# Initial non-invasive in vivo sensing of the lung using time domain diffuse
optics

Antonio Pifferi  1,2 Massimo Miniati  3 Andrea Farina  2,* Sanathana Konugolu
Venkata Sekar  4 Pranav Lanka  1 Alberto Dalla Mora  1 and Paola Taroni1,2
1Politecnico di Milano, Dipartimento di Fisica, Piazza Leonardo da Vinci 32,
Milano, Italy
2INFN-CNR, Consiglio Nazionale delle Ricerche, Istituto di Fotonica e
Nanotecnologie, Piazza Leonardo da Vinci 32, Milano, Italy
3University of Florence, Department of Experimental and Clinical Medicine,
Italy
4Biophotonics@Tyndall, IPIC, Tyndall National Institute, Lee Maltings, Dyke
Parade, Cork, Ireland<EMAIL_ADDRESS>
††journal: boe††articletype: Research Article

Non-invasive in vivo sensing of the lung with light would help diagnose and
monitor pulmonary disorders (caused by e.g. COVID-19, emphysema, immature lung
tissue in infants). We investigated the possibility to probe the lung with
time domain diffuse optics, taking advantage of the increased depth (few cm)
reached by photons detected after a long (few ns) propagation time. An initial
study on 5 healthy volunteers included time-resolved broadband diffuse optical
spectroscopy measurements at 3 cm source-detector distance over the 600-1100
nm range, and long-distance (6-9 cm) measurements at 820 nm performed during a
breathing protocol. The interpretation of the in vivo data with a simplified
homogeneous model yielded a maximum probing depth of 2.6-3.9 cm, suitable to
reach the lung. Also, signal changes related to the inspiration act were
observed, especially at high photon propagation times. Yet, intra- and inter-
subject variability and inconsistencies, possibly alluring to competing
scattering and absorption effects, prevented a simple interpretation. Aspects
to be further investigated to gain a deeper insight are discussed.

##  1 Introduction

The lungs can be regarded as a mixture of air and tissue with a water-
equivalent density. Accordingly, they can be visualised noninvasively through
the attenuation of x-rays from an external source. In clinical practice, both
conventional chest radiography (CXR) and computed tomography (CT) are widely
used to detect pulmonary abnormalities. Thanks to its inherent tomographic
properties, CT is superior to conventional CXR in visualising the fine
structure of the lung. Modern CT scanners are capable of detecting small solid
lesions down to a size of 5 mm in width. However, neither CXR nor CT provide
any direct information as regards the chemical composition of an abnormal lung
density. The same limitation applies to thoracic ultrasound (US) which is
increasingly used by the clinicians to evaluate pleural effusions or
subpleural consolidations at bedside.

Assessing the lung composition and density can be useful to diagnose or
monitor patients with emphysema, a pathologic condition characterized by an
increased fraction of air per unit of lung volume. In patients with left heart
failure, lung density may increase substantially due to the accumulation of
water in the extravascular space of the lung (pulmonary edema). Also, lung
inflammations with fluid accumulation (caused e.g. by COVID-19 or other
infections) leads to changes in lung density and composition. Lung density can
be equally increased by an abnormal deposition of collagen in the lung
interstitium (fibrosis).

Time-Domain Diffuse Optical Spectroscopy (TD-DOS) is a photonics technique
that permits to derive information on the characteristics of a biological
tissue through the absorption ($\mu_{a}$) and reduced scattering
($\mu_{s}^{\prime}$) coefficient via the study of photon propagation of short
(ps) laser pulses within the medium[1, 2]. So far, TD-DOS had been
successfully applied to characterise nonivasively the tissue composition or
function of organs, like the brain[3] or the breast[4], which are in close
proximity to the surface of the body or can be assessed in transmittance
geometry.

From the optical standpoint, the lungs are accessible over a large portion of
the anterior and lateral chest wall. However, the varying thickness of the
chest wall and its heterogeneous composition (collagen, elastin, fat, and
muscular tissue) pose a major challenge when using TD-DOS transcutaneously. In
addition, the varying dimensions of the alveoli and distal conducting airways
over the respiratory cycle may significantly affect the scattering of photons,
making the interpretation of experimental data specially complex[5].

As of now, optical techniques have been applied mostly for the diagnosis of
lung cancer through invasive (bronchoscopy), or minimally invasive (needle
biopsy) techniques[6]. They include diffuse reflectance spectroscopy[7, 8],
endogenous[9] or exogenous fluorescence spectroscopy[10, 11], optical coherent
tomography [12], and Raman spectroscopy [13]. Much effort is also being
devoted to model the photon propagation through the lung by means of
simulations[14, 15] anthropomorphic lung phantoms[16, 17], and ex vivo bovine
lung tissues[18]. Conversely, there are not many attempts to probe the lung
optically non-invasively transcutaneously. In a first clinical study on 3
full-term babies, an optical technique termed "Gas in Scattering Media
Absorption spectroscopy" (GASMAS) was used for noninvasive monitoring of water
vapor in the lungs[19]. A larger study on 29 newborn infants demonstrated
transcutaneous detection of oxygen in the lungs using GASMAS[20]. This
approach was further validated also on a piglet animal model[21]. Yet, to the
best of our knowledge, so far no non-invasive optical approach has been
applied in vivo on the lungs in adults.

In the present study, we investigated the feasibility to sense the lung
transcutaneously by simulations – instructed by in vivo spectroscopy
measurements on the thorax – and by applying TD-DOS in healthy volunteers who
were examined at suspended full inspiration and suspended full expiration.

##  2 Materials and methods

###  2.1 System set-up

Two TD-DOS systems were used for in vivo measurements on volunteers, namely a
broadband spectrometer and a highly sensitive single-wavelength setup.

The broadband spectrometer was a laboratory workstation exploiting
supercontinuum generation to automatically perform time-domain measurements of
the Distribution of Time-of-Flight (DTOF) over a wide spectral range, from 500
up to 1700 nm encompassing different detectors[22]. In the actual embodiment,
an 80 MHz, 10 ps pulsed supercontinuum (SuperK EXTREME, NKT photonics) was
sliced using a rotating Pellin-Broca prism followed by a 50 $\mu m$-core
graded index fiber acting as spectral selection. Light was injected to and
collected from the tissue using 1 mm core step index fibers coupled to a
custom-made Silicon Photomultiplier (SiPM) module[23] connected to a Time-
Correlated Single-Photon Counting (TCSPC) board (SPC130, Becker & Hickl). A
variable optical attenuator set on the source path controlled photon counting
rate within the 1% single-photon statistics. A complete spectral acquisition
from 600 nm to 1100 nm in steps of 10 nm with 4 s acquisitions required around
5 min.

The single-wavelength setup was optimised for maximal light harvesting. It was
based on a prototypal 40 MHz, 25 ps high-power Four-Wave Mixing laser (Fianium
Ltd, Southampton, UK). The tissue was illuminated at 820 nm by a maximum power
of 100 mW expanded to comply with the maximum permissible skin exposure. Re-
emitted light was harvested using 1-mm core step index fiber coupled to a
hybrid photomultiplier (HPM-100-50, Becker & Hickl GmbH, Germany), and
connected with a TCSPC board (SPC130, Becker & Hickl GmbH, Germany).

###  2.2 Simulations

To simulate the DTOFs in a layered tissue, a time-resolved GPU-accelerated
Monte Carlo code [24] was implemented making use of the microscopic Lambert-
Beer approach [25]: for each detected photon the path-length spent in each
layer was saved. The simulation was run without absorption, weighting each
trajectory afterwards depending on the desired absorption in each layer. A
total of 10M photons reaching the receiver were collected for each simulation.

###  2.3 In vivo protocol

Five healthy volunteers were recruited for the study, after signing informed
consent and complying with the authorization from the institutional Ethical
Review Board of Politecnico di Milano. Demographic data are reported in Table
1 presented in Section 3.

For the broadband study, the subject was lying supine breathing normally and
with the probe positioned on the thorax in the upper-right region using a
source-detector distance $\rho=3$ cm.

For the single-wavelength study, the subject was lying supine, with the probe
set in two locations on the thorax, namely on the anterior surface of the
right hemithorax between the second (UR) and fifth (DR) intercostal space
along the mid clavicular line. The source-detector distance $\rho$ was set to
the maximum value ensuring a count-rate around 1 M counts/s, resulting in
$\rho=6-9$ cm depending on the subject and location. The subjects were asked
to follow a suspended full inspiration and suspended full expiration at a
given pace to provide two intervals with the lung inflated (IN) or deflated
(OUT). Two protocols were applied on each subject with 5 repetitions of the
two phases lasting 10 s each (Prot10) or 10 repetitions with 5 s for each
phase (Prot5). Only for Prot5 the first repetition was perfored at normal
breathing to get a reference state. Each protocol was repeated twice in each
location, resulting in 8 complete series of acquisitions per subject.

###  2.4 Data analysis

The spectral measurements were analysed using a homogeneous solution of the
Diffusion Equation under the extrapolated boundary conditions using $\mu_{a}$
and $\mu_{s}^{\prime}$ as free fitting parameters. The Instrument Response
Function (IRF) obtained facing the injection and collection fibers was
convoluted with the theoretical model. The fitting range included points with
a number of counts >80% and >1% of the peak value on the leading and falling
edge of the DTOF, respectively. Tissue composition was derived from the
retrieved absorption and reduced scattering spectra using the linear
combination of 5 key tissue absorbers, namely oxy- (02Hb), and deoxy-
hemoglobin (Hb), water, lipids, and collagen.

The single-wavelength measurements were analysed using a fit with the
homogeneous model similarly to the spectral measurements. For better
visualization, a folding average over the 5 (10) repetitions of the protocol
was applied. Further, the gated intensity was calculated for different time-
windows of the DTOFs. The relative contrast of the gated reflectance signal
$R(t)$ with respect to a reference state $R_{0}(t)$ was derived as:

$C(t)=\frac{\int_{t}^{t+\Delta t}R(t^{\prime})dt^{\prime}-\int_{t}^{t+\Delta
t}R_{0}(t^{\prime})dt^{\prime}}{\int_{t}^{t+\Delta
t}R_{0}(t^{\prime})dt^{\prime}}$ (1)

where $t^{\prime}$ is the photon arrival time, while $t$ and $\Delta t$ are
the starting edge and the width of the gated temporal window. For the in vivo
protocols, $R_{0}$ is assumed to be the average of $R$ over the whole
exercise, therefore $C$ represents the relative change in gated signal with
respect to the mean value.

Equation 1 was used also to calculate the relative contrast for the Monte
Carlo simulations. In that case, for $R_{0}$ we assumed the unperturbed state.

##  3 Spectroscopy of the human chest

As a first step to estimate the optical properties of the tissues overlaying
the lung, we measured the absorption and reduced scattering spectra on the
chest region of healthy volunteers using an intermediate source-detector
distance $\rho=3$ cm. Figure 1 displays the absorption (left pane) and reduced
scattering spectra (right pane) of 5 subjects (see Table 1 for demographics)
obtained assuming a homogeneous model. Some inter-subject variability in the
absorption spectrum is observed, with a pronounced lipid contribution for
subject #5, and a high muscular presence for subject #1 and #3. The clear peak
around 980 nm is ascribed to water, while the lipid peak at 930 is
progressively covered by the dominant water contribution, when moving from #5
to #1. Differences in the region <800 nm are mostly due to blood content.
Overall, around 800 nm – where we performed large-$\rho$ measurements as
presented in the following Section 4 – $\mu_{a}\approx 0.1-0.2$ cm-1 and
$\mu_{s}^{\prime}\approx 7-9$ cm-1. Table 1 reports the average tissue
composition assuming 5 absorbers (Hb, HbO2, water, lipids, and collagen) and
an empirical power law dependence for the scattering spectrum choosing
$\lambda_{0}=600$ nm[26]:

$\lambda=a(\lambda/\lambda_{0})^{-b}$ (2)

The mean tissue composition of different subjects confirms the initial guess
on the fat – or muscle – predominance.

Figure 1: Absorption (left) and reduced scattering (right) spectra on the
chest of 5 healthy volunteers using a source-detector distance $\rho=3$ cm and
a homogeneous model.

To gain an initial guess on the maximum detectable photon traveling time,
which determines the maximum explored depth in the medium[27, 28], in Fig. 2
we show the DTOFs measured on the chest of the same 5 subjects using quite a
large $\rho=7-9$ cm. We adopted here the single-wavelength system while
choosing the maximum $\rho$ permitting to reach the max count rate for the
instrumentation (1 M counts/s). Increasing $\rho$ beyond that limit would
simply decrease the number of collected photons at any times[27]. As expected,
the temporal profile is quite broad and extends to late times, with early
photons arriving not earlier than 1 ns after the pulse injection time ($t=0$
ns). The maximum photon arrival time, or better the max $t$ when the DTOF
stands out of the noise, ranges from 4 ns for subject #1 up to 8 ns for
subject #5. These values can provide a first hint to interpret the simulations
presented below, and to guess the maximum investigation depth of the
measurements. The absorption and scattering properties derived using a
homogeneous model for this large $\rho$ at 820 nm are also reported in Table
1.

Table 1: Subjects’ demographics, mean composition, and optical properties at 820 nm. subject | age | BMI | Hb | HbO2 | water | lipid | collagen | a | b |  $\mu_{a}$(*) |  $\mu_{s}^{\prime}$(*)
---|---|---|---|---|---|---|---|---|---|---|---
| (y) | (Kg/m2) | $\mu M$ | $\mu M$ | g/cm3 | g/cm3 | g/cm3 | cm-1 |  | cm-1 | cm-1
#1 |  |  | 1.5 | 25.1 | 0.34 | 0.43 | 0.21 | 11.1 | 0.63 | 0.32 | 11.7
#2 | 65 | 23.1 | 3.0 | 15.5 | 0.29 | 0.63 | 0.11 | 9.0 | 0.78 | 0.14 | 7.3
#3 | 52 | 19.1 | 6.9 | 26.4 | 0.49 | 0.67 | 0.14 | 10.5 | 0.75 | 0.34 | 11.6
#4 | 34 | 24.1 | 0.8 | 11.4 | 0.12 | 0.69 | 0.04 | 10.0 | 0.59 | 0.15 | 8.2
#5 | 49 | 25.8 | 1.3 | 15.8 | 0.14 | 0.85 | 0.04 | 9.0 | 0.61 | 0.09 | 5.3
(*)820nm |  |  |  |  |  |  |  |  |  |  |
Figure 2: Distribution of photon times of flight (DTOFs) for the 5 volunteers
on the chest in reflectance geometry using the largest source-detector
distance ($\rho=7\sim 9$ cm) yielding a full count-rate collection statistics
of $\approx 1$ Mcounts/s. The maximum photon travelling time is up to $4\sim
8$ ns depending on the subject.

##  4 Simulations

As an initial step to answer the question whether it is possible to reach the
lung, we addressed the problem from a theoretical point of view, studying the
maximum depth reached by photons in a homogeneous diffusive medium in
reflectance geometry. Here, we adopted the approach proposed in Ref.[28] that
derives the mean value of the maximum depth $z_{max}$ of all possible photon
trajectories in a semi-infinite medium (see Material and Methods for details).
Figure 3 displays $z_{max}$ as a function of photon travelling time $t$ for
the $\mu_{s}^{\prime}$ derived for each subject at 820 nm with $\rho=3$ cm
(data from Fig. 1), which reasonably represents the mean scattering properties
in the superficial chest layers than need to be traversed to reach the lung.
For a homogeneous medium, the mean depth is rigorously independent of both
$\mu_{a}$ and $\rho$, which affect the number of collected photons but do not
alter the depth distribution of photon trajectories for a given $t$.
Calculating the maximum photon arrival time $t_{max}$ yielding >10,000
counts/s for a late gate (shot noise = 1%) we derive $t_{max}\approx 3.2-5.7$
ns, which leads to $z_{max}\approx 2.6-3.9$ cm for different subjects, as
represented by lines in Fig. 3.

Figure 3: Mean value of the maximum depth of photon trajectories ($z_{max}$)
as a function of photon arrival time derived for the $\mu_{s}^{\prime}$
properties of the 5 subjects. The straight lines indicate the maximum $t$
yielding a count-rate >10,000 counts/s over a late gate for data in Fig. 2,
and the corresponding value of $z_{max}$

As a more realistic scenario, we assumed a two-layer medium composed of the
lung (bottom layer) and all the overlaying tissues (top layer). Using a Monte
Carlo code, we simulated a 10% reduction in the lung absorption. Figure 4
shows the relative contrast $C$ between the perturbed and unperturbed states
plotted as a function of the photon arrival time $t$ for different depths of
the lung layer $z_{lung}$ (rows) and for different choices of
$\mu_{s}^{\prime}$ in the lung (columns), while assuming $\mu_{s}^{\prime}=7$
cm-1 for the upper layer. A relative contrast $C\approx 2-3\%$ is achieved
down to $z_{lung}\approx 3$ cm for $t=4$ ns, and even down to $z_{lung}\approx
4$ cm for $t=8$ ns. Quite surprisingly, the lung scattering does not affect
much $C$. Yet this is in agreement with the previous argument that the photon
travelling depth is mostly affected by the upper layer. This result relaxes
the threat that an extremely high $\mu_{s}^{\prime}$ in the lung could
completely prevent lung sensing. Indeed, there is not yet a sound knowledge in
literature of the lung scattering properties, and we opted to span a wide
range. The scattering of the upper layer could be an additional variable,
although we observed limited inter-subject variability (see Fig. 1 and Table
1).

Figure 4: Monte Carlo simulations of the relative contrast $C$ as a function
of photon arrival time $t$ for an absorption reduction of 10% in the lung
absorption for different values of $\mu_{s}^{\prime}$ assumed homogeneous
(columns) and of the lung depth (rows).

##  5 In vivo breathing protocol

As an in vivo trial of the actual chances to sense the lung, we performed in
vivo measurements on the 5 subjects following a breathing protocol as
described in Section 2. The aim of this protocol was to produce a decrease in
both $\mu_{a}$ and $\mu_{s}^{\prime}$ of the lung during the inhalation phase,
due to the reduction of lung density, and check the sensitivity of the optical
measurement to such changes.

Figure 5 reports the evolution of $\mu_{a}$ (red, left axis) and
$\mu_{s}^{\prime}$ (blue, right axis) as obtained using a homogeneous fit over
the time of the task for the 5 subjects (columns) and 2 positions on the
chest, namely an up-right location (UR, top row) and a down-right location
(DR, bottom row). The protocol in use was Prot10 and the folding average over
the exercise time was applied. In general, there is a task-related change in
the fitted optical properties. Yet, the results are somehow contradictory with
large inter- and intra-subject differences. In some cases, (e.g. #5-DR), both
$\mu_{a}$ and $\mu_{s}^{\prime}$ decrease with inhalation as expected from the
reduction in lung density, but in other cases the behavior is just the
opposite (e.g. #3-UR), or even a parameter increases while the other decreases
(e.g. #1-DR). Also intra-subject differences among the two positions are
observed (e.g. #1). The task-related alteration is observed also in the not-
refolded time evolution (see Supplementary Fig. S1), yet again with quite
different patterns. According to our own experience in TD-DOS, a homogeneous
fit of a large-$\rho$ measurement typically enhances the optical properties of
the lower (few cm depth) layer[29]. This is particularly true for $\mu_{a}$,
while information on $\mu_{s}^{\prime}$ is more superficial[29]. The other
protocol (Prot5) shows substantially similar results (see Supplementary Fig.
S2 and Fig. S3 for the re-folded and not-folded representation, respectively).

Figure 5: In vivo time evolution of the absorption (red) and reduced
scattering (blue) coefficients at 820 nm during the inhalation protocol Prot10
for the 5 volunteers (rows) and 2 locations (columns) over the lung. At $t=0$
s, the subject is asked to inhale, while at $t=10$ s the subject starts to
exhale.

As a different view to the same data, we analyzed the relative contrast $C$ in
photon counts for different time gates over the IN and OUT phases,
respectively. Figure 6 displays $C(t)$ for 5 different gates, with $t$ ranging
from 0.5 to 4.5 ns and $\Delta t=0.5$ ns as a function of the refolded time
for the 5 subjects (columns) and the 2 positions (rows). The Prot10 protocol
with folding average is displayed here. In all 10 plots, apart from #5-UR and
possibly #4-DR, we observe a reduction in $C$ for late gates, that is a
reduction in signal intensity when the lung tissue gets denser (OUT phase).
The other protocol Prot5 shows substantially similar results (see
Supplementary Fig. S4 for the not-folded Prot10, and Fig. S5 and Fig. S6 for
the Prot5 in re-folded and not-folded visualization, respectively).

Figure 6: In vivo time evolution of the relative contrast for different time
gates (see legend) at 820 nm during the breathing protocol Prot10 for the 5
volunteers (columns) and 2 locations (rows) over the lung. At $t=0$ s, the
subject is asked to inhale, while at $t=10$ s the subject starts to exhale.

In order to better visualize the signal perturbation as a function of the
photon arrival time $t$, we calculated the average contrast with respect to
the plateau state of the inspiration and expiration phases (breath hold).
Figure 7 shows the mean value of $C(t)$ during the IN phase (blue line), the
OUT phase (green line) and the IN-OUT difference (orange line) as a function
of the gate time $t$. As for Fig. 5 and Fig. 6, columns correspond to the
subject, while rows to the position. In general terms, the IN-OUT line is
positive for late gates for most subjects as expected. Yet, following
simulations in Fig. 4, $C(t)$ should be monotonically increasing with $t$ due
to the increase in mean probed depth and therefore in lung contribution. The
situation depicted in Fig. 7 is more complex. There seem to be two opposite
competing factors causing the IN-OUT curve to cross the null contrast at a
given $\bar{t}$ and then to diverge for larger $t$ values. Similar results are
observed for the Prot5 protocol (see Supplementary Fig. S7).

Figure 7: Dependence of the relative contrast $C$ $vs$ photon propagation time
during the plateau of the inhale (IN) or exhale (OUT) phases, together with
the difference (IN-OUT) for the 5 volunteers (columns) and 2 locations (rows).
In most cases, $C$ (i.e. the collected signal) increases from the OUT to the
IN phase, yet with strong inter- and intra-subject variability and complex
evolution for increasing photon propagation time.

All these in vivo results seem to point out that we are possibly reaching the
lung, but there are competing factors, making the interpretation of the
breathing protocol not straightforward. In the following, we analyse five
issues to be considered.

The first factor is the interplay of $\mu_{a}$ and $\mu_{s}^{\prime}$ in the
lung during inhalation. Both of them should decrease, due to the lower tissue
density, yet the forecast of scattering reduction is not necessarily linear
with tissue density due to the contribution of air-filled alveoli as
independent scattering centers[30]. Even more intriguing, complex phenomena
such as anomalous diffusion due to the air gaps could be invoked[31]. As a
very simple, extreme case, we plot in Fig. 8 the Monte Carlo simulation of the
contrast $C(t)$ assuming an identical reduction by 20% of $\mu_{a}$ and
$\mu_{s}^{\prime}$ in the lung. A reduction only in absorption (red line)
yields a linear increase of $C$ vs $t$. A reduction only in scattering (blue
line) yields a linear decrease of $C$ vs $t$ of a comparable amount. The
combined effect of both parameters (green line) dramatically reduces the
overall contrast.

Figure 8: Monte Carlo simulation of the relative contrast as a function of the
photon arrival time ($t$) assuming an increase in lung air content leading to
a 20% reduction only in $\mu_{a}$ (ABS) in $\mu_{s}^{\prime}$ (SCA) or rather
the combined reduction of both (ABS+SCA).

A second troubling factor resides in the lung scattering, which could be
extremely high, thus preventing its exploration. There is still poor knowledge
on the lung optical properties, particularly in vivo. Some ex vivo studies
report different values for $\mu_{s}^{\prime}$ reaching up to $20$ cm-1 at 800
nm (calculated using Eq. 2 and tabulated $a$ and $b$ values)[32], but this
high value could be due to the room temperature adopted in measurements
leading to overestimation of $\mu_{s}^{\prime}$ with respect to the in vivo
environment. At any rate, simulations presented in Fig. 4 indicate that the
scattering of the lower (lung) layer is not so critical in determining the
sensitivity to lung properties beyond a given photon travelling time over a
large range of lung $\mu_{s}^{\prime}$ values.

A third factor to be considered is the pleura, which is the two-foil membrane
enveloping the lung and the chest wall, separated by a lubrication fluid,
which permits smooth sliding and expansion of the lung during the inhalation
act. From the optical point of view, the pleura could cause a light-guiding
effect, creating contamination from more superficial photons even at a large
$t$ values, thus covering deep photon signals. Also in the brain, the
cerebrospinal fluid (CSF) was questioned to cause light-guiding effects in
functional Near Infrared Spectroscopy measurements (fNIRS). This issue was
long discussed in the initial ages of fNIRS with opposite conclusions, but now
the question is definitely settled by strong evidence of cortex-related
activation in many studies and agreement with fMRI scans[33]. The pleura
thickness - extrapolated from animal data - could be in the $30-40\mu m$
range[34], so definitely lower than the CBF layer.

A forth aspect is the inadequacy of the simple analysis tools used in this
study to describe the complex chest structure. The homogeneous fit proposed in
Fig. 5 is a clear oversimplification of the actual geometry. Yet, in previous
works confirmed with phantom experiments, we demonstrated that $\mu_{a}$
spectra obtained for reasonable large $t$ values using a homogeneous fit tend
to adhere to the lower (few cm deep) layer[29]. The photon pathlength in the
upper layer reaches quickly a stable plateau after a given $t$, while the
pathlength in the lower layer linearly increases with $t$ above the same
threshold. Thus, the fit on $\mu_{a}$ is mainly affected by the lower layer
properties being related to the temporal slope on the tail of the DTOF. Still,
this simple argument does not hold true for $\mu_{s}^{\prime}$, and in our
case the variation in both $\mu_{a}$ and $\mu_{s}^{\prime}$ might not be
properly captured by this simple model.

A fifth concern is related to the changes in optical properties in the lung
during the inhalation phase and the conversion of Hb to O2Hb. Still, the
adopted wavelength (820 nm) is close to the isosbestic point and should be
weakly dependent on the oxygenation status. Also, in the long breath holding
protocol (Prot10) the observed changes are quite rapid and possibly related to
the inhalation act and not continuously increasing with laboratory time as if
the reason were the progressive uptake of O2.

##  6 Conclusion

We have presented the first study on the transdermal optical accessibility of
the lung using time domain diffuse optics based on Monte Carlo simulations and
in vivo measurements on 5 healthy volunteers. In vivo broadband absorption and
reduced scattering spectra in the 600-1100 nm range at $\rho=3$ cm analyzed
using a homogeneous model permitted to estimate a possible range of optical
properties mostly ascribed to the tissues to be traversed to reach the lung.
In particular, around 800 nm we obtained $\mu_{a}=0.1-0.2$ cm-1, and
$\mu_{s}^{\prime}=7-9$ cm-1. Even at a larger $\rho$ ($7-9$ cm), the measured
optical properties at 820 nm fall in a similar range. The same large-$\rho$ in
vivo measurements showed that a photon travelling time up to $t\approx
3.2-5.7$ ns with at least 10,000 counts/s (shot noise=1%) in a late gate could
be reached for different subjects. Also, two-layer Monte Carlo simulations
with different choices of lung $\mu_{s}^{\prime}$ in a wide range of values
($3-48$ cm-1) showed a relative contrast $C\approx 2-3\%$ for a 10% reduction
in lung $\mu_{a}$ down to a depth of $3$ $(4)$ cm for $t=4$ $(8)$ ns, which is
well beyond the average lung depth. The in vivo measurements on a paced
breathing protocol showed clear task-related changes in time domain signals.
Yet, the analysis of DTOFs using a homogeneous model showed contradictory
results, not in agreement with the expected decrease in both $\mu_{a}$ and
$\mu_{s}^{\prime}$ during inhalation. Conversely, the plot of the relative
contrast $C$ for the photon counts in temporal gates showed a more consistent
trend with a general increase in reflectance signal during the inhalation
phase, as expected from diffuse optics models. Still, the contrast is not
always increasing upon increasing photon arrival times, as foreseen by a two-
layered model, and there seems to be competition between opposite factors.

In conclusion, we do not have yet a definitive sound evidence to be able to
detect lung-related properties in vivo. Yet, we contribute to the knowledge
base on this intriguing quest by providing optical properties from the chest,
clear simulation scenarios and a potentially powerful protocol for in vivo
validation. We plan to deploy the whole dataset in an open data paper for
future analysis with more refined models. Technology is advancing at a fast
pace in TD-DOS, both in terms of reduction in cost-size and improvement in
performances. In particular, the increase in area of new SiPMs detectors[35]
as well as the gating capabilities[36] means that larger photon travelling
times and consequently penetration depths will be reached in the near future.
What we still lack is a clear understanding of the physics and physiology of
chest diffuse optics and further studies are needed to add additional insight
and in vivo validation.

## Acknowledgments

The authors acknowledge financial support from the European Union’s Horizon
2020 programme under the BITMAP (n.675332) and LaserLab Europe (n.654148)
grants.

## Disclosure

The authors declare no conflicts of interest.

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##  7 Supplementary Figures

The following figures will be attached as supplementary figures to the main
figures, showing the other protocol (Prot5) with 5s duration of each phase and
10 repetitions (the first one with normal breathing), and the non-refolded
analysis.

Figure 1: Supplementary Figure S1 to Fig. 5 Figure 2: Supplementary Figure S2
to Fig. 5 Figure 3: Supplementary Figure S3 to Fig. 5 Figure 4: Supplementary
Figure S4 to Fig. 6 Figure 5: Supplementary Figure S5 to Fig. 6 Figure 6:
Supplementary Figure S6 to Fig. 6 Figure 7: Supplementary Figure S7 to Fig. 7


construct the complete basis of the $Q$-exact operators, which have the same
level $j$ and $SU(3)$ representation with the target trace relation. (The
$Q$-action does not change $j$ and $SU(3)$ representation.) However, in
practice, we can make some ansätze for the $Q$-exact form to reduce the
dimension of $Q$-exact basis. One of our working assumptions is that the
maximal number of $f$’s appearing before the $Q$-action is the same as that of
the trace relation. There is a priori no reason to assume that but it turns
out to be true for our examples. After imposing this assumption (and a couple
of extra practical assumptions), we find a particular linear combination of
the $Q$-exact operators in our basis for the target trace relation. In
general, when we write $R_{I}\sim Qr_{I}$ for a trace relation $R_{I}$, there
exist ambiguities of $r_{I}$ since we can add arbitrary $Q$-closed operators
to $r_{I}$. We partly fix them by requiring $r_{I}$ to vanish when
$\phi,\psi,f$ are restricted to diagonal matrices. We do not know whether such
a requirement can be satisfied in general, but it does work for our examples.
The purpose of this requirement will be explained later. The other ambiguities
are fixed by hand to get compact expressions.

Below, we list the operators $r_{j}^{(n_{f},n_{\psi})}$ related to the
fundamental trace relations $R_{j}^{(n_{f},n_{\psi}-1)}$ by
$i\,Qr_{j}^{(n_{f},n_{\psi})}=R_{j}^{(n_{f},n_{\psi}-1)}$. We will not list
all $r_{j}^{(n_{f},n_{\psi})}$’s, but only those which are used in section
4.1. For the relations without $f$’s in (A), we obtain

$\displaystyle(r_{10}^{(0,1)})^{i}_{jk}=-2\,\epsilon_{a_{1}a_{2}(j}\textrm{tr}\left(\phi^{a_{1}}\phi^{a_{2}}\phi^{i}\psi_{k)}\right)\
,$ $\displaystyle
r_{12}^{(0,1)}=\epsilon_{a_{1}a_{2}a_{3}}\left[6\textrm{tr}\left(\psi_{b}\phi^{a_{1}}\right)\textrm{tr}\left(\phi^{b}\phi^{a_{2}}\phi^{a_{3}}\right)-\textrm{tr}\left(\psi_{b}\phi^{a_{1}}\phi^{a_{2}}\right)\textrm{tr}\left(\phi^{b}\phi^{a_{3}}\right)\right]$
$\displaystyle\qquad\quad-3\epsilon_{a_{1}a_{2}a_{3}}\left[\textrm{tr}\left(\psi_{b}\phi^{b}\phi^{a_{1}}\phi^{a_{2}}\phi^{a_{3}}\right)\\!+\\!\textrm{tr}\left(\psi_{b}\phi^{a_{1}}\phi^{b}\phi^{a_{2}}\phi^{a_{3}}\right)\\!+\\!\textrm{tr}\left(\psi_{b}\phi^{a_{1}}\phi^{a_{2}}\phi^{b}\phi^{a_{3}}\right)\\!+\\!\textrm{tr}\left(\psi_{b}\phi^{a_{1}}\phi^{a_{2}}\phi^{a_{3}}\phi^{b}\right)\right]\
,$
$\displaystyle(r_{12}^{(0,1)})^{ij}_{kl}=-2\epsilon_{a_{1}a_{2}(k}\left[\textrm{tr}\left(\psi_{l)}\phi^{(i}\phi^{j)}\phi^{a_{1}}\phi^{a_{2}}\right)+7\textrm{tr}\left(\psi_{l)}\phi^{(i|}\phi^{a_{1}}\phi^{|j)}\phi^{a_{2}}\right)\right]\
,$
$\displaystyle(r_{12}^{(0,2)})_{ijk}=\frac{1}{2}\epsilon_{a_{1}a_{2}(i}\textrm{tr}\left(\phi^{a_{1}}\psi_{j}\phi^{a_{2}}\psi_{k)}\right)\
,$
$\displaystyle(r_{12}^{(0,2)})^{i}_{j}=6\textrm{tr}\left(\phi^{(i}\phi^{a)}\psi_{(a}\psi_{j)}\right)-5\textrm{tr}\left(\phi^{[i}\psi_{a}\phi^{a]}\psi_{j}\right)\
,$
$\displaystyle(r_{12}^{(0,2)})^{ij}_{kl}=\textrm{tr}\left(\phi^{(i}\phi^{j)}\psi_{(k}\psi_{l)}\right)\
,$
$\displaystyle(r_{14}^{(0,2)})^{i}=3\,\textrm{tr}\left(\phi^{i}\psi_{a_{1}}\phi^{a_{1}}\phi^{a_{2}}\psi_{a_{2}}\right)+2\,\textrm{tr}\left(\phi^{i}\phi^{a_{1}}\right)\textrm{tr}\left(\phi^{a_{2}}\psi_{(a_{1}}\psi_{a_{2})}\right)$
$\displaystyle\qquad\qquad-6\,\textrm{tr}\left(\phi^{i}\psi_{a_{1}}\right)\textrm{tr}\left(\phi^{[a_{1}}\phi^{a_{2}]}\psi_{a_{2}}\right)-\,\textrm{tr}\left(\phi^{i}\psi_{a_{1}}\psi_{a_{2}}\right)\textrm{tr}\left(\phi^{a_{1}}\phi^{a_{2}}\right)\
,$
$\displaystyle(r_{14}^{(0,2)})_{ij}=\frac{5}{9}\epsilon_{a_{1}a_{2}a_{3}}\left[2\textrm{tr}\left(\psi_{(i}\psi_{j)}\phi^{a_{1}}\phi^{a_{2}}\phi^{a_{3}}\right)+\textrm{tr}\left(\psi_{(i}\phi^{a_{1}}\psi_{j)}\phi^{a_{2}}\phi^{a_{3}}\right)\right]$
$\displaystyle\qquad\quad+\epsilon_{a_{1}a_{2}(i}\left[\textrm{tr}\left(\psi_{j)}\psi_{a_{3}}\phi^{a_{1}}\phi^{a_{2}}\phi^{a_{3}}\right)+\textrm{tr}\left(\psi_{j)}\psi_{a_{3}}\phi^{a_{1}}\phi^{a_{3}}\phi^{a_{2}}\right)+\textrm{tr}\left(\psi_{j)}\psi_{a_{3}}\phi^{a_{3}}\phi^{a_{1}}\phi^{a_{2}}\right)\right]$
$\displaystyle\qquad\quad+\epsilon_{a_{1}a_{2}(i}\left[\textrm{tr}\left(\psi_{j)}\phi^{a_{1}}\psi_{a_{3}}\phi^{a_{2}}\phi^{a_{3}}\right)+\textrm{tr}\left(\psi_{j)}\phi^{a_{1}}\psi_{a_{3}}\phi^{a_{3}}\phi^{a_{2}}\right)+\textrm{tr}\left(\psi_{j)}\phi^{a_{3}}\psi_{a_{3}}\phi^{a_{1}}\phi^{a_{2}}\right)\right]$
$\displaystyle\qquad\quad+\epsilon_{a_{1}a_{2}(i}\left[\textrm{tr}\left(\psi_{j)}\phi^{a_{1}}\phi^{a_{2}}\psi_{a_{3}}\phi^{a_{3}}\right)+\textrm{tr}\left(\psi_{j)}\phi^{a_{1}}\phi^{a_{3}}\psi_{a_{3}}\phi^{a_{2}}\right)+\textrm{tr}\left(\psi_{j)}\phi^{a_{3}}\phi^{a_{1}}\psi_{a_{3}}\phi^{a_{2}}\right)\right]$
$\displaystyle\qquad\quad+\epsilon_{a_{1}a_{2}(i}\left[\textrm{tr}\left(\psi_{j)}\phi^{a_{1}}\phi^{a_{2}}\phi^{a_{3}}\psi_{a_{3}}\right)+\textrm{tr}\left(\psi_{j)}\phi^{a_{1}}\phi^{a_{3}}\phi^{a_{2}}\psi_{a_{3}}\right)+\textrm{tr}\left(\psi_{j)}\phi^{a_{3}}\phi^{a_{1}}\phi^{a_{2}}\psi_{a_{3}}\right)\right]$
$\displaystyle\qquad\quad-\frac{1}{3}\epsilon_{a_{1}a_{2}(i}\left[5\textrm{tr}\left(\psi_{j)}\phi^{a_{1}}\phi^{a_{2}}\right)\textrm{tr}\left(\psi_{a_{3}}\phi^{a_{3}}\right)+2\textrm{tr}\left(\psi_{j)}\phi^{(a_{1}}\phi^{a_{3})}\right)\textrm{tr}\left(\psi_{a_{3}}\phi^{a_{2}}\right)-2\textrm{tr}\left(\psi_{j)}\phi^{a_{2}}\right)\textrm{tr}\left(\psi_{a_{3}}\phi^{(a_{1}}\phi^{a_{3})}\right)\right]\
,$
$\displaystyle(r_{14}^{(0,2)})^{ij}_{k}=12\textrm{tr}\left(\phi^{(i}\phi^{a}\phi^{j)}\psi_{(a}\psi_{k)}\right)+12\textrm{tr}\left(\phi^{(i|}\phi^{a}\phi^{|j)}\psi_{(a}\psi_{k)}\right)+54\textrm{tr}\left(\phi^{(i}\phi^{j}\psi_{(a}\phi^{a)}\psi_{k)}\right)-36\textrm{tr}\left(\phi^{(i}\phi^{j)}\psi_{(a}\phi^{a}\psi_{k)}\right)\
,$
$\displaystyle(r_{14}^{(0,2)})^{i}_{jkl}=2\epsilon_{a_{1}a_{2}(j}\left[\textrm{tr}\left(\phi^{i}\phi^{a_{1}}\phi^{a_{2}}\psi_{k}\psi_{l)}\right)+3\textrm{tr}\left(\phi^{i}\phi^{a_{1}}\psi_{k}\phi^{a_{2}}\psi_{l)}\right)-2\textrm{tr}\left(\phi^{i}\psi_{k}\phi^{a_{1}}\phi^{a_{2}}\psi_{l)}\right)\right]\
,$
$\displaystyle(r_{14}^{(0,3)})^{i}_{jkl}=-\frac{1}{2}\textrm{tr}\left(\phi^{i}\psi_{(j}\psi_{k}\psi_{l)}\right)\
,$
$\displaystyle(r_{16}^{(0,3)})_{i}=\frac{39}{4}\textrm{tr}\left(\psi_{i}\\{\psi_{b_{1}}\psi_{b_{2}},\phi^{b_{1}}\phi^{b_{2}}\\}\right)+2\textrm{tr}\left(\psi_{i}\psi_{b_{1}}\phi^{b_{1}}\psi_{b_{2}}\phi^{b_{2}}\right)-\frac{61}{4}\textrm{tr}\left(\psi_{i}\psi_{b_{1}}\phi^{b_{2}}\psi_{b_{2}}\phi^{b_{1}}\right)+\frac{97}{4}\textrm{tr}\left(\psi_{i}\phi^{b_{1}}\psi_{b_{1}}\psi_{b_{2}}\phi^{b_{2}}\right)$
$\displaystyle\qquad\quad-\frac{41}{4}\textrm{tr}\left(\psi_{i}\phi^{b_{2}}\psi_{b_{1}}\psi_{b_{2}}\phi^{b_{1}}\right)-5\textrm{tr}\left(\psi_{i}\psi_{b_{1}}\phi^{b_{1}}\phi^{b_{2}}\psi_{b_{2}}\right)-\frac{25}{2}\textrm{tr}\left(\psi_{i}\psi_{b_{1}}\phi^{b_{2}}\phi^{b_{1}}\psi_{b_{2}}\right)+2\textrm{tr}\left(\psi_{i}\phi^{b_{1}}\psi_{b_{1}}\phi^{b_{2}}\psi_{b_{2}}\right)$
$\displaystyle\qquad\quad-\frac{61}{4}\textrm{tr}\left(\psi_{i}\phi^{b_{2}}\psi_{b_{1}}\phi^{b_{1}}\psi_{b_{2}}\right)-\frac{11}{4}\textrm{tr}\left(\phi^{b_{1}}\phi^{b_{2}}\right)\textrm{tr}\left(\psi_{i}\psi_{b_{1}}\psi_{b_{2}}\right)-\frac{27}{2}\textrm{tr}\left(\psi_{b_{1}}\psi_{b_{2}}\right)\textrm{tr}\left(\psi_{i}\phi^{b_{1}}\phi^{b_{2}}\right)$
$\displaystyle\qquad\quad+\frac{29}{4}\textrm{tr}\left(\phi^{b_{2}}\psi_{b_{2}}\right)\textrm{tr}\left(\psi_{i}[\psi_{b_{1}},\phi^{b_{1}}]\right)\
,$
$\displaystyle(r_{16}^{(0,3)})^{i}_{jk}=2\textrm{tr}\left(\psi_{(j}\psi_{k)}\psi_{b}\phi^{b}\phi^{i}\right)-4\textrm{tr}\left(\psi_{(j}\psi_{k)}\psi_{b}\phi^{i}\phi^{b}\right)-\textrm{tr}\left(\psi_{(j|}\psi_{b}\psi_{|k)}\\{\phi^{b},\phi^{i}\\}\right)-4\textrm{tr}\left(\psi_{(j}\psi_{k)}\phi^{(b}\psi_{b}\phi^{i)}\right)$
$\displaystyle\qquad\qquad+7\textrm{tr}\left(\psi_{(j|}\\{\psi_{b},\phi^{b}\\}\psi_{|k)}\phi^{i}\right)-11\textrm{tr}\left(\psi_{(j|}\\{\psi_{b},\phi^{i}\\}\psi_{|k)}\phi^{b}\right)-4\textrm{tr}\left(\psi_{(j}\psi_{k)}\phi^{b}\phi^{i}\psi_{b}\right)+2\textrm{tr}\left(\psi_{(j}\psi_{k)}\phi^{i}\phi^{b}\psi_{b}\right)$
$\displaystyle\qquad\qquad+3\textrm{tr}\left(\psi_{(j|}\psi_{b}\right)\textrm{tr}\left(\psi_{|k)}[\phi^{b},\phi^{i}]\right)+6\textrm{tr}\left(\psi_{(j}\phi^{[b}\right)\textrm{tr}\left(\\{\psi_{k)},\psi_{b}\\}\phi^{i]}\right)\
.$ (A.6)

For the relations involving $f$’s in (A), we find

$\displaystyle(r_{14}^{(1,1)})_{ij}=5\epsilon_{a_{1}a_{2}(i}\textrm{tr}\left(f\phi^{a_{1}}\psi_{j)}\phi^{a_{2}}\right)+\textrm{tr}\left(\phi^{a}\left\\{\psi_{a},\psi_{(i}\psi_{j)}\right\\}\right)-4\,\textrm{tr}\left(\phi^{a}\psi_{(i|}\psi_{a}\psi_{|j)}\right)\
,$
$\displaystyle(r_{14}^{(1,1)})^{ij}_{k}=3\,\textrm{tr}\left(f\phi^{(i}\phi^{j)}\psi_{k}\right)-3\,\textrm{tr}\left(f\psi_{k}\phi^{(i}\phi^{j)}\right)+\epsilon^{a_{1}a_{2}(i}\textrm{tr}\left(\phi^{j)}\psi_{k}\psi_{a_{1}}\psi_{a_{2}}\right)-\epsilon^{a_{1}a_{2}(i}\textrm{tr}\left(\phi^{j)}\psi_{a_{1}}\psi_{a_{2}}\psi_{k}\right)\
,$
$\displaystyle(r_{16}^{(1,1)})_{i}=13\epsilon_{a_{1}a_{2}a_{3}}\textrm{tr}\left(f\psi_{i}\right)\textrm{tr}\left(\phi^{a_{1}}\phi^{a_{2}}\phi^{a_{3}}\right)+\frac{10}{3}\epsilon_{a_{1}a_{2}a_{3}}\textrm{tr}\left(f\phi^{a_{1}}\right)\textrm{tr}\left(\psi_{i}\phi^{a_{2}}\phi^{a_{3}}\right)+\frac{10}{3}\epsilon_{a_{1}a_{2}a_{3}}\textrm{tr}\left(f\phi^{a_{1}}\phi^{a_{2}}\right)\textrm{tr}\left(\psi_{i}\phi^{a_{3}}\right)$
$\displaystyle+46\epsilon_{ia_{1}a_{2}}\textrm{tr}\left(f\phi^{b}\right)\textrm{tr}\left(\psi_{b}\phi^{a_{1}}\phi^{a_{2}}\right)-7\epsilon_{ia_{1}a_{2}}\textrm{tr}\left(f\phi^{a_{1}}\right)\textrm{tr}\left(\psi_{b}\phi^{a_{2}}\phi^{b}\right)-7\epsilon_{ia_{1}a_{2}}\textrm{tr}\left(f\phi^{b}\phi^{a_{1}}\right)\textrm{tr}\left(\psi_{b}\phi^{a_{2}}\right)$
$\displaystyle+6\epsilon_{ia_{1}a_{2}}\textrm{tr}\left(f\phi^{a_{1}}\phi^{a_{2}}\right)\textrm{tr}\left(\psi_{b}\phi^{b}\right)-\frac{115}{3}\epsilon_{a_{1}a_{2}a_{3}}\textrm{tr}\left(f\psi_{i}\phi^{a_{1}}\phi^{a_{2}}\phi^{a_{3}}\right)-\frac{95}{3}\epsilon_{a_{1}a_{2}a_{3}}\textrm{tr}\left(f\phi^{a_{1}}\psi_{i}\phi^{a_{2}}\phi^{a_{3}}\right)$
$\displaystyle+5\epsilon_{a_{1}a_{2}a_{3}}\textrm{tr}\left(f\phi^{a_{1}}\phi^{a_{2}}\psi_{i}\phi^{a_{3}}\right)+36\epsilon_{ia_{1}a_{2}}\textrm{tr}\left(f\psi_{b}\phi^{a_{1}}\phi^{a_{2}}\phi^{b}\right)-43\epsilon_{ia_{1}a_{2}}\textrm{tr}\left(f\psi_{b}\phi^{a_{1}}\phi^{b}\phi^{a_{2}}\right)$
$\displaystyle+39\epsilon_{ia_{1}a_{2}}\textrm{tr}\left(f\phi^{a_{1}}\psi_{b}\phi^{a_{2}}\phi^{b}\right)-68\epsilon_{ia_{1}a_{2}}\textrm{tr}\left(f\phi^{a_{1}}\phi^{a_{2}}\psi_{b}\phi^{b}\right)+39\epsilon_{ia_{1}a_{2}}\textrm{tr}\left(f\phi^{a_{1}}\phi^{b}\psi_{b}\phi^{a_{2}}\right)$
$\displaystyle+13\textrm{tr}\left(\psi_{i}\\{\psi_{b_{1}}\psi_{b_{2}},\phi^{b_{1}}\phi^{b_{2}}\\}\right)-31\textrm{tr}\left(\psi_{i}\\{\psi_{b_{1}}\psi_{b_{2}},\phi^{b_{2}}\phi^{b_{1}}\\}\right)+14\textrm{tr}\left(\psi_{i}\psi_{b_{1}}\phi^{b_{1}}\psi_{b_{2}}\phi^{b_{2}}\right)$
$\displaystyle-22\textrm{tr}\left(\psi_{i}\psi_{b_{1}}\phi^{b_{2}}\phi^{b_{1}}\psi_{b_{2}}\right)+14\textrm{tr}\left(\psi_{i}\phi^{b_{1}}\psi_{b_{1}}\phi^{b_{2}}\psi_{b_{2}}\right)\
,$
$\displaystyle(r_{16}^{(1,1)})^{i}_{jk}=\epsilon_{a_{1}a_{2}(j}\left[-4\textrm{tr}\left(f\phi^{i}\right)\textrm{tr}\left(\psi_{k)}\phi^{a_{1}}\phi^{a_{2}}\right)-\textrm{tr}\left(\phi^{i}\phi^{a_{2}}\right)\textrm{tr}\left(f[\psi_{k)},\phi^{a_{1}}]\right)\right]$
$\displaystyle+\epsilon_{a_{1}a_{2}(j}\left[3\textrm{tr}\left(f\phi^{a_{1}}\\{\psi_{k)},\phi^{i}\\}\phi^{a_{2}}\right)+5\textrm{tr}\left(f\\{\psi_{k)},\phi^{a_{1}}\phi^{i}\phi^{a_{2}}\\}\right)-4\textrm{tr}\left(f\psi_{k)}\phi^{i}\phi^{a_{1}}\phi^{a_{2}}\right)-4\textrm{tr}\left(f\phi^{a_{1}}\phi^{a_{2}}\phi^{i}\psi_{k)}\right)\right]$
$\displaystyle+2\textrm{tr}\left(\psi_{(j}\psi_{k)}\psi_{b}[\phi^{b},\phi^{i}]\right)-3\textrm{tr}\left(\psi_{(j|}\psi_{b}\psi_{|k)}\\{\phi^{b},\phi^{i}\\}\right)+6\textrm{tr}\left(\psi_{(j|}\\{\psi_{b},\phi^{b}\\}\psi_{|k)}\phi^{i}\right)-9\textrm{tr}\left(\psi_{(j|}\\{\psi_{b},\phi^{i}\\}\psi_{|k)}\phi^{b}\right)$
$\displaystyle-2\textrm{tr}\left(\psi_{(j}\psi_{k)}\left[\phi^{b},\phi^{i}\right]\psi_{b}\right)+\textrm{tr}\left(\psi_{(j|}\psi_{b}\right)\textrm{tr}\left(\psi_{|k)}[\phi^{b},\phi^{i}]\right)+\textrm{tr}\left(\psi_{(j|}\phi^{b}\right)\textrm{tr}\left(\\{\psi_{|k)},\psi_{b}\\}\phi^{i}\right)\
,$
$\displaystyle(r_{16}^{(1,2)})^{i}_{jk}=-\frac{1}{2}\textrm{tr}\left(f\phi^{i}\psi_{(j}\psi_{k)}\right)-\frac{1}{2}\textrm{tr}\left(f\psi_{(j}\phi^{i}\psi_{k)}\right)-\frac{1}{2}\textrm{tr}\left(f\psi_{(j}\psi_{k)}\phi^{i}\right)-\frac{1}{4}\epsilon^{ia_{1}a_{2}}\textrm{tr}\left(\psi_{a_{1}}\psi_{a_{2}}\psi_{(j}\psi_{k)}\right)\
,$
$\displaystyle(r_{18}^{(1,2)})^{i}_{j}=-4\textrm{tr}\left(f\phi^{i}\phi^{a}\right)\textrm{tr}\left(\psi_{j}\psi_{a}\right)-5\textrm{tr}\left(f\phi^{a}\phi^{i}\right)\textrm{tr}\left(\psi_{j}\psi_{a}\right)$
$\displaystyle-\frac{53}{2}\textrm{tr}\left(f\phi^{i}\psi_{j}\right)\textrm{tr}\left(\phi^{a}\psi_{a}\right)+7\textrm{tr}\left(f\phi^{i}\psi_{a}\right)\textrm{tr}\left(\phi^{a}\psi_{j}\right)+\frac{15}{2}\textrm{tr}\left(f\phi^{a}\psi_{j}\right)\textrm{tr}\left(\phi^{i}\psi_{a}\right)+12\textrm{tr}\left(f\phi^{a}\psi_{a}\right)\textrm{tr}\left(\phi^{i}\psi_{j}\right)$
$\displaystyle+2\textrm{tr}\left(f\psi_{j}\phi^{i}\right)\textrm{tr}\left(\phi^{a}\psi_{a}\right)-13\textrm{tr}\left(f\psi_{a}\phi^{i}\right)\textrm{tr}\left(\phi^{a}\psi_{j}\right)+4\textrm{tr}\left(f\psi_{j}\phi^{a}\right)\textrm{tr}\left(\phi^{i}\psi_{a}\right)$
$\displaystyle+6\textrm{tr}\left(f\psi_{j}\psi_{a}\right)\textrm{tr}\left(\phi^{i}\phi^{a}\right)+\frac{13}{2}\textrm{tr}\left(f\psi_{a}\psi_{j}\right)\textrm{tr}\left(\phi^{i}\phi^{a}\right)$
$\displaystyle-4\textrm{tr}\left(f\phi^{i}\right)\textrm{tr}\left(\phi^{a}\psi_{j}\psi_{a}\right)+14\textrm{tr}\left(f\phi^{i}\right)\textrm{tr}\left(\phi^{a}\psi_{a}\psi_{j}\right)-8\textrm{tr}\left(f\phi^{a}\right)\textrm{tr}\left(\phi^{i}\psi_{j}\psi_{a}\right)-8\textrm{tr}\left(f\phi^{a}\right)\textrm{tr}\left(\phi^{i}\psi_{a}\psi_{j}\right)$
$\displaystyle-4\textrm{tr}\left(f\psi_{j}\right)\textrm{tr}\left(\psi_{a}\phi^{i}\phi^{a}\right)-9\textrm{tr}\left(f\psi_{a}\right)\textrm{tr}\left(\psi_{j}\phi^{i}\phi^{a}\right)+6\textrm{tr}\left(f\psi_{a}\right)\textrm{tr}\left(\psi_{j}\phi^{a}\phi^{i}\right)$
$\displaystyle+3\textrm{tr}\left(f\phi^{i}\phi^{a}\psi_{j}\psi_{a}\right)-\frac{31}{2}\textrm{tr}\left(f\phi^{i}\phi^{a}\psi_{a}\psi_{j}\right)+3\textrm{tr}\left(f\phi^{a}\phi^{i}\psi_{j}\psi_{a}\right)+\frac{5}{2}\textrm{tr}\left(f\phi^{a}\phi^{i}\psi_{a}\psi_{j}\right)$
$\displaystyle+12\textrm{tr}\left(f\phi^{i}\psi_{j}\phi^{a}\psi_{a}\right)-\frac{13}{2}\textrm{tr}\left(f\phi^{i}\psi_{a}\phi^{a}\psi_{j}\right)-6\textrm{tr}\left(f\phi^{a}\psi_{j}\phi^{i}\psi_{a}\right)-\frac{13}{2}\textrm{tr}\left(f\phi^{a}\psi_{a}\phi^{i}\psi_{j}\right)$
$\displaystyle+18\textrm{tr}\left(f\phi^{i}\psi_{j}\psi_{a}\phi^{a}\right)$
$\displaystyle-12\textrm{tr}\left(f\psi_{j}\phi^{i}\phi^{a}\psi_{a}\right)+\frac{17}{2}\textrm{tr}\left(f\psi_{a}\phi^{i}\phi^{a}\psi_{j}\right)-\frac{43}{2}\textrm{tr}\left(f\psi_{a}\phi^{a}\phi^{i}\psi_{j}\right)$
$\displaystyle+\frac{1}{3}\epsilon^{a_{1}a_{2}a_{3}}\textrm{tr}\left(\phi^{i}\psi_{j}\right)\textrm{tr}\left(\psi_{a_{1}}\psi_{a_{2}}\psi_{a_{3}}\right)-2\epsilon^{a_{1}a_{2}i}\textrm{tr}\left(\phi^{b}\psi_{a_{1}}\right)\textrm{tr}\left(\psi_{b}\psi_{j}\psi_{a_{2}}\right)$
$\displaystyle-10\epsilon^{a_{1}a_{2}a_{3}}\textrm{tr}\left(\phi^{i}\psi_{j}\psi_{a_{1}}\psi_{a_{2}}\psi_{a_{3}}\right)+8\epsilon^{a_{1}a_{2}a_{3}}\textrm{tr}\left(\phi^{i}\psi_{a_{1}}\psi_{j}\psi_{a_{2}}\psi_{a_{3}}\right)-2\epsilon^{a_{1}a_{2}a_{3}}\textrm{tr}\left(\phi^{i}\psi_{a_{1}}\psi_{a_{2}}\psi_{j}\psi_{a_{3}}\right)\
,$ $\displaystyle(r_{18}^{(1,2)})_{ijk}$
$\displaystyle=-\epsilon_{a_{1}a_{2}(i}\left[\textrm{tr}\left(f\phi^{a_{1}}\right)\textrm{tr}\left(\phi^{a_{2}}\psi_{j}\psi_{k)}\right)-\frac{3}{2}\textrm{tr}\left(f\psi_{j}\right)\textrm{tr}\left(\psi_{k)}\phi^{a_{1}}\phi^{a_{2}}\right)+3\textrm{tr}\left(f\phi^{a_{1}}\psi_{j}\phi^{a_{2}}\psi_{k)}\right)-3\textrm{tr}\left(f\psi_{j}\phi^{a_{1}}\psi_{k)}\phi^{a_{2}}\right)\right]$
$\displaystyle-\frac{1}{2}\textrm{tr}\left(\phi^{a}\psi_{a}\right)\textrm{tr}\left(\psi_{(i}\psi_{j}\psi_{k)}\right)+\frac{3}{2}\textrm{tr}\left(\phi^{a}\psi_{(i|}\right)\textrm{tr}\left(\psi_{a}\psi_{|j}\psi_{k)}\right)+\frac{1}{2}\textrm{tr}\left(\phi^{a}\psi_{(i}\psi_{j|}\right)\textrm{tr}\left(\psi_{a}\psi_{|k)}\right)$
$\displaystyle+\frac{3}{2}\textrm{tr}\left(\phi^{a}\psi_{(i|}\psi_{a}\psi_{|j}\psi_{k)}\right)-\frac{3}{2}\textrm{tr}\left(\phi^{a}\psi_{(i}\psi_{j|}\psi_{a}\psi_{|k)}\right)\
,$ $\displaystyle(r_{20}^{(2,1)})_{ij}$
$\displaystyle=-\epsilon_{a_{1}a_{2}(i}\left[\textrm{tr}\left(ff\right)\textrm{tr}\left(\phi^{a_{1}}\phi^{a_{2}}\psi_{j)}\right)+\frac{1}{2}\textrm{tr}\left(f\psi_{j)}\right)\textrm{tr}\left(f\phi^{a_{1}}\phi^{a_{2}}\right)+2\textrm{tr}\left(f\phi^{a_{1}}\right)\textrm{tr}\left(f[\phi^{a_{2}},\psi_{j)}]\right)\right]$
$\displaystyle+\epsilon_{a_{1}a_{2}(i}\left[4\textrm{tr}\left(ff\phi^{a_{1}}\phi^{a_{2}}\psi_{j)}\right)-\textrm{tr}\left(f\phi^{a_{1}}\phi^{a_{2}}f\psi_{j)}\right)\right]$
$\displaystyle+2\textrm{tr}\left(f\phi^{a}\psi_{(i}\right)\textrm{tr}\left(\psi_{j)}\psi_{a}\right)-4\textrm{tr}\left(f\psi_{(i}\phi^{a}\right)\textrm{tr}\left(\psi_{j)}\psi_{a}\right)$
$\displaystyle-\frac{1}{2}\textrm{tr}\left(f\psi_{(i}\right)\left(\phi^{a}\psi_{j)}\psi_{a}\right)-\frac{5}{2}\textrm{tr}\left(f\psi_{(i|}\right)\left(\phi^{a}\psi_{a}\psi_{|j)}\right)+2\textrm{tr}\left(f\psi_{a}\right)\left(\phi^{a}\psi_{(i}\psi_{j)}\right)$
$\displaystyle-4\textrm{tr}\left(f\psi_{(i|}\psi_{a}\right)\left(\phi^{a}\psi_{|j)}\right)$
$\displaystyle+2\textrm{tr}\left(f\phi^{a}\psi_{(i}[\psi_{j)},\psi_{a}]\right)+4\textrm{tr}\left(f\phi^{a}\psi_{a}\psi_{(i}\psi_{j)}\right)$
$\displaystyle+4\textrm{tr}\left(f\psi_{(i}\phi^{a}\psi_{j)}\psi_{a}\right)-3\textrm{tr}\left(f\psi_{(i|}\phi^{a}\psi_{a}\psi_{|j)}\right)-2\textrm{tr}\left(f\psi_{a}\phi^{a}\psi_{(i}\psi_{j)}\right)$
$\displaystyle-\textrm{tr}\left(f\psi_{(i|}\psi_{a}\phi^{a}\psi_{|j)}\right)+4\textrm{tr}\left(f\psi_{a}\psi_{(i}\phi^{a}\psi_{j)}\right)$
$\displaystyle+\frac{2}{5}\epsilon^{a_{1}a_{2}a_{3}}\left[2\textrm{tr}\left(\psi_{a_{1}}\psi_{a_{2}}\right)\textrm{tr}\left(\psi_{a_{3}}\psi_{(i}\psi_{j)}\right)-3\textrm{tr}\left(\psi_{(i|}\psi_{a_{1}}\psi_{|j)}\psi_{a_{2}}\psi_{a_{3}}\right)\right]\
.$ (A.7)

Finally, we construct relations of these trace relations. Consider a linear
combination of the trace relations with coefficients being the graviton
cohomologies. If it vanishes identically, we call it a relation of relations.
While the trace relations are identities that can be seen at the level of
‘gluons’ $\phi,\psi,f$, the relations of relations are the identities of
mesons $u_{2},u_{3},v_{2},v_{3},w_{2},w_{3}$. We do not need to know how
$u_{2},u_{3},v_{2},v_{3},w_{2},w_{3}$ are made of $\phi,\psi,f$ to obtain the
relations of relations. After constructing relations of relations, one can
write them as the $Q$-action on certain operators using (A), (A). They are the
$Q$-closed operators since their $Q$-actions vanish due to the relations of
relations. This is the way we obtain the $Q$-closed operators in section 4.1.
They can be either $Q$-exact or not and there is no trivial way to judge it
easily. If they are not $Q$-exact, they are the non-graviton cohomologies
since they are made of the linear combinations of $r_{I}$’s, which vanish with
diagonal $\phi,\psi,f$. For the check of the (non-)$Q$-exactness, refer to
section 4.2.

Now we will construct relations of relations at the threshold level $j=24$
which are singlets under $SU(3)\subset SU(4)_{R}$, from the trace relations
(A), (A). There are 5 choices of $(R,J)$ in this sector in which relations of
relations exist.

#### i) $(R,J)=(2,2)$

Let us first enumerate all $SU(3)\subset SU(4)_{R}$ singlets in this sector
made by the product of the trace relations in (A), (A) and the graviton
cohomologies. There are following 6 singlets:

$\displaystyle s_{1}^{(2,0)}=u^{ij}(R_{20}^{(2,0)})_{ij}\ ,\ \
s_{2}^{(2,0)}=w^{ij}(R_{14}^{(1,0)})_{ij}\ ,\ \
s_{3}^{(2,0)}=w^{i}(R_{16}^{(1,0)})_{i}\ ,$ (A.8) $\displaystyle
s_{1}^{(1,2)}={v^{jk}}_{i}(R_{16}^{(1,1)})^{i}_{jk}\ ,\ \
s_{2}^{(1,2)}={v^{j}}_{i}(R_{18}^{(1,1)})^{i}_{j}\ ,\ \
s_{3}^{(1,2)}=w^{i}(R_{16}^{(0,2)})_{i}\ .$

The superscripts denote ($n_{f},n_{\psi}$) of the terms with maximal $n_{f}$
in the operator, as before. There is one relation of these relations given by

$i\,QO^{(2,1)}\equiv
65s_{1}^{(2,0)}-39s_{2}^{(2,0)}+5s_{3}^{(2,0)}-312s_{1}^{(1,2)}-26s_{2}^{(1,2)}+6s_{3}^{(1,2)}=0\
.$ (A.9)

This is the $Q$-action on the $Q$-closed operator (4.5).

#### ii) $(R,J)=(\frac{5}{2},\frac{3}{2})$

There exist 12 $SU(3)$ singlets in this sector given by

$\displaystyle s_{1}^{(1,1)}=u^{a(i}{v^{j)}}_{a}(R_{14}^{(1,0)})_{ij}\ ,\ \
s_{2}^{(1,1)}=\epsilon_{a_{1}a_{2}(i}u^{a_{1}k}{v^{a_{2}}}_{j)}(R_{14}^{(1,0)})^{ij}_{k}\
,\ \ s_{3}^{(1,1)}={v^{jk}}_{i}(R_{16}^{(1,0)})^{i}_{jk}\ ,$ (A.10)
$\displaystyle s_{4}^{(1,1)}=u^{ijk}(R_{18}^{(1,1)})_{ijk}\ ,\ \
s_{5}^{(1,1)}={v^{(j}}_{i}w^{k)}(R_{10}^{(0,0)})^{i}_{jk}\ ,\ \
s_{6}^{(1,1)}=u^{(ij}w^{k)}(R_{12}^{(0,1)})_{ijk}\ ,$ $\displaystyle
s_{7}^{(1,1)}=\epsilon_{a_{1}a_{2}i}u^{a_{1}j}w^{a_{2}}(R_{12}^{(0,1)})^{i}_{j}\
,\ \ s_{8}^{(1,1)}=w^{ij}(R_{14}^{(0,1)})_{ij}\ ,$ $\displaystyle
s_{1}^{(0,3)}=\epsilon^{a_{1}a_{2}(i}{v^{j}}_{a_{1}}{v^{k)}}_{a_{2}}(R_{12}^{(0,1)})_{ijk}\
,\ \ s_{2}^{(0,3)}={v^{j}}_{a}{v^{a}}_{i}(R_{12}^{(0,1)})^{i}_{j}\ ,$
$\displaystyle s_{3}^{(0,3)}=u^{(jk}{v^{k)}}_{i}(R_{14}^{(0,2)})^{i}_{jkl}\ ,\
\ s_{4}^{(0,3)}={v^{jk}}_{i}(R_{16}^{(0,2)})^{i}_{jk}\ .$

There are 4 relations of these relations, given by

$\displaystyle i\,QO_{1}^{(1,2)}\equiv
3s_{5}^{(1,1)}-3s_{6}^{(1,1)}+s_{7}^{(1,1)}=0\ ,$ (A.11) $\displaystyle
i\,QO_{2}^{(1,2)}\equiv
9s_{1}^{(1,1)}-10s_{2}^{(1,1)}-30s_{5}^{(1,1)}-60s_{3}^{(0,3)}=0\ ,$
$\displaystyle i\,QO_{3}^{(1,2)}\equiv
3s_{1}^{(1,1)}-6s_{2}^{(1,1)}+4s_{4}^{(1,1)}-14s_{5}^{(1,1)}-6s_{8}^{(1,1)}-12s_{1}^{(0,3)}-4s_{2}^{(0,3)}=0\
,$ $\displaystyle i\,QO_{4}^{(1,2)}\equiv
3s_{1}^{(1,1)}-14s_{2}^{(1,1)}-8s_{3}^{(1,1)}-42s_{5}^{(1,1)}+12s_{6}^{(1,1)}-24s_{8}^{(1,1)}-36s_{1}^{(0,3)}+8s_{4}^{(0,3)}=0\
.$

They are the $Q$-action on (4.6).

#### iii) $(R,J)=(3,1)$

There exist 16 $SU(3)$ singlets in this sector given by

$\displaystyle
s_{1}^{(1,0)}=\epsilon_{a_{1}a_{2}i}\epsilon_{b_{1}b_{2}j}u^{a_{1}b_{1}}u^{a_{2}b_{2}k}(R_{14}^{(1,0)})^{ij}_{k}\
,\ \
s_{2}^{(1,0)}=\epsilon_{a_{1}a_{2}i}u^{a_{1}(j}w^{k)a_{2}}(R_{10}^{(0,0)})^{i}_{jk}\
,$ (A.12) $\displaystyle
s_{3}^{(1,0)}=\epsilon_{a_{1}a_{2}i}u^{a_{1}jk}w^{a_{2}}(R_{10}^{(0,0)})^{i}_{jk}\
,$ $\displaystyle
s_{1}^{(0,2)}={v^{a}}_{i}{v^{jk}}_{a}(R_{10}^{(0,0)})^{i}_{jk}\ ,\ \
s_{2}^{(0,2)}={v^{(j}}_{a}{v^{k)a}}_{i}(R_{10}^{(0,0)})^{i}_{jk}\ ,\ \
s_{3}^{(0,2)}=u^{a(i}{v^{jk)}}_{a}(R_{12}^{(0,1)})_{ijk}\ ,$ $\displaystyle
s_{4}^{(0,2)}=u^{a(ij}{v^{k)}}_{a}(R_{12}^{(0,1)})_{ijk}\ ,\ \
s_{5}^{(0,2)}=\epsilon_{a_{1}a_{2}i}u^{a_{1}b}{v^{a_{2}j}}_{b}(R_{12}^{(0,1)})^{i}_{j}\
,\ \
s_{6}^{(0,2)}=\epsilon_{a_{1}a_{2}i}u^{a_{1}bj}{v^{a_{2}}}_{b}(R_{12}^{(0,1)})^{i}_{j}\
,$ $\displaystyle
s_{7}^{(0,2)}=\epsilon_{a_{1}a_{2}(i}u^{a_{1}(k}{v^{l)a_{2}}}_{j)}(R_{12}^{(0,1)})^{ij}_{kl}\
,\ \
s_{8}^{(0,2)}=\epsilon_{a_{1}a_{2}(i}u^{a_{1}kl}{v^{a_{2}}}_{j)}(R_{12}^{(0,1)})^{ij}_{kl}\
,$ $\displaystyle
s_{9}^{(0,2)}=\epsilon_{a_{1}a_{2}i}u^{a_{1}(j}u^{kl)a_{2}}(R_{14}^{(0,2)})^{i}_{jkl}\
,\ \
s_{10}^{(0,2)}=\epsilon_{a_{1}a_{2}i}u^{a_{1}b}{v^{a_{2}}}_{b}(R_{14}^{(0,1)})^{i}\
,\ \ s_{11}^{(0,2)}=u^{a(i}{v^{j)}}_{a}(R_{14}^{(0,1)})_{ij}\ ,$
$\displaystyle
s_{12}^{(0,2)}=\epsilon_{a_{1}a_{2}(i}u^{a_{1}k}{v^{a_{2}}}_{j)}(R_{14}^{(0,1)})^{ij}_{k}\
,$ $\displaystyle
s_{13}^{(0,2)}=u^{(jk}{v^{l)}}_{i}(R_{14}^{(0,1)})^{i}_{jkl}\ .$

There are 13 relations of these relations, given by

$\displaystyle i\,QO_{1}^{(1,1)}\equiv s_{2}^{(1,0)}=0\ ,$ $\displaystyle
i\,QO_{2}^{(1,1)}\equiv s_{3}^{(1,0)}=0\ ,$ $\displaystyle
i\,QO_{3}^{(1,1)}\equiv s_{1}^{(1,0)}+5s_{1}^{(0,2)}-2s_{2}^{(0,2)}=0\ ,$
$\displaystyle i\,QO_{1}^{(0,3)}\equiv
4s_{5}^{(0,2)}+3s_{6}^{(0,2)}=(R_{12}^{(0,1)})^{i}_{j}(R_{12}^{(0,1)})^{j}_{i}=i\,Q\left[\frac{1}{2}i\,Q((r_{12}^{(0,2)})^{i}_{j}(r_{12}^{(0,2)})^{j}_{i})\right]=0\
,$ $\displaystyle i\,QO_{2}^{(0,3)}\equiv
s_{7}^{(0,2)}+s_{8}^{(0,2)}=(R_{12}^{(0,1)})^{ij}_{kl}(R_{12}^{(0,1)})^{kl}_{ij}=i\,Q\left[\frac{1}{2}i\,Q((r_{12}^{(0,2)})^{ij}_{kl}(r_{12}^{(0,2)})^{kl}_{ij})\right]=0\
,$ $\displaystyle i\,QO_{3}^{(0,3)}\equiv s_{3}^{(0,2)}=0\ ,$ $\displaystyle
i\,QO_{4}^{(0,3)}\equiv s_{10}^{(0,2)}=0\ ,$ $\displaystyle
i\,QO_{5}^{(0,3)}\equiv 6s_{1}^{(0,2)}-6s_{4}^{(0,2)}-s_{6}^{(0,2)}=0\ ,$
$\displaystyle i\,QO_{6}^{(0,3)}\equiv
24s_{2}^{(0,2)}-6s_{11}^{(0,2)}+s_{12}^{(0,2)}=0\ ,$ $\displaystyle
i\,QO_{7}^{(0,3)}\equiv
s_{1}^{(0,2)}-10s_{2}^{(0,2)}-6s_{4}^{(0,2)}-10s_{8}^{(0,2)}=0\ ,$
$\displaystyle i\,QO_{8}^{(0,3)}\equiv
5s_{1}^{(0,2)}-2s_{2}^{(0,2)}-9s_{4}^{(0,2)}+6s_{9}^{(0,2)}=0\ ,$
$\displaystyle i\,QO_{9}^{(0,3)}\equiv
6s_{1}^{(0,2)}+12s_{2}^{(0,2)}-18s_{4}^{(0,2)}+s_{12}^{(0,2)}=0\ ,$
$\displaystyle i\,QO_{10}^{(0,3)}\equiv
38s_{1}^{(0,2)}+4s_{2}^{(0,2)}-24s_{4}^{(0,2)}-5s_{13}^{(0,2)}=0\ .$ (A.13)

They are the $Q$-action on (4.7). Here, $O^{(0,3)}_{1}$ and $O^{(0,3)}_{2}$
are explicitly shown to be $Q$-exact.

#### iv) $(R,J)=(\frac{7}{2},\frac{1}{2})$

There exist 8 $SU(3)$ singlets in this sector given by

$\displaystyle
s^{(0,1)}_{1}=\epsilon_{a_{1}a_{2}i}u^{a_{1}b}u^{jk}{v^{a_{2}}}_{b}(R_{10}^{(0,0)})^{i}_{jk}\
,\ \
s^{(0,1)}_{2}=\epsilon_{a_{1}a_{2}i}u^{a_{1}b}u^{a_{2}(j}{v^{k)}}_{b}(R_{10}^{(0,0)})^{i}_{jk}\
,$ (A.14) $\displaystyle
s^{(0,1)}_{3}=\epsilon_{a_{1}a_{2}i}u^{a_{1}b(j}{v^{k)a_{2}}}_{b}(R_{10}^{(0,0)})^{i}_{jk}\
,\ \
s^{(0,1)}_{4}=\epsilon_{a_{1}a_{2}(i}u^{a_{1}(k}{v^{l)a_{2}}}_{j)}(R_{12}^{(0,0)})^{ij}_{kl}\
,$ $\displaystyle
s^{(0,1)}_{5}=\epsilon_{a_{1}a_{2}(i}u^{a_{1}kl}{v^{a_{2}}}_{j)}(R_{12}^{(0,0)})^{ij}_{kl}\
,\ \
s^{(0,1)}_{6}=\epsilon_{a_{1}a_{2}(i}\epsilon_{j)b_{1}b_{2}}u^{a_{1}b_{1}}u^{a_{2}b_{2}}u^{kl}(R_{12}^{(0,1)})^{ij}_{kl}\
,$ $\displaystyle
s^{(0,1)}_{7}=\epsilon_{a_{1}a_{2}(i}\epsilon_{j)b_{1}b_{2}}u^{a_{1}b_{1}}u^{a_{2}b_{2}k}(R_{14}^{(0,1)})^{ij}_{k}\
,\ \
s^{(0,1)}_{8}=\epsilon_{a_{1}a_{2}i}u^{a_{1}(j}u^{kl)a_{2}}(R_{14}^{(0,1)})^{i}_{jkl}\
.$

There are 6 relations of these relations, given by

$\displaystyle i\,QO^{(0,2)}_{1}\equiv s_{1}^{(0,1)}-2s_{2}^{(0,1)}=0\ ,$
(A.15) $\displaystyle i\,QO^{(0,2)}_{2}\equiv 6s_{3}^{(0,1)}+s_{4}^{(0,1)}=0\
,$ $\displaystyle i\,QO^{(0,2)}_{3}\equiv s_{1}^{(0,1)}+s_{5}^{(0,1)}=0\ ,$
$\displaystyle i\,QO^{(0,2)}_{4}\equiv s_{1}^{(0,1)}+s_{6}^{(0,1)}=0\ ,$
$\displaystyle i\,QO^{(0,2)}_{5}\equiv
4s_{1}^{(0,1)}+24s_{3}^{(0,1)}-s_{7}^{(0,1)}=0\ ,$ $\displaystyle
i\,QO^{(0,2)}_{6}\equiv s_{1}^{(0,1)}-12s_{3}^{(0,1)}+3s_{8}^{(0,1)}=0\ .$

They are the $Q$-action on (4.9).

#### v) $(R,J)=(4,0)$

There exist 4 $SU(3)$ singlets in this sector given by

$\displaystyle
s^{(0,0)}_{1}=\epsilon_{a_{1}a_{2}a_{3}}\epsilon_{b_{1}b_{2}i}u^{a_{1}b_{1}}u^{a_{2}b_{2}}u^{a_{3}jk}(R_{10}^{(0,0)})^{i}_{jk}\
,$ (A.16) $\displaystyle s^{(0,0)}_{2}=R_{12}^{(0,0)}R_{12}^{(0,0)}\ ,$
$\displaystyle
s^{(0,0)}_{3}=\epsilon_{a_{1}a_{2}(i}\epsilon_{j)b_{1}b_{2}}u^{a_{1}b_{1}}u^{a_{2}b_{2}}u^{kl}(R_{12}^{(0,0)})^{ij}_{kl}\
,$ $\displaystyle
s^{(0,0)}_{4}=\epsilon_{a_{1}a_{2}(i}\epsilon_{j)b_{1}b_{2}}u^{a_{1}b_{1}(k}u^{l)a_{2}b_{2}}(R_{12}^{(0,0)})^{ij}_{kl}\
.$

There is 1 relation of these relations, given by

$i\,QO^{(0,1)}\equiv 36s_{1}^{(0,0)}+5s_{2}^{(0,0)}-6s_{3}^{(0,0)}=0\ .$
(A.17)

This is the $Q$-action on (4.10).

It is straightforward to generate relations of relations in other charge
sectors. We present relations of relations at $j=30$ and $(R,J)=(3,2)$ which
are singlets under $SU(3)$ global symmetry and do not involve $f$’s. They will
yield the fermionic $Q$-closed operators with
$(R,J)=\left(\frac{5}{2},\frac{5}{2}\right)$, which may be good ansätze for
the non-graviton cohomology detected by the index at $j=30$. The $SU(3)$
singlets in this sector are given by

$\displaystyle p_{1}$
$\displaystyle=\epsilon^{a_{1}a_{2}a_{3}}{v^{(i}}_{a_{1}}{v^{j}}_{a_{2}}{v^{k)}}_{a_{3}}(R_{12}^{(0,1)})_{ijk}\;,\
\ p_{2}={v^{j}}_{a}{v^{a}}_{b}{v^{b}}_{i}(R_{12}^{(0,1)})^{i}_{j}\;,\ \
p_{3}={v^{(k}}_{a}{v^{l)}}_{(i}{v^{a}}_{j)}(R_{12}^{(0,1)})^{ij}_{kl}\;,$
(A.18) $\displaystyle p_{4}$
$\displaystyle=u^{(jk}{v^{l)}}_{a}{v^{a}}_{i}(R_{14}^{(0,2)})^{i}_{jkl}\;,\ \
p_{5}=u^{a(j}{v^{k}}_{a}{v^{l)}}_{i}(R_{14}^{(0,2)})^{i}_{jkl}\;,\ \
p_{6}={v^{(jk}}_{a}{v^{l)a}}_{i}(R_{14}^{(0,2)})^{i}_{jkl}\;,$ $\displaystyle
p_{7}$
$\displaystyle={v^{(k}}_{(i}{v^{lm)}}_{j)}(R_{16}^{(0,2)})^{ij}_{klm}\;,\ \
p_{8}={v^{(j}}_{a}{v^{k)a}}_{i}(R_{16}^{(0,2)})^{i}_{jk}\;,\ \
p_{9}={v^{a}}_{i}{v^{jk}}_{a}(R_{16}^{(0,2)})^{i}_{jk}\;,$ $\displaystyle
p_{10}$ $\displaystyle={v^{ia}}_{b}{v^{b}}_{a}(R_{16}^{(0,2)})_{i}\;,\ \
p_{11}=\epsilon_{ia_{1}a_{2}}u^{a_{1}b}{v^{a_{2}}}_{b}(R_{20}^{(0,3)})^{i}\;.$

The relations of relations of the above singlets are given as follows:

$\displaystyle 5p_{1}-10p_{2}-30p_{4}+8p_{11}=0\ ,$ (A.19) $\displaystyle
15p_{1}+6p_{2}-40p_{3}-30p_{4}+90p_{5}=0\ ,$ $\displaystyle
105p_{1}-336p_{2}+140p_{3}-1050p_{4}+10080p_{6}-900p_{7}=0\ ,$ $\displaystyle
15p_{1}-138p_{2}-160p_{3}-570p_{4}+864p_{6}-48p_{8}=0\ ,$ $\displaystyle
375p_{1}+6p_{2}+1760p_{3}+150p_{4}+4320p_{6}-120p_{9}=0\ ,$ $\displaystyle
55p_{1}-266p_{2}-160p_{3}-1050p_{4}+2880p_{6}+40p_{10}=0\ .$

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