We incorporate a practical data assimilation methodology into our previously established experimental-computational framework to predict the heterogeneous response of glioma cells receiving fractionated radiation treatment. Replicates of 9L and C6 glioma cells grown in 96-well plates were irradiated with six different fractionation schemes and imaged via time-resolved microscopy to yield 360- and 286-time courses for the 9L and C6 lines, respectively. These data were used to calibrate a biology-based mathematical model and then make predictions within two different scenarios. For Scenario 1, 70% of the time courses are fit to the model and the resulting parameter values are averaged. These average values, along with the initial cell number, initialize the model to predict the temporal evolution for each test time course (10% of the data). In Scenario 2, the predictions for the test cases are made with model parameters initially assigned from the training data, but then updated with new measurements every 24 hours via four versions of a data assimilation framework. We then compare the predictions made from Scenario 1 and the best version of Scenario 2 to the experimentally measured microscopy measurements using the concordance correlation coefficient (CCC). Across all fractionation schemes, Scenario 1 achieved a CCC value (mean ± standard deviation) of 0.845 ± 0.185 and 0.726 ± 0.195 for the 9L and C6 cell lines, respectively. For the best data assimilation version from Scenario 2 (validated with the last 20% of the data), the CCC values significantly increased to 0.954 ± 0.056 (p = 0.002) and 0.901 ± 0.061 (p = 8.9e-5) for the 9L and C6 cell lines, respectively. Thus, we have developed a data assimilation approach that incorporates an experimental-computational system to accurately predict the in vitro response of glioma cells to fractionated radiation therapy.
Citation: Junyan Liu, David A. Hormuth II, Jianchen Yang, Thomas E. Yankeelov. A data assimilation framework to predict the response of glioma cells to radiation[J]. Mathematical Biosciences and Engineering, 2023, 20(1): 318-336. doi: 10.3934/mbe.2023015
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We incorporate a practical data assimilation methodology into our previously established experimental-computational framework to predict the heterogeneous response of glioma cells receiving fractionated radiation treatment. Replicates of 9L and C6 glioma cells grown in 96-well plates were irradiated with six different fractionation schemes and imaged via time-resolved microscopy to yield 360- and 286-time courses for the 9L and C6 lines, respectively. These data were used to calibrate a biology-based mathematical model and then make predictions within two different scenarios. For Scenario 1, 70% of the time courses are fit to the model and the resulting parameter values are averaged. These average values, along with the initial cell number, initialize the model to predict the temporal evolution for each test time course (10% of the data). In Scenario 2, the predictions for the test cases are made with model parameters initially assigned from the training data, but then updated with new measurements every 24 hours via four versions of a data assimilation framework. We then compare the predictions made from Scenario 1 and the best version of Scenario 2 to the experimentally measured microscopy measurements using the concordance correlation coefficient (CCC). Across all fractionation schemes, Scenario 1 achieved a CCC value (mean ± standard deviation) of 0.845 ± 0.185 and 0.726 ± 0.195 for the 9L and C6 cell lines, respectively. For the best data assimilation version from Scenario 2 (validated with the last 20% of the data), the CCC values significantly increased to 0.954 ± 0.056 (p = 0.002) and 0.901 ± 0.061 (p = 8.9e-5) for the 9L and C6 cell lines, respectively. Thus, we have developed a data assimilation approach that incorporates an experimental-computational system to accurately predict the in vitro response of glioma cells to fractionated radiation therapy.
Matrix decomposition can catch some characteristic information of matrix, in which low-rank and sparse components are two of the most interesting. In the past decades, the theory of matrix decomposition has been developed rapidly, especially low-rank or sparse matrix decomposition have been used for many popular research areas, such as machine learning, image/signal processing and network analysis etc[19,22,24,20]. The problem of low-rank and sparse matrix decomposition (LRSMD) is highlighted and intensively studied recently in [4,6], especially in robust principle component analysis (PCA).
As described in the literature [4], we present a matrix
D=L0+S0, | (1) |
where
minimize (rank(L0),card(S0))subject to L0+S0=D. | (2) |
It is known that the LRSMD problem is in general ill-posed and NP-hard [4], and the easiest way is finding a convex optimization problem to approximate original problem. The heuristics of using the
minimize ‖L‖∗+λ‖S‖1subject to L+S=D, | (3) |
where
Nowadays, many methods based on different strategies have been proposed. Iterative thresholding (IT) algorithms [1,2] regularize the problem (3) and construct a Lagrange function, and then two objective variables are updated alternately. The iterative form of IT algorithms is simple and convergent, but the rate of convergence is slow and it is difficult to select the appropriate length of step. Accelerated proximal gradient (APG) algorithm [12] is another iterative approach, which building partial quadratic approximation by using Lipschitz gradient to solve the Lagrange function, iterating and getting solution. Although it is similar as IT algorithms, it reduces the number of iterations greatly. The dual method [12] is proposed because the nuclear norm is dual to spectral norm, the dual problem of problem (3) is easier to solve. Compared with APG algorithm, it has better scalability because it do not need complete singular decomposition in each iteration. One of the most influential method is augmented Lagrange multipliers (ALM) method [13]. The ALM method constructs augmented Lagrange function firstly, and then using alternating iteration method to optimize the problem. It is called EALM when using an exact augmented Lagrange multipliers while it is named IALM with inexact multipliers. IALM is also known as alternating direction methods (ADM)[21]. ALM method is more efficient than APG method, and can achieve higher accuracy with lower storage space.
For low-rank and sparse matrix decomposition problem, most of methods consider the ratio of low-rank degree and sparsity as a certain parameter, however different given matrices or requirements are always with different parameter
The remainder of this paper is organized as follows. Section 2 introduces the related background and motivation about the proposed method. In Section 3, the proposed model and algorithm are presented in detail. Experimental study is described in Section 4. Finally, concluding remarks are given in Section 5.
In this section, in order to make it easier to understand the proposed method, we will give a brief introduction to multiobjective optimization. Then we will describe the motivation of multiobjective low-rank and sparse matrix decomposition.
In general, it will be called multiobjective optimization (MOO), when the number of objective function is more than one and need to optimize simultaneously [8,7,14]. A MOP can be stated as:
min F(x)=(f1(x),⋯,fm(x))Tsubject to x∈Ω, | (4) |
where
Let
∀i=1,2,⋯,m fi(xA)≤fi(xB)∃j=1,2,⋯,m fj(xA)<fj(xB), | (5) |
and mark it as
In practical applications, most objective functions have many or even infinite solutions, and we can not get all of them [23]. The purpose of multiobjective optimization is that finding a uniformly distributed Pareto front under a certain amount, which can represent the whole Pareto front. Evolutionary algorithm is one of the most popular methods for solving multiobjective optimization [11,18,25,9,23], such as VEGA [18], SPEA-Ⅱ [25], NSGA-Ⅱ [9] and MOEA/D [23]. Zhang and Li [23] proposed the MOEA/D which has a good performance in MOPs. MOEA/D decomposes the MOPs into a series scalar optimization subproblems with different weights. Due to the solution of each subproblem is similar to neighboring subproblems, the solution can be optimized by neighboring subproblems information. In this paper, due to the problem can be decomposed naturally with different parameter
For matrix decomposition with constraints of rank and sparsity, as described in problem 2, most of methods convert NP-hard problem into convex optimization. However these algorithms have to set a value of parameter
Multiobjective optimization is a popular way to avoiding the difficulty of parameter
For LRSMD, we focus on a more original problem, which thinks LRSMD as a multiobjective problem. The essence of the MOO is to optimize several conflicting objectives simultaneously. As described in problem (2), there are two existing conflicting objectives, which represent low-rank degree and sparsity respectively. We can obtain different solutions by a evolutionary multiobjective algorithm, such as MOEA/D. And each solution in the Pareto front represents a trade-off between reducing rank of low-rank component and decreasing the sparsity of sparse part, which can satisfy different requirements of decision makers.
First of all, we introduce the model of multiobjective low-rank and sparse matrix decomposition based on matrix singular value decomposition. Then details of multiobjective low-rank and sparse matrix decomposition algorithm will be described, which show how the MOO approach is applied in matrix decomposition.
As is mentioned in Section 1, problem (2) is a standard multiobjective problem, however it is hard to optimize even with evolutionary algorithm. The reasons can summary as follow: 1) both two objective functions are NP-hard and the low-rank component and sparse component are tightly correlative. 2) the decision variable of problem (2) is complex whatever it is low-rank component
Let the singular value vector
f1(x)=n∑i=1|xi|. | (6) |
Similarly, sparsity of sparse component is defined as
f2(x)=m∑i=1n∑j=1|Sij(x)|, | (7) |
where
At this point, the two objective functions can be combined into a MOP. It has the following form:
minxF(x)=minx(f1,f2)subject to x∈Ω. | (8) |
For low rank component, it is hard to measure and optimize, but the singular value vector can approximate it. When
Algorithm 1 Multiobjective LRSMD Model |
Input: |
Output: |
1: Step 1) |
2: Step 2) Generate a set of nondominated solutions ( |
3: Step 3) Recovery |
4: for |
5: Step 3.1) |
6: Step 3.2) |
7: Step 3.2) |
8: end for |
In this paper, the framework of MOEA/D [23] is applied in MOLRSMD because of the characteristic of MOEA/D which decomposing the MOP into several scalar optimization subproblems. In MOLRSMD we use Tchebycheff approach to decompose the multiobjective optimization problem (8), and the scalar optimization subproblem is in the form
min gte(x|w)=max{w1|f1−z∗1|,w2|f2−z∗2|}subject to x∈Ω, | (9) |
where
The procedure of the proposed algorithm for low-rank and sparse matrix decomposition is given in Algorithm 2. The MOLRSMD problem can be decomposed into
Algorithm 2 Procedure of MOLRSMD based on MOEA/D |
Input: stopping criterion: |
Output: Pareto front solution; matrix decomposition results. |
1: Step 1) Initialization |
2: Step 1.1) Generate a uniform distributed of |
3: Step 1.2) Generate an initial population |
4: Step 1.3) Initialize two reference points |
5: Step 1.4) For each |
6: Step 2) Set |
7: Step 3) Update |
8: for |
9: Step 3.1) Randomly select two indexes |
10: Step 3.2) Apply a thresholding method to improve |
11: Step 3.3) For each |
12: Step 3.4) For each index |
13: end for |
14: Step 4) Stopping criteria: If |
In this algorithm, some ideas of Tchebycheff decomposition and genetic operators are used which make solutions distribute more uniform and algorithm convergence faster. Next, some details about these ideas will be introduced.
In this paper, due to the difference of orders of magnitude between two objective functions value, the strategy of objective functions normalization is employed. In general, the value of objective function is large especially the range of two objectives have a huge difference, which make Pareto front is not uniform and most of points cluster in corner. A simple function normalization method formed as follow
¯fi=fi−zlizui−zli | (10) |
where
¯gws(x|w,zl,zu)=max{w1|f1−z∗1|,w2|f2−z∗2|}=max{w1|¯f1−¯z∗1|,w2|¯f2−¯z∗2|}=max{w1|f1−zl1zu1−zl1|,w2|f2−zl2zu2−zl2|}, | (11) |
Genetic operator is the key of evolutionary algorithm, with genetic operator combined characteristic of optimization problem, the convergence speed of algorithm can be improved. In this paper, we use a difference strategy and double local mutation method to speed up algorithm.
In crossover operator, difference optimization strategy is applied. We select two individuals randomly from the neighborhood of
yj={xij+F(xkj−xlj) if r<pcxij otherwise | (12) |
where
We apply a scale controllable mutation with two different mutations to balance diversity and precision of population. In the paper, we select Gauss mutation and polynomial mutation. Gauss mutation is simple than polynomial mutation and well done in global searching. Polynomial mutation for local search can improve the quality of solution. Therefore, if a random number
y′={Gauss(y,σ)if rand<psPolynomial(y)otherwise | (13) |
where
In this section, setting singular value vector as decision variable, makes problem simpler and easier to optimize. On the one hand, the complexity of variable is
In this section, we will represent experiments on artificial generated datasets and some nature images to evaluate the performance of the proposed method. The experiment focus on how the multiobjective low-rank and sparse matrix decomposition algorithm works with different type test data, and analyzing the performance with the convergence, stability and robustness.
Let
In order to illustrate the result of experiment intuitively, we define some indexes to measure results and the performance of the proposed method. The relative error to the original low-rank matrix
ErrLR:=||L−L∗||F||L∗||F. | (14) |
Similarly, the relative error to the original sparse matrix
ErrSP:=||S−S∗||F||S∗||F. | (15) |
The total error to original matrix is
ErrT:=||L+S−D∗||F||D∗||F. | (16) |
Due to the characteristic of multiobjective matrix decomposition method, there are many different decomposition results and some results are widely different with original matrices. On the one hand, in order to analysis the accuracy of matrix decomposition in artificial datasets, we just focus on solution with smallest recovery error and the distribute of Pareto front. On the other hand, we could select solution with actual demand when all decomposed solutions would be represented with images in nature images.
The parameters in the proposed method are given in Table 1.
Parameter | Meaning | Value |
The number of subproblems | 100 | |
The number of neighbors | 20 | |
| The maximum of generations | 300 |
| The probability of crossover | 0.8 |
| The differential multiplier | 0.5 |
| The probability of mutation | 0.2 |
| The probability of mutation selection | 0.85 |
For artificial generated datasets, we select three different size test matrices, which the sizes are
Furthermore, we also do experiments on some data with noise to test the robustness of the proposed method. For data with size:
Figure 4 displays the box-plot of recovering error in five noisy situations, which is described as above. We can know that not only the error of low-rank recovering but also sparse recovering is changed in a small range with different types noise. With the improved of number of noise points or
In the nature images, the proposed method also works well. The images that we used is consisted of Lena and orl-faces datasets. Due to no standard decomposed results of datasets, we just analysis the Pareto front and practical results to evaluate performance of the proposed method. Some results are displayed in Figure 6 and Figure 7. Figure 6 is about image Lena and Figure 7 is result in one random face image from orl-faces. We can see that the Pareto front of the proposed method is smooth and equally distributed, which means the method converges to PF stably. The (b) of Figure 6 and 7 show three different decomposed results with different locations in the PF. The first row of part (b) means sparse component has an absolute advantages, on the contrary, the third row shows low-rank part with more advantage. The middle one is a balance between low-rank and sparse components.
The performance of the proposed MOLRSMD have been examined in the previous subsection. In this subsection, simple comparison with some advanced LRSMD algorithms is tested on artificial datasets. ADM [21] and ALM [13] are two mainstream and efficient algorithms in exiting research. Both of them need set parameter
Methods | | | |
MOLRSMD | 0.0100 | 0.1986 | 0 |
ADM | 0.0299 | 0.1331 | 1.9135e-17 |
ALM | 0.0148 | 0.0661 | 2.2412e-10 |
Each row of Figure 8 presents the Pareto front by different algorithms. The Pareto front of MOLRSMD is complete and smooth as compared with ADM and ALM. The results of ADM and ALM are with uneven distribution, most of Pareto optimal solutions cluster in low sparsity region even the region with sparsity close to zero. In particular, from the second row of Figure 8, solutions of ADM and ALM with a sudden change and most of solutions cluster in two regions (the value of two objective functions are about 7500,2500 and 4500, 0 respectively) while solutions of the proposed MOLRSMD with uniform and smooth distribution. Table 2 represents the smallest recovery error from 100 solutions of the proposed MOLRSMD, ADM and ALM on artificial datasets (size: 50
In this paper, an evolutionary multiobjective low-rank and sparse matrix decomposition method has been proposed, for finding optimal trade-off solutions between the rank of low-rank component and the sparsity of sparse part. We focus on more original problem of low-rank and sparse matrix decomposition, with the theorem of matrix singular value decomposition, modifying the model by replacing decision variable with singular value vector. We have experiments on different type datasets, including artificial datasets, datasets with noise and nature images, to indicate that the proposed MOLRSMD is always with good convergence, reliable stability and acceptable robustness. It also show that it can be applied to some practical problem.
Future work will be attention on decreasing the accuracy of recovering. In the proposed method, the orthogonal matrices
[1] | C. Fernandes, A. Costa, L. Osório, R. C. Lago, P. Linhares, B. Carvalho, et al., Current standards of care in glioblastoma therapy, in Glioblastoma, Codon Publications, (2017), 197–241. doi: 10.15586/codon.glioblastoma.2017.ch11 |
[2] |
M. E. Davis. Glioblastoma: Overview of disease and treatment, Clin. J. Oncol. Nurs., 20 (2016), S2–S8. doi: 10.1188/16.CJON.S1.2-8 doi: 10.1188/16.CJON.S1.2-8
![]() |
[3] |
K. M. Walsh, H. Ohgaki, M. R. Wrensch, Chapter 1-Epidemiology, Handb. Clin. Neurol., 134 (2016), 3–18. doi: 10.1016/B978-0-12-802997-8.00001-3 doi: 10.1016/B978-0-12-802997-8.00001-3
![]() |
[4] |
C. Ke, K. Tran, Y. Chen, A. T. Di Donato, L. Yu, Y. Hu, et al., Linking differential radiation responses to glioma heterogeneity, Oncotarget, 5 (2014), 1657–1665. doi: 10.18632/oncotarget.1823 doi: 10.18632/oncotarget.1823
![]() |
[5] |
D. A. Jaffray, Image-guided radiotherapy: From current concept to future perspectives, Nat. Rev. Clin. Oncol., 9 (2012), 688–699. doi: 10.1038/nrclinonc.2012.194 doi: 10.1038/nrclinonc.2012.194
![]() |
[6] |
B. Wang, X. Zou, J. Zhu, Data assimilation and its applications, Proc. Natl. Acad. Sci. U.S.A., 97 (2000), 11143–11144. doi: 10.1073/pnas.97.21.11143 doi: 10.1073/pnas.97.21.11143
![]() |
[7] |
W. A. Lahoz, P. Schneider, Data assimilation: Making sense of earth observation, Front. Environ. Sci., 2 (2014), 16. doi: 10.3389/fenvs.2014.00016 doi: 10.3389/fenvs.2014.00016
![]() |
[8] |
M. Ghil, P. Malanotte-Rizzoli, Data assimilation in meteorology and oceanography, Adv. Geophys., 33 (1991), 141–266. doi: 10.1016/S0065-2687(08)60442-2 doi: 10.1016/S0065-2687(08)60442-2
![]() |
[9] | S. M. Bentzen, C. M. Joiner, The linear-quadratic approach in clinical practice, in Basic Clinical Radiobiology, CRC Press, (2009), 112–124. doi: 10.1201/9780429490606-10 |
[10] |
D. A. Hormuth, A. M. Jarrett, E. A. B. F. Lima, M. T. McKenna, D. T. Fuentes, T. E. Yankeelov, Mechanism-based modeling of tumor growth and treatment response constrained by multiparametric imaging data, JCO Clin. Cancer Inf., 3 (2019), 1–10. doi: 10.1200/CCI.18.00055 doi: 10.1200/CCI.18.00055
![]() |
[11] |
D. A. Hormuth, M. Farhat, C. Christenson, B. Curl, C. Chad Quarles, C. Chung, et al., Opportunities for improving brain cancer treatment outcomes through imaging-based mathematical modeling of the delivery of radiotherapy and immunotherapy, Adv. Drug Deliv. Rev., 187 (2022), 114367. doi: 10.1016/j.addr.2022.114367 doi: 10.1016/j.addr.2022.114367
![]() |
[12] |
D. A. Hormuth, C. M. Phillips, C. Wu, E. A. B. F. Lima, G. Lorenzo, P. K. Jha, et al., Biologically-based mathematical modeling of tumor vasculature and angiogenesis via time-resolved imaging data, Cancers, 13 (2021), 3008. doi: 10.3390/cancers13123008 doi: 10.3390/cancers13123008
![]() |
[13] |
A. S. Kazerouni, M. Gadde, A. Gardner, D. A. Hormuth, A. M. Jarrett, K. E. Johnson, et al., Integrating quantitative assays with biologically based mathematical modeling for predictive oncology, Iscience, 23 (2020), 101807. doi: 10.1016/j.isci.2020.101807 doi: 10.1016/j.isci.2020.101807
![]() |
[14] |
R. C. Rockne, A. D. Trister, J. Jacobs, A. J. Hawkins-Daarud, M. L. Neal, K. Hendrickson, et al., A patient-specific computational model of hypoxia-modulated radiation resistance in glioblastoma using 18F-FMISO-PET, J. R. Soc. Interface., 12 (2015), 20141174. doi: 10.1098/rsif.2014.1174 doi: 10.1098/rsif.2014.1174
![]() |
[15] |
D. A. Hormuth, J. A. Weis, S. L. Barnes, M. I. Miga, V. Quaranta, T. E. Yankeelov, Biophysical modeling of in vivo glioma response after whole-brain radiation therapy in a murine model of brain cancer, Int. J. Radiat. Oncol. Biol. Phys., 100 (2018), 1270–1279. doi: 10.1016/j.ijrobp.2017.12.004 doi: 10.1016/j.ijrobp.2017.12.004
![]() |
[16] |
J. Liu, D. A. Hormuth, T. Davis, J. Yang, M. T. McKenna, A. M. Jarrett, et al., A time-resolved experimental-mathematical model for predicting the response of glioma cells to single-dose radiation therapy, Integr. Biol., 13 (2021), 167–183. doi: 10.1093/intbio/zyab010 doi: 10.1093/intbio/zyab010
![]() |
[17] |
S. Brüningk, G. Powathil, P. Ziegenhein, J. Ijaz, I. Rivens, S. Nill, et al., Combining radiation with hyperthermia: A multiscale model informed by in vitro experiments, J. R. Soc. Interface, 15 (2018), 20170681. doi: 10.1098/rsif.2017.0681 doi: 10.1098/rsif.2017.0681
![]() |
[18] |
E. J. Kostelich, Y. Kuang, J. M. McDaniel, N. Z. Moore, N. L. Martirosyan, M. C. Preul, Accurate state estimation from uncertain data and models: an application of data assimilation to mathematical models of human brain tumors, Biol. Direct., 6 (2011), 64. doi: 10.1186/1745-6150-6-64 doi: 10.1186/1745-6150-6-64
![]() |
[19] |
M. U. Zahid, N. Mohsin, A. S. R. Mohamed, J. J. Caudell, L. B. Harrison, C. D. Fuller, et al., Forecasting individual patient response to radiation therapy in head and neck cancer with a dynamic carrying capacity model, Int. J. Radiat. Oncol. Biol. Phys., 111 (2021), 693–704. doi: 10.1016/j.ijrobp.2021.05.132 doi: 10.1016/j.ijrobp.2021.05.132
![]() |
[20] | J. Liu, D. A. Hormuth, J. Yang, T. E. Yankeelov, A multi-compartment model of glioma response to fractionated radiation therapy parameterized via time-resolved microscopy data, Front. Oncol., 12 (2022), 811415. https://www.frontiersin.org/article/10.3389/fonc.2022.811415 |
[21] |
Z. Neufeld, W. von Witt, D. Lakatos, J. Wang, B. Hegedus, A. Czirok, The role of Allee effect in modelling post resection recurrence of glioblastoma, PLoS Comput. Biol., 13 (2017), e1005818. doi: 10.1371/journal.pcbi.1005818 doi: 10.1371/journal.pcbi.1005818
![]() |
[22] |
R. Huang, P. K. Zhou, DNA damage repair: historical perspectives, mechanistic pathways and clinical translation for targeted cancer therapy, Signal Transduction Targeted Ther., 6 (2021), 254. doi: 10.1038/s41392-021-00648-7 doi: 10.1038/s41392-021-00648-7
![]() |
[23] |
D. R. Green, Apoptotic pathways: Ten minutes to dead, Cell, 121 (2005), 671–674. doi: 10.1016/j.cell.2005.05.019 doi: 10.1016/j.cell.2005.05.019
![]() |
[24] |
J. A. Nickoloff, N. Sharma, L. Taylor, Clustered DNA double-strand breaks: Biological effects and relevance to cancer radiotherapy, Genes, 11 (2020), 99. doi: 10.3390/genes11010099 doi: 10.3390/genes11010099
![]() |
[25] | D. S. Chang, F. D. Lasley, I. J. Das, M. S. Mendonca, J. R. Dynlacht, Cell death and survival assays, in Basic Radiotherapy Physics and Biology (eds. D. S. Chang, F. D. Lasley, I. J. Das, M. S. Mendonca and J. R. Dynlacht), Springer, Cham, (2014), 211–219. doi: 10.1007/978-3-319-06841-1_20 |
[26] |
H. Vakifahmetoglu, M. Olsson, B. Zhivotovsky, Death through a tragedy: mitotic catastrophe, Cell Death Differ., 15 (2008), 1153–1162. doi: 10.1038/cdd.2008.47 doi: 10.1038/cdd.2008.47
![]() |
[27] |
Z. Chen, K. Cao, Y. Xia, Y. Li, Y. Hou, L. Wang, et al., Cellular senescence in ionizing radiation (Review), Oncol. Rep., 42 (2019), 883–894. doi: 10.3892/or.2019.7209 doi: 10.3892/or.2019.7209
![]() |
[28] | B. Wang, Analyzing cell cycle checkpoints in response to ionizing radiation in mammalian cells, in Cell Cycle Control (eds. E. Noguchi and M. Gadaleta), Humana Press, 1170 (2014), 313–320. doi: 10.1007/978-1-4939-0888-2_15 |
[29] | L. Lin, L. D. Torbeck, Coefficient of accuracy and concordance correlation coefficient: New statistics for methods comparison, PDA J. Pharm. Sci. Technol., 52 (1998), 55–59. |
[30] |
S. J. McMahon, The linear quadratic model: Usage, interpretation and challenges, Phys. Med. Biol., 64 (2018), 01TR01. doi: 10.1088/1361-6560/aaf26a doi: 10.1088/1361-6560/aaf26a
![]() |
[31] |
D. J. Brenner, The linear-quadratic model is an appropriate methodology for determining isoeffective doses at large doses per fraction, Semin. Radiat. Oncol., 18 (2008), 234–239. doi: 10.1016/j.semradonc.2008.04.004 doi: 10.1016/j.semradonc.2008.04.004
![]() |
[32] | B. G. Wouters, Cell death after irradiation: How, when and why cells die, in Basic Clinical Radiobiology, CRC Press, (2009), 27–40. doi: 10.1201/b13224-4 |
[33] |
M. Kuznetsov, J. Clairambault, V. Volpert, Improving cancer treatments via dynamical biophysical models, Phys. Life Rev., 39 (2021), 1–48. doi: 10.1016/j.plrev.2021.10.001 doi: 10.1016/j.plrev.2021.10.001
![]() |
[34] |
H. Enderling, J. C. L. Alfonso, E. Moros, J. J. Caudell, L. B. Harrison, Integrating mathematical modeling into the roadmap for personalized adaptive radiation therapy, Trends in Cancer, 5 (2019), 467–474. doi: 10.1016/j.trecan.2019.06.006 doi: 10.1016/j.trecan.2019.06.006
![]() |
[35] |
J. Yang, J. Virostko, D. A. H. Ii, J. Liu, A. Brock, J. Kowalski, et al., An experimental-mathematical approach to predict tumor cell growth as a function of glucose availability in breast cancer cell lines, PLoS One, 16 (2021), e0240765. doi: 10.1371/journal.pone.0240765 doi: 10.1371/journal.pone.0240765
![]() |
[36] |
K. E. Johnson, G. R. Howard, D. Morgan, E. A. Brenner, A. L. Gardner, R. E. Durrett, et al., Integrating transcriptomics and bulk time course data into a mathematical framework to describe and predict therapeutic resistance in cancer, Phys. Biol., 18 (2020), 016001. doi: 10.1088/1478-3975/abb09c doi: 10.1088/1478-3975/abb09c
![]() |
[37] | I. M. Navon, Data assimilation for numerical weather prediction: A review, in Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications (eds. S. K. Park and L. Xu), Springer, (2009), 21–65. doi: 10.1007/978-3-540-71056-1_2 |
[38] |
J. S. Liu, R. Chen, Sequential monte carlo methods for dynamic systems, J. Am. Stat. Assoc., 93 (1998), 1032–1044. doi: 10.1080/01621459.1998.10473765 doi: 10.1080/01621459.1998.10473765
![]() |
[39] |
P. L. Houtekamer, F. Zhang, Review of the ensemble kalman filter for atmospheric data assimilation, Mon. Weather Rev., 144 (2016), 4489–4532. doi: 10.1175/MWR-D-15-0440.1 doi: 10.1175/MWR-D-15-0440.1
![]() |
[40] |
L. J. Forker, A. Choudhury, A. E. Kiltie, Biomarkers of tumour radiosensitivity and predicting benefit from radiotherapy, Clin. Oncol., 27 (2015), 561–569. doi: 10.1016/j.clon.2015.06.002 doi: 10.1016/j.clon.2015.06.002
![]() |
[41] |
J. Herrmann, M. Babic, M. Tölle, K. U. Eckardt, M. van der Giet, M. Schuchardt, A novel protocol for detection of senescence and calcification markers by fluorescence microscopy, Int. J. Mol. Sci., 21 (2020), 3475. doi: 10.3390/ijms21103475 doi: 10.3390/ijms21103475
![]() |
[42] |
D. A. Hormuth, A. M. Jarrett, G. Lorenzo, E. A. B. F. Lima, C. Wu, C. Chung, et al., Math, magnets, and medicine: enabling personalized oncology, Expert Rev. Precis. Med. Drug Dev., 6 (2021), 79–81. doi: 10.1080/23808993.2021.1878023 doi: 10.1080/23808993.2021.1878023
![]() |
[43] |
D. A. Hormuth, A. M. Jarrett, X. Feng, T. E. Yankeelov, Calibrating a predictive model of tumor growth and angiogenesis with quantitative MRI, Ann. Biomed. Eng., 47 (2019), 1539–1551. doi: 10.1007/s10439-019-02262-9 doi: 10.1007/s10439-019-02262-9
![]() |
[44] |
D. A. Hormuth, K. A. Al Feghali, A. M. Elliott, T. E. Yankeelov, C. Chung, Image-based personalization of computational models for predicting response of high-grade glioma to chemoradiation, Sci. Rep., 11 (2021), 8520. doi: 10.1038/s41598-021-87887-4 doi: 10.1038/s41598-021-87887-4
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Parameter | Meaning | Value |
The number of subproblems | 100 | |
The number of neighbors | 20 | |
| The maximum of generations | 300 |
| The probability of crossover | 0.8 |
| The differential multiplier | 0.5 |
| The probability of mutation | 0.2 |
| The probability of mutation selection | 0.85 |
Methods | | | |
MOLRSMD | 0.0100 | 0.1986 | 0 |
ADM | 0.0299 | 0.1331 | 1.9135e-17 |
ALM | 0.0148 | 0.0661 | 2.2412e-10 |
Parameter | Meaning | Value |
The number of subproblems | 100 | |
The number of neighbors | 20 | |
| The maximum of generations | 300 |
| The probability of crossover | 0.8 |
| The differential multiplier | 0.5 |
| The probability of mutation | 0.2 |
| The probability of mutation selection | 0.85 |
Methods | | | |
MOLRSMD | 0.0100 | 0.1986 | 0 |
ADM | 0.0299 | 0.1331 | 1.9135e-17 |
ALM | 0.0148 | 0.0661 | 2.2412e-10 |