by large animals, including (A) elephant, (B) cow, (C) goat, and (D) dog. Inset of cow is reprinted from the public domain and cited in
SI Appendix. (E) Schematic of the urinary system. (F) Ultrasound image of the bladder and urethra of a female human. The straight arrow indicates the
urethra, and the curved arrow indicates the bladder. Reproduced with permission from ref. 20, (Copyright 2005, Radiological Society of North America). (G)
Transverse histological sections of the urethra from a female pig. Reproduced with permission from ref. 9, (Copyright 2001, Elsevier). (H) The relationship
between body mass and urination time.
Table 1 shows the corresponding allometric relationships to be
used in numerical predictions for flow rate and urination time.
We begin by showing that the urinary system is isometric (i.e.,
it has constant proportions across animal size). Fig. 3A shows
the relation between body mass M and urethral dimensions
(length L and diameter D). As shown by the nearly parallel
trends for L and D (L = 35M0.43 and D = 2M0.39), the aspect
ratio of the urethra is 18. Moreover, the exponents are close to
the expected isometric scaling of M1/3. Fig. 3B shows the relationship between body mass and bladder capacity. The bladder’s
capacity is V ∼ M0.97, and the exponent of near unity indicates
isometry.
In ultrasonic imaging (Fig. 1F), the urethra appears circular
(20). However, in histology (Fig. 1G), the urethra is clearly
corrugated, which decreases its cross-sectional area (9). The
presence of such corrugation has been verified in studies in
which flow is driven through the urethra (51, 52), although the
precise shape has been too difficult to measure. We proceed
by using image analysis to measure cross-sectional area A
from urethral histological diagrams of dead animals in the
absence of flow (9, 53, 54). We define a shape factor α = 4A/
πD2, which relates the urethral cross-sectional area with respect to that of a cylinder of diameter D. Fig. 3C shows the
shape factor α = 0.2 ± 0.05 (n = 5) for which the corrected
cross-sectional area is 20% of the original area considered.
Yang et al.
This shape factor is nearly constant across species and body
mass and consistent with the value of 0.17 found by Wheeler
et al. (55).
Peak bladder pressure is difficult to measure in vivo, and instead, it is estimated using pressure transducers placed within the
bladders of anesthetized animals. Pressure is measured when the
bladder is filled to capacity by the injection of fluid. This technique yields a nearly constant bladder pressure across animal
size: Pbladder = 5.2 ± 0.86 kPa (n = 8), which is shown in Fig. 3D.
The constancy of bladder pressure at 5.2 kPa is consistent with
other systems in the body. One prominent example is the respiratory system, which generates pressures of 10 kPa for animals
spanning from a mosquito to an elephant (56).
Steady-State Equation of Urine Flow. We model the flow as steady
state and the urine as an incompressible fluid of density ρ,
viscosity μ, and surface tension σ. The energy equation relates the pressures involved, each of which has units of energy
per cross-sectional area of the urethra per unit length down
the urethra:
Pbladder + Pgravity = Pinertia + Pviscosity + Pcapillary :
[1]
Each term in Eq. 1 may be written simply by considering the
pressure difference between the entrance and exit of the
PNAS | August 19, 2014 | vol. 111 | no. 33 | 11933
Fig. 2. Dripping urination by small animals. (A) A
rat’s excreted urine drop. (B) A urine drop released
by the lesser dog-faced fruit bat Cynopterus brachyotis. Courtesy of Kenny Breuer and Sharon
Swartz. (C) A rat’s urine drop grows with time. Inset
is reprinted from the public domain and cited in SI
Appendix. (D) Time course of the drop radii of the
rat (triangles) along with prediction from our model
(blue dotted line, α = 0.5; green solid line, α = 1; red
dashed line, α = 0.2).
urethra. The combination of bladder and hydrostatic pressure
drives urine flow. Bladder pressure Pbladder is a constant given
in Fig. 3D. We do not model the time-varying height in the
bladder, because bladders vary greatly in shape (57). Thus,
hydrostatic pressure scales with urethral length: Pgravity ∼ ρgL,
where g is the acceleration of gravity. Dynamic pressure Pinertia
scales as ρu2/2 and is associated with the inertia of the flow.
The viscous pressure drop in a long cylindrical pipe is given by
the Darcy–Weisbach
equation (58): Pviscosity = fD(Re)ρLu2/2αD.
pffiffiffi
We use αD as the effective diameter of the pipe to keep the crosssectional area of the pipe consistent with experiments. The
Darcy friction factor fD is a function of the Reynolds number
Re = ρuD/μ, such that fD(Re) = 64/Re for laminar flow and
5
fD(Re) = 0.316/Re1/4 for turbulent flow (104 < Re
pffiffiffi< 10 ). Drops
generated from an orifice of effective
diameter
α
D
experience
pffiffiffi
a capillary force (59) of Pcapillary = 4σ= αD. Substituting these terms
into Eq. 1, we arrive at
ρu2
ρLu2
4σ
+ fD ðReÞ
+ pffiffiffi :
Pbladder + ρgL =
2
2αD
αD
[2]
The relative magnitudes of the five pressures enumerated in
Eq. 2 are prescribed by six dimensionless groups, including the
aforementioned Reynolds number
pffiffiffiffiffiffiand Darcy friction factor
and well-known Froude Fr = u= gL and Bond Bo = ρgD2/σ
numbers (60) as well as dimensionless groups pertaining to the
urinary system, the urethra aspect ratio As = D/L, and pressure
ratio Pb = Pbladder/ρgL. Using these definitions, we nondimensionalize Eq. 2 to arrive at
Pb + 1 =
Fr2
Fr2
4As
+ fD ðReÞ
+ pffiffiffi :
2
2αAs
αBo
[3]
In the following sections, we solve Eq. 3 in the limits of large and
small animals.
In SI Appendix, we apply a variation of the Washburn law
(61) to show that the steady-state model given in Eq. 2 is
accurate for most animals. Animals lighter than 100 kg achieve 90% of their flow velocity within 4 s; however, for animals
such as elephants, the transient phase can be substantial. For
11934 | www.pnas.org/cgi/doi/10.1073/pnas.1402289111
our derivations here, however, we assume that the transient
phase is negligible.
Large Animals Urinate for Constant Duration. Bladder pressure,
gravity, and inertia are dominant for large animals, which can be
shown by consideration of the dimensionless groups in SI Appendix. Eq. 2 reduces to
Pbladder + ρgL =
ρu2
:
2
[4]
The urination time T, the time to completely empty the bladder,
may be written as the ratio of bladder capacity to time-averaged
flow rate, T = V/Q. We define the flow rate as Q = uA, where A =
απD2/4 is the cross-sectional area of the urethra. Using Eq. 4 to
substitute for flow speed yields
T=
απD2
4V
2Pbladder
ρ
1=2 :
+ 2gL
[5]
By isometry, bladder capacity V ∼ M and urethral length and
diameter both scale with M1/3; substitution of these scalings into
Eq. 5 yields urination time T ∼ M1/6 ≈ M0.16 in the limit of increasing body mass. Agreement between predicted and measured scaling exponents is very good (0.13 compared with 0.16).
We, thus, conclude that our scaling has captured the observed
invariance in urination time.
We go beyond a simple scaling by substituting the measured
allometric relationships from Table 1 for L, D, α, V, and Pbladder
into Eq. 5, yielding a numerical prediction for urination time.
This prediction (Fig. 1H, solid line) is shown compared with experimental values (Fig. 1H, dashed line). The general trend is
captured quite well. We note that numerical values are underpredicted by a factor of three across animal masses, likely because
of the angle and cross-section of the urethra in vivo.
How can an elephant empty its bladder as quickly as a cat?
Larger animals have longer urethras and therefore, greater hydrostatic pressure driving flow. Greater pressures lead to higher
flow rates, enabling the substantial bladders of larger animals to
be emptied in the same duration as those of their much smaller
counterparts.
Yang et al.
10000
100000
A
10000
1000
1
B
1000
100
0.1
100
10
10
1
1
0.1
0.01
100
C
0.1
1
10
100
0.1
0.01
1000 10000
10000
D
0.1
1
10
100
1000 10000
0.01
0.01
10000
E
1000
1000
100
100
10
10
1
1
0.1
1
10
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1000
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1000
10000
F
0.1
1
10
100
1000
0.1
0.1
1
10
100
1000
10000
0.1
0.1
Fig. 3. The relation between body mass M and properties of the urinary system. (A) Urethral length L (green triangles) and diameter D (blue circles). (B)
Bladder capacity V. (C) Shape factor α associated with the urethral cross-section. (D) Bladder pressure Pbladder. (E) Flow rate of males. (F) Flow rate of females.
Symbols represent experimental measurements, dashed lines represent best fits to the data, and solid lines represent predictions from our model.
Our model provides a consistent physical picture on consideration of flow rate. Combining Eq. 4 and the definition of flow
rate (Q = uA) yields
Q=
1=2
απD2 2Pbladder
+ 2gL
:
4
ρ
[6]
Our model gives insight into the distinct flow-rate scalings
observed for both male and female mammals. Male mammals
generally stand on four legs and have a penis that extends
downward, enabling them to urinate vertically. Assuming isometry
(D ∼ M1/3 and L ∼ M1/3), flow rate scales as Q ∼ M5/6 ≈ M0.83
in the limit of large body mass. This predicted exponent is within 10% of the observed scaling for males: QM ∼ M 0.92. By substituting the allometric relations from Table 1 into Eq. 6, we
compute a numerical prediction for flow rate (Fig. 3E, solid line)
that is five times higher than experimental flow rates (Fig. 3E,
dashed line). This overprediction is roughly consistent with our
underprediction for urination time.
Female mammals have anatomy such that the urethral exit is
near the anus: thus, many female animals urinate horizontally.
The scaling of Eq. 6 without the gravitational term is Q ∼ M2/3 ≈
M0.67, and the exponent is in correspondence to that found in our
experiments for females: QF ∼ M0.66. Substituting allometric
relations from Table 1 yields a numerical prediction (Fig. 3F,
solid line) that remains in good agreement with experiments.
Small Animals Urinate Quickly and for Constant Duration. Bladder
pressure, viscous pressure, and capillary pressure are dominant
for small animals, which is shown by the associated dimensionless groups in SI Appendix. In this limit, Eq. 2 reduces to
Pbladder =
ρu2 32μLu
4σ
+
+ pffiffiffi ;
αD2
2
αD
[7]
which we solve numerically for flow speed u. To predict the flow
speed of a rat, inputs to this equation include the rat’s bladder
Yang et al.
pressures and urethral anatomy (15, 16, 50) (Pbladder = 6.03 kPa,
L = 20 mm, D = 0.8 mm).
To determine urination time, we turn to the dynamics of drop
filling. A spherical drop is filled by the influx of urine, Q = απD2u/4.
By conservation of mass, dVdrop/dt = Q, a first-order differential
equation that may be easily integrated to obtain the drop volume
V(t). We assume that the initial drop
pffiffifficorresponds to a sphere of the
same diameter as the urethra, αD. Thus, the radius of the
spherical urine drop may be written
!1=3
3
α2 D3 3αD2 ut
RðtÞ =
:
[8]
+
16
8
Combining Eq. 8 and the numerical solution for Eq. 7, we compute the time course of the drop radius. This prediction is compared with experimental values in Fig. 2D. We find that the
prediction is highly sensitive to the value of α. Without consideration of the corrugated cross-section, a prediction of α = 1
(Fig. 2D, green solid line) yields a flow rate that is too high.
Using the shape factor α = 0.2 (Fig. 2D, red dashed line), our
model predicts a flow speed of u = 1.2 m/s, which fits the data
fairly well. Using nonlinear least-squares fitting in Matlab, the
best fit to the experimental data yields an intermediate value of
α = 0.5 (Fig. 2D, blue dotted line).
The drop does not grow without limit but falls when its gravitational force, scaling as 4πR3f ρg=3, overcomes
its attaching cappffiffiffi
illary force to the urethra, scaling as π αDσ. Equating these two
forces yields the final drop radius before detachment,
pffiffiffi 1=3
3σ αD
;
Rf =
4ρg
[9]
which does a fair job of predicting the drop size. We predict drop
radii for rats and mice of 1.3 and 1.1 mm, respectively, which are
two times as large as experimental values of 2.0 ± 0.1 (n = 5) and
PNAS | August 19, 2014 | vol. 111 | no. 33 | 11935
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0.01
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10
Table 1. Measured allometric relationships for the urinary
system of animals
Variable
Duration of urination
Urethral length
Urethral diameter
Shape factor
Bladder capacity
Bladder pressure
Flow rate for females
Flow rate for males
T
L
D
α
V
Pbladder
QF
QM
Unit
Best fit
s
mm
mm
—
mL
kPa
mL/s
mL/s
0.13
8.2
35
2.0
0.2
4.6
5.2
1.8
0.3
M
M0.43
M0.39
M−0.05
M0.97
M−0.01
M0.66
M0.92
R2
N
0.2
0.9
0.9
0.5
0.9
0.02
0.9
0.9
32
47
22
5
9
8
16
15
Body mass M given in kilograms. Duration of urination corresponds to
animals heavier than 3 kg. Urethral length and diameter, shape factor, bladder capacity, bladder pressure, and flow rates correspond to animals heavier
than 0.02 kg.
2.2 ± 0.4 mm (n = 5), respectively. We suspect this difference is
caused by our underestimation of urethral perimeter at the exit.
For such a large drop to remain attached, we require the attachment diameter to be larger by a factor of two, which is quite
possible, because the urethral exit is elliptical.
Substituting Eq. 9 into Eq. 8, the time to eject one drop may
be written
!
pffiffiffi
16 αD 3 cos θ 1
4D cos θ 1
Tdrop =
≈ pffiffiffi
:
[10]
−
3
3u
8
αBo u
2
4α Bo
The predictions of maximum drop size and time to fall are in
excellent correspondence with observed values for rats and mice.
Using Eq. 10, we predict drop falling times of 0.05 and 0.15 s for rats
and mice, respectively, which are nearly identical to experimental
values of 0.06 ± 0.05 (n = 5) and 0.14 ± 0.1 s (n = 14), respectively.
A scaling for urine duration for small animals is not straightforward because of the nonlinearity of Eq. 7. We conduct a scaling
analysis in the limit of decreasing animal size for which the Reynolds number approaches zero. Because of isometry, V ∼ M and D ∼
M1/3. Rewriting Eq. 7 at low Reynolds number, we have u ∼ D,
and therefore, the time to eject one drop from Eq. 10 scales as
Tdrop ∼ Bo−1 ∼ M−2/3. Using Eq. 9, the final drop size is Rf ∼ D ∼ M1/3.
By conservation of mass, a full bladder of volume V can produce
N spherical drops, where N = 3V =4πR3f ∼ M 2=3 . Thus, the urination time for small animals T = NTdrop is constant and therefore, independent of animal size. This prediction indicates
that small animals urinate for different durations than large
animals, which is in correspondence with experiments. Our
experiments indicate that mammals of mass 0.03–0.3 kg urinate for durations of 0.1–2 s. We have insufficient range of
masses for small animals to conclude our prediction that
urination time is constant in this regime.
The model yields insight into the challenges facedpby
ffiffiffi small
animals. In Eq. 7, flow speed is positive only if Pbladder αD ≥ 4σ,
where σ is the surface tension of urine, which for humans is comparable with the surface tension of water (62). Thus, we predict
thatpthe
ffiffiffi smallest urethra to expel urine has a diameter of
4σ= αPbladder ∼ 0:1 mm. According to our allometric trends, the
smallest animal that can urinate independently corresponds to
a body mass of 0.8 g and urethral length of 1.7 mm. This mass
corresponds to that of altricial mice (0.5–3 g), which are dependent on their mother’s licking of excreted urine drops (63).
urethra. In the medical literature, the function of the urethra was
previously unknown. It was simply defined as a conduit between
bladder and genitals. In this study, we find that the urethra is
analogous to Pascal’s Barrel: by providing a water-tight pipe to
direct urine downward, the urethra increases the gravitational
force acting on urine and therefore, the rate at which urine is
expelled from the body. Thus, the urethra is critical to the
bladder’s ability to empty quickly as the system is scaled up.
Engineers may apply this result to design a system of pipes and
reservoirs for which the drainage time does not depend on system
size. This concept of a scalable hydrodynamic system may be used
in the design of portable reservoirs, such as water towers, water
backpacks, and storage tanks.
Why is urination time 21 s, and why is this time constant across
animal sizes? The numerical value of 21 s arises from the underlying physics involving the physical properties of urine as well
as the dimensions of the urinary system. Our model shows that
differences in bladder capacity are offset by differences in flow
rate, resulting in a bladder emptying time that does not change
with system size. Such invariance has been observed in a number
of other systems. Examples include the height of a jump (64) and
the number of heartbeats in a lifetime (65). Many of these examples
arise from some aspect of isometry, such as with our system.
From a biological perspective, the invariance of urination time
suggests its low functional significance. Because bladder volume
is 4.6 mL/kg body mass and daily urine voided is 26 mL/kg body
mass (66), mammals urinate 5.6 times/d. Because the time to
urinate once is 21 s, the daily urination time is 2 min, which can
be translated to 0.2% of an animal’s day, a negligible portion
compared with other daily activities, such as eating and sleeping,
for which most animals take care to avoid predation. Thus, urination time likely does not influence animal fitness. The geometry
of the urethra, however, may play a role in other activities of
high functional significance, such as ejaculation.
In our study, we found that urination time is highly sensitive
to urethral cross-section. This dependency is particularly high for
small animals for which urine flow is resisted by capillary and
viscous forces, which scale with the perimeter of the urethra. More
accurate predictions for small animals require measurements of
the urethral exit perimeter and the urethral cross-section, which is
known to vary with distance down the urethra (67). Current
models of noncircular pipe flow are not applicable, because they
only account for infinitesimal corrugations (68). Additional mathematical techniques as well as accurate urethral measurements
are needed to increase correspondence with experiments.
Materials and Methods
We filmed urination of animals using both Sony HDR-XR200 and high-speed
cameras (Vision Research v210 and Miro 4). The masses of animals are taken
from annual veterinary procedures or measured using an analytical balance.
Flow rate Q is measured by simultaneous high-speed videography and
manual urine collection using containers of appropriate size and shape. We
use the open-source software Tracker to measure the time course of the
radius of urine drops produced by rats and mice.
Discussion
The urinary system works effectively across a range of length scales.
This robustness is caused by the hydrodynamic contribution of the
ACKNOWLEDGMENTS. We acknowledge photographer C. Hobbs and our
hosts at Zoo Atlanta (R. Snyder), the University of Georgia (L. Elly), the
Atlanta Humane Society (A. Lopez), and the animal facilities at Georgia Tech
(L. O’Farrell). We thank YouTube contributors, including Alex Cobb, Cole
Onyx, demondragon115, drakar2835, ElMachoPrieto83, Ilze Darguze, Joe
BERGMANN, Joey Ponticello, krazyboy35, laupuihang, MegaTobi89, Mixetc,
mpwhat, MrTitanReign, relacsed, RGarrido121, ronshausen63, Sandro Puelles,
Silvia Lugli, and Tom Holloway. Our funding sources were National Science
Foundation Faculty Early Career Development Program (Division of Physics)
Grant 1255127 for the modeling and Georgia Tech President’s Undergraduate
Research Awards for the experiments.
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Uroflowmetry, filling cystometry, and pressure-flow studies. Neurourol Urodyn 21(3):
261–274.
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Yang et al.
Yang et al.
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Supporting Videos
Video
Video
Video
Video
S1.
S2.
S3.
S4.
Urination
Urination
Urination
Urination
of
of
of
of
a rat, mass of 0.24 kg. Time slowed by 33 X.
a goat, mass of 70 kg. Time slowed by 17 X.
a cow, mass of 640 kg. Time slowed by 33 X.
an elephant, mass of 3540 kg. Time slowed by 33 X.
Appendix
Unsteady hydrodynamic urination model for large and small animals
In this section, we estimate the time for the urine flow to achieve steady state. The column of urine
descends due to gravitational and bladder pressure forces. This descent is resisted by viscosity,
fluid inertia and capillary pressure.
At time t = 0 the urethra is empty. We parametrize the height of urine in the urethra with a
height z(t), measured from the bladder. Consider Figure 1E in the main text. We consider a control volume including the mass of urine with the bladder and urethra. Conservation of momentum
for this control volume may be written
(m + ma ) z̈ = A (Pbladder + Pgravity − Pinertia − Pviscosity − Pcapillary ) .
(1)
Each terms in Equation (1) has units of force and may be written simply. The mass of urine in the
urethra is m = ρπαD2 z/4. Using techniques presented by Bush∗ , the added mass associated with
acceleration of fluid in the bladder may be written ma = 7ρπα3/2 D3 /48 . Bladder pressure Pbladder
is constant. Hydrostatic pressure Pgravity scales as ρgz where g is the gravitational acceleration.
Dynamic pressure Pinertia scales as ρż 2 /2, and is associated with inertia of the flow. Assuming
laminar flow, the pressure drop due to viscosity in a long cylindrical pipe is Pviscosity = 32µz ż/αD2 .
√
The capillary force of drops generated from an orifice of effective diameter αD is Pcapillary =
√
4σ/ αD. Substituting these terms into Equation (1), we arrive at
7√
ż 2 32µżz
4σ
Pbladder
z̈ z +
+ gz −
−
− √
αD =
(2)
2
12
ρ
2
ραD
ρ αD
We compute velocity ż in Equation (2) using a Runge-Kutta single-step solver (ode45) in Matlab.
Inputs to this equation include allometric relationships of bladder pressure, urethral diameter, and
shape factor, given in Table 1. Figure S1 shows the time course of urine velocity. For animal
lighter than 100 kg, the flow reaches 90% of its final velocity in 4 seconds, which is 15% of the the
time to empty the bladder, 21 seconds. We thus conclude our steady state model reported in the
main text is accurate for animals lighter than 100 kg. For larger animals such as elephants, the
transition can be substantial.
∗
Bush JWM (2010) 18.357 Interfacial Phenomena, Fall 2010. Avaliable at http://ocw.mit.edu Accessed 2 May,
2014.
14
M = 1 kg
12
M = 10 kg
M = 100 kg
Velocity (m/s)
10
M = 200 kg
8
6
4
2
0
0
1
2
3
4
5
Time (s)
Figure S1. Time course of urine velocity.
100000
Fr
Bo
Re
As
Pb
Values of Dimensionless Groups
10000
1000
100
10
1
0.1
0.01
0.1
1
10
100
1000
10000
Mass (kg)
Figure S2. Values of dimensionless groups including the Froude Fr, Bond Bo, Reynolds Re, aspect ratio
As, and ratio of bladder and hydrostatic pressure Pb.
Table S1. Duration of urination.
Animal
Sex
Mass (kg)
Duration (s)
Source
Mouse (N=5)
F
0.03
1.50
Experiment at Georgia Tech
Bat
F
0.03
0.32
K. Breuer and S. Swartz, Brown University
Rat (N=5)
F
0.20
0.10
Experiment at Georgia Tech
Applehead Chihuahua
M
3
4
Experiment at local park
Cat
F
5
18
Youtube, Mattern (2000)
Goat
F
18
8
Experiment at Zoo Atlanta
Goat
M
67.8
9
Experiment at Zoo Atlanta
Great Dane
M
71
24
Youtube, Wilcox (1997)
Great Dane
M
71
17
Youtube, Wilcox (1997)
Great Dane
M
71
17
Youtube, Wilcox (1997)
Jaguar
F
76
15
Youtube, Wilson (2001)
Gorilla
F
100
20
Youtube, Miller (1997)
Panda
F
125
11
Youtube, Brown (1996)
Donkey
M
140
8
Youtube, Starkey (1992)
Lion
M
200
36
Youtube, Nowak (1999)
Tapir
F
318
9
Youtube, Lynette (2013)
Tapir
M
318
59
Youtube, Lynette (2013)
Elk
M
331
46
Youtube, Linnaeus (1758)
Zebra
M
430
8
Youtube, Kingdon (1988)
White horse
F
470
10
Youtube, Marvin (1992)
Horse
M
470
17
Youtube, Marvin (1992)
Race Horse
M
600
27
Youtube, Marvin (1992)
Horse
M
850
19
Youtube, Bongianni (1988)
Bison
M
907
20
Youtube, Potts (1997)
Indian Rhinoceros
M
2150
20
Youtube, Toon (2002)
Rhino
F
2200
49
Youtube, Toon (2002)
Rhino
F
2200
12
Youtube, Toon (2002)
Rhino
M
2200
17
Youtube, Toon (2002)
Elephant
F
3538
28
Experiment at Zoo Atlanta
Elephant
F
5000
17
Youtube, Shoshani (1982)
Elephant
F
5000
17
Youtube, Shoshani (1982)
Elephant
F
5000
15
Youtube, Shoshani (1982)
Elephant
M
8000
35
Youtube, Shoshani (1982)
Elephant
M
8000
29
Youtube, Shoshani (1982)
Elephant
M
8000
22
Youtube, Shoshani (1982)
Table S2. Urethral length.
Animal
Sex
Mass (kg)
Length (mm)
Source
Mouse
F
0.02
10
Clair (1999)
Wister Rat (N=20)
F
0.2
9.5
Chang (2000)
Rat
F
0.2
17
Clair (1999)
Sprague-Dawley Rat (N=61)
F
0.3
20.0
Kamo (2004)
Dunkin Hartley Guinea Pig
M
0.4
20.0
Vogel (2007)
Normal Adult Cat
F
2.3
49.4
Johnston (1985)
Normal Adult Cat
F
2.3
50.0
Johnston (1985)
Normal Adult Cat
F
2.7
66.7
Johnston (1985)
Normal Adult Cat
F
2.7
54.2
Johnston (1985)
Normal Adult Cat
F
3.2
68.3
Johnston (1985)
Normal Adult Cat
F
3.6
65.6
Johnston (1985)
Normal Adult Cat
M
3.6
116.6
Johnston (1985)
Normal Adult Cat
M
4.1
107.8
Johnston (1985)
Normal Adult Cat
M
4.1
99.4
Johnston (1985)
Normal Adult Cat
M
4.6
108.3
Johnston (1985)
Normal Adult Cat
M
4.6
118.6
Johnston (1985)
Normal Adult Cat
M
4.6
113.8
Johnston (1985)
Normal Adult Dog
F
6.8
57.3
Johnston (1985)
Normal Adult Dog
F
8.2
56.5
Johnston (1985)
Normal Adult Dog
F
9.1
84.8
Johnston (1985)
Normal Adult Dog
F
10.5
96.0
Johnston (1985)
Normal Adult Dog
F
12.3
94.0
Johnston (1985)
Mongrel Dog (N=10)
F
12.5
55.0
Takeda (1995)
Normal Adult Dog
F
12.7
57.8
Johnston (1985)
Normal Adult Dog
F
15.9
86.5
Johnston (1985)
Normal Adult Dog
F
19.1
77.5
Johnston (1985)
Woman
F
74.0
40
Prasad (2005), Ogden (2004)
Man (N=109)
M
92.0
223
Kohler (2008)
African Lion (N=7)
M
200.0
300
Lueders (2012), Nowak (1999)
African Elephant (5 years)
F
803.8
680
Balke (1988), Krumrey (1968)
African Elephant (5 years)
F
803.8
800
Balke (1988), Krumrey (1968)
African Elephant (6 years)
F
933.1
840
Balke (1988), Krumrey (1968)
African Elephant (6 years)
F
933.1
760
Balke (1988), Krumrey (1968)
African Elephant (6 years)
F
933.1
680
Balke (1988), Krumrey (1968)
African Elephant (7 years)
F
1058.6
820
Balke (1988), Krumrey (1968)
African Elephant (9 years)
F
1300.3
980
Balke (1988), Krumrey (1968)
African Elephant (9 years)
F
1300.3
800
Balke (1988), Krumrey (1968)
African Elephant (9 years)
F
1300.3
940
Balke (1988), Krumrey (1968)
African Elephant (10 years)
F
1417.4
880
Balke (1988), Krumrey (1968)
African Elephant (10 years)
F
1417.4
1180
Balke (1988), Krumrey (1968)
African Elephant (10 years)
F
1417.4
880
Balke (1988), Krumrey (1968)
African Elephant (11 years)
F
1532.4
860
Balke (1988), Krumrey (1968)
African Elephant (14 years)
F
1866.7
830
Balke (1988), Krumrey (1968)
Elephant
F
2700.0
1200
Hildebrandt (2000), Nowak (1999)
African Elephant (23 years)
F
2802.4
900
Balke (1988), Krumrey (1968)
African Elephant (25 years)
F
3000.3
1100
Balke (1988), Krumrey (1968)
Elephant
M
5400.0
1000
Fowler (2006), Nowak (1999)
Table S3. Urethral diameter.
Animal
Sex
Mass (kg)
Diameter (mm)
Source
Wister Rat
F
0.2
0.8
Chang (2000)
Wister Rat (N=10)
F
0.3
0.6
de Souza (2008)
Hybrid Rat (12 months) (N=5)
F
0.3
1.4
Russell (1996), Tasaki (2009)
Hybrid Rat (32 months) (N=5)
F
0.3
1.5
Russell (1996), Tasaki (2009)
Adult Rat (N=176)
M
0.6
1.8
Kunstyvr (1982), Perrin (2003)
Short Hair Cat (7 weeks) (N=6)
F
0.9
2.0
Root (1996), Sturman (1985)
Short Hair Cat (7 weeks) (N=6)
M
0.9
3.3
Root (1996), Sturman (1985)
Short Hair Cat (7 months) (N=5)
F
2.4
3.2
Root (1996), Scott (1970)
Short Hair Cat (7 months) (N=5)
M
3.5
5.4
Root (1996), Lein (1983)
Woman
F
74.0
6.0
Gray (1918), Ogden (2004)
Man
M
86.1
6.0
Gray (1918), Ogden (2004)
Man (71.7 years) (N=32)
M
87.1
7.3
Tsujimoto (2003), Ogden (2004)
Miniature Horse (N=7)
M
102.5
15.0
Pozor (2002)
Pony (N=27)
M
171.5
19.0
Pozor (2002)
Hereford X Angus Bull (N=96)
M
300.0
6.5
Bailey (1975)
Pozor (2002)
Light Horse (N=53)
M
452.5
26.0
Heavy Horse (N=15)
M
687.0
30.0
Pozor (2002)
Elephant
F
2700.0
35.0
Hildebrandt (2000), Nowak (1999)
African Elephant (N=6)
M
4000.0
45.0
Hildebrandt (1998)
Asian Elephant (N=2)
M
5000.0
52.5
Hildebrandt (1998)
Asian Elephant
M
5000.0
48.0
Hildebrandt (1998)
Asian Elephant
M
5400.0
180.0
Fowler (2006)
Table S4. Shape factor, bladder capacity, and bladder pressure.
Animal
Sex
Mass (kg)
Shape factor α
Source
Mouse
F
0.03
0.25
Treuting (2011)
Rat
F
0.28
0.24
Praud (2003)
dog
M
11.8
0.17
Caceci, Johnston (1985)
White Pig (N=4)
F
70
0.22
Dass (2001)
Man
M
86.1
0.14
Skarva, Ogden (2004)
Animal
Sex
Mass (kg)
Bladder capacity (mL)
Source
Mouse (N=10)
F
0.02
0.15
Pandita (2000)
Mouse
M
0.03
0.22
Birder (2002)
Sprague Dawley Rat
M
0.25
1
Herrera (2010)
Cat (N=3)
F
3
5.4
Thor (1995)
Applehead Chihuahua
M
3.45
4.2
Experiment at local park
Mongrel Dog (N=14)
M
22.5
327
Abdel (2001)
Dog (N=52)
-
28.7
45.5
Atalan (1998)
Equine
-
500
4250
Higgins (2006)
Elephant
-
5000
18000
Fowler (2006), Shoshani (1982)
Animal
Sex
Mass (kg)
Bladder pressure (kPa)
Source
Mice (N=10)
F
0.02
6.25
Pandita (2000)
Sprague Dawley Rat (N=7)
F
0.24
6.03
Ishizuka (1996)
Wister Rat (N=3)
M
0.45
4.02
Van (1995)
Sprague Dawley Rat (N=18)
M
0.54
4.40
Schmidt (2003)
Cat (N=3)
F
3
4.31
Thor (1995)
Cat (N=4)
F
3.5
5.54
Walter (2005)
Dog
F
10
4.88
Hinman (1971)
Man (N=17)
M
75.5
5.78
Schmidt (2003)
Table S5. Urine flow rate.
Animal
Sex
Mass (kg)
Flow rate (mL/s)
Source
Rat 1
F
0.24
1.11
Experiment at Georgia Tech
Rat 5
F
0.25
2.39
Experiment at Georgia Tech
Rat 3
F
0.26
0.38
Experiment at Georgia Tech
Rat 4
F
0.27
1.43
Experiment at Georgia Tech
Rat 2
F
0.29
0.67
Experiment at Georgia Tech
Cat (N=4)
F
3.5
2.40
Walter (2005)
Woman (3-4 years)
F
16.60
8.42
Segura (1997), Ogden (2004)
Woman (5-6 years)
F
21.46
9.43
Segura (1997), Ogden (2004)
Woman (7-8 years)
F
28.81
10.26
Segura (1997), Ogden (2004)
Woman (9-11 years)
F
41.28
13.49
Segura (1997), Ogden (2004)
Woman (12-14 years)
F
56.57
14.79
Segura (1997), Ogden (2004)
Dog
F
18.14
17.07
Experiment at local park
Woman (59 years) (N = 183)
F
76.90
9.4
Madersbacher (1998) , Ogden (2004)
Woman (55 years) (N = 185)
F
76.90
20.10
Nitti (1999) , Ogden (2004)
Cow
F
635.00
450.00
L. Ely, University of Georgia
L. Ely, University of Georgia
Cow
F
635.00
453.00
Wister Rat (N=2)
M
0.45
0.15
Van (1995)
Sprague Dawley Rat (N=18)
M
0.54
0.13
Schmidt (2003)
Dunkin Hartley Guinea Pig (N=4)
M
0.94
0.27
Van (1995)
Applehead Chihuahua
M
3.45
1.04
Experiment at local park
Man (3-4 years)
M
17.1
7.90
Segura (1997), Ogden (2004)
Man (5-6 years)
M
22.5
8.43
Segura (1997), Ogden (2004)
Man (7-8 years)
M
30.1
9.30
Segura (1997), Ogden (2004)
Man (9-11 years)
M
39.5
10.10
Segura (1997), Ogden (2004)
Man (12-14 years)
M
55.8
12.76
Segura (1997), Ogden (2004)
Nigerian Dwarf Goat
M
37.8
3.11
Experiment at Zoo Atlanta
Nubian Goat
M
71.00
6.25
Experiment at Zoo Atlanta
Japanese Man (N=271)
M
63.30
16.70
Masumori (1996)
Man (N=17)
M
75.70
24.40
Schmidt (2003), Schmidt (2002)
American Man (N=467)
M
87.60
19.80
Masumor (1996)
Man (53.8 years) (N=58)
M
88.80
17.60
Folkestad (2004), Ogden (2004)
Table S6. Bibliography for images and animal masses.
Online Image
Source
Inset in Figure 1(c)
Man vyi. (2005) Jersey cattle in Jersey, Wikimedia Commons
http://commons.wikimedia.org/wiki/File:Jersey_cattle_in_Jersey.jpg
Inset in Figure 2(b)
Stephens, J. (1992) Lobund-Wistar Rat
https://visualsonline.cancer.gov/details.cfm?imageid=2568
Animal silhouettes
in Figure 1(h)
Public Domain Pictures
http://www.publicdomainpictures.net/
Caceci, T. (2008) Canine penis; H&E stain, paraffin section (decalcified), 20x, VM8054: Veterinary Histology
Histology in Table S4 http://www.vetmed.vt.edu/education/curriculum/vm8304/lab_companion/histo-path/vm8054/labs/Lab23/EXAMPLES/
EXURETH.HTM
Skarva, F. Cross-Section of a Normal Human Penis Showing the Urethra and Corpora Spongiosum, H&E Stain, LM X12
Histology in Table S4 http://www.allposters.com/-sp/Cross-Section-of-a-Normal-Human-Penis-Showing-the-Urethra-and-Corpora-Spongiosum-H-EStain-LM-X12-Posters_i9005304_.htm
Body masses of animals considered
Nowak RM, Paradiso JL (1999) Walker’s mammals of the world (The Johns Hopkins University Press, Baltimore, MD) Vol. 1, 6th Ed.
Mattern MY, McLennan DA (2000) Phylogeny and speciation of felids. Cladistics 16(2):232–253.
Wilcox C (1997) The Great Dane (Capstone, North Mankato, MN).
Wilson D, Burnie D (2001) Animal: the definitive visual guide to the world’s wildlife (DK Publishing, New York).
Miller-Schroeder P (1997) Gorillas (Raintree Steck-Vaughn, Austin, TX).
Brown G (1996) The Great Bear Almanac (Globe Pequot, Guilford, CT).
Starkey P, Mwenya E, Stares J (1994) Improving animal traction technology. Proceedings of the First Workshop of the Animal Traction Network for
Eastern and Southern Africa held 18-23 January 1992, Lusaka, Zambia (Technical Centre for Agricultural and Rural Cooperation, Wageningen, The
Netherlands).
Lynette R (2013) South American Tapirs (Bearport Publishing, New York).
Linnaeus C (1758) Systema naturae per regna tria naturae, secundum classes, ordines, genera, species, cum characteribus, differentiis, synonymis,
locis. (Laurentii Salvii, Stockholm), 10th Ed.
Kingdon J (1988) East African Mammals: An Atlas of Evolution in Africa, Part A: Carnivores (University of Chicago Press, Chicago) Vol. 3.
Hall MH, Comerford PM (1992) Pasture and hay for horses. Cooperative Extension, The Pennsylvania State University 32:1–4.
Bongianni M (1988) Simon and Schuster’s guide to horses and ponies of the world (Simon and Schuster, New York).
Potts S (1997) The American Bison (Capstone, Mankato, MN).
Toon A, Toon S (2002) Rhinos (Voyageur Press, Minneapolis, MN).
Shoshani J, Eisenberg JF (1982) Elephas maximus. Mammalian Species 182:1–8.
Ogden CL, Fryar CD, Carroll MD, Flegal KM (2004) Mean Body Weight, Height, and Body Mass Index: United States 1960-2002 (Department of
Health and Human Services, Centers for Disease Control and Prevention, National Center for Health Statistics).
Krumrey WA, Buss IO (1968) Age estimation, growth, and relationships between body dimensions of the female African elephant. Journal of
Mammalogy 49(1):22–31.
Tasaki M, et al. (2009) Simultaneous induction of non-neoplastic and neoplastic lesions with highly proliferative hepatocytes following dietary exposure
of rats to tocotrienol for 2 years. Archives of Toxicology 83(11):1021–1030.
Perrin D, Soulage C, Pequignot J, Geloen A (2003) Resistance to obesity in Lou/C rats prevents ageing-associated metabolic alterations. Diabetologia
46(11):1489–1496.
Sturman J, Moretz R, French J, Wisniewski H (1985) Postnatal taurine deficiency in the kitten results in a persistence of the cerebellar external granule
cell layer: correction by taurine feeding. Journal of Neuroscience Research 13(4):521–528.
Scott P, Hafez E (1970) Reproduction and Breeding Techniques for Laboratory Animals. (Lea and Febiger, Philadelphia).
Lein D, Concannon P (1983) Infertility and fertility treatments and management in the queen and tomcat. Current Therapy VIII. Kirk, R.(ed). (Elsevier
Saunders, Philadelphia) 936–987.
Schmidt F, Shin P, Jorgensen TM, Djurhuus JC, Constantinou CE (2002) Urodynamic patterns of normal male micturition: Influence of water
consumption on urine production and detrusor function. The Journal of Urology 168(4 Pt 1):1458–1463.
Supporting Information
Yang et al. 10.1073/pnas.1402289111
Movie S1. Urination of a rat (mass of 0.24 kg). Time slowed by 33 times.
Movie S1
Movie S2. Urination of a goat (mass of 70 kg). Time slowed by 17 times.
Movie S2
Yang et al. www.pnas.org/cgi/content/short/1402289111
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Movie S3.
Urination of a cow (mass of 640 kg). Time slowed by 33 times.
Movie S3
Movie S4. Urination of an elephant (mass of 3,540 kg). Time slowed by 33 times.
Movie S4
Other Supporting Information Files
SI Appendix (PDF)
Yang et al. www.pnas.org/cgi/content/short/1402289111
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