an inverse modi ed helmholtz problem for identifying ... · the putative morphogen where the...
TRANSCRIPT
An Inverse Modified Helmholtz
Problem for Identifying Morphogen
Sources from Sliced Biomedical Image
Data
Marcus PensaSupervised by Mark Flegg
Monash University
February 26, 2018
Vacation Research Scholarships are funded jointly by the Department of Education and Training
and the Australian Mathematical Sciences Institute.
1 Introduction
During both female meiosis (cell division) I and II, an oocyte (egg cell) divides asymmetrically resulting in the
production of a small cell, called a polar body, which ultimately breaks down [Schmerler and Wessel, 2011].
An important early event in this division is asymmetrical accumulation of a molecular polymer called actin at
the oocyte cortex, where the greatest actin accumulation occurs at a location which coincides with the location
of polar body formation and extrusion [Longo and Chen, 1985]. The actin thickening forms adjacent to the
meiotic spindle, a structure which helps separate the cell’s chromosomes into a set(s) which will stay in the
oocyte, and a set(s) which will enter the polar body. It is known that DNA, a molecule which is a constituent
part of a chromosome, can act at a distance to induce the formation of the actin cap, however, the way in which
the DNA communicates with the cell surface remains unknown [Deng et al., 2007].
In attempt to explain the observed actin thickening, we hypothesis that a morphogen (diffusible signalling
molecule) is produced with an asymmetrical distribution in the oocyte, possibly from a location which coincides
with the location of DNA in the cell, and that the morphogen promotes actin formation when it reaches the
cell membrane. Furthermore, we hypothesise that the morphogen is processed or broken down upon reaching
the cell membrane. In order to create a model for our hypothesis, we assume that morphogen is generated from
one or more point sources at a fixed (but to be determined) location in the oocyte and that the morphogen
generation, decay as well as diffusion rates are also fixed parameters which are yet to be determined. We assume
that the concentration of morphogen is zero at the cell surface, due to processing of the morphogen at the cell
surface, and that morphogen levels reach equilibrium. The mathematical model of morphogen concentration
is represented by an inhomogeneous modified Helmholtz equation with zero boundary conditions. We assume
that actin thickness is in pseudo-equilibrium and that actin thickness depends only on a linear production-
degradation process whereby production is caused by signalling from the putative morphogen. Equivalently,
these assumptions can be stated as actin thickness being proportional to the flux (amount broken down) of
morphogen through the cell membrane.
This model is motivated by a desire to determine unknown parameters and location of production for
the putative morphogen where the available data for analysis is the distribution of cortical actin. Therefore,
algorithms were generated which solve the inverse problem; the algorithms determines what model parameters
should be used such that the the measured/known actin distribution is the outcome.
To solve the inverse problem, Duxbury [2016] has previously presented a least square approach which employs
a look-up table. In this report, we describe an iterative approach that does not require interpolation of a look-up
table and still uses a least square method. Furthermore, we present a weighted least squared approach.
Using our model, we were able to generate control data by finding theoretical actin distributions for sets
of possible parameters and sampling the resultant actin distribution on concentric rings of the cell surface,
then adding noise. This control data emulates real data which is available from confocal microscopy images of
oocytes where actin thickness is visible in each slice image. Our un-weighted least square inverse algorithm was
validated using this control data.
1
2 Model
Let u denote the morphogen concentration at any point in space r ∈ Ω, as described using spherical co-ordinates
where θ is the azimuthal angle; r = (r, θ, φ). We will assume a spherical cell volume, Ω, centred at the origin,
with radius R. Let the surface of the cell be given by ∂Ω, and because moprhogen molecules striking the cell
surface are absorbed, u = 0 on ∂Ω. We model the transport of morphogen using the continuity equation with a
generation term, f(r), and with a degradation term, µu, where µ represents break down rate of the morphogen.
By assuming our system reaches equilibrium and by applying Fick’s first law, the continuity equation with our
boundary condition results in the BVP
−D∇2u(r) + µu(r) = f(r), r ∈ Ω (1a)
u(r) = 0, r ∈ ∂Ω (1b)
where D is the diffusivity of the morphogen.
We can non-dimensionalise the BVP given by (1a) and (1b) by rescaling Ω isotropically by a factor of R.
Using this rescaling, the BVP becomes
−D∇2u(r) +R2µu(r) = R2f(r), r ∈ Ω (2a)
u(r) = 0, r ∈ ∂Ω (2b)
where our new domain, Ω, is the origin-centred unit sphere, r = rR and r = r
R . To express the DE (2a) in the
form of a standard inhomogeneous Helmholtz equation, we will divide through by D and introduce κ = R2µ/D
as well as F (r) = R2f(r)/D:
−∇2u(r) + κ2u(r) = F (r), r ∈ Ω (3a)
u(r) = 0, r ∈ ∂Ω (3b)
.
To solve the BVP given by (3a) and (3b), we introduce the Greens function, G, defined such that
−∇2G(r; r0) + κ2G(r; r0) = δ3(r− r0), r ∈ Ω (4a)
G(r) = 0, r ∈ ∂Ω (4b)
where r0 = (r0, θ0, φ0).
The solution to the BVP described by (4a) and (4b) is
G(r; r0) =κ
2π2
∞∑l=0
(2l + 1)
(il(κr<)kl(κr>)− kl(κ)il(κr0)il(κr)
il(κ)
)Pl(cosφ′0), 0 ≤ r ≤ 1 (5)
Where Pl are Legendre polynomials, il and kl are Modified Spherical Bessel functions of the first and second
kind, r< = min(r, r0), r> = max(r, r0) and
cos(φ′i) = cos θ sinφ cos θi sinφi + sin θ sinφ sin θi sinφi + cosφ cosφi.
2
See the Appendix for a derivation of this solution.
By construction of the Greens function, we can express u in terms of the Greens function as
u =
Ω
G(r; r0)F (r0) dr0. (6)
We can model the case where morphogen generation occurs from n point sources by setting F (r) such that
F (r) =
n∑i=1
λiδ3(r− ri).
By substituting this definition of the function F into equation (6), we can write u as a linear combination of
Green’s functions:
u =
n∑i=1
λiG(r; ri). (7)
If we let a(θ, φ) represent the thickness of actin on the cell boundary for a unit sphere cell and assume that
actin generation is proportional to morphogen flux through the cell boundary, then we can model the rate of
actin generation by the equation∂a
∂t= κa∇u · n |r=1 − µaa
where n is an outward pointing normal unit vector on the unit sphere, κa is a constant of actin generation rate
and µa is a constant of actin breakdown rate.
However, the outward pointing unit normal vector of a sphere is r, and the flux in the radial outward
direction can alternatively be expressed as the rate of change with respect to radius. Hence we can express the
rate of actin thickness change with respect to time as
∂a
∂t= κa
∂u
∂r
∣∣∣∣r=1
− µaa. (8)
If we assume that actin thickness is in pseudo-equilibrium, then we can re-write equation (8) as
a ∝ ∂u
∂r
∣∣∣∣r=1
. (9)
Recalling both (5) and (7), the actin distribution is given by
a ∝n∑i=1
λi∂G(r; ri)
∂r
∣∣∣∣r=1
=
n∑i=1
λi
(κ2
2π2
∞∑l=0
(2l + 1)
(il(κri)k
′l(κ)− kl(κ)il(κri)i
′l(κ)
il(κ)
)Pl(cosφ′i)
). (10)
We can simplify this expression by using Wronskian identity i′l(κ)kl(κ) = π/(2κ2) + il(κ)k′l(κ):
a ∝ 1
4π
n∑i=1
(λi
∞∑l=0
(2l + 1)
(il(κri)
il(κ)Pl(cosφ′i)
)). (11)
Relationship (11) gives us a compact way to express theoretical actin thickness in terms of parameters
relating to normalised degradation (κ), morphogen production intensity (λ) and source locators (ri, θi, φi) of
the putative morphogen.
3
2.1 Modelling a single point source
Consider the case when the generation of morphogen occurs only as a single point source. Then relation (11)
becomes
a ∝ 1
4π
∞∑l=0
(2l + 1)
(il(κr1)
il(κ)Pl(cosφ′1)
). (12)
It should also be noted that calculating l consecutive Legendre polynomials using Matlab’s inbuilt Legendre
polynomial function is slow. It is faster to use the recurrence relationship
Pl+1 =(2l + 1) cosφ′1Pl − lPl−1
l + 1.
Because all Legendre polynomials over the interval [−1, 1] have a magnitude with an upper bound of one, the
best point to truncate the series such that (12) has a small error is dependent on the Modified Spherical Bessel
quotientil(κr1)
il(κ).
The code used to generate the algorithms discussed in this report truncate the series when the lth Bessel quotient
is smaller than a one millionth of the first Bessel quotient. For extreme values of κ where κ→ 0,∞ or r1 where
r1 → 0, 1, Matlab is unable to calculate many terms of the Bessel quotient. This is because both il(κr1) and
il(κ) are individually extremely large or small, even though the quotient is well behaved. Fortunately, a code
does not need to be generated which can evaluate equation (11) for the cases when κ or r1 are large. This is
because a large κ value means that the morphogen is broken down much faster than it diffuses and a large r1
value means that the point source of morphogen production is very close to the cell surface. Either of these
two cases would result in a very sharp peak of actin on the membrane which by inspection of the experimental
data is clearly not the case. For the cases when κ or r1 are small, it is useful to consider the following bounds
on the Bessel quotient [Paris, 1984]:
rl1 expκ(r1−1) <il(κr1)
il(κ)< rl1, (13)
where the upper bound also approximates the long term behaviour for κ l. For the case when r1 is small, the
Bessel quotient needs to be evaluated for few terms as the Bessel quotient is dominated by rl1 which will quickly
approach zero. For the case when κ is small, the bound becomes very tight and rl1 quickly becomes a good
approximation of the Bessel quotient. The algorithms discussed in this report begin approximating the Bessel
quotient using the long term behaviour approximation after one of two cases: (1) the difference between the long
term behaviour approximation and Bessel quotient is less than a one millionth of the size of the approximation
or (2) the difference between the long term behaviour approximation and Bessel quotient is less than a one
thousandth of the size of the approximation and the lth Bessel quotient is smaller than a one thousandth of the
first Bessel quotient.
4
3 Experimental data
We have obtained images in 2-D cross sections, spaced 2µ and oriented perpendicular to the z-axis, of mouse
oocytes undergoing their second asymmetrical division. The z-stack images are taken without prior knowledge
of the orientation of the actin cap or the adjacent spindle structures, however, a sufficient number of images are
taken to sample the entire spindle. The series of images begin before the spindle enters the field of view and
end after the spindle leave the field of view. Because the actin cap is adjacent to the spindles, the collection of
images are centred about the location of the cell where actin is thickest. The number of images available per
cell ranges from 14 to 24. Given that a typical oocyte being analysed has a diameter of approximately 70µm,
the amount of the oocyte being sampled from these images ranges from approximately 37% (26/70) to 66 %
(46/70). Figure 1 shows a sample of images from a series of 21 successive z-stack images. The actin can be seen
in green and the mitotic spindle in red and blue. The images are displayed only for some slices; (a) z0, (b) z7
(14µm above z0), (c) z17 (34µm above z0) and (d) z21 (42µm above z0).
(a) z0. (b) z7. (c) z17. (d) z21.
Figure 1: Confocal microscopy images of meiosis II oocytes showing cortical actin (green) and the meiotic
spindle (chromosomes in blue, microtubules in red). Image supplied by Dr Wai Shan Yuen.
Notice in Figure 1 that the actin is thickest adjacent to the meiotic spindle. Also, in the slice z21 a polar
body formed from meiosis I is visible. Discriminating between the actin from the polar body and actin from the
egg cell is a challenge which we are yet to resolve. A possible solution is to analyse meiosis I oocytes as these
cells have not yet extruded a polar body.
4 Control data
Control data was generated to validate an algorithm which attempts to match actual data to the closest
theoretical distribution. The control data was generated to emulate the experimental data where approximately
half of the cell is sampled with evenly spaced z stack images, and the slices are centred about the thickest region
of actin in the cell. For a set of possible parameters, r1, θ1, φ1, κ and λ the forward solution is calculated using
equation (12) giving a theoretical actin distribution. Centring the emulated data around the thickest region of
actin in the cell was achieved using the fact that if there is a single point source of signalling molecule, then peak
5
actin intensity on the surface theoretically occurs at the same orientation as the point source relative to the
origin. For a theoretical 70µm cell with an actin distribution centred at φ = 1, slices were taken such that they
were approximately centred about φ = 1. Seventeen slices, each 2µm apart on the z axis, were taken beginning
from the middle of the cell and the final slice was 3µm from the top of the cell. On each slice, actin intensity
was sampled at 20 evenly spaced θ values. Noise was added both multiplicatively and additively at each data
point, j, to the actin data, aj , using the formula
aj = aj + ζMaj + ζA
where aj is control data with simulated noise at a given data point and ζ ∼ N(0, 12 expPk/5) for k = M,A.
We call Pk the noise power. Furthermore, because actin thickness is non-negative, if adding noise results in
aj < 0, then we set aj = 0. Figure 2 illustrates typical control data.
Figure 2: Actin distribution for a typical control problem
5 Inverse problem and results
The inverse problem involves finding parameters r1, φ1, θ1, κ and λ which lead to an actin distribution which
best matches the data. Due to φ1 and θ1 being the most robust parameters and able to be independently
evaluated using a centre of mass approach, as presented by Duxbury [2016], we generated an inverse algorithm
which requires known θ1 and φ1 as inputs. Thus the inverse problem becomes a 3-parameter optimisation in
r1, κ and λ.
6
5.1 Un-weighted least square approach
For the un-weighted approach, an algorithm was created which uses the sum of square residuals as an objective
function; a least square regression was used. Matlab’s fminsearchbnd function, which uses a Nelder-Mead
method, was used to minimise the objective function. Additionally, this function allows bounds to be placed on
parameters, thus preventing impossible parameter spaces from being interrogated (eg κ < 0) and the algorithm
breaking down. This approach is similar to that presented by Duxbury [2016], however, the new approach
notably differs in that calculation of a look-up table is not required. When applied to noise free data the new
approach returns actin distributions and parameters that fit the control problem with very high accuracy and
precision. This is an improvement on the previous method which in some cases returned a bad fit. Figure 3
illustrates a case when the look-up table approach failed to return a good fit while the new approach succeeded.
(a) Solution using look-up table method. (b) Solution using new approach.
Figure 3: Comparison of the least square optimised solution returned using an algorithm with and without a
look-up table
Notice that in Figure 3 (b) the sampled vales of φ′1 do not extend to π due to the fact the new algorithm
was tested using control data where data was only collected on slices spaced such that they do not span the
entire cell.
The un-weighted least square algorithm was also tested on noise containing control data. More specifically,
the algorithm was run on on 5 sets of 50 control problems where each set has a different level of noise power
but the same set of parameters such that r1 = 0.5, φ1 = 1, θ1 = 3, κ = 2 and λ = 1. The errors returned by
the algorithm for each of the parameters (a) r1, (b) κ and (c) λ are presented in Figure 4. For each parameter,
when the noise level was low the average error is close to zero, however, with increasing noise the average error
deviated from zero in a way which appears to be systematic. This change in average error may be due to noise
not being normally distributed which is a result of rejecting a proportion of noise if it resulted in aj becoming
7
negative.
(a) (b)
(c)
Figure 4: Comparison of the least square optimised solution returned using an algorithm with and without a
look-up table
5.2 Weighted least square
For the weighted approach, an algorithm was created which uses the weighted sum of square residuals as
an objective function where the weighting is proportional to the theoretical actin distribution’s gradient at
the interrogated data point. It was decided to heavily weigh data points corresponding to regions on the
distribution where the gradient is steep because the shape of the distribution is determined most strongly by
regions of the steepest gradient. The gradient was calculated using the gradient formula where the two actin
intensities selected corresponded to (1) the φ′1 value of the data point and (2) the φ′1 value which is one degree
(1) larger.
8
This algorithm returned parameters with very high precision and accuracy for noise free control data,
however, each iteration is slower and many more iterations are required using this approach relative to the
un-weighted algorithm. Furthermore, this method returned large errors for all three parameters when being
run on control problems with noise.
6 Discussion
The un-weighted algorithm performed significantly better than the weighted algorithm. Of the three parameters
being optimised, r1, κ and λ, the most robust (able to be returned most accurately) parameter was r1 while
large error was associated with the κ and λ values returned. Thus, for control data our un-weighted algorithm
is well suited to locating the theoretical point source of morphogen production, but not describing morphogen
characteristics (modified breakdown rate or production intensity). The instability of the 3-parameter optimi-
sation exists because very similar actin distributions can result from significantly different parameters. Moving
the point source of morphogen closer to the cell surface, corresponding to ↑ r1, or decreasing the modified
breakdown rate, ↓ κ, or increasing source intensity, ↑ λ, can all increase the amplitude of the distribution in
such a way that change in one variable can be mostly compensated by changes in the other two variables. Figure
5 shows two similar curves corresponding to vastly different parameters.
Figure 5: Similar actin distributions returned for different parameters
A two parameter optimisation of κ and λ should be more reliable as of these two parameters only κ can
effect the shape of the actin distribution. A different limitation of the method we present is that the morphogen
production source is likely from each chromosome rather than from a single point source in the cell (as assumed).
Fortunately, a biological method exists where the chromosomes can be removed from the cell and a glass bead
coated in DNA can be injected and ectopically induce an actin thickening [Deng et al., 2007]. Using data
generated form this approach could allow us to control the location of morophogen production and limit this
9
location to a small volume of the cell, the glass bead, resembling a point source. Furthermore, having a known
location of the morphogen point source could reduce our problem to a two parameter optimisation, therefore
removing the instability associated with the 3 parameter approach and also removing the need to apply a centre
of mass approach to find φ1 and θ1.
7 Conclusion
This report has vigorously developed the mathematics to find the theoretical actin distribution on the oocyte
surface under the assumption that there are any finite number of point sources of morphogen production. For
the case where there is a single point source located at a known angle from the centre of the cell, a 3-parameter
optimisation algorithm was generated which can identify the most likely point source location (radius) as well
as the modified breakdown rate and production intensity of the morphogen. Based on testing with control
data, the error associated with the predicted location (radius) of the point source is small while large error is
associated with the predicted modified breakdown rate and production intensity of the morphogen.
8 Appendix - Deriving the Greens function solution
To find the solution to BVP described by (4a) and (4b), we will transform the BVP using a transformation
mapping r : (r, φ, θ) → r′0 : (r′0, φ′0, θ′0) where the transformation is a rotation such that r′00 sits on the new
z axis as given by φ′0 = 0. By performing this transformation, we simplify the problem so that the solution is
independent of θ′0. The angle from the new z axis to any other point in the domain can be described by
cosφ′0 = (r′0 · r0)/(|r′0||r0|) = cos θ sinφ cos θ0 sinφ0 + sin θ sinφ sin θ0 sinφ0 + cosφ cosφ0
The new radius is given by r′0 = r. Because they are interchangeable, and due to simplicity of notion, we will
avoid using r′0 and instead use r where possible. In our new co-ordinate system, the PDE (4a) can be written
as
−∇2G+ κ2G = δ3(r′0 − r0′0) (14)
Consider equation (14) when r′0 6= r0′0. In this case, equation (14) reduces to the homogeneous Helmholtz
equation which we can solve using a separation of variables approach. Lets look for solutions of the form
G = R(r)Φ(φ′0). Substituting this form of solution into the homogeneous Helmholtz equation gives
−(
Φ
r2
∂
∂r
(r2 ∂G
∂r
)+
R
r2 sinφ′0∂
∂φ′0
(sinφ′0
∂Φ
∂φ′0
))+ κ2RΦ = 0
Notice that the θ′0 dependent term in the Laplacian were omitted and this is because the solution is
independent of θ′0, so the θ′0 dependent term vanish.
Divide through by RΦ/r2, then move all r dependent terms to RHS and keep φ′0 dependent terms on LHS:
− 1
Φ sinφ′0∂
∂φ′0
(sinφ′0
∂Φ
∂φ′0
)=
1
R
∂
∂r
(r2 ∂G
∂r
)− κ2r2
10
Because LHS is independent of r, and RHS is independent of φ′0, but RHS=LHS, then both LHS and RHS
must each be independent of both r and φ′0. That is, we can set both sides of the equation equal to a constant.
In particular, we will choose the constant l(l + 1) for l = 0, 1, 2, 3... This process yields two equations for us to
solve:
1
R
∂
∂r
(r2 ∂G
∂r
)− κ2r2 = l(l − 1) (R)
AND
− 1
Φ sinφ′0∂
∂φ′0
(sinφ′0
∂Φ
∂φ′0
)= l(l − 1) (Φ)
8.1 Solving equation (R)
We can nondimensionalise (R) via rescaling by a factor of κ. Using this rescaling, then moving all terms to
LHS, (R) becomes the Modified Spherical Bessel Differential Equation
τ∂2R
∂τ2+ 2τ
∂R
∂τ− (τ2 + l(l + 1))R = 0
where τ = κr. This has the general solution
Rl = Alil(τ) + Blkl(τ) = Alil(κr) + Blkl(κr) (15)
Where Al and Bl are real constants, il and kl are respectively Modified Spherical Bessel functions of the first
and second kind.
8.2 Solving equation (Φ)
Apply the product rule to equation (Φ), multiply through by −Φ, expand and move all terms to LHS so that
the equation in a from that is recognisable as the Legendre DE:
∂2Φ
∂(φ′0)2+
cosφ′0
sinφ′0∂Φ
∂φ′0+ Φl(l + 1) = 0
This has the general solution
Φl = ClPl(cosφ′0) +DlQl(cosφ′0)
Where Cl and Dl are real constants, Pl and Ql are respectively Legendre polynomials of the first and second
kind. However, we require that our solution is bounded, so we must have Dl = 0 ∀l. Thus
Φl = ClPl(cosφ′0) (16)
8.3 Putting it together
We have now found Rl and Φl, so we can describe all solutions of the form Gl = RlΦl. Furthermore, by the
principle of superposition, a general solution can can be described as the sum of all Gl solutions:
11
G =
∞∑l=0
(Alil(κr) +Blkl(κr))Pl(cosφ′0) (17)
where Al = AlCl and Bl = BlCl
8.4 Determining Al and Bl
Lets divide our solution into a component where ro < r and r < r0 by introducing G±, g±l , A±l and B±l such
that:
G+ =
∞∑l=0
g+l Pl(cosφ′0) =
∞∑l=0
(A+l il(κr) +B+
l kl(κr))Pl(cosφ′0), r0 ≤ r ≤ 1 (18)
G− =
∞∑l=0
g−l Pl(cosφ′0) =
∞∑l=0
(A−l il(κr) +B−l kl(κr))Pl(cosφ′0), 0 ≤ r ≤ r0 (19)
We have four unknowns, A±l and B±l , but only 3 equations, two boundary conditions and the continuity of
Green’s functions. To obtain a fourth equation lets look at the Green’s function ”jump” condition.
8.5 Green’s function jump condition
Multiply equation (14) through by Pn(cosφ′0), where Pn is the nth Legendre polynomial. Then integrate with
respect to r′0 over the origin centred infinitesimally thin spherical shell with radius r which we will denote as
Ωr0
−
Ωr0
Pn(cosφ′0)∇2Gdr +
Ωr0
Pn(cosφ′0)κ2Gdr =
Ωr0
Pn(cosφ′0)δ3(r′0 − r0′0)dr (20)
Using the fact that Pn∇2G = ∇ · (Pn∇G) − ∇Pn∇G, we can decompose the first integral into two further
integrals, one of which vanishes because we are integrating a finite function over an infinitesimally small domain.
We will apply divergence theorem to the remaining integral coming out of the decomposition. The second integral
from equation (20) also vanishes due to being an integration of a finite function over an infinitesimally small
domain. Using properties of the dirac delta function, we can evaluate the LHS integral. Therefore, Equation
(20) becomes
−
Ωr0
∇(Pn(cosφ′0)∇G) · ndr = Pn(cosφ′00 ) (21)
where n is an outward pointing normal unit vector on the shell Ωr0 . On the outside of the shell, which we shall
denote as Ω+r0 , the normal is given by n = r′0. On the inside of the shell, Ω−r0 , n = −r′0. Notice that
∇(Pn(cosφ′0)∇G) · ±r
is simply a scaled directional derivative of G in the inward or outward direction, so can alternatively be expressed
as a scaled partial derivative of G with respect change in radius. Also recall that φ′0 = 0. We can therefore
12
write equation (21) as
Ω+r0
∂G+
∂r
∣∣∣∣r=r0
Pn(cosφ′0)dr−
Ω−r0
∂G−
∂r
∣∣∣∣r=ro
Pn(cosφ′0)dr = −Pn(cos 0) = −1 (22)
Using the definition of G± from equations (18) and (19) in equation (22), then factorising, gives
∞∑l=0
((∂g+
l
∂r
∣∣∣∣r=ro
−∂g−l∂r
∣∣∣∣r=ro
)Ωr0
Pl(cosφ′0)Pn(cosφ′0)dr
)= −1
Integrating in spherical co-ordinates gives
∞∑l=0
((∂g+
l
∂r
∣∣∣∣r=ro
−∂g−l∂r
∣∣∣∣r=ro
) 2π
0
π
0
Pl(cosφ′0)Pn(cosφ′0)r20 sinφ′0dφ′0dθ′0
)= −1
Notice that the expression being integrated is independent of θ′0. Perform a the substitution x = cosφ′0.
Under this change of variables, the equation becomes
−1 = 2πr20
∞∑l=0
(∂g+
l
∂r
∣∣∣∣r=ro
−∂g−l∂r
∣∣∣∣r=ro
) 1
−1
Pl(x)Pn(x)dx
Using orthogonality of Legendre polynomials and rearranging the equation gives the jump condition
−2l + 1
4πr20
=∂g+
l
∂r
∣∣∣∣r=ro
−∂g−l∂r
∣∣∣∣r=ro
(23)
8.6 Using the boundary conditions
Because our solution must be bounded at the origin, we require that B−l = 0. Enforcing the condition (4b),
which can alternatively be expressed as G(1, θ′0φ′0) = 0, gives
B+l = −A+
l
il(κ)
kl(κ)
Using this definition of B+l and the fact that A−l = 0, we can write
g−l = A−l il(κr) (24)
AND
g+l = A+
l
(il(κr)−
il(κ)kl(κr)
kl(κ)
)(25)
8.7 Using continuity of the Greens function
By definition, the Green’s function is continuous, so g+l = g−l . Equating g+
l and g−l from equations (24) and
(25), then solving for A−l yields
A−l = A+l
(1− il(κ)kl(κr0)
kl(κ)il(κr0)
)(26)
As seen in equation (25), g+l can be expressed with A+
l as the only unknowon. We can now express also
express g−l where the only unknown is A+l :
g−l = A+l
(1− il(κ)kl(κr0)
kl(κ)il(κro)
)il(κr) (27)
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If we differentiate (24) and (25) with respect to r at r0, then use these definitions in the jump condition as
described by equation (23) we get:
− (2l + 1)
4πr20
= κA+l
(il(κr)−
il(κ)kl(κr)
kl(κ)
)− κA+
l
(1− il(κ)kl(κr0)
kl(κ)il(κro)
)il(κr) (28)
Solving equation (28) for A+l gives
A+l =
(2l + 1)il(κr0)kl(κ)
4πr2oκil(κ)
∗ 1
il(κr0)k′l(κr0)− i′l(κr0)kl(κr0)
Using the definition of the Wronskian for Modified Spherical Bessel functions, we can simplify our definition of
A+l to
A+l = − (2l + 1)il(κr0)kl(κ)κ
2π2il(κ)(29)
Using this definition A+l in equations (25) and (27), then rearranging RHS for each equation yields
g+l =
(2l + 1)κ
2π2
(il(κr0)kl(κr)−
kl(κ)il(κr0)il(κr)
il(κ)
), r0 ≤ r ≤ 1 (30)
AND
g−l =(2l + 1)κ
2π2
(il(κr)kl(κr0)− kl(κ)il(κr0)il(κr)
il(κ)
), 0 ≤ r ≤ r0 (31)
Using these definitions of g+l and g−l in equations (18) and (19) we can rewrite G+ and G− as
G+ =
∞∑l=0
(2l + 1)κ
2π2
(il(κr0)kl(κr)−
kl(κ)il(κr0)il(κr)
il(κ)
)Pl(cosφ′0), r0 ≤ r ≤ 1 (32)
AND
G− =
∞∑l=0
(2l + 1)κ
2π2
(il(κr)kl(κr0)− kl(κ)il(κr0)il(κr)
il(κ)
)Pl(cosφ′0), 0 ≤ r ≤ r0 (33)
We can alternatively express G as
G =
∞∑l=0
(2l + 1)κ
2π2
(il(κr<)kl(κr>)− kl(κ)il(κr0)il(κr)
il(κ)
)Pl(cosφ′0), 0 ≤ r ≤ 1 (34)
where r< = min(r, r0) and r> = max(r, r0)
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