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Two simple calculations on the gross movements of human spermatozoa on the cervical surface
461
MATHEMATICAL BIOSCIENCES
Two Simple Calculations on the Gross Movements of Human Spermatozoa on the Cervical Surface W. W. CHOW, A. G. HARDEE,
AND
KNOX MILLSAPS
Colorado State University, Fort Coliins, Colorado
Communicated by Richard Bellman
ABSTRACT In connection with the Sims-Huhner postcoital test for determining minimal levels of male fertility, calculations for the number of human spermatozoa passing through the cervical OS are made for two models of cervical geometry and for the diffusion of spermatozoa in cervical mucus. The simpler one of the two models in which linear trajectories of noninteracting bodies are assumed leads to closed form solutions, and the more complicated model which is based on random walking and the diffusion process yields moderately involved Bessel expansions whose evaluation is relatively interesting. Numerical examples and graphs of the solutions are presented.
INTRODUCTION
One of the more widely used postcoital evaluations of male fertility is the Sims-Huhner [4] test which is essentially a microscopic determination of the surface density of motile spermatozoa in cervical mucus. The quantitative value of the test is controversial because the precise relationship between surface insemination of the cervix and actual fertilization of the ovum is not known; however, as a partial delineation of this relationship, it seems worthwhile to examine in some detail the connection between the surface density of spermatozoa on the cervix and the eventual passage of some of the spermatozoa through the cervical OS. Of course, the use of such an investigation to study natural and artificial cervical insemination is clear, and it is really the latter problem of artificial insemination with processed semen that originally attracted our attention Mathematical Biosciences 5 (1969), 461-469
Copyright @ 1969 by American Elsevier Publishing Company, Inc. 81
W. W. CHOW, A. G. HARDEE. AND k. MILLSAPS
462
in cormection -withengineering calculations of minimal quantities. Also, the pertinence to a comparison of artificial uterine and cervical insemination is obvious. 1. A SIMPLE MODEL WITH NUMERICAL RESULTS
One may model the cervix as a flat circular disk of outer radius r, ith a circular opening, the OS,or radius r, which is centrally and concentrically located. It is assumed that a uniformly thick surface coating of uniformly constituted semen containing a Iconstant density of spermatozoa is initially deposited on the cervix, an;,1it is now required to calculate the number of spermatozoa that eventually pass through the OS. A rough calculation can be made using a set of simplifying assumptions: (as) all spermatozoa travel at a constant velocity in homoschedastic directions; (b) the spermatozoa do not interact; and (c) any spermatozoon that reaches the cervical-vaginal boundary at r = r, is lost to the process. A schematic diagram is shown in Fig. 1.
FIG. 1. Geometrical model of the cervix.
From the diagram and the assumptions it is clear that f-c
Iv hm.dkal
BIbsciences 5 (1969), 461-469
U.1)
MOVEMENTS OF SPERMATQZOA
463
Noting that 0 = arc csc p and performing the simple quadrature with an integration by parts, an integral form of (1.1) is V =
/I”
arc csc ~9+ (/I” - l)lj2 - 5
.
(1.2)
The graph of v as a function of B is shown in Fig. 2; it should be noted that the asymptotic value of v is 28 - r/2. A numerical example in which
100
80 60 d 40 20 0 0
IO
30
20
40
50
.P FIG.
2. Graph of Yas a function of b for linear trajectory model.
r. and r, are taken to be 1.25 mm and 25.00 mm, respectively, and in which s is taken to be 58 spermatozoa/mm2 (the customarily assumed Sims-Huhner value of 5 motile spermatozoa per high-powered field that indicates the minimal level for male fertility) shows that N is 35 x lo2 spermatozoa or 3.1% of the initial deposit. N is an important quantity in the engineering design of devices in which the modus operandi involves the processing of the individual spermatozoon and for which processed semen is to be used for artificial cervical insemination. Mathematical Biosciences 5 (1969), 46 l-469
W. W, CHOW, A. G. HARDEE, AND K. MILLSAPS
464
MECHANICAL GATE
SPERM
CERVICAL
AT INITIAL
M WCUS
X=3L X=ZL X=0 X=L Pm.3, Schematicdiagramfor determinationof the coefficientof diffusivity.
I
0a
5
44
9
s3
.2
.I
0
0
5
1.0
I.5
2.0
2.5
3.0
Fm. 4. Graph of q as a function of 0 for parametricallyintegralvalues of 6. Mathemafical Biosciences 5 (1969),461-469
MOVEMENTS
465
OF SPERMATOZOA
,I2
~_
~_
_
.IO *08 Y
*06
.
.O4
,
.02 I 0 0
.5
1.0
2.0
30 .
CT FIG.
2.
4 (continued)
A MORE REALISTIC
MODEL
It is a familiar result of probability theory that the random walk 0~ sufficiently large population is mathematically equivalent in the asymptotic sense to the partial differential equation that describes the diffusion process [3]. Hence the problem of calculating the passage of spermatozoa through the OSwith the same geometrical model as before can be formulated as the study of the differential system:
(2.1) q(rO, t) = 0, q(rC, t) = 0
and
q(r, 0) = s
or, in terms of dimensionless quantities, the system becomes (2.2) Mathematical Biosciences 5 (1969),461-469
W. W.
466
CHOW,
A. G. :HARDEE,
AND
K. MILLSAPS
where q(l, T) = 0,
@, 7) = 0, and
~(p,
0) =
1.
the quantity of principal interest is 2@5~/ap)~=, or which will give the number of spermatozoa that pass 2%(+@1~& through the OS as a function of the time, the initial state, and the geometry. 2.1. DETERMINATIQN OF THE COEFFICIENT OF DIFFUSIVITY. One notes that the random walk formulation contains the coefficient of diffusivity, d, and it is pertinent to outline a simple method for an experimental measurement of this quantity. A direct determination of d can be made by using the essentially one-dimensional situation that is shown schematically in Fig. 3. The partial differential system for the situation is Of
Course,
84 Z=
d a24 ax2
(2.3)
with q(x9 0) = s qk
0) = 0
for 0 < x < I, for
x > I,
q(0, t) = 0.
The physical situation can be described verbally in the following manner: In a semi-infinite channel aligned with the x-axis, whole semen is contained in a fixed length 2 when t < 0 and is separated from seminal plasma contained in the remainder of the channel by a mechanical gate at x = I which is instantaneously removed at t = 0; the ratio q/s is measured at I, 21, 3/,41, etc., as a function of time, d being subsequently determined by the method that is to be delineated. The coefficient of diffusivixy of spermatozoa into cervical mucus can be determined in the same manner. In dimensionless form the differential equation is transformed into
with r(& 0) = 1 for O
Cd,
The analytical solution [l, Chap. 2.4, (iv), p. 621 for this problem is known to be
5: t-1 q = -2(I2erf- 20 - erf- 20
1
1
Mathematical Biosciences 5 ( 1969),461-469
cc-1 A+ - 2a
1-
MOVEMENTS
OF SPERMATOZOA
0
5
467
IO
15
20
25
FIG. 5. Graph of (ar~/ap),,=~ as a function of T for p = 20.
Since the horizontal positions of the microscopes are easily fixed, Fig. 4 shows a plot of q vs. 0 for parametrically integral values of E; the value of d must be calculated by interpolation. Experimental values will be published in another paper; however, an order of magnitude may be calculated as 25 x 1O-4mm2/sec by noting that an average value for the velocity of spermatozoa in cervical mucus is approximately 5 X 10m3 mm/set and that a frequently observed path length before change of direction is about 50 x 1O-21-m [5]. 2.2. THE SOLUTION WITH NUMERICAL RESULTS. Equation (2.2) occurs in the mathematical theory of the conduction of heat in solids, and the solution [l, Chap. ‘7.10, (13), p. 2071 that satisfies the stated boundary conditions is
Mat?tematical @osciences 5 (1969)) 46 l-469
w. w. CHOW, A. G, HARDEE, AND K. MILLSAPS
1000 900 800 700 600 500 400 300 200 too 1) -0
5
10 15 20 25 30 35 40
45 50
ES 6. Graph of P as a function of p for the random walk model.
Since one is interested in the passage of spermatozoa through the OS, a&_ is a significant quantity that can be found by differentiating (2.6) with respect to p and by evaluating the resulting expression for p = 1 (see Fig. 5), recalling that J,@) = --Jx(p) and that Y;(p) = -Y,(p), this simple differentiation gives
Figure 5 shows (a&+,, as a function of T for /? = 20. When the random walk model is used for the same conditions as the numerical example in Section 1 (see Fig. 6) N i:i f&.rndto be 18 x lo3 spermatozoa or 16% of the initial deposit, provided that the hypotheses under which (2.6) was derived remain valid for a sufficiently long period. N is given by the formula
MOVEMENTS OF SPER.MATOZOA SYMBOLS
d
coefficient of diffusivity erf(x) the error function Jzdx) the Bessel function of the first kind of orderp 1 a fixed length N number of spermatozoa passing through the cervical OS surface density of spermatozoa at 4 time t and at location r r radial coordinate radius of the cervix rc radius of the cervical OS r. S initial surface density of spermatozoa t temporal coordinate
linear coordinate the Bessel function of the second kind of order p the kth root of the transcendental equation, Jo(a) &
PI the dimensionless ratio rc/ro the dimensionless ratio q/s arc csc p the dimensionless ratio N/Sri the dimensionless ratio x/1 the dimensionless ratio r/r0 the dimensionless ratio
REFERENCES 1 H. S. Carslaw and J. C. Jaeger, Conductionof Heat in Solids, 2nd ed., Oxford Univ. Press, London, 1959. 2 H. B. Dwight, Table of roots for natural frequencies in coaxial type cavities, J. Math. Phys., 27(1948), 85. 3 W. Feller, An Introduction to A*obabilityTheory and Its Applications, Vol. 2, John
Wiley and Sons, New York, 1966, Chap. X.4, p. 320. 4 M. Huhner, Sexual Disordersin 1he Male and Female includingSterility and Impotence, 2nd ed., F. A. Davis, Philadelphia, 1939. 5 E. Odeblad, Motility patterns and propagation rate of spermatozoa in human cervical mucus, Ferfility and Skrility, Excerpta Medica Foundation, Amsterdam, 1967, p. 655.
Mathematical Biusciences 5 (1969), 461-469
MATHEMATICAL BIOSCIENCES
Two Simple Calculations on the Gross Movements of Human Spermatozoa on the Cervical Surface W. W. CHOW, A. G. HARDEE,
AND
KNOX MILLSAPS
Colorado State University, Fort Coliins, Colorado
Communicated by Richard Bellman
ABSTRACT In connection with the Sims-Huhner postcoital test for determining minimal levels of male fertility, calculations for the number of human spermatozoa passing through the cervical OS are made for two models of cervical geometry and for the diffusion of spermatozoa in cervical mucus. The simpler one of the two models in which linear trajectories of noninteracting bodies are assumed leads to closed form solutions, and the more complicated model which is based on random walking and the diffusion process yields moderately involved Bessel expansions whose evaluation is relatively interesting. Numerical examples and graphs of the solutions are presented.
INTRODUCTION
One of the more widely used postcoital evaluations of male fertility is the Sims-Huhner [4] test which is essentially a microscopic determination of the surface density of motile spermatozoa in cervical mucus. The quantitative value of the test is controversial because the precise relationship between surface insemination of the cervix and actual fertilization of the ovum is not known; however, as a partial delineation of this relationship, it seems worthwhile to examine in some detail the connection between the surface density of spermatozoa on the cervix and the eventual passage of some of the spermatozoa through the cervical OS. Of course, the use of such an investigation to study natural and artificial cervical insemination is clear, and it is really the latter problem of artificial insemination with processed semen that originally attracted our attention Mathematical Biosciences 5 (1969), 461-469
Copyright @ 1969 by American Elsevier Publishing Company, Inc. 81
W. W. CHOW, A. G. HARDEE. AND k. MILLSAPS
462
in cormection -withengineering calculations of minimal quantities. Also, the pertinence to a comparison of artificial uterine and cervical insemination is obvious. 1. A SIMPLE MODEL WITH NUMERICAL RESULTS
One may model the cervix as a flat circular disk of outer radius r, ith a circular opening, the OS,or radius r, which is centrally and concentrically located. It is assumed that a uniformly thick surface coating of uniformly constituted semen containing a Iconstant density of spermatozoa is initially deposited on the cervix, an;,1it is now required to calculate the number of spermatozoa that eventually pass through the OS. A rough calculation can be made using a set of simplifying assumptions: (as) all spermatozoa travel at a constant velocity in homoschedastic directions; (b) the spermatozoa do not interact; and (c) any spermatozoon that reaches the cervical-vaginal boundary at r = r, is lost to the process. A schematic diagram is shown in Fig. 1.
FIG. 1. Geometrical model of the cervix.
From the diagram and the assumptions it is clear that f-c
Iv hm.dkal
BIbsciences 5 (1969), 461-469
U.1)
MOVEMENTS OF SPERMATQZOA
463
Noting that 0 = arc csc p and performing the simple quadrature with an integration by parts, an integral form of (1.1) is V =
/I”
arc csc ~9+ (/I” - l)lj2 - 5
.
(1.2)
The graph of v as a function of B is shown in Fig. 2; it should be noted that the asymptotic value of v is 28 - r/2. A numerical example in which
100
80 60 d 40 20 0 0
IO
30
20
40
50
.P FIG.
2. Graph of Yas a function of b for linear trajectory model.
r. and r, are taken to be 1.25 mm and 25.00 mm, respectively, and in which s is taken to be 58 spermatozoa/mm2 (the customarily assumed Sims-Huhner value of 5 motile spermatozoa per high-powered field that indicates the minimal level for male fertility) shows that N is 35 x lo2 spermatozoa or 3.1% of the initial deposit. N is an important quantity in the engineering design of devices in which the modus operandi involves the processing of the individual spermatozoon and for which processed semen is to be used for artificial cervical insemination. Mathematical Biosciences 5 (1969), 46 l-469
W. W, CHOW, A. G. HARDEE, AND K. MILLSAPS
464
MECHANICAL GATE
SPERM
CERVICAL
AT INITIAL
M WCUS
X=3L X=ZL X=0 X=L Pm.3, Schematicdiagramfor determinationof the coefficientof diffusivity.
I
0a
5
44
9
s3
.2
.I
0
0
5
1.0
I.5
2.0
2.5
3.0
Fm. 4. Graph of q as a function of 0 for parametricallyintegralvalues of 6. Mathemafical Biosciences 5 (1969),461-469
MOVEMENTS
465
OF SPERMATOZOA
,I2
~_
~_
_
.IO *08 Y
*06
.
.O4
,
.02 I 0 0
.5
1.0
2.0
30 .
CT FIG.
2.
4 (continued)
A MORE REALISTIC
MODEL
It is a familiar result of probability theory that the random walk 0~ sufficiently large population is mathematically equivalent in the asymptotic sense to the partial differential equation that describes the diffusion process [3]. Hence the problem of calculating the passage of spermatozoa through the OSwith the same geometrical model as before can be formulated as the study of the differential system:
(2.1) q(rO, t) = 0, q(rC, t) = 0
and
q(r, 0) = s
or, in terms of dimensionless quantities, the system becomes (2.2) Mathematical Biosciences 5 (1969),461-469
W. W.
466
CHOW,
A. G. :HARDEE,
AND
K. MILLSAPS
where q(l, T) = 0,
@, 7) = 0, and
~(p,
0) =
1.
the quantity of principal interest is 2@5~/ap)~=, or which will give the number of spermatozoa that pass 2%(+@1~& through the OS as a function of the time, the initial state, and the geometry. 2.1. DETERMINATIQN OF THE COEFFICIENT OF DIFFUSIVITY. One notes that the random walk formulation contains the coefficient of diffusivity, d, and it is pertinent to outline a simple method for an experimental measurement of this quantity. A direct determination of d can be made by using the essentially one-dimensional situation that is shown schematically in Fig. 3. The partial differential system for the situation is Of
Course,
84 Z=
d a24 ax2
(2.3)
with q(x9 0) = s qk
0) = 0
for 0 < x < I, for
x > I,
q(0, t) = 0.
The physical situation can be described verbally in the following manner: In a semi-infinite channel aligned with the x-axis, whole semen is contained in a fixed length 2 when t < 0 and is separated from seminal plasma contained in the remainder of the channel by a mechanical gate at x = I which is instantaneously removed at t = 0; the ratio q/s is measured at I, 21, 3/,41, etc., as a function of time, d being subsequently determined by the method that is to be delineated. The coefficient of diffusivixy of spermatozoa into cervical mucus can be determined in the same manner. In dimensionless form the differential equation is transformed into
with r(& 0) = 1 for O
Cd,
The analytical solution [l, Chap. 2.4, (iv), p. 621 for this problem is known to be
5: t-1 q = -2(I2erf- 20 - erf- 20
1
1
Mathematical Biosciences 5 ( 1969),461-469
cc-1 A+ - 2a
1-
MOVEMENTS
OF SPERMATOZOA
0
5
467
IO
15
20
25
FIG. 5. Graph of (ar~/ap),,=~ as a function of T for p = 20.
Since the horizontal positions of the microscopes are easily fixed, Fig. 4 shows a plot of q vs. 0 for parametrically integral values of E; the value of d must be calculated by interpolation. Experimental values will be published in another paper; however, an order of magnitude may be calculated as 25 x 1O-4mm2/sec by noting that an average value for the velocity of spermatozoa in cervical mucus is approximately 5 X 10m3 mm/set and that a frequently observed path length before change of direction is about 50 x 1O-21-m [5]. 2.2. THE SOLUTION WITH NUMERICAL RESULTS. Equation (2.2) occurs in the mathematical theory of the conduction of heat in solids, and the solution [l, Chap. ‘7.10, (13), p. 2071 that satisfies the stated boundary conditions is
Mat?tematical @osciences 5 (1969)) 46 l-469
w. w. CHOW, A. G, HARDEE, AND K. MILLSAPS
1000 900 800 700 600 500 400 300 200 too 1) -0
5
10 15 20 25 30 35 40
45 50
ES 6. Graph of P as a function of p for the random walk model.
Since one is interested in the passage of spermatozoa through the OS, a&_ is a significant quantity that can be found by differentiating (2.6) with respect to p and by evaluating the resulting expression for p = 1 (see Fig. 5), recalling that J,@) = --Jx(p) and that Y;(p) = -Y,(p), this simple differentiation gives
Figure 5 shows (a&+,, as a function of T for /? = 20. When the random walk model is used for the same conditions as the numerical example in Section 1 (see Fig. 6) N i:i f&.rndto be 18 x lo3 spermatozoa or 16% of the initial deposit, provided that the hypotheses under which (2.6) was derived remain valid for a sufficiently long period. N is given by the formula
MOVEMENTS OF SPER.MATOZOA SYMBOLS
d
coefficient of diffusivity erf(x) the error function Jzdx) the Bessel function of the first kind of orderp 1 a fixed length N number of spermatozoa passing through the cervical OS surface density of spermatozoa at 4 time t and at location r r radial coordinate radius of the cervix rc radius of the cervical OS r. S initial surface density of spermatozoa t temporal coordinate
linear coordinate the Bessel function of the second kind of order p the kth root of the transcendental equation, Jo(a) &
PI the dimensionless ratio rc/ro the dimensionless ratio q/s arc csc p the dimensionless ratio N/Sri the dimensionless ratio x/1 the dimensionless ratio r/r0 the dimensionless ratio
REFERENCES 1 H. S. Carslaw and J. C. Jaeger, Conductionof Heat in Solids, 2nd ed., Oxford Univ. Press, London, 1959. 2 H. B. Dwight, Table of roots for natural frequencies in coaxial type cavities, J. Math. Phys., 27(1948), 85. 3 W. Feller, An Introduction to A*obabilityTheory and Its Applications, Vol. 2, John
Wiley and Sons, New York, 1966, Chap. X.4, p. 320. 4 M. Huhner, Sexual Disordersin 1he Male and Female includingSterility and Impotence, 2nd ed., F. A. Davis, Philadelphia, 1939. 5 E. Odeblad, Motility patterns and propagation rate of spermatozoa in human cervical mucus, Ferfility and Skrility, Excerpta Medica Foundation, Amsterdam, 1967, p. 655.
Mathematical Biusciences 5 (1969), 461-469