sizes clearly pertain to the size-regime of the non-crystalline (amorphous)
phase, and cannot be repesented by top-down derived (crystalline) energies and
structures.

##  5 Conclusions

We modeled the nucleation of alumina dust particles from a bottom-up
perspective, starting with atomic and molecular precursors and using a
chemical-kinetic approach applied to different circumstellar gas trajectories
representing two model stars, a Semi-Regular Variable (SRV) and a regular
variable (MIRA). In both sets of circumstellar models, SRV and MIRA, alumina
cluster particles form efficiently, independent of their initial gas-phase
compositions. The results of the non-pulsating monotonic outflow
($\beta$-velocity law) and the pulsating (ballistic trajectory) models show
similar trends. The non-pulsating models predict a later onset of cluster
formation at larger radii (2$-$2.5 R⋆) in the SRV envelope, as compared with
the MIRA model star, where the clusters form already at 1.5 R⋆. The pulsating
models show that alumina clusters form yet at 1 R⋆ in the late post-shock gas.
In the SRV pulsating model, the amount of alumina clusters formed at 1 R⋆ is
already substantial ($\sim$ 72 % of the total aluminium content). However, in
the MIRA pulsating envelope, the cluster synthesis is even more fruitful
showing that more than 90 % of the total Al content is in condensable clusters
after one pulsation period. These results reflect the evolutionary sequence
along the AGB showing an increase of the stellar mass-loss rate. Furthermore,
the models consistently show larger AlO abundances in the SRV case, as
compared with the MIRA case, reflecting the trends of the most recent ALMA
observations. Still, the modelled AlOH abundances exceed those derived from
observations by a about two orders of magnitude. A possible process,
destroying the stable singlet AlOH molecule, is the photolysis of AlOH.
However, in AGB stars with rather low temperatures ($<2500$ K) the AlOH
photolysis is of secondary importance. Therefore, it is predominantly the
bimolecular reaction AlO+H2 $\leftrightarrow$ AlOH+H determining the AlO/AlOH
ratio. The effectiveness of the latter process depends on the concentration of
molecular hydrogen, which is generally larger in the denser MIRA models.
Therefore, the AlO/AlOH ratio is primarly controlled by the H/H2 in
circumstellar envelopes.
Larger-sized (Al2O3)n clusters ($n=4-10$) and their interpolation to the bulk-
limit ($n\rightarrow\infty$) are described by size-dependent properties, as a
chemical-kinetic treatment becomes increasingly expensive and prohibitive. We
find that the cluster energy can be approximated by a fit of the form
E(n)=$-$1687.82 +1075.38 n-1/3 kJ mol-1. The offset of $-$1687.82 kJ mol-1 is
resonably close to experimental values for bulk alumina (corundum).
Nevertheless, it is very likely that the nucleation in oxygen-rich AGB stars
proceeds heterogeneously involving additional species, that are different from
pure aluminium-oxygen clusters. Moreover, we note that the small size regime
under consideration here is still deeply in the non-crystalline (amorphous)
size regime and that phase transitions could affect our bottom-up nucleation
fit.

##  6 Acknowledgements

We acknowledge the referee H.-P. Gail for his valuable comments that helped
improving the quality of this manuscript. D.G., J.M.C.P, and L.D. acknowledge
support from the ERC consolidator grant 646758 “AEROSOL” and STFC grant
ST/T000287/1. S.T.B. acknowledges Spanish MICIUN/FEDER RTI2018-095460-B-I00
and María de Maeztu MDM-2017-0767 grants and a Generalitat de Catalunya
2017SGR13 grant. We acknowledge the CINECA award under the ISCRA initiative,
for the availability of high performance computing resources and support for
the projects IsC65 “SPINEL” and IsC76 “DUSTY”. We acknowledge the “Accordo
Quadro” INAF-CINECA MoU 2017$-$2020 for the projects INA17$-$C5A28 and
INA20$-$C7B35.

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##  Appendix A Equilibrium constant and detailled balance:

$\centering K_{eq}=\frac{k_{f}}{k_{r}}=\exp\left(-\frac{\Delta
G_{r}(T)}{RT}\right)\@add@centering$ (21)

where Keq is the dimensionless equilibrium constant, kf and kr are the forward
and the reverse reaction rates, respectively.

##  Appendix B Chemical-kinetic network

Table 10: Kinetic rate network listing (1) the reaction number, (2) the reaction, (3) the CBS-QB3 heat of reaction (enthalpy) at T= 0 K, (4) the critical temperature Tc, where the free energy of reaction, $\Delta$Gr(T) changes its sign, (5) the reaction rate with the pre-exponential factor A given as $a(-b)\equiv a\times 10^{-b}$(for unimolecular reactions (photodissociations in units of s-1, for bimolecular reactions in units of cm3s-1, for termolecular reactions in units of cm6s-1), and (6) the reference or method of calculation. Number | Reaction |  $\Delta$ Hr(0K) | Tc | Rate k | Reference / remark
---|---|---|---|---|---
1 | H + H + H2 $\rightarrow$ H2 \+ H2 | -437 | 3800 | 8.85(-33)(T/300)-0.6 | NIST
2 | H2 \+ H2 $\rightarrow$ H + H + H2 | +437 | 3800 | 1.5(-9)(T/300)${}^{0.34}\exp(-48346.4/T)$ | NIST
3 | H + H + H $\rightarrow$ H2 \+ H | -437 | 3800 | 8.82(-33) | NIST
4 | H2 \+ H $\rightarrow$ H + H + H | +437 | 3800 | 2.54(-8)(T/300)${}^{-0.10}\exp(-52555.6/T)$ | NIST
5 | H + H + He $\rightarrow$ H2 \+ He | -437 | 3800 | 4.96(-33) | NIST
6 | H2 \+ He $\rightarrow$ H + H + He | +437 | 3800 | 8.85(-10)$\exp(-48346.4/T)$ | NIST
7 | OH + OH $\rightarrow$ H2O + O | -67.4 | 4400 | 1.65(-12)(T/300)${}^{1.1}\exp(-50.5/T)$ | NIST
8 | H2O + O $\rightarrow$ OH + OH | +67.4 | 4400 | 1.84(-11)(T/300)${}^{0.95}\exp(-8573.7/T)$ | NIST
9 | CO + OH $\rightarrow$ CO2 \+ H | -105 | 3000 | 3.52(-12)$\exp(-2630.2/T)$ | NIST
10 | CO2 \+ H $\rightarrow$ CO + OH | +105 | 3000 | 2.51(-10)$\exp(-13229.1/T)$ | NIST
11 | OH + H + H2O $\rightarrow$ H2O + H2O | -494 | 4000 | 1.19(-30)(T/300)-2.1 | NIST
12 | H2O + H2O $\rightarrow$ OH + H + H2O | +494 | 4000 | 2.66(-7)$\exp(-7500.0/T)$ | NIST
13 | OH + H $\rightarrow$ H2 \+ O | -10.9 | 1100 | 6.86(-14)(T/300)${}^{2.8}\exp(-1949.5/T)$ | NIST
14 | H2 \+ O $\rightarrow$ OH + H | +10.9 | 1100 | 3.44(-13)(T/300)${}^{2.67}\exp(-3159.3/T)$ | NIST
15 | H2 \+ OH $\rightarrow$ H2O + H | -56.5 | no | 1.55(-12)(T/300)${}^{1.6}\exp(-1659.7/T)$ | NIST
16 | H2O + H $\rightarrow$ H2 \+ OH | 56.5 | no | 6.82(-12)(T/300)${}^{1.6}\exp(-9719.8/T)$ | NIST
17 | C + O $\rightarrow$ CO + h$\nu$ | -1073 | no | 1.58(-17)(T/300)${}^{0.34}\exp(-1297.4/T)$ | Fit to Dalgarno et al. (1990)
18 | C + O + M $\rightarrow$ CO + M | -1073 | no | 2.00(-34) | NIST
19 | CO + M $\rightarrow$ C + O + M | 1073 | no | 4.40(-10)$\exp(-98600.0/T)$ |  Appleton et al. (1970), eq. 4.7
20 | H + O + M $\rightarrow$ OH + M | -426 | 4000 | 4.36(-32)(T/300)-1.00 | NIST
21 | OH + M $\rightarrow$ H + O + M | 426 | 4000 | 4.00(-9)$\exp(-50000/T)$ | NIST
22 | CO + O + M $\rightarrow$ CO2 \+ M | -532 | 3800 | 1.20(-32)$\exp(-2160/T)$ | NIST
23 | CO2 \+ M $\rightarrow$ CO + O + M | 532 | 3800 | 8.02(-11)$\exp(-26900/T)$ | Willacy et al. (1998)
24 | OH + H + M $\rightarrow$ H2O + M | -494 | 4000 | 2.59(-31)(T/300)-2.00 | NIST
25 | H2O + M $\rightarrow$ OH + H + M | 494 | 4000 | 5.80(-9)$\exp(-52920/T)$ | NIST
26 | Al + O + M $\rightarrow$ AlO + M | -498 | 4900 | 3.34(-32)(T/300)-1.79 | RRKM fit
27 | AlO + M $\rightarrow$ Al + O + M | 498 | 4900 | 4.01(-10)$\exp(-57391/T)$ |  Gómez Martín et al. (2017) RRKM fit
28 | AlO + H + M $\rightarrow$ AlOH + M | -482 | 4900 | 1.32(-28)(T/300)-3.08 | RRKM fit
29 | AlOH + M $\rightarrow$ AlO + H + M | 482 | 4900 | 1.47(-7)$\exp(-55985/T)$ | RRKM fit
30 | Al + CO 2 $\rightarrow$ Al + CO | 33.6 | 800 | 6.32(-10)$\exp(-4522.0/T)$ | RRKM fit
31 | AlO + CO $\rightarrow$ Al + CO2 | -33.6 | 800 | 10${}^{-16.6318+3.43135\log(T)-0.5786\log(T)^{2}}$ |  Mangan et al. (2020) RRKM fit
32 | Al + O2 $\rightarrow$ AlO + O | -0.9 | no | 1.68(-10)(T/300)-0.26 | Gómez Martín et al. (2017)
33 | AlO + O $\rightarrow$ Al + O2 | 0.9 | no | 7.15(-12)$\exp(-945./T)$ | RRKM fit
34 | Al + H2O $\rightarrow$ AlOH + H | -60.4 | no | 1.7(-12)$\frac{-422.}{T}$+1.45(-10)$\exp(-2657./T)$ | Mangan et al. (2021)
35 | AlOH + H $\rightarrow$ Al + H2O | 60.4 | no | 4.31(-11)$\exp(-9457./T)$ | Mangan et al. (2021)
36 | AlO + H2 $\rightarrow$ AlOH + H | -45.0 | no | 5.37(-13)(T/300)${}^{2.77}\exp(-2190./T)$ | Mangan et al. (2021)
37 | AlOH + H $\rightarrow$ AlO + H2 | 45.0 | no | 8.89(-11)$\exp(-9092/T)$ | Mangan et al. (2021)
38 | AlO + O + M $\rightarrow$ AlO2 \+ M | -391 | 2900 | 6.74(-30)(T/300)-2.53 |  Plane et al. (2021) RRKM fit
39 | AlO2 \+ M $\rightarrow$ AlO + O + M | 391 | 2900 | 4.51(-7)$\exp(-44166/T)$ |  Plane et al. (2021) RRKM fit
40 | AlO + CO2 $\rightarrow$ AlO2 \+ CO | 140 | no | 1.81(-10)(T/300)${}^{-0.81}\exp(-18138.2/T)$ | detailed balance
41 | AlO2 \+ CO $\rightarrow$ AlO + CO2 | -140 | no | 2.55(-12)(T/300)0.17 |  Mangan et al. (2020) RRKM fit
42 | AlO + O2 $\rightarrow$ AlO2 \+ O | 106 | no | 3.54(-11)(T/300)${}^{0.19}\exp(-13140/T)$ | detailed balance
43 | AlO2 \+ O $\rightarrow$ AlO + O2 | -106 | no | 1.90(-10)(T/300)0.17 |  Mangan et al. (2020) RRKM fit
44 | AlO + AlO $\rightarrow$ Al2O + O | -70.2 | 1800 | 1.42(-9)(T/300)0.17 | capture
45 | Al2O + O $\rightarrow$ AlO + AlO | 70.2 | 1800 | 1.42(-9)(T/300)${}^{0.17}\exp(-8447.4/T)$ | reverse capt
46 | AlOH + AlOH $\rightarrow$ Al2O + H2O | -25.9 | 600 | 9.85(-10)(T/300)0.17 | capture
47 | Al2O + H2O $\rightarrow$ AlOH + AlOH | 25.9 | 600 | 9.85(-10)(T/300)${}^{0.17}\exp(-3593.2/T)$ | reverse capt
48 | AlO + AlOH $\rightarrow$ Al2O + OH | -14.3 | 500 | 1.00(-9)(T/300)0.17 | capture
49 | Al2O + OH $\rightarrow$ AlO + AlOH | 14.3 | 500 | 1.00(-9)(T/300)${}^{0.17}\exp(-1727.3/T)$ | reverse capture
50 | AlO + AlO + M $\rightarrow$ Al2O2 \+ M | -547 | 3500 | 2.35(-33) | capture
51 | Al2O2 \+ M $\rightarrow$ AlO + AlO + M | 547 | 3500 | 1.45(-10)(T/300)${}^{-0.63}\exp(-65887.9/T)$ | detailed balance
52 | AlOH + AlOH $\rightarrow$ Al2O2 \+ H2 | -21.0 | 400 | 9.8(-10)(T/300)0.17 | capture
53 | Al2O2 \+ H2 $\rightarrow$ AlOH + AlOH | 21.0 | 400 | 9.8(-10)(T/300)${}^{0.17}\exp(-2524.6/T)$ | reverse capture
54 | AlO + AlOH $\rightarrow$ Al2O2 \+ H | -65.9 | 1000 | 1.00(-9)(T/300)0.17 | capture
55 | Al2O2 \+ H $\rightarrow$ AlO + AlOH | 65.9 | 1000 | 1.00(-9)(T/300)${}^{0.17}\exp(-7926.3/T)$ | reverse capture
56 | Al2O + O2 $\rightarrow$ Al2O2 \+ H | 19.5 | no | 8.11(-12)(T/300)${}^{1.69}\exp(-1965.8/T)$ | detailed balance
57 | Al2O2 \+ O $\rightarrow$ Al2O + O2 | -19.5 | no | 7.46(-10)(T/300)0.17 | capture
58 | Al2O + H2O $\rightarrow$ Al2O2 \+ H2 | 4.9 | no | 5.90(-11)(T/300)${}^{1.37}\exp(-1391.5/T)$ | detailed balance (capture)
59 | Al2O2 \+ H2 $\rightarrow$ Al2O + H2O | -4.9 | no | 1.20(-11)$\exp(-17285/T)$ | TST calculation
60 | Al2O + OH $\rightarrow$ Al2O2 \+ H | -51.5 | no | 5.69(-10)(T/300)0.17 | capture
---|---|---|---|---|---
61 | Al2O2 \+ H $\rightarrow$ Al2O + OH | 51.5 | no | 1.86(-07)(T/300)${}^{-1.55}\exp(-6310.9/T)$ | detailed balance
62 | Al2O2 \+ H2O $\rightarrow$ Al2O3 \+ H2 | 85.0 | no | 1.78(-09)(T/300)${}^{0.77}\exp(-11797.1/T)$ | detailed balance
63 | Al2O3 \+ H2 $\rightarrow$ Al2O2 \+ H2O | -85.0 | no | 1.52(-9)(T/300)0.17 | capture
64 | Al2O2 \+ OH $\rightarrow$ Al2O3 \+ H | 28.2 | 4200 | 2.57(-11)(T/300)${}^{1.78}\exp(-1977.4/T)$ | detailed balance
65 | Al2O3 \+ H $\rightarrow$ Al2O2 \+ OH | -28.2 | 4200 | 1.13(-9)(T/300)0.17 | capture
66 | AlO2 \+ AlOH $\rightarrow$ Al2O3 \+ H | -72.5 | 2700 | 1.24(-9)(T/300)0.17 | capture
67 | Al2O3 \+ H $\rightarrow$ AlO2 \+ AlOH | 72.5 | 2700 | 1.24(-9)(T/300)${}^{0.17}\exp(-8714.8/T)$ | reverse capture
68 | AlO + H2O $\rightarrow$ OAlOH + H | 8.6 | 5400 | 2.03(-11)$\exp(-1360.0/T)$ | Mangan RRKM fit
69 | OAlOH + H $\rightarrow$ AlO + H2O | -8.6 | 5400 | 3.91(-09)(T/300)${}^{-1.94}\exp(-725.6/T)$ | detailed balance
70 | AlO + H2O $\rightarrow$ AlOH + OH | 11.5 | 1000 | 3.89(-10)$\exp(-1295.0/T)$ | Mangan et al. (2021)
71 | AlOH + OH $\rightarrow$ AlO + H2O | -11.5 | 1000 | 6.05(-10)(T/300)0.17 | capture
72 | AlO + H2O + M $\rightarrow$ Al(OH)2 \+ M | -314 | 3100 | 2.40(-26)(T/300)-3.87 | Mangan et al. (2021)
73 | Al(OH)2 \+ M $\rightarrow$ AlO + H2O + M | 314 | 3100 | 1.00(-10)$\exp(-17205/T)$ | Estimate
74 | OAlOH + SiO $\rightarrow$ AlSiO3 \+ H | -6.0 | 100 | 1.22(-9)(T/300)0.17 | capture
75 | AlSiO3 \+ H $\rightarrow$ OAlOH + SiO | 6.0 | 100 | 1.22(-9)(T/300)${}^{0.17}\exp(-724.1/T)$ | reverse capture
76 | AlSiO3 \+ AlO $\rightarrow$ Al2O3 \+ SiO | -28.3 | no | 1.64(-9)(T/300)-0.17111dipole-dipole up to 1700K | capture
77 | Al2O3 \+ SiO $\rightarrow$ AlSiO3 \+ AlO | 28.3 | no | 6.28(-10)(T/300)${}^{-0.18}\exp(-3110.2/T)$ | detailed balance
78 | OAlOH + SiO + M $\rightarrow$ HAlSiO3 \+ M | -356 | 1500 | 2.03(-33) | capture
79 | HAlSiO3 \+ M $\rightarrow$ OAlOH + SiO + M | 356 | 1500 | 2.72(-10)(T/300)${}^{2.37}\exp(-35190.3/T)$ | detailed balance
80 | HAlSiO3 \+ H $\rightarrow$ AlSiO3 \+ H2 | -87.3 | no | 1.13(-9)(T/300)0.17 | capture
81 | AlSiO3 \+ H2 $\rightarrow$ HAlSiO3 \+ H | 87.3 | no | 5.07(-13)(T/300)${}^{-1.04}\exp(-11390.8/T)$ | detailed balance
82 | Al2O3 \+ SiO + M $\rightarrow$ Al2O3SiO + M | -460 | 3000 | 1.38(-33) | capture
83 | Al2O3SiO + M $\rightarrow$ Al2O3 \+ SiO + M | 460 | 3000 | 4.10(-11)(T/300)${}^{-0.8}\exp(-53641.8/T)$ | detailed balance
84 | Al2O3SiO + AlO $\rightarrow$ Al3O4 \+ SiO | -200 | no | 1.13(-9)(T/300)0.17 | capture
85 | Al3O4 \+ SiO $\rightarrow$ Al2O3SiO + AlO | 200 | no | 1.67(-8)(T/300)${}^{0.40}\exp(-19243/T)$ | detailed balance
86 | Al3O4 \+ AlOH $\rightarrow$ Al4O5 \+ H | -148 | 3200 | 1.22(-9)(T/300)0.17 | capture
87 | Al4O5 \+ H $\rightarrow$ Al3O4 \+ AlOH | 148 | 3200 | 1.22(-9)(T/300)${}^{0.17}\exp(-17777.0/T)$ | reverse capture
88 | Al2O2 \+ AlO + M $\rightarrow$ Al3O3 \+ M | -518 | 5000 | 2.38(-33) | capture
89 | Al3O3 \+ M $\rightarrow$ Al2O2 \+ AlO + M | 518 | 5000 | 1.65(-12)(T/300)${}^{-1.16}\exp(-62359.5/T)$ | detailed balance
90 | Al2O2 \+ AlOH $\rightarrow$ Al3O3 \+ H | -36.5 | no | 1.40(-9)(T/300)0.17 | capture
91 | Al3O3 \+ H $\rightarrow$ Al2O2 \+ AlOH | 36.5 | no | 3.20(-11)$\exp(-4713/T)$ | TST calculation
92 | Al2O3 \+ OH $\rightarrow$ Al2O4 \+ H | -119 | 4200 | 6.63(-10)(T/300)0.17222dipole-dipole up to 300K | capture
93 | Al2O4 \+ H $\rightarrow$ Al2O3 \+ OH | 119 | 4200 | 3.18(-7)(T/300)${}^{-0.96}\exp(-13422.8/T)$ | detailed balance
94 | Al2O3 \+ H2O $\rightarrow$ Al2O4 \+ H2 | -62.2 | 2000 | 7.84(-10)(T/300)0.17 333dipole-dipole up to 400K | capture
95 | Al2O4 \+ H2 $\rightarrow$ Al2O3 \+ H2O | 62.2 | 2000 | 7.84(-10)(T/300)${}^{0.17}\exp(-7478.2/T)$ | reverse capture
96 | Al2O3 \+ AlOH $\rightarrow$ Al3O4 \+ H | -139 | 1600 | 8.96(-10))(T/300)0.17 | capture
97 | Al3O4 \+ H $\rightarrow$ Al2O3 \+ AlOH | 139 | 1600 | 8.96(-10))(T/300)${}^{0.17}\exp(-16655.4/T)$ | reverse capture
98 | Al3O3 \+ H2O $\rightarrow$ Al3O4 \+ H2 | -17.1 | 250 | 7.83(-10)(T/300)0.17 | capture
99 | Al3O4 \+ H2 $\rightarrow$ Al3O3 \+ H2O | 17.1 | 250 | 7.83(-10)(T/300)${}^{0.17}\exp(-2077.8/T)$ | reverse capture
100 | Al3O3 \+ OH $\rightarrow$ Al3O4 \+ H | -73.6 | 1000 | 7.01(-10)(T/300)0.17 | capture
101 | Al3O4 \+ H $\rightarrow$ Al3O3 \+ OH | 73.6 | 1000 | 7.01(-10)(T/300)${}^{0.17}\exp(-8869.2/T)$ | reverse capture
102 | Al3O3 \+ AlO + M $\rightarrow$ Al4O4 \+ M | -558 | 4400 | 1.76(-33) | capture
103 | Al4O4 \+ M $\rightarrow$ Al3O3 \+ AlO + M | 558 | 4400 | 1.88(-11)(T/300)${}^{-1.18}\exp(-67158.6/T)$ | detailed balance
104 | Al3O3 \+ AlOH $\rightarrow$ Al4O4 \+ H | -76.4 | 2000 | 1.03(-9)(T/300)0.17 | capture
105 | Al4O4 \+ H $\rightarrow$ Al3O3 \+ AlOH | 76.4 | 2000 | 1.03(-9)(T/300)${}^{0.17}\exp(-9181.9/T)$ | reverse capture
106 | Al4O4 \+ H2O $\rightarrow$ Al4O5 \+ H2 | -88.8 | 1100 | 8.46(-10)(T/300)0.17 | capture
107 | Al4O5 \+ H2 $\rightarrow$ Al4O4 \+ H2O | 88.8 | 1100 | 8.46(-10)(T/300)${}^{0.17}\exp(-10673.0/T)$ | reverse capture
108 | Al4O4 \+ OH $\rightarrow$ Al4O5 \+ H | -145 | 1700 | 7.57(-10)(T/300)0.17 | capture
109 | Al4O5 \+ H $\rightarrow$ Al4O4 \+ OH | 145 | 1700 | 7.57(-10)(T/300)${}^{0.17}\exp(-17464.4/T)$ | reverse capture
110 | Al3O4 \+ AlO + M $\rightarrow$ Al4O5 \+ M | -630 | 4500 | 2.07(-33) | capture
111 | Al4O5 \+ M $\rightarrow$ Al3O4 \+ AlO + M | 630 | 4500 | 1.04(-10)(T/300)${}^{-1.15}\exp(-75773.8/T)$ | detailed balance
112 | Al4O5 \+ H2O $\rightarrow$ Al4O6 \+ H2 | -172 | 2000 | 1.56(-9)(T/300)0.17 | capture
113 | Al4O6 \+ H2 $\rightarrow$ Al4O5 \+ H2O | 172 | 2000 | 3.00(-9)$\exp(-21435.0/T)$ | detailed balance (truncated)
114 | Al4O5 \+ OH $\rightarrow$ Al4O6 \+ H | -229 | 2500 | 1.40(-9)(T/300)0.17 | capture
115 | Al4O6 \+ H $\rightarrow$ Al4O5 \+ OH | 229 | 2500 | 1.40(-9)(T/300)${}^{0.17}\exp(-27501.7/T)$ | reverse capture
116 | AlO + AlH $\rightarrow$ Al2O + H | -267 | no | 1.12(-9)(T/300)0.17 | capture
117 | Al2O + H $\rightarrow$ AlO + AlH | 267 | no | 1.59(-7)(T/300)${}^{0.15}\exp(-31560.5/T)$ | detailed balance
118 | AlO2 \+ AlH $\rightarrow$ Al2O2 \+ H | -353 | no | 1.40(-9)(T/300)0.17 | capture
119 | Al2O2 \+ H $\rightarrow$ AlO2 \+ AlH | 353 | no | 1.29(-6)(T/300)${}^{-1.35}\exp(-42708.5/T)$ | detailed balance
120 | Al2O2 \+ AlH $\rightarrow$ Al3O2 \+ H | -73.0 | no | 1.62(-9)(T/300)0.17 | capture
---|---|---|---|---|---
121 | Al3O2 \+ H $\rightarrow$ Al2O2 \+ AlH | 73.0 | no | 1.68(-10)(T/300)${}^{-0.87}\exp(-7920.0/T)$ | detailed balance
122 | Al2O3 \+ AlH $\rightarrow$ Al3O3 \+ H | -317 | no | 1.04(-9)(T/300)0.17 | capture
123 | Al3O3 \+ H $\rightarrow$ Al2O3 \+ AlH | 317 | no | 4.67(-8)(T/300)${}^{-0.85}\exp(-37825.9/T)$ | detailed balance
124 | Al2O4 \+ AlH $\rightarrow$ Al3O4 \+ H | -272 | no | 8.27(-10)(T/300)0.17 | capture
125 | Al3O4 \+ H $\rightarrow$ Al2O4 \+ AlH | 272 | no | 3.02(-6)(T/300)${}^{-0.85}\exp(-33036.1/T)$ | detailed balance
126 | Al3O3 \+ AlH $\rightarrow$ Al4O3 \+ H | -117 | no | 1.22(-9)(T/300)0.17 | capture
127 | Al4O3 \+ H $\rightarrow$ Al3O3 \+ AlH | 117 | no | 2.72(-9)(T/300)${}^{-0.86}\exp(-13171.8/T)$ | detailed balance
128 | Al3O4 \+ AlH $\rightarrow$ Al4O4 \+ H | -256 | no | 1.44(-9)(T/300)0.17 | capture
129 | Al4O4 \+ H $\rightarrow$ Al3O4 \+ AlH | 256 | no | 1.77(-10)(T/300)${}^{-0.84}\exp(-29882.8/T)$ | detailed balance
130 | Al3O5 \+ AlH $\rightarrow$ Al4O5 \+ H | -387 | no | 1.15(-10)(T/300)0.17 | capture
131 | Al4O5 \+ H $\rightarrow$ Al3O5 \+ AlH | 387 | no | 1.44(-07)(T/300)${}^{-0.84}\exp(-46031.3/T)$ | detailed balance
132 | Al3O6 \+ AlH $\rightarrow$ Al4O6 \+ H | -682 | 5100 | 1.15(-9)(T/300)0.17 | capture
133 | Al4O6 \+ H $\rightarrow$ Al3O6 \+ AlH | 682 | 5100 | 1.00(-10)$\exp(-81959.0/T)$ | capture reverse
134 | Al + F + M $\rightarrow$ AlF + M | -682 | no | 3.32(-32)(T/300)-1.66 | RRKM fit
135 | AlF + M $\rightarrow$ Al + F + M | 682 | no | 5.76(-10)$\exp(-80937.0/T)$ | RRKM fit
136 | Al + Cl + M $\rightarrow$ AlCl + M | -515 | 5400 | 3.31(-31)(T/300)0.2 | RRKM fit
137 | AlCl + M $\rightarrow$ Al + Cl + M | 515 | 5400 | 1.00(-9)$\exp(-619370/T)$ | RRKM fit
138 | Al + H + M $\rightarrow$ AlH + M | -301 | 3200 | 4.42(-31)(T/300)0.34 | Swihart et al. (2003)
139 | AlH + M $\rightarrow$ Al + H + M | 301 | 3200 | 1.00(-9)$\exp(-36206/T)$ | RRKM fit
140 | AlCl + H $\rightarrow$ Al + HCl | 82.8 | no | 1.00(-10)$\exp(-9959/T)$ | RRKM fit
141 | Al + HCl $\rightarrow$ AlCl + H | -82.8 | no | 1.52(-10)$\exp(-803/T)$ | Rogowski et al. (1989)
142 | AlCl + H $\rightarrow$ AlH + Cl | 214 | no | 8.20(-12)(T/300)${}^{0.36}\exp(-25738.0/T)$ | detailed balance
143 | AlH + Cl $\rightarrow$ AlCl + H | -214 | no | 7.19(-12)(T/300)0.5 | collision
144 | F + H2 $\rightarrow$ HF + H | -133 | 3200 | 1.10(-10)$\exp(-450/T)$ | NIST
145 | HF + H $\rightarrow$ F + H2 | 133 | 3200 | 8.37(-11)(T/300)${}^{0.6}\exp(-16357/T)$ | NIST
146 | Cl + H2 $\rightarrow$ HCl + H | 4.3 | 400 | 3.87(-12)(T/300)${}^{1.58}\exp(-1610/T)$ | NIST
147 | HCl + H $\rightarrow$ Cl + H2 | -4.3 | 400 | 2.41(-12)(T/300)${}^{1.44}\exp(-1240/T)$ | NIST
148 | Al + HF $\rightarrow$ AlH + F | 269 | no | 6.34(-10)(T/300)${}^{0.57}\exp(-31487.8/T)$ | detailed balance
149 | AlH + F $\rightarrow$ Al + HF | -269 | no | 3.92(-10)(T/300)0.17 | capture
150 | Al + HF $\rightarrow$ AlF + H | -112 | no | 4.96(-10)(T/300)0.17 | capture
151 | AlF + H $\rightarrow$ Al + HF | 112 | no | 1.79(-10)(T/300)${}^{0.02}\exp(-12303.0/T)$ | detailed balance
152 | AlO + Cl $\rightarrow$ AlCl + O | -17.4 | no | 6.79(-12)(T/300)0.5 | collision
153 | AlCl + O $\rightarrow$ AlO + Cl | 17.4 | no | 3.18(-12)(T/300)${}^{0.48}\exp(-1833.5/T)$ | detailed balance
154 | AlOH + Cl $\rightarrow$ AlCl + OH | 38.5 | no | 6.90(-9)(T/300)${}^{-0.79}\exp(-5043.7/T)$ | detailed balance
155 | AlCl + OH $\rightarrow$ AlOH + Cl | -38.5 | no | 6.35(-9)(T/300)0.17 | capture
156 | AlOH + Cl $\rightarrow$ AlO + HCl | 49.3 | no | 4.65(-9)(T/300)${}^{-0.74}\exp(-6143.9/T)$ | detailed balance
157 | AlO + HCl $\rightarrow$ AlOH + Cl | -49.3 | no | 7.45(-10)(T/300)0.17 | capture
158 | Al4O6 \+ Al4O6 $\rightarrow$ Al8O12 | -834 | no | 2.00(-9) | coagulation
159 | Al8O12 $\rightarrow$ Al4O6 \+ Al4O6 | 834 | no | 6.5(-3)$\exp(-100000./T)$ | detailed balance
160 | AlO + OH $\rightarrow$ AlOH + O | -55.9 | no | 3.0(-12) | Estimate from trajectory calculation 222Having a late barrier, a successful reaction on the triplet potential energy surface (PES) requires that AlO is vibrationallly excited, and a translational collision energy less than 7.5 kJ mol-1. The resulting rate coefficient is estimated to be 3.0-3.7(-12) cm3 s-1 for T=1000-2000 K.
161 | AlOH + O $\rightarrow$ AlO + OH | 55.9 | no | 1.50(-11)(T/300)${}^{-0.97}\exp(-6861.8/T)$ | detailed balance
162 | Al + OH $\rightarrow$ AlO + H | -71.9 | no | 1.40(-10)$\exp(-9640./T)$ | Estimate from trajectory calculation
163 | AlO + H $\rightarrow$ Al + OH | 71.9 | no | 7.35(-11)(T/300)${}^{-0.14}\exp(-17406.3/T)$ | RRKM fit
164 | Al2O + h$\nu_{\lambda<213nm}$ $\rightarrow$ AlO + Al | -568 | 4100 | 8.20(-7)$\exp(0.00273T)$ | (Mangan et al. 2021)
165 | AlO + h$\nu_{\lambda<252nm}$ $\rightarrow$ Al + O | -498 | 4900 | 9.68(-7)$\exp(0.00102T)$ | (Mangan et al. 2021)
166 | AlOH + h$\nu_{\lambda<238nm}$ $\rightarrow$ AlO + H | -482 | 4900 | 1.53(-4)($\exp(0.0014T)$) | (Mangan et al. 2021)
Table 11: The cartesian coordinates x,y,z of (Al2O3)n, $n=$1$-$10, GM clusters (in units of Å)444Tables 11 are only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/. 1A | B3LYP/6-311+G(d) | -710.72608 a.u. |
---|---|---|---
Al | 0.00000 | 0.00000 | 1.69489
.. | .. | .. | …

##  Appendix C Thermo-chemical tables

Table 12: The thermochemical tables of (Al2O3)n, $n=$1$-$10, GM clusters as a function of temperature.666Tables 12 are only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/ 1A |  |  |  |  |  |
---|---|---|---|---|---|---
T(K) | S (J/molK) | Cp (J/molK) | ddH (kJ/mol) | dHf (kJ/mol) | dGf (kJ/mol) | log Kf
0.00 | 0.000 | 0.000 | 0.000 | -546.572 | -546.572 | $\infty$
.. | .. | .. | .. | .. | .. | ..

##  Appendix D Thermochemistry: Fit parameters

Table 13: Fitting parameters a,b,c,d,e for the computation of $\Delta$G${}_{f}^{0}$(T) used in equilibrium calculations. Species | a | b | c | d | e
---|---|---|---|---|---
AlO | 5.99300E+04 | -1.37970E+00 | -1.73737E+00 | 3.91773E-04 | -2.51845E-08
AlOH | 1.18027E+05 | -2.05964E+00 | -8.69160E+00 | 5.90024E-04 | -3.04906E-08
OAlOH | 1.69811E+05 | -3.52430E+00 | -1.43673E+01 | 1.25241E-03 | -7.34479E-08
HAlOH | 1.34662E+05 | -4.57736E+00 | -4.35980E+00 | 1.66011E-03 | -9.78547E-08
AlO2H2 | 2.08760E+05 | -5.10482E+00 | -1.50931E+01 | 1.92278E-03 | -1.11241E-07
AlO2 | 1.07130E+05 | -2.61203E+00 | -8.89728E+00 | 7.50030E-04 | -4.93246E-08
Al2O | 1.28411E+05 | -2.40921E+00 | -1.05621E+00 | 6.49789E-04 | -4.25611E-08
Al2O2 | 1.86105E+05 | -3.31380E+00 | -1.99433E+01 | 1.58854E-03 | -1.06058E-07
Al2O3 | 2.34139E+05 | -3.95729E+00 | -2.82239E+01 | 2.11009E-03 | -1.40019E-07
Al2O4 | 2.99834E+05 | -4.30886E+00 | -4.27871E+01 | 2.51478E-03 | -1.66932E-07
Al3O3 | 3.08682E+05 | -4.20634E+00 | -3.99769E+01 | 2.44693E-03 | -1.61816E-07
Al3O4 | 3.69176E+05 | -5.56346E+00 | -5.20710E+01 | 3.40984E-03 | -2.28468E-07
Al4O4 | 4.36049E+05 | -4.97450E+00 | -6.35660E+01 | 3.25225E-03 | -2.14255E-07
Al4O5 | 5.05163E+05 | -6.53645E+00 | -7.58250E+01 | 4.29600E-03 | -2.85685E-07
Al4O6 | 5.84165E+05 | -8.82800E+00 | -8.42817E+01 | 5.63090E-03 | -3.74702E-07
AlSiO3 | 2.67025E+05 | -4.60668E+00 | -2.59855E+01 | 2.35614E-03 | -1.54434E-07
HAlSiO3 | 2.75443E+05 | -1.00583E+01 | -7.73659E+00 | 6.85770E-03 | -8.55013E-07
Al2O3SiO | 3.80803E+05 | -9.85869E+00 | -2.79375E+01 | 9.23519E-03 | -1.18218E-06
AlF | 8.19970E+04 | -1.31009E+00 | -2.43594E+00 | 4.12562E-04 | -3.03640E-08
AlCl | 6.20720E+04 | -9.43409E-01 | -4.14453E+00 | 2.44222E-04 | -1.87099E-08
Al2 | 1.09331E+04 | -7.39015E-01 | -5.44758E+00 | 1.35276E-04 | -1.05029E-08
AlH | 3.62088E+04 | -1.68480E+00 | 1.07115E+00 | 4.62791E-04 | -2.91925E-08
Al3O5 | 4.22272E+05 | -6.47463E+00 | -6.17019E+01 | 4.04645E-03 | -2.69671E-07
Al3O6 | 4.65554E+05 | -5.97223E+00 | -7.53001E+01 | 3.95721E-03 | -2.60889E-07
Al4O3 | 3.59063E+05 | -3.47918E+00 | -5.55172E+01 | 2.29771E-03 | -1.50608E-07
Al3O2 | 2.31189E+05 | -2.55393E+00 | -3.25776E+01 | 1.41095E-03 | -9.25233E-08
Al8O12 | 1.16740E+06 | -1.57762E+01 | -1.99568E+02 | 1.12760E-02 | -7.49050E-07
Al10O15 | 1.48494e+06 | -1.94214E+01 | -2.57899E+02 | 1.41833E-02 | -9.41950E-07
Al12O18(B) | 1.80753E+06 | -2.37453E+01 | -3.17029E+02 | 1.75208E-02 | -1.16698E-06
Al14O21 | 2.12820E+06 | -2.75694E+01 | -3.73757E+02 | 2.04905E-02 | -1.36329E-06
Al16O24 | 2.44711E+06 | -3.23660E+01 | -4.30676E+02 | 2.40589E-02 | -1.60386E-06
Al18O27 | 2.78902E+06 | -3.60898E+01 | -4.90629E+02 | 2.70106E-02 | -1.79999E-06
Al20O30 | 3.10574e+06 | -4.06379E+01 | -5.45915E+02 | 3.04129E-02 | -2.02790E-06

##  Appendix E Cluster geometries

Table 14: Cluster volumes (in Å3) and radii (in Å) as derived from atomic core coordinates (Delauney triangulation) and from van-der-Waals interactions. The radii are calculated assuming spherical volumes. $n$ | Delauney | van der Waals
---|---|---
| Volume | Radius | Volume | Radius
1 | 0.0 | 0.0 | 73.73 | 2.60
2 | 13.88 | 1.49 | 128.59 | 3.13
3 | 29.63 | 1.92 | 185.49 | 3.54
4 | 51.16 | 2.30 | 249.61 | 3.91
5 | 80.00 | 2.67 | 308.37 | 4.19
6 | 102.00 | 2.90 | 358.37 | 4.41
7 | 142.74 | 3.24 | 347.3 | 4.36
8 | 153.15 | 3.32 | 473.94 | 4.84
9 | 154.17 | 3.33 | 520.73 | 4.99
10 | 226.18 | 3.78 | 594.16 | 5.22

##  Appendix F Cluster coordinates


# MOAB: Multi-modal Outer Arithmetic Block for fusion of histopathological
images and genetic data for brain tumor grading

###### Abstract

Brain tumors are an abnormal growth of cells in the brain. They can be
classified into distinct grades based on their growth. Often grading is
performed based on a histological image and is one of the most significant
predictors of a patient’s prognosis; the higher the grade, the more aggressive
the tumor. Correct diagnosis of the tumor’s grade remains challenging. Though
histopathological grading has been shown to be prognostic, results are subject
to interobserver variability, even among experienced pathologists. Recently,
the World Health Organization reported that advances in molecular genetics
have led to improvements in tumor classification. This paper seeks to
integrate histological images and genetic data for improved computer-aided
diagnosis. We propose a novel Multi-modal Outer Arithmetic Block (MOAB) based
on arithmetic operations to combine latent representations of the different
modalities for predicting the tumor grade (Grade ii@, iii@ and iv@). Extensive
experiments evaluate the effectiveness of our approach. By applying MOAB to
The Cancer Genome Atlas (TCGA) glioma dataset, we show that it can improve
separation between similar classes (Grade ii@ and iii@) and outperform prior
state-of-the-art grade classification techniques.

Index Terms—  Multi-modal fusion, Outer-arithmetic fusion, Cancer grade
classification, Channel fusion, Brain tumor

##  1 Introduction

With the advent of artificial neural networks, many advanced medical imaging
algorithms have been proposed to analyze histopathological cancer images for
grade classification [1, 2] and survival prediction [3, 4]. Digitized
histopathology provides whole slide images (WSI) and related phenotypic
information. Literature suggests both single-modality [5] and multi-modality
[6] methods for histopathological image analysis. However, approaches based on
a single modality are limited and unable to exploit the complementary
information present in other modalities for cancer prognosis prediction [7,
8]. This provides an opportunity to integrate multi-modal data for more
precise diagnosis.

A recent study [9] integrated ’omic data with image modalities using attention
gating and a combination of concatenation and arithmetic operation-based
fusion approaches. Braman et al. [9] presented an intermediate fusion deep
multi-modality network that integrated radiology, pathology, genetics, and
clinical data to predict glioma patients. Deep learning models were used in
[9] to extract features from each modality and generate its corresponding
embedding layer. Using the attention gated mechanism and an outer product
operator, the authors were able to fuse significant features to improve
prediction.

A multi-modal fusion method was proposed by [10] to learn the combined feature
representations of histopathological images and mRNA expression, through
ResNet-152 and a sparse graph convolutional network respectively. The features
were merged using a shared multilayer perceptron to predict survival analysis
and cancer grade. Simailarly, Pathomic Fusion [6], an integrative image-omic
analysis method used a gated attention mechanism and a Kronecker product to
combine features of different modalities. The image features were extracted
using a combination of convolutional and graph convolutional neural networks,
and a feed-forward network was trained separately to learn genomics features.
Mobadersanya et al. [11] proposed survival convolutional neural networks
(SCNNs) to merge a histology image and genomic biomarkers.

Fig. 1: Architecture of our proposed approach Multi-modal Outer Arithmetic
Block (MOAB) fusion model. (a) MOAB (b) Generic outer arithmetic approach.

We note that previous work [6, 10] focused on Grade iv@ and did not consider
Grade ii@ and<EMAIL_ADDRESS>These two grades are minorities in the dataset compared to
Grade iv@ and their similarity hinders typical classification techniques from
detecting them accurately. However, Grade iii@ is considered malignant and as
invasive as Grade iv@ [12] which could make its classification as crucial as
Grade<EMAIL_ADDRESS>Hence, in this study, we show how to integrate rich features from
genetic data and histology images to improve grading of tumors.

We propose a novel fusion method that captures the interrelated features
between disparate modalities for improved classification. An overview of our
approach is presented in Fig. 1. Our paper makes the following contributions:

  * •

We introduce a novel Multi-modal Outer Arithmetic Block (MOAB) that fuses
latent representations of different modalities using arithmetic operations.

  * •

We demonstrate MOAB’s ability to capture rich representations of fused data in
a brain cancer application using a histology image and genetic data. Our
method outperforms the previous state-of-art, by producing more discriminative
fused features for classification. This is also demonstrated with t-SNE plots
which visualize the improved separability of classes thanks to MOAB.

  * •

Source code for all proposed fusion methodologies are available at
https://github.com/omniaalwazzan/MOAB

##  2 METHOD

From paired histology and genetic data with known cancer grades, we aim to
learn and combine informative features that outperform single-modality
classifiers in supervised learning. For this purpose, we propose a novel
method for effective integration of multi-modal data.

We first provide a brief overview of the single modality backbone that we use,
followed by in-depth explanations of the architecture of two components: outer
arithmetic fusion methodologies and our proposed MOAB.

###  2.1 Single modality classifiers

We initially created a single modality classifier for each data type (i.e.
histology image and genetic data). For the histology images, a pre-trained
ConvNeXt initialized with ImageNet weights was used. ConvNeXt is a recently
proposed [13], state-of-the-art CNN demonstrating how convolutional backbones
can outperform transformers in image classification [14]. We replace the
LayerNorm2D of the ConvNeXt classifier layer with a dropout = 0.2 followed by
a fully connected (FC) layer that outputs three values representing the brain
tumor grades (ii@, iii@ and iv@). We fine-tune ConvNeXt on the histology
images. We observe that ConvNeXt is more robust than the VGG19 backbone
implemented in [6] in terms of generalizing well to the histology images after
a few iterations.

A 3 consecutive blocks of FC layers with dimensions [80, 40, 32] forming our
multi-layer perceptron (MLP) was utilized to classify the genomic data. Each
layer in the MLP is followed by a rectified linear unit (ReLU) activation and
a layer normalization. After the second and third layers, a gentle dropout of
0.2 was applied. According to the results presented in Table 1, the MLP
achieves comparable performance to the ConvNeXt applied to the histological
images.

###  2.2 Outer Arithmetic Operations

Our work is based on Outer Product Fusion (OPF), which has been used
previously [9] to combine feature vectors of different modalities. Consider
two feature vectors $\mathbf{a}\in\mathbb{R}^{N\times 1}$ and
$\mathbf{b}\in\mathbb{R}^{M\times 1}$. We first append a 1 to each feature
vector, i.e. $\mathbf{a}_{1}=[1;\mathbf{a}]$ and
$\mathbf{b}_{1}=[1;\mathbf{b}]$. The OPF is expressed as

$(\mathbf{a}_{1}\otimes\mathbf{b}_{1})_{ij}=a_{1i}*b_{1j},$ (1)

for $i\in[1...N+1]$ and $j\in[1...M+1]$ resulting in an $(N+1)\times(M+1)$
matrix combining all pairs of features. The appended 1 in both
$\mathbf{a}_{1}$ and $\mathbf{b}_{1}$ ensures the original $\mathbf{a}$ and
$\mathbf{b}$ vectors appear in the outer product matrix.

We extend the OPF by proposing the use of three new additional arithmetic
operations: _Outer Addition Fusion (OAF)_ , _Outer Subtraction Fusion (OSF)_
and _Outer Division Fusion (ODF)_. The ODF is expressed as:

$(\mathbf{a}_{1}\oslash\mathbf{b}_{1})_{ij}=a_{1i}/(b_{1j}+\epsilon),$ (2)

where $\epsilon$ is a small number to avoid division by zero.

For the outer addition and outer subtraction fusion, we append a 0 to each
feature vector, i.e. $\mathbf{a}_{0}=[0;\mathbf{a}]$ and
$\mathbf{b}_{0}=[0;\mathbf{b}]$. Then, the OAF is defined as

$(\mathbf{a}_{0}\oplus\mathbf{b}_{0})_{ij}=a_{0i}+b_{0j},$ (3)

and OSF as

$(\mathbf{a}_{0}\ominus\mathbf{b}_{0})_{ij}=a_{0i}-b_{0j},$ (4)

As before, the appended 0 in both $\mathbf{a}_{0}$ and $\mathbf{b}_{0}$
ensures the original $\mathbf{a}$ and $\mathbf{b}$ vectors appear in the outer
addition and subtraction matrices. Each of the four outer arithmetic fusion
operators will produce a matrix of size $(N+1)\times(M+1)$.

###  2.3 Multi-modal Outer Arithmetic Block Fusion

The MOAB fusion model uses the previously explained outer arithmetic fusion
operations to capture essential features from the genomic and histology data.
We perform the fusion on a four branch scheme. First, we obtain latent
representation feature vectors $\mathbf{a}$ and $\mathbf{b}$ from histology
and genomic data, respectively. As illustrated in Fig 1(a), $\mathbf{a}$ was
extracted from the customized ConvNeXt network used previously for histology
image classification. Inspired by [6], we obtained 32 features by modifying
the number of the output dimension of the last FC layer from the ConvNeXt.
Similarly, 32 features were extracted from the last FC layer from our MLP to
represent $\mathbf{b}$.

Next, each branch will take $\mathbf{a}$ and $\mathbf{b}$, perform the four
arithmetic operations to produce four multi-modal feature maps, $A_{i,j}$,
$S_{i,j}$, $P_{i,j}$, and $D_{i,j}$, that represent the OAF, OSF, OPF and ODF
respectively. Moreover, we utilized a sigmoid function to compress our feature
maps between [0,1]. The sigmoid is employed due to the addition of the
$\epsilon$ = 1.2e−20 avoiding zero division in the ODF branch. The sigmoid was
applied to all four branches to maintain compatible fusion between all
branches.

The four matrices $A_{i,j}$, $S_{i,j}$, $P_{i,j}$, and $D_{i,j}$, are
concatenated along the channel dimension into a multi-modal tensor
$M_{ij}\in\mathbb{R}^{4\times 33\times 33}$. Then, a 2D convolution layer is
applied with a filter $f=1$ and a stride $s=1$ to take advantage of
interrelated interactions and produce one condensed multi-modal feature tensor
$M_{ij}^{*}\in\mathbb{R}^{1\times 33\times 33}$. We hypothesize that channel
fusion will maintain the proximity of closer points and will use fewer
parameters compared to a typical concatenation. By combining features across
the channel dimension, we massively decrease the dimension of the final FC
layer from $4356$ to $1089$ in terms of concatenation. It has been noted that
$f={3,5}$ leads to a decrease in the model’s performance. Hence, we avoid
using filter $f>1$.

Finally, $M_{ij}^{*}$ is passed to two FC layers to perform the final
classification. A mild dropout$=$0.1 is employed before the final layer to
avoid overfitting. We also adopt the orthogonal weight initialization
introduced in [6] for the MLP component in the proposed architecture.

##  3 EXPERIMENTS AND RESULTS

Fig. 2: t-Distributed Stochastic Neighbor Embedding (t-SNE) visualization for
fusion models. Note in (a), the MOAB fusion provides better separability
between brain tumor grades.

.

###  3.1 Dataset

The glioma dataset was preprocessed as proposed by Chen et al. [6], and
provides paired histology images and gene expression derived from The Cancer
Genome Atlas (TCGA) repository. The TCGA is a cancer database containing
paired high-throughput genome diagnostic whole slide images with ground-truth
histological grade labelling. There are 769 patients belong to 1505
histological region-of-interests (ROIs) for astrocytomas and glioblastomas in
the TCGA-GBMLGG project. The selection of ROI derived from histopathological
whole slide images (WSIs) was generated, revised, and reviewed by [11]. Each
patient had 1-3 ROIs from diagnostic WSI. For the genomic data, 80 features
include 79 copy number variations (CNVs) and one mutation status. Similar to
[6], we removed cases with missing grades, resulting in 396, 408, and 654 ROIs
for Grade ii@, iii@ and iv@, respectively, and corresponding to 736 cases. As
in [6], ROIs are used for training models, while 9 overlapping patches
extracted per each ROI derived from the testing cohort were used for testing.
Each image was regarded as a single data point as in [6], with genetic and
ground-truth label data copied over.

###  3.2 Implementation Details

Proposed algorithms were implemented in PyTorch using the Adam optimizer and a
cross-entropy loss function for all experiments. For the single modalities, we
empirically found the best learning rate of 0.001. While for fusion models, to
avoid overfitting, we regularized our networks with a weight decay of size
0.0005 and a 0.005 learning rate. A batch size of 8 was set for all
experiments. All models were trained for 10 epochs. The parameters for
ConvNeXt, MLP and MOAB are 87M, 11K and 88M respectively. A cluster of NVidia
A100 GPUs was utilised for all experiments.

###  3.3 Ablation Study

As shown in Table 1, we validate the performance of MOAB fusion through
multiple ablation studies. In an identical Monte Carlo 15-fold cross-
validation train-test split obtained from [6, 11], our ablation studies
compare different model configurations and fusion methodologies. In addition,
as highlighted in Fig. 2, a (t-SNE) algorithm [15] was applied to the
inference split for ablated fusion models to visualize the high-dimensional
relationships between the three learned grades (ii@, iii@ and iv@) in three-
dimensional space. In t-SNE plots, nearby dots represent comparable samples,
whereas distant dots reflect dissimilar ones. We empirically set an optimal
perplexity and a max iteration to 80 and 1000.

Table 1: Ablation studies. Method | F1 (Grade IV) | F1-Micro | F1-Macro
---|---|---|---
CNN (Image) | 0.888 | 0.715 | 0.586
MLP (Genes) | 0.901 | 0.700 | 0.626
Concatenation | 0.910 | 0.726 | 0.658
OAF | 0.944 | 0.740 | 0.675
DBF | 0.916 | 0.727 | 0.660
Standard Addition* | 0.933 | 0.730 | 0.665
MOAB (VGG19_bn+MLP)) | 0.945 | 0.752 | 0.689
MOAB (ConvNext+MLP) | 0.956 | 0.766 | 0.697

At first, single classifiers for both gene and image data were constructed to
draw a fair comparison to multi-modality fusion. Next, we show results for a
simple concatenation fusion model to further investigate the ability of other
fusion methodologies. To explore the capability of MOAB, we produced two
ablated fusion models: the Outer Addition Fusion (OAF) and a Dual-branch
channel Fusion (DBF). Note that OAF has no channel fusion, features are
combined as shown in Fig. 1(b) performing $A_{ij}\in\mathbb{R}^{33\times 33}$
and followed by FC layers similar to MOAB. However, DBF imitates the same
configuration as MOAB but with two branches: OAF and OPF. From Table 1 and
Fig. 2(b) we observe that OAF performs better than the DBF even with a simple
integration. This might occur due to the redundant features obtained from OAF,
and OPF. Yet, DBF achieves similar metric scores (F1 and F1-Micro) to that of
concatenation, the separated cluster shown in Fig. 2(c) is slightly better
than those of Fig. 2(d), which might reflect the effect of OAF operation on
capturing correlated features.

As demonstrated in Table 1, the best results were obtained utilising MOAB.
This is further supported by Fig. 2(a), which shows MOAB fusion has the most
separated classes among all other ablated fusion models. By comparing Fig.
2(a) with Fig. 2(f) we can clearly inspect the positive effect of integrating
features from genetic data and histology images. Table 1 summarizes that
utilizing all four branches in the channel fusion has significant improvements
on the classifier performance namely F1-Micro and F1-Macro. F1-micro score is
calculated from the micro-averaged precision (miPr), and a micro-averaged
recall (miRe) and defined as the following:

$\displaystyle miPr$
$\displaystyle=\frac{\sum_{i=1}^{r}TP_{i}}{\sum_{i=1}^{r}(TP_{i}+FP_{i})}$ (5)
$\displaystyle miRe$
$\displaystyle=\frac{\sum_{i=1}^{r}TP_{i}}{\sum_{i=1}^{r}(TP_{i}+FN_{i})}$

The $TP_{i}$, $FP_{i}$ $FN_{i}$ denote true positive, false positive and false
negative rates at a cell $i$ in the confusion matrix. Then the micro-averaged
F1 score (miF1) is defined as the mean of $miPr$, $miRe$ values.

We use the F1-micro score as the primary metric to evaluate our results,
following [10],[6], as it is an appropriate evaluation metric for imbalanced
class data. Notably, F1-Grade iv@ is an easier classification task given that
it dominates the majority of our dataset; still, we include results for
comparison purposes. Furthermore, to quantify the performance of MOAB, we
calculate the F1-macro average which treats classes equally and gives a sense
of MOAB’s effectiveness on minority classes. The macro averaging method is
more challenging for our task because it is more strongly influenced by the
performance of rare classes [16]. Table 1 proves that MOAB manages to achieve
strong performance. Finally, to conclusively say OAF is preserving the
interlarded features between modalities, we conduct an experiment for a
standard addition fusion model shown in Table 1, where we increase the
parameters of the standard addition (denoted as an *) to have roughly the same
learnable parameters as the OAF. Results of the standard addition exhibited in
Table 1 and Fig. 2(e) indicate that OAF outperforms standard addition fusion*.

We have also switched the backbone of the MOAB from ConvNxet to the VGG19 with
batch normalization used in [6], again, our obtained results outperform the
Pathomic fusion [6] approach. We conclude that MOAB is improving the
classifier thoroughly compared to all other configurations presented in Fig.
2. We observed that MOAB is simple to implement and extracts rich interaction
combining modalities.

###  3.4 Comparison Experiments

We compared our method with the Pathomic fusion [6] and a MultiCoFusion [10]
on the glioma dataset. The results are shown in Table 2. The TCGA glioma
dataset has been used with 15-folds in both [6] and [10], however, the
approach used in [10] is slightly different than ours, in which they
additionally use mRNA expression data using a graph CNN.

Table 2: Comparison experiments. Model | F1 (Grade IV) | F1-Micro
---|---|---
Pathomic SNN [6] |  0.857 $\pm$ 0.017 |  0.652 $\pm$ 0.015
Pathomic (CNN+SNN) [6] |  0.913 $\pm$ 0.011 |  0.730 $\pm$ 0.019
MultiCoFusion [10] |  0.998 $\pm$ 0.005 |  0.759 $\pm$ 0.032
MLP (Genes) |  0.901 $\pm$ 0.051 |  0.700 $\pm$ 0.042
MOAB (VGG19_bn+MLP)) |  0.945 $\pm$ 0.012 |  0.752 $\pm$ 0.002
MOAB (ConvNext+MLP) |  0.956 $\pm$ 0.000 | 0.766 $\pm$ 0.001

According to the ablation studies shown in Table 1 and the comparison
presented in Table 2, the F1-Micro of MOAB is superior to that of the other
models. Experiments were performed 15 times; the results shown in the tables
represent the mean of the total of all evaluation metrics for 15-folds, as
demonstrated in [6] and [10].

##  4 CONCLUSION

In this paper, we proposed a novel Multi-modal Outer Arithmetic Block (MOAB)
that fuses features from histology and genetic data. Involving features from
all arithmetic operations encourages the classifier to capture intrinsic
relations between modalities, ultimately enhancing the overall prediction. The
channel fusion strategy enables MOAB to reduce the parameters of conventional
fusion methods and enhances the classifier’s knowledge. All conducted
experiments demonstrate the effectiveness of proposed approach. Last, we note
that the architecture of MOAB is transferrable to other multi-modal problems
as it can combine modalities of any type with any CNN backbones. In the
future, We plan to experiment MOAB with additional datasets and expand it to
integrate more modalities.

##  5 Acknowledgments

The authors appreciate the support of the University of Jeddah and the Saudi
Arabia Cultural Bureau. This paper utilised Queen Mary’s Andrena HPC facility.
This work also acknowledges the support of the National Institute for Health
and Care Research Barts Biomedical Research Centre (NIHR203330), a delivery
partnership of Barts Health NHS Trust, Queen Mary University of London, St
George’s University Hospitals NHS Foundation Trust and St George’s University
of London.

##  6 COMPLIANCE WITH ETHICAL STANDARDS

The licence related to the open-access data ensured no ethical approval was
necessary.

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