# Enantioselective chiral orientation induced by a combination of a long and a
short laser pulse

Long Xu<EMAIL_ADDRESS>AMOS and Department of Chemical and Biological
Physics, The Weizmann Institute of Science, Rehovot 7610001, Israel

###### Abstract

Enantioselective orientation of chiral molecules excited by a shaped
picosecond laser pulse and a delayed femtosecond pulse is considered. Using
quantum mechanical simulations, we demonstrate a strong field-free
enantioselective orientation along the laser propagation direction. In
addition, we use a classical model to reproduce the enantioselective
orientation. Moreover, the analysis of the corresponding classical system
allows understanding the qualitative features of the induced enantioselective
orientation. The strong enantioselective orientation may be used for the
separation of chiral enantiomers using inhomogeneous electrostatic fields.

##  I Introduction

Chiral molecules exist in two species — left- and right-handed enantiomers.
These enantiomers are mirror images of each other and they cannot be
superimposed on each other by translations and rotations [1]. Since its
discovery by Louis Pasteur in 1848 [2], the phenomenon of molecular chirality
has gained immense importance in biology, physics, chemistry, and the
pharmaceutical industry. Various methods for chiral discrimination have been
proposed and implemented over the years. Examples involving laser fields
include photoelectron circular dichroism [3, 4, 5, 6, 7], microwave three-wave
mixing spectroscopy [8, 9, 10, 11, 12, 13, 14, 15], Coulomb explosion imaging
[16, 17, 18], high-order harmonic generation [19], and enantiospecific
interaction with achiral magnetic substrates [20].

In the last few years, the enantioselective orientation of chiral molecules
excited by laser pulses with twisted polarization has been investigated [21,
22, 23, 24, 25, 26]. Examples of fields with twisted polarization include the
delayed cross-polarized laser pulses [27, 28, 29], the optical centrifuge for
molecules [30, 31, 32, 33, 34], chiral pulse trains [35, 36], and
polarization-shaped pulses [37, 38, 39, 40, 41]. In addition, it was
theoretically shown that terahertz (THz) pulses with twisted polarization [42]
and two-color laser pulses [43, 44] are effective for inducing
enantioselective orientation.

When molecules are exposed to linearly polarized non-resonant laser fields,
the most polarizable molecular axis tends to align along the polarization
direction. When molecules are excited by laser pulses with twisted
polarization, the most polarizable molecular axis tends to follow the rotating
polarization vector, resulting in unidirectional rotation [27, 28, 29, 45, 46,
47]. Furthermore, in the case of chiral molecules, such twisted fields induce
an orienting torque along the most polarizable axis [22, 23]. This additional
torque enantioselectively orients the chiral molecules perpendicular to the
plane of polarization twisting. The orientation direction (along or against
the laser propagation direction) depends on the sense of twisting and the
handedness of the molecule.

In this paper, we study the orientation of chiral molecules excited by a
linearly polarized shaped picosecond laser pulse [48, 49, 50, 51, 52] and a
delayed co-propagating cross-polarized femtosecond pulse. A slowly rising and
sharply truncated picosecond pulse is used to produce a highly aligned
molecular state. With long pulses, a high degree of alignment can be reached
without significant molecular ionization. Achieving the same degree of
alignment with femtosecond laser pulse would require a much higher peak power,
which inevitably results in sizable molecular ionization. After the shaped
picosecond pulse is suddenly switched off, a cross-polarized femtosecond laser
pulse is applied, leading to enantioselective orientation along a direction
perpendicular to the plane of polarization twisting, that is, along the
propagation direction of the pulses. The highly aligned state induced by the
picosecond pulse enables a high degree of enantioselective orientation. A
classical model reproduces well the enantioselective orientation, and is used
to analyze the various qualitative features of the effect, including the
vanishing orientation along any direction in the plane of laser polarizations,
the enantioselective orientation along the laser propagation direction, and
the roles played by the picosecond and the femtosecond laser pulses.

Table 1: Molecular properties of (_R_)-PPO molecule: moments of inertia (in atomic units), elements of the dipole moment (in Debye), polarizability tensors (in atomic units) expressed in the frame of principal axes of inertia.  Molecule |  Moments of inertia |  Dipole components |  Polarizability components
---|---|---|---
(_R_)-PPO |  $I_{a}=180386$ $I_{b}=493185$ $I_{c}=553513$ |  $\mu_{a}=0.965$ $\mu_{b}=-1.733$ $\mu_{c}=0.489$ |  $\alpha_{aa}=45.63$, $\alpha_{ab}=2.56$ $\alpha_{bb}=37.96$, $\alpha_{ac}=0.85$ $\alpha_{cc}=37.87$, $\alpha_{bc}=0.65$

##  II Numerical methods

For the quantum mechanical simulations, we consider the chiral molecule as a
rigid rotor. The Hamiltonian describing the molecular rotation driven by the
laser field is given by $H(t)=H_{r}+H_{\mathrm{int}}(t)$ [53, 54], where
$H_{r}$ is the field-free Hamiltonian and
$H_{\mathrm{int}}(t)=-\mathbf{E}\cdot(\bm{\alpha}\mathbf{E})/2$ is the
interaction of molecule with the laser field. Here $\bm{\alpha}$ is the
molecular polarizability tensor and $\mathbf{E}$ is the external electric
field. The wave function is expressed in the basis of field-free symmetric-top
eigenfunctions $|JKM\rangle$ [55]. Here $J$ is the total angular momentum,
while $K$ and $M$ are its projections on the molecule-fixed $z$ and the
laboratory-fixed $Z$ axes respectively. The nonzero matrix elements of the
field-free Hamiltonian are [55]

$\displaystyle\langle
JKM|H_{r}|JKM\rangle=\frac{B+C}{2}\left[J(J+1)-K^{2}\right]+AK^{2},$
$\displaystyle\langle JKM|H_{r}|J,K\pm 2,M\rangle=\frac{B-C}{4}f(J,K\pm 1),$
(1)

where $f(J,K)=\sqrt{(J^{2}-K^{2})[(J+1)^{2}-K^{2}]}$, $A=\hbar^{2}/2I_{a}$,
$B=\hbar^{2}/2I_{b}$, $C=\hbar^{2}/2I_{c}$ with the moments of inertia
$I_{a}<I_{b}<I_{c}$. The initial state is the ground state, corresponding to a
rotational temperature of $0\,\mathrm{K}$. The time-dependent Schrödinger
equation $i\hbar\partial_{t}|\Psi(t)\rangle=H(t)|\Psi(t)\rangle$ is solved by
numerical exponentiation of the Hamiltonian matrix (see Expokit [56]). A
detailed description of our numerical scheme can be found in [25].

##  III strong enantioselective orientation

Here we consider propylene oxide molecule (PPO, $\mathrm{CH_{3}CHCH_{2}O}$) as
a typical example of a chiral molecule. Table 1 summarizes the molecular
properties of (_R_)-PPO (right-handed enantiomer), which were computed using
GAUSSIAN software package (method: CAM-B3LYP/aug-cc-pVTZ) [57]. The molecules
are excited by a shaped picosecond laser pulse followed by a femtosecond laser
pulse. Both pulses propagate in the same direction, the $Z$ direction. The
picosecond laser pulse is polarized along the laboratory-fixed $X$ axis and it
is used to produce a highly aligned molecular state, while the femtosecond
laser pulse, polarized at angle $\pi/4$ to the polarization of the picosecond
pulse, is used to induce the enantioselective orientation. The electric field
of the shaped picosecond laser pulse (see Fig. 1) is given by

$\mathbf{E}^{p}(t)=E^{p}_{0}\exp\left(-2\ln
2\frac{t^{2}}{\sigma^{2}}\right)\cos(\omega t)\mathbf{e}_{X},$ (2)

where

$\begin{cases}\sigma=\sigma_{\mathrm{on}},&t<0,\\\
\sigma=\sigma_{\mathrm{off}},&t\geq 0.\end{cases}$

The electric field of the femtosecond laser pulse is

$\displaystyle\mathbf{E}^{f}(t)=\frac{E^{f}_{0}}{\sqrt{2}}\exp\left[-2\ln
2\frac{(t-\tau)^{2}}{\sigma_{f}^{2}}\right]\cos(\omega
t)(\mathbf{e}_{X}+\mathbf{e}_{Y}).$ (3)

Here $E^{p}_{0}\,(E^{f}_{0})$ is the peak amplitude of the picosecond
(femtosecond) field, $\omega$ is the carrier frequency, and
$\mathbf{e}_{X}\,(\mathbf{e}_{Y})$ is a unit vector along the laboratory
$X\,(Y)$ axis. $\sigma_{\mathrm{on}}$ and $\sigma_{\mathrm{off}}$ represent
the full width at half maximum (FWHM) of the slow switch-on and the rapid
switch-off, respectively. $\sigma_{f}$ is the FWHM of the femtosecond laser
pulse and $\tau$ is defined as the time at which the femtosecond laser pulse
is applied.

Figure 1: Quantum mechanically calculated time-dependent degree of molecular
alignment along the polarization direction of the picosecond pulse. The
initial temperature is $0\,\mathrm{K}$. Note that the degrees of alignment for
the two enantiomers, (_R_)- and (_S_)-PPO are the same. The green dashed line
represents the envelope of the shaped picosecond field [see Eq. (2)]. The
inset shows the magnified portion of
$\braket{\cos^{2}\,(\theta_{\alpha_{1}X})}$.

Figure 1 shows the time-dependent degree of alignment, quantified by the
average of the squares of the directional cosines,
$\braket{\cos^{2}\,(\theta_{aX})}(t)=\langle\Psi(t)|(\bm{a}\cdot\mathbf{e}_{X})^{2}|\Psi(t)\rangle$
and
$\braket{\cos^{2}\,(\theta_{\alpha_{1}X})}(t)=\langle\Psi(t)|(\bm{\alpha}_{1}\cdot\mathbf{e}_{X})^{2}|\Psi(t)\rangle$.
$\bm{a}=(0,0,1)$ is the molecule-fixed $a$ axis, while $\bm{\alpha}_{1}$ is
the most polarizable molecular axis and for (_R_)-PPO molecule,
$\bm{\alpha}_{1}=(0.293,0.115,0.949)$. The vectors are expressed in the
molecule-fixed frame using the basis consisting of the three inertia principal
axes $b,\,c,\,a$, namely $x\rightarrow b$, $y\rightarrow c$, $z\rightarrow a$,
such that an arbitrary vector $\bm{r}$ has the Cartesian components
$\bm{r}=(r_{b},r_{c},r_{a})$ in the molecule-fixed frame. $\theta_{aX}$
($\theta_{\alpha_{1}X}$) denotes the angle between the molecule-fixed $a$
($\alpha_{1}$) axis and the laboratory-fixed $X$ axis (polarization direction
of the picosecond pulse). Here the initial state is the ground rotational
state (temperature is $0\,\mathrm{K}$). The peak intensity of the picosecond
laser pulse is $I^{p}=10^{12}\,\mathrm{W/cm^{2}}$, the FWHMs are
$\sigma_{\mathrm{on}}=50\,\mathrm{ps}$ and
$\sigma_{\mathrm{off}}=2\,\mathrm{ps}$. In the presence of the picosecond
pulse, the most polarizable molecular $\alpha_{1}$ axis is aligned along the
polarization direction. Figure 1 shows high degrees of alignment of the
molecular $a$ and $\alpha_{1}$ axes building up during the picosecond laser
pulse. The maximal degrees of alignment are
$\braket{\cos^{2}\,(\theta_{aX})}\simeq 0.82$ and
$\braket{\cos^{2}\,(\theta_{\alpha_{1}X})}\simeq 0.88$.

Figure 2: Quantum expectation value of $Z$ projection of the dipole signal,
$\braket{\mu_{Z}}(t)$ or the degree of orientation,
$\braket{\cos\,(\theta_{\mu Z})}\equiv\braket{\mu_{Z}}(t)/\mu$. Here
$\theta_{\mu Z}$ is the angle between the molecular dipole moment and the
laboratory $Z$ axis, $\mu$ is the magnitude of the dipole moment. (_R_)- and
(_S_)-PPO denote the right- and left-handed enantiomers of the PPO molecule,
respectively. Note that the signals are $\pi$ out of phase for the two
enantiomers.

During the turn-off stage of the shaped picosecond pulse, around the time when
the degree of alignment at maximum (see the inset in Fig. 1), a cross-
polarized femtosecond laser pulse is applied. The femtosecond laser pulse is
polarized at angle $\pi/4$ to the $X$ axis. The peak intensity of the
femtosecond laser pulse is $I^{f}=10^{13}\,\mathrm{W/cm^{2}}$, its FWHM is
$\sigma_{f}=0.1\,\mathrm{ps}$, and $\tau=0.77\,\mathrm{ps}$, see Eq. (3).
Figure 2 shows the time-dependent dipole signal along the laboratory $Z$ axis,
$\braket{\mu_{Z}}(t)=\langle\Psi(t)|\bm{\mu}\cdot\mathbf{e}_{Z}|\Psi(t)\rangle$
($\bm{\mu}$ is the permanent dipole moment). As expected, shortly after the
pulse, enantioselective orientation along the laser propagation direction
appears, namely the sign of $\braket{\mu_{Z}}(t)$ is positive (negative) for
(_R_)-PPO and negative (positive) for (_S_)-PPO. The degree of orientation
along any axis in the $XY$ plane is identically zero. The general behavior is
consistent with previous studies on the enantioselective orientation induced
by laser pulses with twisted polarization [21, 22, 23, 24, 25, 26]. The scheme
presented here, which is based on the shaped picosecond and femtosecond laser
pulses, achieves comparable degree of orientation, resulting in a field-free
transient dipole signal of about 0.47 Debye corresponding to a degree of
orientation $\braket{\cos\,(\theta_{\mu Z})}\simeq 0.23$. The high degree of
alignment generated by the picosecond field (see Fig. 1) serves as a basis for
inducing the high degree of enantioselective orientation using the femtosecond
laser pulse polarized at angle $\pi/4$ to the polarization direction of the
picosecond pulse.

##  IV Qualitative description — Classical model

To understand the origin of the enantioselective orientation shown in Fig. 2
and the role of each of the laser pulses, we resort to a classical model. We
consider a collection of classical non-interacting chiral molecules, where
each molecule is modeled by a rigid polarizable asymmetric-top. The picosecond
field prepares a highly aligned molecular ensemble, where the most polarizable
molecular $\alpha_{1}$ axis points, on average, along the field polarization
direction. To simplify the analysis, we assume that initially (immediately
after the end of the picosecond pulse) all the molecular $a$ axes (which are
close to the $\alpha_{1}$ axis) are perfectly aligned along the laboratory $X$
axis, and we take the initial temperature to be $0\,\mathrm{K}$. The aligned
molecular ensemble is then excited, at $t=\tau=0.77\,\mathrm{ps}$, by a
femtosecond laser pulse polarized at angle $\pi/4$ to the $X$ axis (in the
$XY$ plane), see Eq. (3).

Classically, the rotation of a rigid body is described by Euler’s equations
[58]

$\mathbf{I}\bm{\dot{\Omega}}=(\mathbf{I}\bm{\Omega})\times\bm{\Omega}+\mathbf{T},$
(4)

where $\bm{\Omega}=(\Omega_{b},\Omega_{c},\Omega_{a})$ is the angular velocity
vector, $\mathbf{I}=\mathrm{diag}(I_{b},I_{c},I_{a})$ is the moment of inertia
tensor, and $\mathbf{T}=(T_{b},T_{c},T_{a})$ is the external torque vector.
All the quantities in Eq. (4) are expressed in the molecule-fixed frame of
reference using the basis consisting of the three principal axes of inertia,
$b,\,c,\,a$. The torque is given by
$\mathbf{T}=\boldsymbol{\alpha}\mathbf{E}\times\mathbf{E}$ and it can be
obtained by expressing the electric field vector in the molecule-fixed frame.
The relation between the laboratory-fixed and the molecule-fixed frames is
described by $(X,Y,Z)^{T}=U(\phi,\theta,\chi)(b,c,a)^{T}$, where the
transformation matrix is given by [55]

$\displaystyle
U(A)=\begin{pmatrix}\cos(\phi)\cos(\theta)\cos(\chi)-\sin(\phi)\sin(\chi)&-\cos(\phi)\cos(\theta)\sin(\chi)-\sin(\phi)\cos(\chi)&\cos(\phi)\sin(\theta)\\\
\sin(\phi)\cos(\theta)\cos(\chi)+\cos(\phi)\sin(\chi)&-\sin(\phi)\cos(\theta)\sin(\chi)+\cos(\phi)\cos(\chi)&\sin(\phi)\sin(\theta)\\\
-\sin(\theta)\cos(\chi)&\sin(\theta)\sin(\chi)&\cos(\theta)\end{pmatrix}.$ (5)

Here $A=(\phi,\theta,\chi)$ is a shorthand notation for the three Euler
angles: $\phi$ and $\theta$ are the azimuthal and polar angles defining the
orientation of the molecular $z$ axis (associated with the molecular $a$ axis
here) in the laboratory-fixed frame, and $\chi$ is the additional rotation
angle about the $z$ axis, see Fig. 3. The rates of change of the Euler angles
are given by [55]

$\displaystyle\dot{\phi}$
$\displaystyle=\frac{-\Omega_{b}\cos(\chi)+\Omega_{c}\sin(\chi)}{\sin(\theta)},$
(6a) $\displaystyle\dot{\theta}$
$\displaystyle=\Omega_{b}\sin(\chi)+\Omega_{c}\cos(\chi),$ (6b)
$\displaystyle\dot{\chi}$
$\displaystyle=\Omega_{a}-\cos(\theta)\frac{-\Omega_{b}\cos(\chi)+\Omega_{c}\sin(\chi)}{\sin(\theta)}.$
(6c)

To describe the ensemble behavior, we use the Monte Carlo approach and
consider the dynamics of $N\gg 1$ chiral molecules. For each molecule, we
numerically solve a system of differential equations consisting of Eqs. (4)
and (6a-6c). The $N$ molecules are divided into two groups, I and II. The
initial state of the molecules in group I is given by
$A(t_{0})=(0,\pi/2,\chi_{0})$, where $\chi_{0}$ is uniformly distributed
between 0 and $2\pi$, corresponding to the initial state in which the
molecular $a$ axes point _along_ the laboratory $X$ axis [see Fig. 3(a)]. The
initial state of the molecules in group II is defined by
$B(t_{0})=(\pi,\pi/2,\chi_{0})$, such that the molecular $a$ axes point
_against_ the laboratory $X$ axis [see Fig. 3(b)]. The initial angular
velocities are set to $\Omega_{i}=0$, $i=b,c,a$, corresponding to a
temperature of $0\,\mathrm{K}$.

Figure 3: A set of three Euler rotations (in accordance with the convention of
[55]) describing the orientation of the molecule-fixed frame ($bca$) relative
to the laboratory-fixed frame ($XYZ$) for (a) $A=(0,\pi/2,\pi/6)$ and (b)
$B=(\pi,\pi/2,\pi/6)$. The blue arrows represent the dipole moment
($\bm{\mu}$) of a (_R_)-PPO molecule.

For convenience, the femtosecond laser pulse is treated as an impulsive
excitation, such that the angular velocity just after the pulse is directly
proportional to the applied torque, $\Omega_{i}\propto T_{i}/I_{i}$.
Considering two molecules, one from each group, in initial states $A(t_{0})$
and $B(t_{0})$, the torque components in the rotating molecular frame are the
same for the two molecules. After averaging over the rapid oscillations of the
laser field (the carrier frequency, $\omega$ is larger than the typical
rotation frequency of the molecule by several orders of magnitude), the torque
components are given by

$\displaystyle T_{b}(\chi_{0})$
$\displaystyle=\frac{(E^{f})^{2}}{4}\Big{[}(\alpha_{cc}-\alpha_{aa})\cos(\chi_{0})+\alpha_{bc}\sin(\chi_{0})$
$\displaystyle+\alpha_{ac}\sin^{2}(\chi_{0})-\alpha_{ab}\sin(\chi_{0})\cos(\chi_{0})\Big{]},$
(7a) $\displaystyle T_{c}(\chi_{0})$
$\displaystyle=\frac{(E^{f})^{2}}{4}\Big{[}(\alpha_{aa}-\alpha_{bb})\sin(\chi_{0})-\alpha_{bc}\cos(\chi_{0})$
$\displaystyle-\alpha_{ab}\cos^{2}(\chi_{0})+\alpha_{ac}\sin(\chi_{0})\cos(\chi_{0})\Big{]},$
(7b) $\displaystyle T_{a}(\chi_{0})$
$\displaystyle=\frac{(E^{f})^{2}}{4}\Big{[}(\alpha_{bb}-\alpha_{cc})\sin(\chi_{0})\cos(\chi_{0})$
$\displaystyle+\alpha_{bc}\cos(2\chi_{0})+\alpha_{ab}\cos(\chi_{0})-\alpha_{ac}\sin(\chi_{0})\Big{]}.\\!\\!$
(7c)

Figure 4 shows the classically calculated ensemble average,
$\braket{\mu_{Z}}(t)$. Here, the initial conditions and the field parameters
of the femtosecond pulse are the same as in Fig. 2. Comparing Figs. 2 and 4,
it is evident that the simplified classical model qualitatively reproduces the
enantioselective orientation. Therefore, in what follows, we analyze the
system using the classical model in order to gain physical insight to the
qualitative features of the enantioselective orientation.

Figure 4: Classically calculated $Z$ projection of the dipole signal or the
degree of orientation. Initially, the molecular $a$ axes are aligned along the
$X$ axis, and then the aligned molecules are excited by a femtosecond laser
pulse polarized at an angle $\pi/4$ to the $X$ axis. The initial temperature
is $0\,\mathrm{K}$. The field parameters of the femtosecond laser pulse are
the same as the case shown in Fig. 2. $N=10^{7}$ molecules are used in the
classical simulation.

###  IV.1 Absence of orientation in the plane of the pulses’ polarizations —
$XY$ plane

In this subsection, we consider the orientation along the laser propagation
direction and the absence thereof of orientation in any perpendicular
direction. Here, we focus on a single enantiomer. The next subsection
addresses the enantioselectivity. As can be seen from Eqs. (6a-6c) and Eqs.
(7a-7c), the torques (the induced angular velocities) and the rate of change
of the Euler angles are independent of $\phi$. Accordingly, the initial states
$A(t_{0})$ and $B(t_{0})$ evolve into
$A(t)=(\delta\phi,\delta\theta+\pi/2,\delta\chi+\chi_{0})$ and
$B(t)=(\delta\phi+\pi,\delta\theta+\pi/2,\delta\chi+\chi_{0})$, respectively.
Here $\delta\phi$, $\delta\theta$, and $\delta\chi$ are the changes in Euler
angles resulting from the rotational dynamics induced by the femtosecond laser
pulse kick. Notice the $\pi$ difference between $\phi$ angles in $A(t)$ and
$B(t)$.

The time-dependent projections of the dipole moment on the laboratory axes can
be obtained using the transformation matrix $U$ [see Eq. (5)]

$\displaystyle U\left[A(t)\right](\mu_{b},\mu_{c},\mu_{a})^{T}$
$\displaystyle=(\mu_{X},\mu_{Y},\mu_{Z})^{T},$ (8a) $\displaystyle
U\left[B(t)\right](\mu_{b},\mu_{c},\mu_{a})^{T}$
$\displaystyle=(-\mu_{X},-\mu_{Y},\mu_{Z})^{T}.$ (8b)

Equations (8a) and (8b) follow from the fact that the initial $\pi$ difference
in the $\phi$ angles of $A$ and $B$ is preserved throughout the motion. The
same relation can also be found in Fig. 7, where we plot the dipole signals as
functions of $\chi_{0}$ for the two molecular groups ($\phi_{0}=0$ and
$\phi_{0}=\pi$). The dipole moment projections on the $X$ and $Y$ axes are of
opposite sign, and when averaged over the entire molecular ensemble (groups I
and II), resulting in zero averaged dipole signal along any direction in the
$XY$ plane, including $\braket{\mu_{X}}=\braket{\mu_{Y}}=0$. In contrast, the
dipole moment projections on the $Z$ axis are of the same sign, giving rise to
the nonzero ensemble-averaged orientation for chiral molecules, while for non-
chiral molecules, $\braket{\mu_{Z}}=0$ (see below for further details).

###  IV.2 Enantioselective orientation along the laser propagation direction —
$Z$ direction

Figure 5: Two initial orientations of the molecular frame for (a)
$A=(0,\pi/2,\chi_{0})$ and (b) $B_{1}=(\pi,\pi/2,\pi-\chi_{0})$. (c) Same as
panel (b) for $B_{1}^{*}=(0,\pi/2,\chi_{0})$, but in terms of the basis
$(b^{*},c^{*},a^{*})=(-b,c,-a)$. Here, $\chi_{0}=\pi/6$. $\bm{\mu}^{(R)}$ and
$\bm{\mu}^{(S)}$ denote dipole moments of (_R_)- and (_S_)-PPO molecules,
respectively. Note that $\bm{\mu}^{(S)}$ in (b) and (c) point in the same
directions, while $\bm{\mu}^{(R)}$ in (a) points in the opposite direction.

In this subsection, we consider the implications of chirality on molecular
orientation. The two enantiomers are related by a reflection transformation
[1], here we set the $ab$-plane as the reflection plane. Accordingly, the
dipole moment components and the polarizability components of the two
enantiomers [(_S_) and (_R_)] are related as $\mu_{c}^{(S)}=-\mu_{c}^{(R)}$,
$\alpha_{ac}^{(S)}=-\alpha_{ac}^{(R)}$, and
$\alpha_{bc}^{(S)}=-\alpha_{bc}^{(R)}$. The other components are the same for
both enantiomers.

To analyze the enantioselective orientation, we begin with two enantiomers of
the propylene oxide molecule, (_R_)- and (_S_)-PPO. Initially, the molecular
$a$ axes are aligned along the $X$ axis, while the dipole moments of the two
enantiomers point in the opposite directions (see Fig. 5). Without loss of
generality, we consider the initial states $A(t_{0})=(0,\pi/2,\chi_{0})$
[panel (a)] and $B_{1}(t_{0})=(\pi,\pi/2,\pi-\chi_{0})$ [panel (b)] for (_R_)-
and (_S_)-PPO, respectively. For these two states,
$\boldsymbol{\mu}^{(R)}=-\boldsymbol{\mu}^{(S)}$. Figure 6 shows the torque
components [see Eqs. (7a-7c)] as functions of $\chi_{0}$ for the two
enantiomers. The relation between the torque components is given by

$T_{i}^{(S)}(\pi-\chi_{0})=(-1)^{\delta_{ci}+1}T_{i}^{(R)}(\chi_{0}),\;i=b,\,c,\,a.$
(9)

To simplify the comparison of the two enantiomers, we introduce a new basis in
the molecular frame of $(S)$-PPO, consisting of the $b^{*},c^{*},a^{*}$ axes,
where $(b^{*},c^{*},a^{*})=C_{2}(c)(b,c,a)=(-b,c,-a)$ and $C_{2}(c)$ is the
two-fold rotation about the $c$ axis [55]. In terms of the new basis, the
molecular orientation in Fig. 5(b) is given by
$B_{1}^{*}(t_{0})=(0,\pi/2,\chi_{0})$ [see Fig. 5(c)]. Notice,
$B_{1}^{*}(t_{0})=A(t_{0})$. The relation between the torque components of
$(R)$-PPO and $(S)$-PPO (in terms of the new basis) is

$T_{i^{*}}^{(S)}(\chi_{0})=T_{i}^{(R)}(\chi_{0}),$ (10)

because
$T_{i^{*}}^{(S)}(\chi_{0})=(-1)^{\delta_{ci}+1}T_{i}^{(S)}(\pi-\chi_{0})$,
$i^{*}=b^{*},\,c^{*},\,a^{*}$.

Figure 6: Torque components [Eqs. (7a-7c)] as functions of $\chi_{0}$ for
(_R_)-PPO (solid) and (_S_)-PPO (dashed) molecules with $E^{f}=1$.

Thus, in this new basis, $(b^{*},c^{*},a^{*})$, the torque components _and_
the initial states for both enantiomers in Fig. 5 are the _same_. As a
consequence, $B_{1}^{*(S)}(t)=A^{(R)}(t)$ [see Eqs. (6a-6c)]. Moreover, since
$\left(\mu_{b^{*}}^{(S)},\mu_{c^{*}}^{(S)},\mu_{a^{*}}^{(S)}\right)=\left(-\mu_{b}^{(S)},\mu_{c}^{(S)},-\mu_{a}^{(S)}\right)=-\left(\mu_{b}^{(R)},\mu_{c}^{(R)},\mu_{a}^{(R)}\right)$,
we obtain the relations

$\displaystyle\begin{pmatrix}\begin{matrix}\mu_{X}^{(S)}[B_{1}(t)]\\\
\mu_{Y}^{(S)}[B_{1}(t)]\\\
\mu_{Z}^{(S)}[B_{1}(t)]\end{matrix}\end{pmatrix}=U\left[B_{1}^{(S)}(t)\right]\begin{pmatrix}\begin{matrix}\mu_{b}^{(S)}\\\
\mu_{c}^{(S)}\\\
\mu_{a}^{(S)}\end{matrix}\end{pmatrix}=U\left[B_{1}^{*(S)}(t)\right]\begin{pmatrix}\begin{matrix}\mu_{b^{*}}^{(S)}\\\
\mu_{c^{*}}^{(S)}\\\
\mu_{a^{*}}^{(S)}\end{matrix}\end{pmatrix}=-U\left[A^{(R)}(t)\right]\begin{pmatrix}\begin{matrix}\mu_{b}^{(R)}\\\
\mu_{c}^{(R)}\\\
\mu_{a}^{(R)}\end{matrix}\end{pmatrix}=-\begin{pmatrix}\begin{matrix}\mu_{X}^{(R)}[A(t)]\\\
\mu_{Y}^{(R)}[A(t)]\\\
\mu_{Z}^{(R)}[A(t)]\end{matrix}\end{pmatrix},\\!\\!\\!\\!\\!$ (11)

which indicates that the opposite sign between $\bm{\mu}^{(S)}$ and
$\bm{\mu}^{(R)}$ is preserved throughout the motion (which can also be seen
from Figs. 7 and 8). As discussed above in Sec. IV.1, the ensemble-averaged
dipole signal of each enantiomer along any direction in the $XY$ plane is
zero. Consequently, the orientation appears in the $Z$ direction and it is
_enantioselective_. Note, however, that in the case of a racemic mixture
(which has equal amounts of left- and right-handed enantiomers), the
orientation in the $Z$ direction vanishes. Similar and simplified arguments
were previously used in [22, 23] to explain the enantioselective orientation.

###  IV.3 Chirality dependence of the orientation along the laser propagation
direction – $Z$ direction

In the special case of non-chiral molecules, having diagonal polarizability
tensor, the torque components in Eqs. (7a-7c) reduce to

$\displaystyle T_{b}(\chi_{0})$
$\displaystyle=\frac{(E^{f})^{2}}{4}(\alpha_{cc}-\alpha_{aa})\cos(\chi_{0}),$
(12a) $\displaystyle T_{c}(\chi_{0})$
$\displaystyle=\frac{(E^{f})^{2}}{4}(\alpha_{aa}-\alpha_{bb})\sin(\chi_{0}),$
(12b) $\displaystyle T_{a}(\chi_{0})$
$\displaystyle=\frac{(E^{f})^{2}}{4}(\alpha_{bb}-\alpha_{cc})\sin(\chi_{0})\cos(\chi_{0}).$
(12c)

In contrast to the case of chiral molecules (see Fig. 6), here the torque
components have a well defined symmetry (symmetric/antisymmetric) about
$\chi_{0}=n\pi/2$, $n=0,1,2,3$. For instance,

$\displaystyle T_{i}(-\chi_{0})=(-1)^{\delta_{bi}+1}T_{i}(\chi_{0}),$ (13a)
$\displaystyle T_{i}(\pi-\chi_{0})=(-1)^{\delta_{ci}+1}T_{i}(\chi_{0}),$ (13b)
$\displaystyle T_{i}(\pi+\chi_{0})=(-1)^{\delta_{ai}+1}T_{i}(\chi_{0}).$ (13c)

Note that the relation in Eq. (13b) is the same as in Eq. (9). When the dipole
moment projection on the molecular $c$ axis is zero ($\mu_{c}=0$), all the
expressions from Sec. IV.2 still apply. Similar to Eq. (11), the relation
between the dipole moments corresponding to the initial states of $A(t_{0})$
and $B_{1}(t_{0})$ is given by

$\displaystyle\begin{pmatrix}\begin{matrix}\mu_{X}[B_{1}(t)]\\\
\mu_{Y}[B_{1}(t)]\\\
\mu_{Z}[B_{1}(t)]\end{matrix}\end{pmatrix}=-\begin{pmatrix}\begin{matrix}\mu_{X}[A(t)]\\\
\mu_{Y}[A(t)]\\\ \mu_{Z}[A(t)]\end{matrix}\end{pmatrix}.$ (14)

Consequently, the ensemble-averaged dipole signal along any laboratory axis is
zero, and this includes $\braket{\mu_{Z}}$ the projection along the laser
propagation direction. Compared to the two enantiomers in Sec. IV.2, the two
non-chiral molecules in initial states $A(t_{0})$ and $B(t_{0})$ are
indistinguishable and can be viewed as a mixture containing equal amounts of
left- and right-handed enantiomers, leading to the vanishing of the ensemble-
averaged orientation. Moreover, notice that
$\mu_{Z}[B_{1}(t)]=-\mu_{Z}[A(t)]=-\mu_{Z}[B(t)]$ [see Eqs. (8a) and (8b)],
such that even within each group, $A(t_{0})$ or $B(t_{0})$, the dipole signal
$\braket{\mu_{Z}}$ vanishes. The symmetries of the torque components [see Eqs.
(13a) and (13c)] allow repeating the same arguments for molecules in which
$\mu_{b}=0$ or $\mu_{a}=0$.

In contrast, in the case of a single enantiomer of chiral molecules, the
torque components do not have symmetry similar to those in Eqs. (13a-13c),
resulting in the nonzero orientation along the $Z$ axis. This orientation
depends on the polarizability tensor being non-diagonal, which is the case for
chiral molecules.

###  IV.4 Roles of the laser pulses

For a collection of isotropically distributed chiral molecules, collinearly
polarized one-color laser pulses can only induce molecular alignment, but no
orientation. Furthermore, orthogonally polarized laser pulses which do not
overlap cannot induce orientation either. The reason is that the field-
polarizability interaction and the spatial distribution of the molecular
ensemble are always symmetric about the axis parallel/perpendicular to the
polarization direction.

For the sake of simplicity, we consider an ensemble with the molecular $a$
axes initially aligned along the $X$ axis ($\phi_{0}=0,\pi$, and
$\theta_{0}=\pi/2$) and being excited by a $Y$-polarized laser pulse. In this
case, the torque components denoted as $T_{i}^{\prime}$ can be written as

$\displaystyle T_{b}^{\prime}(\chi_{0})$
$\displaystyle=-\frac{(E^{f})^{2}}{2}\big{[}\alpha_{ab}\sin(\chi_{0})\cos(\chi_{0})+\alpha_{ac}\cos^{2}(\chi_{0})\big{]},$
$\displaystyle T_{c}^{\prime}(\chi_{0})$
$\displaystyle=\frac{(E^{f})^{2}}{2}\big{[}\alpha_{ab}\sin^{2}(\chi_{0})+\alpha_{ac}\sin(\chi_{0})\cos(\chi_{0})\big{]},$
$\displaystyle T_{a}^{\prime}(\chi_{0})$
$\displaystyle=\frac{(E^{f})^{2}}{4}\big{[}2\alpha_{bc}\cos(2\chi_{0})+(\alpha_{bb}-\alpha_{cc})\sin(2\chi_{0})\big{]}.$

Accordingly, $T_{i}^{\prime}(\pi+\chi_{0})=T_{i}^{\prime}(\chi_{0})$ and
shortly after the pulse,
$\Omega_{i}^{\prime}(\pi+\chi_{0})=\Omega_{i}^{\prime}(\chi_{0})$. From Eqs.
(6a-6c), it follows that for the same $\theta_{0}$,

$\displaystyle\delta\theta^{\prime}(\pi+\chi_{0})=-\delta\theta^{\prime}(\chi_{0}),\quad\delta\chi^{\prime}(\pi+\chi_{0})$
$\displaystyle=\delta\chi^{\prime}(\chi_{0}),$

such that $\mu_{Z}^{\prime}(\pi+\chi_{0})=-\mu_{Z}^{\prime}(\chi_{0})$ [see
Eq. (5)]. As a result, because the contributions from the molecules at
$\pi+\chi_{0}$ and $\chi_{0}$ cancel each other (can also be seen from Fig.
9), the ensemble-averaged dipole signal along the $Z$ axis is zero. Since the
torque components depend on $\chi_{0}$ only, the same argument as in Sec. IV.1
can also be applied, meaning that the dipole signal along any direction in the
$XY$ plane vanishes. Thus, there is no orientation in this case.

In contrast, in the case of a pair of delayed crossed-polarized pulses, when
the second laser pulse is polarized at an angle ($\neq n\pi/2,\,n=0,1,2,3$)
with respect to the first one, the first pulse effectively breaks the symmetry
of spatial distribution about the polarization direction of the second one.
Consequently, the induced asymmetric torques [see Eqs. (7a-7c)] induce
molecular orientation along the direction perpendicular to the polarization of
the two exciting pulses. It is required that the time scales of polarization
twisting and the molecular rotation should be comparable, otherwise, the
spatial symmetry about the polarization direction prevails, leading to zero
orientation. In our qualitative discussions we did not consider the initial
angular velocities, but the conclusions remain valid even when the molecules
have finite initial angular velocities.

A similar effect of enantioselective orientation can be achieved using THz
pulses with twisted polarization. However, since the THz field couples to the
molecular permanent dipole moment, unlike the case of laser pulses, two
delayed orthogonally polarized THz pulses can also induce the enantioselective
orientation. [42].

##  V Conclusions

In this work, we have investigated the enantioselective orientation of chiral
molecules excited by a shaped picosecond laser pulse and a delayed femtosecond
pulse. We show that orientation is induced along the laser propagation
direction, and the sign of orientation is opposite for the two enantiomers.
This is similar to previous works on enantioselective orientation induced by
laser pulses with twisted polarization [21, 22, 23, 24, 25, 26]. The use of a
relatively weak, truncated picosecond pulse to induce alignment may be
compared to the use of a strong femtosecond laser pulse because it avoids
molecular ionization while still allowing higher degree of alignment. Another
method for achieving a high degree of alignment for a given pulse energy while
minimizing molecular ionization is to use a pulse sequence consisting of
multiple low-intensity pulses [59, 60, 61, 62, 63, 64]. This approach allows
the distribution of the energy, which would otherwise be delivered in a single
intense pulse, over multiple low-intensity pulses. However, for the optimal
degree of alignment, optimization of the sequence parameters is required.
Despite the relatively low intensities of the exciting laser pulses, a strong
enantioselective orientation (the degree of orientation $\approx 0.23$) was
demonstrated. The qualitative features of the effect, including the vanishing
orientation in the plane of laser polarizations, the chirality dependence of
the enantioselective orientation along the laser propagation direction, and
the roles of the two delayed cross-polarized laser pulses, may be understood
in terms of the classical model. The shaped picosecond laser pulse prepares a
highly aligned molecular ensemble, which is followed by a cross-polarized
femtosecond laser pulse. This induces asymmetric torques (depending on the
off-diagonal polarizability elements which exist in chiral molecules),
resulting in the nonzero orientation along the direction perpendicular to the
polarizations of the two exciting pulses. During this process, the orientation
along any direction in the plane of pulses’ polarizations remains zero. The
enantioselective orientation may be useful for the enantiomeric excess
analysis and potentially for separating mixtures of enantiomers using
inhomogeneous electromagnetic fields [65].

###### Acknowledgements.

The author appreciates many useful discussions with Ilia Tutunnikov, Yehiam
Prior, and Ilya Sh. Averbukh. This work was supported by the Israel Science
Foundation (Grant No. 746/15).

##  Appendix A Time-dependent dipole signal as a function of $\chi_{0}$

Figure 7: (a, d) $X$, (b, e) $Y$, and (c, f) $Z$ projections of the dipole
signal as functions of $\chi_{0}$ and time for the case of (_R_)-PPO molecules
shown in Fig. 4. (a)-(c) $\phi_{0}=0,\,\theta_{0}=\pi/2$. (d)-(f)
$\phi_{0}=\pi,\,\theta_{0}=\pi/2$. The color scales are in units of Debye.

Figure 8: (a, d) $X$, (b, e) $Y$, and (c, f) $Z$ projections of the dipole
signal as functions of $\pi-\chi_{0}$ and time for the case of (_S_)-PPO
molecules shown in Fig. 4. (a)-(c) $\phi_{0}=0,\,\theta_{0}=\pi/2$. (d)-(f)
$\phi_{0}=\pi,\,\theta_{0}=\pi/2$. The color scales are in units of Debye.
Figure 9: $Z$ projection of the dipole signal as a function of $\chi_{0}$ and
time for (_R_)-PPO molecules. The color scales are in units of Debye. The
initial conditions and the field parameters of the femtosecond laser pulse are
the same as the case shown in Fig. 4 except that the femtosecond pulse is
polarized along the $Y$ direction. Note that the results for $\phi_{0}=0$ and
$\phi_{0}=\pi$ are the same.

Figure 7 depicts the time-dependent dipole signal as a function of $\chi_{0}$
for the case of (_R_)-PPO shown in Fig. 4. As can be seen, the dipole signals
along the laboratory $X$ and $Y$ axes have opposite signs for
$A(t_{0})=(0,\pi/2,\chi_{0})$ [panels (a) and (b)] and
$B(t_{0})=(\pi,\pi/2,\chi_{0})$ [panels (d) and (e)]. Consequently, the
ensemble-averaged dipole signals $\braket{\mu_{X}}$ and $\braket{\mu_{Y}}$ are
zero. On the other hand, Figs. 7(c) and 7(f) show that for $A(t)$ and $B(t)$,
the $Z$ projections of the dipole signal are the same. These results are
consistent with the analysis presented in Sec. IV.1.

Figure 8 shows the case of (_S_)-PPO molecules. The dipole signal of (_S_)-PPO
along the $X$ and $Y$ directions are the same as that of (_R_)-PPO
[$\mu_{X}^{(S)}(B_{1})=\mu_{X}^{(R)}(B)$ and
$\mu_{Y}^{(S)}(B_{1})=\mu_{Y}^{(R)}(B)$], while
$\mu_{Z}^{(S)}(B_{1})=-\mu_{Z}^{(R)}(B)$. Here $B=(\pi,\pi/2,\chi_{0})$ and
$B_{1}=(\pi,\pi/2,\pi-\chi_{0})$. According to Eq. (11),
$\mu_{j}^{(S)}(B_{1})=-\mu_{j}^{(R)}(A)$, $j=X,Y,Z$. In addition,
$\mu_{j}(B)=(-1)^{\delta_{jZ}+1}\mu_{j}(A)$ (see Sec. IV.1), such that
$\mu_{j}^{(S)}(B_{1})=(-1)^{\delta_{jZ}}\mu_{j}^{(R)}(B)$. The numerical
results are consistent with the analysis, and
$\mu_{Z}^{(S)}(B_{1})=-\mu_{Z}^{(R)}(B)$ implies the enantioselective
orientation.

Taking (_R_)-PPO molecule as an example, we plot $\mu_{Z}(t)$ as a function of
$\chi_{0}$ in Fig. 9. Initially, $\phi_{0}=0,\pi$ and $\theta_{0}=\pi/2$,
corresponding to the molecular $a$ axis pointing along or against the
laboratory $X$ axis. The initial temperature is $0\,\mathrm{K}$. The
femtosecond laser pulse is polarized along the laboratory $Y$ axis. As shown
in Fig. 9, $\mu_{Z}(\pi+\chi_{0})=-\mu_{Z}(\chi_{0})$, in agreement with the
discussion in Sec. IV.4. Consequently, in this case, the ensemble-averaged
dipole signal along the $Z$ axis, $\braket{\mu_{Z}}$ vanishes.

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# NCL-SM: A Fully Annotated Dataset of Images from Human Skeletal Muscle
Biopsies
††thanks: EPSRC Centre for Doctoral Training in Cloud Computing for Big Data,
Newcastle University ††thanks: This is a full paper based on the poster and
findings presented at ML4H Symposium 2023.

Atif Khan  School of Computing, WCMR
Newcastle University
Newcastle, UK
<EMAIL_ADDRESS>Conor Lawless  WCMR
Newcastle University
Newcastle, UK
<EMAIL_ADDRESS>Amy E Vincent  WCMR
Newcastle University
Newcastle, UK
<EMAIL_ADDRESS>Charlotte Warren *Corresponding Author WCMR
Newcastle University
Newcastle, UK
<EMAIL_ADDRESS>Valeria Di Leo  WCMR
Newcastle University
Newcastle, UK
<EMAIL_ADDRESS>Tiago Gomes  WCMR
Newcastle University
Newcastle, UK
<EMAIL_ADDRESS>A. Stephen McGough  School of Computing
Newcastle University
Newcastle, UK
<EMAIL_ADDRESS>
###### Abstract

Single cell analysis of human skeletal muscle (SM) tissue cross-sections is a
fundamental tool for understanding many neuromuscular disorders. For this
analysis to be reliable and reproducible, identification of individual fibres
within microscopy images (segmentation) of SM tissue should be automatic and
precise. Biomedical scientists in this field currently rely on custom tools
and general machine learning (ML) models, both followed by labour intensive
and subjective manual interventions to fine-tune segmentation. We believe that
fully automated, precise, reproducible segmentation is possible by training ML
models. However, in this important biomedical domain, there are currently no
good quality, publicly available annotated imaging datasets available for ML
model training. In this paper we release NCL-SM: a high quality bioimaging
dataset of 46 human SM tissue cross-sections from both healthy control
subjects and from patients with genetically diagnosed muscle pathology. These
images include $>$ 50k manually segmented muscle fibres (myofibres). In
addition we also curated high quality myofibre segmentations, annotating
reasons for rejecting low quality myofibres and low quality regions in SM
tissue images, making these annotations completely ready for downstream
analysis. This, we believe, will pave the way for development of a fully
automatic pipeline that identifies individual myofibres within images of
tissue sections and, in particular, also classifies individual myofibres that
are fit for further analysis.

###### Index Terms:

Skeletal muscle fibre segmentation, Segmentation by machine learning, Skeletal
muscle fibre annotations, Immunofluorescence imaging, Imaging mass cytometry,
Biomedical imaging, Mitochondrial disease, Tissue sections, Human tissue
samples

††publicationid: pubid:

##  I Introduction

Microscopy imaging and cytometry-based pseudo imaging techniques allow us to
observe protein expressions within individual cells within tissue images, when
a single cell segmentation approach is used. For many biological and disease
processes, this is the most appropriate spatial scale for understanding
mechanisms. Further, the spatial arrangement of cells of different classes
within tissues in vitro is often informative about biology and disease
pathology. There are many diseases affecting SM tissue, including amyotrophic
lateral sclerosis [1], multiple sclerosis [2], muscular dystrophy [3] and a
wide range of mitochondrial diseases [4]. The dataset we present here is
collected from healthy human control subjects and from patients suffering from
genetically diagnosed muscle pathology, including mitochondrial diseases.
Mitochondrial diseases are individually uncommon but are collectively the most
common metabolic disorder affecting 1 in 5,000 people [5]. They can cause
severe disabilities and adversely affect the life expectancy of patients [6].
Mitochondrial disease pathology is complex and highly heterogeneous [5]. Some
of the latest approaches to classifying mitochondrial diseases, particularly
previously unknown mitochondrial diseases, and quantifying disease severity
are based on the analysis of single myofibre protein expression profiles and
clinical information from patients [7]. These same approaches are currently
being successfully applied to other types of muscle pathology [8].
Any analysis of SM images at the single myofibre level requires 1) precise
segmentation of individual myofibres, as regions close to the myofibre
membrane are known to exhibit differential features [9]. 2) removal of
myofibres damaged by freezing that might occur in the process of storage and
thawing, as the subcellular patterns in protein expression, or indeed per-cell
mean expression, will be impacted by the technical artefact, masking the
target biology. 3) removal of SM myofibres that are not sliced in transverse
orientation or are partially sectioned, as the presence of such myofibres does
not allow for a standard or uniform comparison across all the myofibres in a
tissue and 4) removal of folded tissue. Tissue can fold in on itself during
tissue handling and slide preparation. Such folding artificially amplifies
apparent protein expression in affected regions and is again a purely
technical artefact, not related to target biology.

Currently most single-myofibre SM analysis is carried out using custom built
semi-automatic pipelines like mitocyto [10], using general image analysis
tools like Ilastik [11] or cellprofiler [12], or using vanilla ML models like
stardist [13] or cellpose [14]. None of these approaches produce segmentation
quality required for analysis of SM out of the box, without suitable training
in our experience. Custom built pipelines like mitocyto are used more often
than general ML models (which are not trained on SM data) as these require
relatively fewer corrections than general ML models. We evaluate and discuss
mitocyto segmentation quality in section V. To improve the segmentation
quality and remove compromised myofibres and SM regions, biomedical scientists
spend hours manually correcting the issues in a tissue section before doing
downstream quantitative analysis. This can be an inefficient use of
scientist’s time but also the corrections are subjective and not reproducible.

The main barrier to the development of a suitable ML tool/model for fully
automatic segmentation and curation of SM myofibres is the lack of any high
quality, manually segmented and curated SM myofibre image data on which to
train appropriate ML models. In this paper we make available a high quality
fully manually segmented and mostly manually curated SM imaging dataset
collected using two different imaging technologies: Imaging Mass Cytometry
(IMC) and ImmunoFluorescence (IF) amounting to 50,434 myofibres in 46 tissue
sections, 30,794 of which are classed as ‘analysable’, 18,102 classed as ‘not-
analysable-due-to-shape’, 1,538 myofibres classed as ‘not-analysable-due-to-
freezing-damage’ and 405 annotations of folded tissue regions. We describe the
Newcastle Skeletal Muscle (NCL-SM) dataset, introduce quantitative quality
assessment metrics relevant to single-myofibre SM tissue segmentation and
compare annotation quality achieved using our existing pipeline with our new,
ground truth manual annotations.

The main contributions of this paper are:

  * •

Publishing a high quality, manually segmented set of IMC and IF images of
human skeletal muscle tissue, curated by expert biomedical scientists: the
NCL-SM dataset. We provide NCL-SM111https://doi.org/10.25405/data.ncl.24125391
and our code222www.github.com/atifkhanncl/NCL-SM that evaluates annotation
quality and classifies non-transverse sliced myofibres.

  * •

Introducing quantitative quality assessment metrics for SM myofibre
segmentation.

  * •

Evaluating an existing, semi-automated pipeline applied to NCL-SM and
comparing results to clearly highlight where improvements are required and to
pave the way for development of an automatic ML pipeline for segmenting single
myofibres that are fit for analysis.

##  II Tissue Collection and Image Curation

The Wellcome Centre for Mitochondrial Research (WCMR)333www.newcastle-
mitochondria.com/ is one of the leading institutes conducting research into
mitochondrial diseases worldwide, and we have an unparalleled repository of
clinical data and tissue from healthy control subjects and patients with
genetically diagnosed muscle pathology, largely mitochondrial disease patients
with mitochondrial myopathy. The NCL-SM dataset includes images of 46 tissue
sections that capture spatial variation in protein expression within tissue
(including within myofibres). We use microscopy-based techniques i.e. IF and
advanced protein expression measurement techniques like IMC that allow us to
observe the spatial variation in the expression of up to 40 proteins in tissue
simultaneously.

The assessment process before ethical approval for collection of SM tissue
from human donors is granted, is rigorous and muscle tissue donations can be
painful. As a result, SM tissue collections are scarce and valuable, requiring
available SM tissue sections to be used efficiently. Biomedical scientists in
our centre follow processes to preserve and efficiently use the available
samples throughout the research cycle. This also applies to the imaging data
captured from these tissues i.e. within the images it is critical to
efficiently curate all usable myofibres. Analysing all available, usable
myofibres is also a way to ensure that no selection bias creeps into the
analysis. We identify all usable/analysable myofibres by precisely segmenting
all available SM fibres and curating the subset that are fit for further
analysis. Currently, segmentation and curation is a heavily manual process.

Machine learning has made some great strides in image segmentation, object
detection and image classification [15, 16, 13, 14]. However, leveraging ML
tools is only possible with large, good quality datasets. To our knowledge
there is no such publicly available dataset for SM fibre segmentation or
classification. Making this dataset available and clearly defining the
challenge involved in myofibre segmentation and curating the usable myofibres
is a crucial first step towards development of an automatic tool for this
problem.

High quality annotated datasets are critical for development of relevant ML
models or pipelines. This has been evident since the early days of modern ML
with datasets such as MNIST [17], COCO [18] and more recently SA-1B [19]
enabling the construction of some seminal ML models like ResNet [20], VGG
[21], vision transformer [22] and SAM [19].

##  III Capturing images

The following is the sequential process of collecting SM tissue images.

TABLE I: Annotation counts in NCL-SM dataset. In the table TS, AM, NTM,FAM and FR stands for Tissue Section, Analysable Myofibre, Non-Transverse Myofibre, Freezing Artefact Myofibre and Folded Region respectively.  Imaging Technique | TS Count | Myofibre Count | AM Count | NTM Count | FAM Count | FR Count
---|---|---|---|---|---|---
IMC | 27 | 22,979 | 14,841 | 7,358 | 780 | 84
IF | 19 | 27,455 | 15,953 | 10,744 | 758 | 321
Total | 46 | 50,434 | 30,794 | 18,102 | 1,538 | 405

###  III-A Biopies

Following ethical approval from the Newcastle and North Tyneside Local
Research Ethics Committee and informed consent from control and patient
subjects, biopsies of SM were collected from patient and healthy control
subjects. Specific ethical approval for this project was not required as it
consisted of a re-analysis of previously captured image data. This original
data was obtained from approved studies of samples from Newcastle Brain Tissue
Resource (Approval Ref:2021031) and Newcastle Biobank (Application Ref: 042).

###  III-B IMC

Imaging mass cytometry is an imaging technique to simultaneously observe the
expression of multiple proteins in a tissue section as described in Figure 1.

#### NCL-SM images

The 46 tissue section images in NCL-SM are made by arranging grayscale images
of a cell membrane protein marker i.e. dystrophin and of a mitochondrial mass
protein marker VDAC1 into an RGB image where R = membrane protein marker, G =
mass protein marker and B = 0. Each channel of the images are contrast
stretched (5 to 95 percentile) to improve image contrast for the segmentation
task but raw images without contrast stretching are also included in NCL-SM.

Figure 1: Imaging Mass Cytometry Experimental Procedure During IMC myofibres
are stained with a panel of antibodies conjugated to heavy metals. Tissue
(made up of myofibres) is scanned by a pulsed laser which ablates a spot of
tissue section. The tissue is vaporised on each laser shot and enters the mass
cytometer where the relative concentration of heavy metals can be quantified.
These measurements are later combined as pseudo images, one for each
metal/protein, where location and intensity of each pixel corresponds to the
amount of metal isotopes at each spot. Myofibres are segmented in these images
for conducting single myofibre data analysis. Figure adapted from [23]

.

###  III-C IF

Immunofluorescence imaging is another technology that uses the antibodies to
their antigen to target fluorescent dyes to specific protein targets within a
cell/myofibres. It allows the capture of high resolution and high bit-depth
microscopy images, but only up to 5 protein targets can be observed
simultaneously [24].

###  III-D Myofibre Segmentation

The protocol for myofibre segmentation was i) include all areas within a
myofibre that had mitochondrial mass signal, ii) exclude any areas within a
myofibre that had myofibre membrane signal and iii) prioritise signal from
within myofibre when membrane signal is weak. Point iii) is necessary because
noise is common in these types of data resulting in some overlapping mass and
membrane pixels. In such scenarios, we consider mitochondrial mass signal
within a myofibre to be the most reliable indicator of myofibre morphology. We
employed annotation specialists from Gamaed444www.gamaed.com/ working under
the close oversight of expert biomedical scientists from WCMR to segment each
myofibre in all 46 tissue sections, amounting to 50,434 myofibres. Gamaed
annotators used the online Apeer platform555www.apeer.com/ for all manual
annotations. All segmentation went through rigorous visual inspection and a
number of random myofibres in the data were segmented separately by expert
biomedical scientists for quality assurance (QA). This is further discussed in
Section IV.

Figure 2: Typical Manual Segmentation of a SM Tissue Section: An IMC SM tissue
section image from subject P17. Consists of 1,068 myofibres manually segmented
following the protocol i.e. include all myofibre mass and exclude membrane.

###  III-E Freezing Artefact Myofibre Classification

The protocol for identifying myofibres with freezing artefacts was i) look for
a leopard spot pattern within myofibres that are typical of freezing damage
and ii) look for partial myofibres i.e. large part of myofibres missing as a
result of freezing. These patterns are demonstrated in Figure 3.

The freezing artefact classification annotation was duplicated by two experts
from WCMR and any disagreement was resolved by discussion between a panel of
experts. This resulted in 1,538 SM fibres with freezing artefacts where both
annotators were in agreement. All annotations which differed between
annotators were reviewed and resolved by a second expert after discussion.

Figure 3: Identifying Myofibre Freezing Artefacts All myofibres from section
P02. A (id:190), B (id:138) and C (id:257) are typical freezing damaged
myofibres resulting in a leopard spot pattern, D (id:448), E (id:117) and F
(id:398) are partial myofibres that are more severely damaged by freezing, G
(id:303), H (id:415) and I (id:288) are examples of myofibres without any
freezing defects.

###  III-F Non-Transverse Sliced Myofibre Classification

The protocol for identifying non-transverse sliced myofibres was to look for
i) myofibres with skewed aspect ratio e.g. elongated ii) all myofibres at the
border of image: these are partial observations iii) segmented objects which
are too small or too big and iv) myofibres with unusual convexity. As
demonstrated in Figure 4, for annotating non-transverse sliced fibres we
employed the following approach 1) two experts working together identified up
to 1,500 such myofibres in the data 2) using these 1,500 myofibres, thresholds
for area, convexity and aspect ratio were calculated 3) these thresholds and a
function to detect any myofibre on the edge were then applied on whole data,
resulting in two classes of myofibres. 4) finally both classes of myofibres
were rigorously visually inspected to detect and correct any mis-
classification. These resulted in 18,102 non-transverse sliced myofibres
(NTM).

Figure 4: Typical Non-Transverse Sliced Myofibres: Myofibres from tissue
section C04. A (ID: 39 ) is a typical elongated myofibre cross-section which
has not been cut in the transverse orientation B is myofibre (ID:6) with
unusual convexity, ‘C’ is myofibre (ID:24) of large area and ‘D’ is a partial
myofibre (ID:12) that has been truncated by the border of the image

###  III-G Folded Tissue Segmentation

Folded tissue regions were segmented by an expert biomedical scientist, these
were identified by looking for overlapping membrane signal that results in
mesh pattern as demonstrated in Figure 5. This resulted in annotations of 405
different tissue regions affecting 37 out of 46 sections.

Figure 5: Typical SM Tissue Folding Segmentation of folded regions in a tissue
section P06

##  IV Quality Evaluation Metrics and Measurements

In order to evaluate the quality of myofibre segmentation annotations we need
quantitative metrics that capture the nuances required during SM analysis. In
this section we first define these nuances and introduce the quality
evaluation metrics, we then present the annotation quality of NCL-SM in terms
of these metrics.

###  IV-A Assessment Metrics

Figure 6: Area Near the Membrane IMC image of a fibre (ID:723) from tissue
section P17 illustrating Area outside the Border (AoB, red) and Area inside
the Border (AiB, green) on either side of the border of an annotated myofibre.
These areas are identified by eroding and dilating the border using a 5x5 and
9x9 pixel kernels for IMC and IF images respectively.

Intersection over Union (IoU) is widely used evaluation metric to measure the
quality of annotation/segmentation in computer vision tasks. But for myofibre
segmentation examining IoU of each myofibre alone will not reveal important
aspects about segmentation quality i.e. the area missed or included matter as
emphasised in Sections I and III-D, in other words we want any automatic
pipeline to have high accuracy segmenting areas on either side of the border
of each myofibre as illustrated in Figure 6. To measure this, we developed two
quantitative metrics: myofibre mass missed correlation ($r_{AoB}$) and
myofibre membrane included correlation ($r_{AiB}$) along with the
aforementioned IoU. All three metrics are defined below:

  * •

Myofibre mass missed correlation ($r_{AoB}$) : This is the Pearson correlation
between the proportion of myofibre mass pixels missed in the Area outside the
myofibre Border ($AoB$) by annotator $x$, compared to annotator $y$, across
all available myofibres $i\in{1...n}$.

$r_{AoB}=\frac{{}\sum_{i=1}^{n}(x_{AoBi}-\overline{x_{AoB}})(y_{AoBi}-\overline{y_{AoB}})}{\sqrt{\sum_{i=1}^{n}(x_{AoBi}-\overline{x_{AoB}})^{2}\sum_{i=1}^{n}(y_{AoBi}-\overline{y_{AoB}})^{2}}}$

where $x\textsubscript{AoBi}$ is the proportion of myofibre mass pixels in AoB
in myofiber $i$ as annotated by annotator $x$; $y\textsubscript{AoBi}$ is the
proportion of myofibre mass pixels in AoB in myofiber $i$ as annotated by
annotator $y$; overbar represents the mean across all $n$ myofibres; and $n$
is the number of myofibres assessed.

  * •

Myofibre membrane included correlation ($r_{AiB}$): Defined as above, but for
$AiB$:

$r_{AiB}=\frac{{}\sum_{i=1}^{n}(x_{AiBi}-\overline{x_{AiB}})(y_{AiBi}-\overline{y_{AiB}})}{\sqrt{\sum_{i=1}^{n}(x_{AiBi}-\overline{x_{AiB}})^{2}\sum_{i=1}^{n}(y_{AiBi}-\overline{y_{AiB}})^{2}}}$

where $x\textsubscript{AiBi}$ is the proportion of myofibre membrane pixels in
AiB in myofiber $i$ as annotated by annotator $x$; $y\textsubscript{AiBi}$ is
the proportion of myofibre membrane pixels in AiB in myofiber $i$ as annotated
by annotator $y$.

  * •

IoU ($IoU_{i}$): This is defined as intersection of overlapping pixels divided
by union of all pixels between two annotations of myofibre $i$. This is
measured per myofibre and $\overline{IoU}$ is the mean across all $n$
myofibres assessed.

TABLE II:  Annotation quality for human-to-human annotation comparison as
mention in Section III-D. In the table MF-A, $r\textsubscript{AoB}$,
$r\textsubscript{AiB}$, $\overline{IoU}$, ( A%(IoU $>$0.80), A%(IoU $>$0.90),
A%(IoU $>$0.95)), QA-IMC, QA-IF stands for ‘Myofibres Assessed’, ‘Myofibre
Mass Missed Correlation’, ‘Myofibre Membrane Included Correlation’, ’Mean
IoU’, (’Accuracy in terms of % of myofibres meeting IoU threshold of 0.8, 0.9
and 0.95’), ‘Quality Assessment for IMC images’ and ‘Quality Assessment for IF
images’ respectively.
Annotations | MF-A | $r\textsubscript{AoB}$ | $r\textsubscript{AiB}$ | $\overline{IoU}$ | A%(IoU $>$0.80) | A%(IoU $>$0.90) | A%(IoU $>$0.95)
---|---|---|---|---|---|---|---
QA-IMC | 53 | 0.99 | 0.77 | 0.96 | 100 | 100 | 77.4
QA-IF | 23 | 0.92 | 0.94 | 0.96 | 100 | 100 | 74
TABLE III: Annotation quality comparison between NCL-SM (considered as ground truth) and 1) mitocyto+ and 2) QA annotation used as benchmark. In the table TS, mitocyto+ and QA stands for ‘Tissue sections’, ‘mitocyto with manual corrections’ and ‘QA annotations’ respectively.  Annotation | TS | MF-A | $r_{A}oB$ | $r_{A}iB$ | $\overline{IoU}$ | A%(IoU$>$0.8) | A%(IoU$>$0.9) | A%(IoU$>$0.95)
---|---|---|---|---|---|---|---|---
mitocyto+ | P02, P06 | 1040 | 0.9 | -0.15 | 0.91 | 95.24 | 74 | 11.3
QA | P02, P06 | 26 | 0.96 | 0.72 | 0.95 | 100 | 100 | 61.5

###  IV-B Assessment Measurements

As mentioned in Section III-D, for QA during manual segmentation, experts at
WCMR routinely annotated random myofibres in various sections to make sure the
quality of annotations by specialists from Gamaed remain high quality. The
observations are listed in Table II. As these are comparison of duplicate
manual annotations, we believe this should be used as benchmark for any
automatic segmentation tool or pipeline.

##  V Evaluation of Existing Tools on NCL-SM

Considering NCL-SM annotations as ‘ground truth’ we evaluated the segmentation
quality produced by mitocyto666www.github.com/CnrLwlss/mitocyto a custom
Python image analysis package for semi-automated SM segmentation, built around
OpenCV [25]. This package requires manual intervention to correct
classifications, which were performed by experts at WCMR. Observations are
listed in Table III. We can clearly see that for the purpose of analysis of SM
at the single myofibre level where precision of segmentation near the membrane
of the myofibre is important as described in Section I the current pipeline
with manual intervention is not fit for purpose. This is highlighted by our
metrics i.e. low values for rAiB and accuracy measured in terms of IoU
thresholds of 0.9 and 0.95 demonstrate lack of precision segmenting near the
membrane.

##  VI Discussion

###  VI-A Metrics

As observed in Section V the metrics we introduced are capable of quantifying
the segmentation quality required for SM analysis at the single myofibre
level. It also demonstrates that IoU on its own can be misleading in this
case.

###  VI-B Problem definition

The challenge of segmenting for single myofibre analysis can be divided into
two tasks

  * •

Segmentation: Each myofibre and folded tissue region needs to be identified in
the tissue section image. Following this all folded regions need to be removed
and an instance segmentation mask of all remaining myofibres should be made.

  * •

Classification: The two types of not analysable myofibres i.e. ‘not-
analysable-due-to-shape’ and ‘not-analysable-due-to-freezing-damage’ need to
be identified and removed. Leaving a final instance segmentation mask of
curated myofibre that are fit for further analysis.

###  VI-C Possible ML Solutions

This paper has focused on the collection and curation of the NCL-SM dataset
along with defining metrics for analysing the quality of annotation
approaches. Although ML approaches for performing segmentation and
classification is out of scope for this work we suggest some possible starting
points:

Workflow: We believe the sequential workflow that worked in the manual
annotation process to create NCL-SM should also work in any automatic
pipeline. That is 1) Segment all myofibres in the image to create an instance
segmentation mask, 2) segment folded regions in the image to create FR mask,
3) classify NTMs using the myofibres instance segmentation mask to create a
NTM mask, 4) use the myofibres instance segmentation mask and the image to
create a FAM mask and 5) Subtract masks created in 2,3,4 from 1 to create
final instance segmentation mask of ’Analysable Myofibres’.

Segmentation: There is a range of ML models like UNETs [26], R-CNNs [27] that
might be useful and the metrics we defined above can be used as loss/accuracy
metrics for model training.
Classification: Similarly object detection or image classification models like
ResNet [20] and vision transformer [22] might make accurate predictions after
training with NCL-SM.

##  VII Conclusion and Future Work

In this paper we release the NCL-SM dataset, a high quality dataset of $>$ 50k
annotated SM myofibre segmentations with a curated subset of myofibres
classified as fit for downstream analysis. We describe how we assess
segmentation quality for single myofibres, introducing the quantitative
metrics relevant for assessment. We demonstrate the high quality of annotation
of NCL-SM in terms of relevant metrics. We define the challenges involved in
segmenting SM myofibres for possible automatic ML solution. We believe this
dataset will enable the development of novel ML solutions to address the
problem of automatic segmentation of SM fibres and classification of
individual myofibres that are fit for downstream analysis.

NCL-SM consists of $>$ 50k myofibres segmentations but we have significantly
fewer FR segmentations i.e. 405 across all images. To address this in our
future work 1) we shall expand NCL-SM to include more high quality annotated
data not only from our centre but establish a process in place by which others
can add their data which meet the quality standards of NCL-SM and 2) we want
to to build an automatic ML pipeline to address the challenges highlighted in
this paper utilising the NCL-SM dataset.

##  VIII Acknowledgements

This work was supported by the EPSRC Centre for Doctoral Training in Cloud
Computing for Big Data and Wellcome Centre for Mitochondrial Research. AEV is
in receipt of a Newcastle University Academic Track Fellowship. Annotation
work was carried out by an independent, specialist data labelling company:
Gamaed.

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