Deep learning enhanced cost-aware multi-fidelity uncertainty quantification of a computational model for radiotherapy

Piermario Vitullo; Nicola Rares Franco; Paolo Zunino

doi:10.3934/fods.2024022

Article Contents

2025, Volume 7, Issue 1: 386-417. Doi: 10.3934/fods.2024022

This issue Previous Article Neural network approaches for parameterized optimal control Next Article Unsupervised physics-informed disentanglement of multimodal data

Deep learning enhanced cost-aware multi-fidelity uncertainty quantification of a computational model for radiotherapy

MOX, Department of Mathematics, Politecnico di Milano, Italy

^*Corresponding author: Paolo Zunino
^*Corresponding author: Paolo Zunino

Received: January 2024

Revised: April 2024

Early access: May 12, 2024

Published online: May 12, 2024

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Abstract

Forward uncertainty quantification (UQ) for partial differential equations is a many-query task that requires a significant number of model evaluations. The objective of this work is to mitigate the computational cost of UQ for a 3D-1D multiscale computational model of microcirculation. To this purpose, we present a deep learning enhanced multi-fidelity Monte Carlo (DL-MFMC) method that integrates the information of a multiscale full-order model (FOM) with that coming from a deep learning enhanced non-intrusive projection-based reduced order model (ROM). The latter is constructed by leveraging on proper orthogonal decomposition (POD) and mesh-informed neural networks (previously developed by the authors and co-workers), integrating diverse architectures that approximate POD coefficients while introducing fine-scale corrections for the microstructures. The DL-MFMC approach provides a robust estimator of specific quantities of interest and their associated uncertainties, with optimal management of computational resources. In particular, the computational budget is efficiently divided between training and sampling, ensuring a reliable estimation process suitably exploiting the ROM speed-up. Here, we apply the DL-MFMC technique to accelerate the estimation of biophysical quantities regarding oxygen transfer and radiotherapy outcomes. Compared to classical Monte Carlo methods, the proposed approach shows remarkable speed-ups and a substantial reduction of the overall computational cost.

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Mathematics Subject Classification: Primary: 65L60, 68T07, 65N22, 65M22, 65Z05.

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Figure 1. The tissue oxygenation map, measured in $ mL_O2/mL_B $, is visually depicted on the left panel through the FOM solution (light blue corresponds to low oxygen). On the right panel, we showc the 1D embedded vascular microstructure, which visibly impacts the oxygen map. Furthermore, oxygen concentration in the blood, indicated as $ C_v $, is also reported

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Figure 2. General layout of the full order model for the whole vascular microenvironment

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Figure 3. Examples of architectures with maximum extravascular distance increasing from left to right and higher vascular density from top to bottom

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Figure 4. A sketch of the POD-MINN+ method. The macroscale parameters and the microscale ones are fed to two separate architectures, whose outputs are later combined to approximate the POD coefficients. The coefficients are then expanded over the POD basis, $ \mathbb{V} $, and the ROM solution is further corrected with a closure term computed by a third network that accounts for the local features related to the high frequencies

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Figure 5. Regression models for the law $ 1 - \rho^2(n_j) \leq c_1 n_j^{- \zeta}+c_{2} $ for each QoI, $ \overline{pO_2} $, $ \Delta pO_2 $, $ TCP $, varying the sample sizes as $ j = 1,...,k $

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Figure 6. Regression model for the law describing the ROM training time $ t(n_j) \leq c_{3} n_j+c_{4} $, varying the sample sizes as $ j = 1,...,k $

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Figure 7. Confidence intervals estimates for all the QoIs, comparing the DL-MFMC estimator with the standard Monte-Carlo FOM-based, fixed $ \gamma = 99\% $

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Figure 8. Computational budget spent to achieve a fixed level of uncertainty for both the considered estimators, considering three different QoIs

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Table 1. In the first three rows, the biophysical parameters of the ROM are presented along with their respective ranges of variation. The last two rows outline the hyper-parameters utilized to initialize the algorithm responsible for generating the vascular network

Symbol	Parameter	Unit	Range of variation
$ P_{O_2} $	$ O_2 $ wall permeability	$ m/s $	$ 0.35 \cdot 10^{-4}-3.00 \cdot 10^{-4} $
$ V_{max} $	$ O_2 $ consumption rate	$ \frac{mL_{O_2}}{cm^3\cdot s} $	$ 0.40 \cdot 10^{-4}-2.40 \cdot 10^{-4} $
$ C_{v,in} $	$ O_2 $ concentration at the inlets	$ \frac{mL_{O_2}}{mL_B} $	$ 2.25 \cdot 10^{-3}-3.75 \cdot 10^{-3} $
$ \% \frac{SEEDS_{(-)}}{SEEDS_{(+)}} $	Seeds for angiogenesis	$ \% $	$ 0-75 $
$ S/V $	Vascular surface per unit volume	$ m^{-1} $	$ 5\cdot10^3-7\cdot10^3 $

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Table 2. Prescribed values of input parameters in the comprehensive computational model for the high-fidelity approximation of the solution through the finite element method

Symbol	Parameter	Unit	Value	Ref.#
$ L $	characteristic length	$ m $	$ 1 \cdot 10^{-3} $	-
$ R $	average radius	$ m $	$ 4 \cdot 10^{-6} $	[40]
$ K $	tissue hydraulic conductivity	$ m^2 $	$ 1\cdot 10^{-18} $	[24,40]
$ \mu_t $	interstitial fluid viscosity	$ cP $	$ 1.2 $	[48]
$ \mu_v $	blood viscosity	$ cP $	$ 3.0 $	[42]
$ L_p $	wall hydraulic conductivity	$ m^{2} $ s kg$ ^{-1} $	$ 1 \cdot 10^{-12} $	[40]
$ \delta \pi $	oncotic pressure gradient	$ mmHg $	25	[40]
$ \sigma $	reflection coefficient	$ - $	$ 0.95 $	[25]
$ D_v $	vascular diffusion coefficient	$ m^2/s $	$ 2.18 \cdot 10^{-9} $	[26]
$ N \cdot MCHC $	max. hemoglobin-bound $ O_2 $	$ \frac{mL_{O_2}}{mL_{RBC}} $	$ 0.46 $	[45]
$ \gamma $	Hill constant	−	$ 2.64 $	[26,53]
$ p_{s_{50}} $	$ O_2 $ at half-saturation	$ mmHg $	$ 27 $	[28,53]
$ \alpha_{t} $	$ O_2 $ solubility coefficient	$ \frac{mL_{O_2}/mL}{mmHg} $	$ 3.89 \cdot 10^{-5} $	[45]
$ D_t $	tissue diffusion coefficient	$ m^2/s $	$ 2.41 \cdot 10^{-9} $	[26]
$ C $	Characteristic $ O_2 $ concentration	$ \frac{mL_{O_2}}{mL_B} $	$ 1.50 \cdot 10^{-3} $	−

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Table 3. Input parameters values assigned in the linear-quadratic model to compute the TCP QoI

Symbol	Parameter	Unit	Value	Ref.#
$ D $	radiation dose	$ Gy $	$ 20 $	–
$ \alpha $	radiosensitivity parameter for 'single' hit	$ Gy^{-1} $	$ 0.178 $	[23]
$ \beta $	radiosensitivity parameter for 'multiple' hits	$ Gy^{-2} $	$ 0.0455 $	[23]
$ \delta $	TIN parameter	-	$ 1.38 $	[41]
$ M $	TIN parameter	-	$ 2.81 $	[41]
$ a $	TIN parameter	$ keV/\mu m $	$ 522.45 $	[41]
$ b $	TIN parameter	$ mmHg $	$ 1.24 $	[41]
$ N_c $	Clonogenic cells in the interstitial volume	-	$ 10^8 $	[12]
$ LET $	Linear Energy Transfer in photons	$ keV/\mu m $	$ 2 $	[13]

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Table 4. Number of FOM and ROM simulations employed for a UQ analysis of the oxygen transfer processes with the DL-MFMC estimator for three fixed reference computational budgets. In particular in the last column we report the percentage of FOM simulations required in the training phase

QoI	Budget $ p $	$ n^* $	$ m_0^* $	$ m_1^* $	$ \%\frac{n^}{n^+m_0^*} $
$ \overline{pO_2} $	$ 12\,h $	136	232	8311	36.96$ \% $
	$ 18\,h $	199	342	13522	36.78$ \% $
	$ 24\,h $	259	451	19082	36.48$ \% $
$ \Delta pO_2 $	$ 12\,h $	126	266	6037	32.14$ \% $
	$ 18\,h $	151	426	10053	26.17$ \% $
	$ 24\,h $	171	590	14202	22.47$ \% $
$ TCP $	$ 12\,h $	98	292	6168	25.13 $ \% $
	$ 18\,h $	116	462	9996	20.07$ \% $
	$ 24\,h $	130	635	13914	16.99$ \% $

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Table 5. Computational cost and UQ for the DL-MFMC estimator. FOM simulations = total number of high-fidelity simulations required by the computational pipeline, namely $ n_*+m_0^* $. Uncertainty = amplitude of the DL-MFMC confidence interval, Eq. (2.11). In parentheses, the comparison with the standard Monte Carlo estimator. E.g.: in the first row, we see that, compared to standard Monte Carlo, the DL-MFMC estimator reduced the uncertainty by 36.26% while simultaneously requiring 19.12% less FOM simulations

QoI	Budget $ p $	FOM simulations	Uncertainty
$ \overline{pO_2} $	$ 12\,h $	368 (-19.12%)	0.54 $ mmHg $	(-36.24%)
	$ 18\,h $	541 (-20.79%)	0.42 $ mmHg $	(-38.68%)
	$ 24\,h $	710 (-22.06%)	0.33 $ mmHg $	(-43.41%)
$ \Delta pO_2 $	$ 12\,h $	392 (-13.85%)	1.03 $ mmHg $	(-6.50%)
	$ 18\,h $	577 (-15.52%)	0.58 $ mmHg $	(-34.98%)
	$ 24\,h $	761 (-16.47%)	0.48 $ mmHg $	(-36.33%)
$ TCP $	$ 12\,h $	390 (-14.29%)	2.75%	(-12.33%)
	$ 18\,h $	578 (-15.37%)	1.84%	(-28.30%)
	$ 24\,h $	765 (-16.03%)	1.35%	(-38.44%)

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