In the realm of medical imaging, the precise segmentation and classification of gliomas represent fundamental challenges with profound clinical implications. Leveraging the BraTS 2018 dataset as a standard benchmark, this study delves into the potential of advanced deep learning models for addressing these challenges. We propose a novel approach that integrates a customized U-Net for segmentation and VGG-16 for classification. The U-Net, with its tailored encoder-decoder pathways, accurately identifies glioma regions, thus improving tumor localization. The fine-tuned VGG-16, featuring a customized output layer, precisely differentiates between low-grade and high-grade gliomas. To ensure consistency in data pre-processing, a standardized methodology involving gamma correction, data augmentation, and normalization is introduced. This novel integration surpasses existing methods, offering significantly improved glioma diagnosis, validated by high segmentation dice scores (WT: 0.96, TC: 0.92, ET: 0.89), and a remarkable overall classification accuracy of 97.89%. The experimental findings underscore the potential of integrating deep learning-based methodologies for tumor segmentation and classification in enhancing glioma diagnosis and formulating subsequent treatment strategies.
Citation: Sonam Saluja, Munesh Chandra Trivedi, Shiv S. Sarangdevot. Advancing glioma diagnosis: Integrating custom U-Net and VGG-16 for improved grading in MR imaging[J]. Mathematical Biosciences and Engineering, 2024, 21(3): 4328-4350. doi: 10.3934/mbe.2024191
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In the realm of medical imaging, the precise segmentation and classification of gliomas represent fundamental challenges with profound clinical implications. Leveraging the BraTS 2018 dataset as a standard benchmark, this study delves into the potential of advanced deep learning models for addressing these challenges. We propose a novel approach that integrates a customized U-Net for segmentation and VGG-16 for classification. The U-Net, with its tailored encoder-decoder pathways, accurately identifies glioma regions, thus improving tumor localization. The fine-tuned VGG-16, featuring a customized output layer, precisely differentiates between low-grade and high-grade gliomas. To ensure consistency in data pre-processing, a standardized methodology involving gamma correction, data augmentation, and normalization is introduced. This novel integration surpasses existing methods, offering significantly improved glioma diagnosis, validated by high segmentation dice scores (WT: 0.96, TC: 0.92, ET: 0.89), and a remarkable overall classification accuracy of 97.89%. The experimental findings underscore the potential of integrating deep learning-based methodologies for tumor segmentation and classification in enhancing glioma diagnosis and formulating subsequent treatment strategies.
In [1] (see [2] for type A), the authors introduced cluster categories which were associated to finite dimensional hereditary algebras. It is well known that cluster-tilting theory gives a way to construct abelian categories from some triangulated and exact categories.
Recently, Nakaoka and Palu introduced extriangulated categories in [3], which are a simultaneous generalization of exact categories and triangulated categories, see also [4,5,6]. Subcategories of an extriangulated category which are closed under extension are also extriangulated categories. However, there exist some other examples of extriangulated categories which are neither exact nor triangulated, see [6,7,8].
When T is a cluster tilting subcategory, the authors Yang, Zhou and Zhu [9, Definition 3.1] introduced the notions of T[1]-cluster tilting subcategories (also called ghost cluster tilting subcategories) and weak T[1]-cluster tilting subcategories in a triangulated category C, which are generalizations of cluster tilting subcategories. In these works, the authors investigated the relationship between C and modT via the restricted Yoneda functor G more closely. More precisely, they gave a bijection between the class of T[1]-cluster tilting subcategories of C and the class of support τ-tilting pairs of modT, see [9, Theorems 4.3 and 4.4].
Inspired by Yang, Zhou and Zhu [9] and Liu and Zhou [10], we introduce the notion of relative cluster tilting subcategories in an extriangulated category B. More importantly, we want to investigate the relationship between relative cluster tilting subcategories and some important subcategories of modΩ(T)_ (see Theorem 3.9 and Corollary 3.10), which generalizes and improves the work by Yang, Zhou and Zhu [9] and Liu and Zhou [10].
It is worth noting that the proof idea of our main results in this manuscript is similar to that in [9, Theorems 4.3 and 4.4], however, the generalization is nontrivial and we give a new proof technique.
Throughout the paper, let B denote an additive category. The subcategories considered are full additive subcategories which are closed under isomorphisms. Let [X](A,B) denote the subgroup of HomB(A,B) consisting of morphisms which factor through objects in a subcategory X. The quotient category B/[X] of B by a subcategory X is the category with the same objects as B and the space of morphisms from A to B is the quotient of group of morphisms from A to B in B by the subgroup consisting of morphisms factor through objects in X. We use Ab to denote the category of abelian groups.
In the following, we recall the definition and some properties of extriangulated categories from [4], [11] and [3].
Suppose there exists a biadditive functor E:Bop×B→Ab. Let A,C∈B be two objects, an element δ∈E(C,A) is called an E-extension. Zero element in E(C,A) is called the split E-extension.
Let s be a correspondence, which associates any E-extension δ∈E(C,A) to an equivalence class s(δ)=[Ax→By→C]. Moreover, if s satisfies the conditions in [3, Definition 2.9], we call it a realization of E.
Definition 2.1. [3, Definition 2.12] A triplet (B,E,s) is called an externally triangulated category, or for short, extriangulated category if
(ET1) E:Bop×B→Ab is a biadditive functor.
(ET2) s is an additive realization of E.
(ET3) For a pair of E-extensions δ∈E(C,A) and δ′∈E(C′,A′), realized as s(δ)=[Ax→By→C] and s(δ′)=[A′x′→B′y′→C′]. If there exists a commutative square,
![]() |
then there exists a morphism c:C→C′ which makes the above diagram commutative.
(ET3)op Dual of (ET3).
(ET4) Let δ and δ′ be two E-extensions realized by Af→Bf′→D and Bg→Cg′→F, respectively. Then there exist an object E∈B, and a commutative diagram
![]() |
and an E-extension δ′′ realized by Ah→Ch′→E, which satisfy the following compatibilities:
(i). Dd→Ee→F realizes E(F,f′)(δ′),
(ii). E(d,A)(δ′′)=δ,
(iii). E(E,f)(δ′′)=E(e,B)(δ′).
(ET4op) Dual of (ET4).
Let B be an extriangulated category, we recall some notations from [3,6].
● We call a sequence Xx→Yy→Z a conflation if it realizes some E-extension δ∈E(Z,X), where the morphism x is called an inflation, the morphism y is called an deflation and Xx→Yy→Zδ⇢ is called an E-triangle.
●When Xx→Yy→Zδ⇢ is an E-triangle, X is called the CoCone of the deflation y, and denote it by CoCone(y); C is called the Cone of the inflation x, and denote it by Cone(x).
Remark 2.2. 1) Both inflations and deflations are closed under composition.
2) We call a subcategory T extension-closed if for any E-triangle Xx→Yy→Zδ⇢ with X, Z∈T, then Y∈T.
Denote I by the subcategory of all injective objects of B and P by the subcategory of all projective objects.
In an extriangulated category having enough projectives and injectives, Liu and Nakaoka [4] defined the higher extension groups as
Ei+1(X,Y)=E(Ωi(X),Y)=E(X,Σi(Y)) for i≥0. |
By [3, Corollary 3.5], there exists a useful lemma.
Lemma 2.3. For a pair of E-triangles Ll→Mm→N⇢ and Dd→Ee→F⇢. If there is a commutative diagram
![]() |
f factors through l if and only if h factors through e.
In this section, B is always an extriangulated category and T is always a cluster tilting subcategory [6, Definition 2.10].
Let A, B∈B be two objects, denote by [¯T](A,ΣB) the subset of B(A,ΣB) such that f∈[¯T](A,ΣB) if we have f: A→T→ΣB where T∈T and the following commutative diagram
![]() |
where I is an injective object of B [10, Definition 3.2].
Let M and N be two subcategories of B. The notation [¯T](M,Σ(N))=[T](M,Σ(N)) will mean that [¯T](M,ΣN)=[T](M,ΣN) for every object M∈M and N∈N.
Now, we give the definition of T-cluster tilting subcategories.
Definition 3.1 Let X be a subcategory of B.
1) [11, Definition 2.14] X is called T-rigid if [¯T](X,ΣX)=[T](X,ΣX);
2) X is called T-cluster tilting if X is strongly functorially finite in B and X={M∈C∣[¯T](X,ΣM)=[T](X,ΣM) and [¯T](M,ΣX)=[T](M,ΣX)}.
Remark 3.2. 1) Rigid subcategories are always T-rigid by [6, Definition 2.10];
2) T-cluster tilting subcategories are always T-rigid;
3) T-cluster tilting subcategories always contain the class of projective objects P and injective objects I.
Remark 3.3. Since T is a cluster tilting subcategory, ∀X∈B, there exists a commutative diagram by [6, Remark 2.11] and Definition 2.1((ET4)op), where T1, T2∈T and h is a left T-approximation of X:
![]() |
Hence ∀X∈B, there always exists an E-triangle
Ω(T1)fX→Ω(T2)→X⇢ with Ti∈T. |
By Remark 3.2(3), P⊆T and B=CoCone(T,T) by [6, Remark 2.11(1),(2)]. Following from [4, Theorem 3.2], B_=B/T is an abelian category. ∀f∈B(A,C), denote by f_ the image of f under the natural quotient functor B→B_.
Let Ω(T)=CoCone(P,T), then Ω(T)_ is the subcategory consisting of projective objects of B_ by [4, Theorem 4.10]. Moreover, modΩ(T)_ denotes the category of coherent functors over the category of Ω(T)_ by [4, Fact 4.13].
Let G: B→modΩ(T)_, M↦HomB_(−,M)∣Ω(T)_ be the restricted Yoneda functor. Then G is homological, i.e., any E-triangle X→Y→Z⇢ in B yields an exact sequence G(X)→G(Y)→G(Z) in modΩ(T)_. Similar to [9, Theorem 2.8], we obtain a lemma:
Lemma 3.4. Denote proj(modΩ(T)_) the subcategory of projective objects in modΩ(T)_. Then
1) G induces an equivalence Ω(T)∼→proj(modΩ(T)_).
2) For N∈modΩ(T)_, there exists a natural isomorphism
HommodΩ(T)_(G(Ω(T)),N)≃N(Ω(T)). |
In the following, we investigate the relationship between B and modΩ(T)_ via G more closely.
Lemma 3.5. Let X be any subcategory of B. Then
1) any object X∈X, there is a projective presentation in mod Ω(T)_
PG(X)1πG(X)→PG(X)0→G(X)→0. |
2) X is a T-rigid subcategory if and only if the class {πG(X)∣X∈X} has property ((S) [9, Definition 2.7(1)]).
Proof. 1). By Remark 3.3, there exists an E-triangle:
Ω(T1)fX→Ω(T0)→X⇢ |
When we apply the functor G to it, there exists an exact sequence G(Ω(T1))→G(Ω(T0))→G(X)→0. By Lemma 3.4(1), G(Ω(Ti)) is projective in mod Ω(T)_. So the above exact sequence is the desired projective presentation.
2). For any X0∈X, using the similar proof to [9, Lemma 4.1], we get the following commutative diagram
![]() |
where α=HommodΩT_(πG(X),G(X0)). By Lemma 3.4(2), both the left and right vertical maps are isomorphisms. Hence the set {πG(X)∣X∈X} has property ((S) iff α is epic iff HomB_(fX,X0) is epic iff X is a T-rigid subcategory by [10, Lemma 3.6].
Lemma 3.6. Let X be a T-rigid subcategory and T1 a subcategory of T. Then X∨T1 is a T-rigid subcategory iff E(T1,X)=0.
Proof. For any M∈X∨T1, then M=X⊕T1 for X∈X and T1∈T1. Let h: X→T be a left T-approximation of X and y: T1→Σ(X′) for X′∈X any morphism. Then there exists the following commutative diagram
![]() |
with P1∈P, f=(h001) and β=(i000i1).
When X∨T1 is a T-rigid subcategory, we can get a morphism g: X⊕T1→Σ(X′)⊕Σ(T′1) such that βg=(10)y(0 1)f. i.e., ∃b: T1→I such that y=i0b. So E(T1,X′)=0 and then E(T1,X)=0.
Let γ=(r11r12r21r22): T⊕T1→Σ(X′)⊕Σ(T′1) be a morphism. As X is T-rigid, r11h: X→Σ(X′) factors through i0. Since E(T,X)=0, r12: T1→Σ(X′) factors through i0. As T is rigid, the morphism r21h: X→T→Σ(T′1) factors through i1, and the morphism r22: T1→Σ(T′1) factors through i1. So the morphism γf can factor through β=(i000i1). Therefore X∨T1 is an T-rigid subcategory.
For the definition of τ-rigid pair in an additive category, we refer the readers to see [9, Definition 2.7].
Lemma 3.7. Let U be a class of T-rigid subcategories and V a class of τ-rigid pairs of modΩ(T)_. Then there exists a bijection φ: U→V, given by : X↦(G(X),Ω(T)∩Ω(X)).
Proof. Let X be T-rigid. By Lemma 3.5, G(X) is a τ-rigid subcategory of mod Ω(T)_.
Let Y∈Ω(T)∩Ω(X), then there exists X0∈X such that Y=Ω(X0). Consider the E-triangle Ω(X0)→P→X0⇢ with P∈P. ∀X∈X, applying HomB(−,X) yields an exact sequence HomB(P,X)→HomB(Ω(X0),X)→E(X0,X)→0. Hence in B_=B/T, HomB_(Ω(X0),X)≅E(X0,X).
By Remark 3.3, for X0, there is an E-triangle Ω(T1)→Ω(T2)→X0⇢ with T1, T2∈T. Applying HomB_(−,X), we obtain an exact sequence HomB_(Ω(T2),X)→HomB_(Ω(T1),X)→E(X0,X)→E(Ω(T2),X). By [10, Lemma 3.6], HomB_(Ω(T2),X)→HomB_(Ω(T1),X) is epic. Moreover, Ω(T2)_ is projective in B_ by [4, Proposition 4.8]. So E(Ω(T2),X)=0. Thus E(X0,X)=0. Hence ∀X∈X,
G(X)(Y)=HomB_(Ω(X0),X)=0. |
So (G(X),Ω(T)∩Ω(X)) is a τ-rigid pairs of modΩ(T)_.
We will show φ is a surjective map.
Let (N,σ) be a τ-rigid pair of modΩ(T)_. ∀N∈N, consider the projective presentation
P1πN→P0→N→0 |
such that the class {πN|N∈N} has Property (S). By Lemma 3.4, there exists a unique morphism fN: Ω(T1)→Ω(T0) in Ω(T)_ satisfying G(fN)=πN and G(Cone(fN))≅N. Following from Lemma 3.5, X1:= {cone(fN)∣N∈N} is a T-rigid subcategory.
Let X=X1∨Y, where Y={T∈T∣Ω(T)∈σ}. For any T0∈Y, there is an E-triangle Ω(T0)→P→T0⇢ with P∈P. For any Cone(fN)∈X1, applying HomB_(−,Cone(fN)), yields an exact sequence HomB_(Ω(T0),Cone(fN))→E(T0,Cone(fN))→E(P,Cone(fN))=0. Since (N,σ) is a τ-rigid pair, HomB_(Ω(T0),Cone(fN))=G(Cone(fM))(Ω(T0))=0. So E(T0,Cone(fN))=0. Due to Lemma 3.6, X=X1∨Y is T-rigid. Since Y⊆T, we get G(Y)=HomB_(−,T)∣Ω(T)=0 by [4, Lemma 4.7]. So G(X)=G(X1)=N.
It is straightforward to check that Ω(T)∩Ω(X1)=0. Let X∈Ω(T)∩Ω(X), then X∈Ω(T) and X∈Ω(X)=Ω(X1)∨σ. So we can assume that X=Ω(X1)⊕E, where E∈σ. Then Ω(X1)⊕E∈Ω(T). Since E∈Ω(T), we get Ω(X1)∈Ω(T)∩Ω(X1)=0. So Ω(T)∩Ω(X)⊆σ. Clearly, σ⊆Ω(T). Moreover, σ⊆Ω(X). So σ⊆Ω(T)∩Ω(X). Hence Ω(T)∩Ω(X)=σ. Therefore φ is surjective.
Lastly, φ is injective by the similar proof method to [9, Proposition 4.2].
Therefore φ is bijective.
Lemma 3.8. Let T be a rigid subcategory and Aa→B→Cδ⇢ an E-triangle satisfying [¯T](C,Σ(A))=[T](C,Σ(A)). If there exist an E-extension γ∈E(T,A) and a morphism t: C→T with T∈T such that t∗γ=δ, then the E-triangle Aa→B→Cδ⇢ splits.
Proof. Applying HomB(T,−) to the E-triangle A→Ii→Σ(A)α⇢ with I∈I, yields an exact sequence HomB(T,A)→E(T,X)→E(T,I)=0. So there is a morphism d∈HomB(T,Σ(A)) such that γ=d∗α. So δ=t∗γ=t∗d∗α=(dt)∗α. So we have a diagram which is commutative:
![]() |
Since [¯T](C,Σ(A))=[T](C,Σ(A)) and dt∈[T](C,Σ(A)), dt can factor through i. So 1A can factor through a and the result follows.
Now, we will show our main theorem, which explains the relation between T-cluster tilting subcategories and support τ-tilting pairs of modΩ(T)_.
The subcategory X is called a preimage of Y by G if G(X)=Y.
Theorem 3.9. There is a correspondence between the class of T-cluster tilting subcategories of B and the class of support τ-tilting pairs of modΩ(T)_ such that the class of preimages of support τ-tilting subcategories is contravariantly finite in B.
Proof. Let φ be the bijective map, such that X↦(G(X),Ω(T∩Ω(X))), where G is the restricted Yoneda functor defined in the argument above Lemma 3.4.
1). The map φ is well-defined.
If Xis T-cluster tilting, then X is T-rigid. So φ(X) is a τ-rigid pair of modΩ(T)_ by Lemma 3.7. Therefore Ω(T)∩Ω(X)⊆KerG(X). Assume Ω(T0)∈Ω(T) is an object of KerG(X). Then HomB_(Ω(T0),X)=0. Applying HomB_(−,X) with X∈X to Ω(T0)→P→T0⇢ with P∈P, yields an exact sequence
HomB_(P,X)→HomB_(Ω(T),X)→E(T0,X)→0. |
Hence we get E(T0,X)≅HomB_(Ω(T0),X)=0.
Applying HomB(T0,−) to X→I→Σ(X)⇢, we obtain
(3.1) [¯T](T0,Σ(X))=[T](T0,Σ(X)). |
For any ba: Xa→Rb→Σ(T0) with R∈T, as T is rigid, we get a commutative diagram:
![]() |
Hence we get (3.2)[¯T](X,Σ(T0))=[T](X,Σ(T0)).
By the equalities (3.1) and (3.2) and X being a T-rigid subcategory, we obtain
[¯T](X,Σ(X⊕T0))=[T](X,Σ(X⊕T0)) and [¯T](X⊕T0,Σ(X))=[T](X⊕T0,Σ(X)). |
As X is T-cluster tilting, we get X⊕T0∈X. So T0∈X. And thus Ω(T0)∈Ω(T)∩Ω(X). Hence KerG(X)=Ω(T)∩Ω(X).
Since X is functorially finte, similar to [6, Lemma 4.1(2)], ∀Ω(T)∈Ω(T), we can find an E-triangle Ω(T)f→X1→X2⇢, where X1, X2∈X and f is a left X-approximation. Applying G, yields an exact sequence
G(Ω(R))G(f)→G(X1)→G(X2)→0. |
Thus we get a diagram which is commutative, where HomB_(f,X) is surjective.
![]() |
By Lemma 3.4, the morphism ∘G(f) is surjective. So G(f) is a left G(X)-approximation and (G(X),Ω(T)∩Ω(X)) is a support τ-tilting pair of modΩ(T)_ by [3, Definition 2.12].
2). φ is epic.
Assume (N,σ) is a support τ-tilting pair of modΩ(T)_. By Lemma 3.7, there is a T-rigid subcategory X satisfies G(X)=N. So ∀Ω(T)∈Ω((T)), there is an exact sequence G(Ω(T))α→G(X3)→G(X4)→0, such that X3, X4∈X and α is a left G(X)-approximation. By Yoneda's lemma, we have a unique morphism in modΩ((T))_:
β: Ω(T)→X3 such that α=G(β) and G(cone(β))≅G(X4). |
Moreover, ∀X∈X, consider the following commutative diagram
![]() |
By Lemma 3.4, G(−) is surjective. So the map HomB_(β,X) is surjective.
Denote Cone(β) by YR and X∨add{YR∣Ω(T)∈Ω(T)} by ˜X.
We claim ˜X is T-rigid.
(I). Assume a: YRa1→T0a2→Σ(X) with T0∈T and X∈X. Consider the following diagram:
![]() |
Since X is T-rigid, ∃f: X3→I such that aγ=if. So there is a morphism g:Ω(T)→X making the upper diagram commutative. Since HomB_(β,X) is surjective, g factors through β. Hence a factors through i, i.e., [¯T](YR,Σ(X))=[T](YR,Σ(X)).
(II). For any morphism b: Xb1→T0b2→Σ(YR) with T0∈T and X∈X. Consider the following diagram:
![]() |
By [3, Lemma 5.9], R→Σ(X3)→Σ(YT)⇢ is an E-triangle. Because T is rigid, b2 factors through γ1. By the fact that X is T-rigid, b=b2b1 can factor through iX. Since γ1iX=iY, we get that b factors through iY. So [¯T](X,Σ(YT))=[T](X,Σ(YT)).
By (I) and (II), we also obtain [¯T](YT,Σ(YT))=[T](YT,Σ(YT)).
Therefore ˜X=X∨add{YT∣Ω(T)∈Ω(T)} is T-rigid.
Let M∈B satisfying [¯T](M,Σ(˜X))=[T](M,Σ(˜X)) and [¯T](˜X,ΣM)=[T](˜X,ΣM). Consider the E-triangle:
Ω(T5)f→Ω(T6)g→M⇢ |
where T5, T6∈T. By the above discussion, there exist two E-triangles:
Ω(T6)u→X6v→Y6⇢ and Ω(T5)u′→X5v′→Y5⇢. |
where X5, X6∈X, u and u′ are left X-approximations of Ω(T6), Ω(T5), respectively. So there exists a diagram of E-triangles which is commutative:
![]() |
We claim that the morphism x=uf is a left X-approximation of Ω(T5). In fact, let X∈X and d: Ω(T5)→X, we can get a commutative diagram of E-triangles:
![]() |
where P∈P. By the assumption, [¯T](M,Σ(X))=[T](M,Σ(X)). So d2h factors through iX. By Lemma 2.3, d factors through f. Thus ∃f1: Ω(T6)→X such that d=f1f. Moreover, u is a left X-approximation of Ω(T6). So ∃u1: X6→X such that f1=u1u. Thus d=f1f=u1uf=u1x. So x=uf is a left X-approximation of Ω(T5).
Hence there is a commutative diagram:
![]() |
By [3, Corollary 3.16], we get an E-triangle X6(yλ)→N⊕X5→Y5x∗δ5⇢
Since u′ is a left X-approximation of Ω(T5), there is also a commutative diagram with P∈P:
![]() |
such that δ5=t∗μ. So x∗δ5=x∗t∗μ=t∗x∗μ. By Lemma 3.8, the E-triangle x∗δ5 splits. So N⊕X5≃X6⊕Y5∈˜X. hence N∈˜X.
Similarly, consider the following commutative diagram with P∈P:
![]() |
and the E-triangle M→N→Yg∗δ6⇢. Then ∃t: Y→T6 such that δ6=t∗δ. Then g∗δ6=g∗t∗δ=t∗(g∗δ). Since [¯T](˜X,ΣM)=[T](˜X,ΣM), the E-triangle g∗δ6 splits by Lemma 3.5 and M is a direct summands of N. Hence M∈˜X.
By the above, we get ˜X is a T-cluster tilting subcategory.
By the definition of YR, G(YR)∈G(X). So G(˜X)≃G(X)≃N. Moreover, σ=Ω(T)∩Ω(X)⊆Ω(T)∩Ω(˜X) and Ω(T)∩Ω(˜X)⊆kerG(X)=σ. So Ω(T)∩Ω(˜X)=σ. Hence φ is surjective.
3). φ is injective following from the proof of Lemma 3.7.
By [4, Proposition 4.8 and Fact 4.13], B_≃modΩ(T)_. So it is easy to get the following corollary by Theorem 3.9:
Corollary 3.10. Let X be a subcategory of B.
1) X is T-rigid iff X_ is τ-rigid subcategory of B_.
2) X is T-cluster tilting iff X_ is support τ-tilting subcategory of B_.
If let H=CoCone(T,T), then H can completely replace B and draw the corresponding conclusion by the proof Lemma 3.7 and Theorem 3.9, which is exactly [12, Theorem 3.8]. If let B is a triangulated category, then Theorem 3.9 is exactly [9, Theorem 4.3].
This research was supported by the National Natural Science Foundation of China (No. 12101344) and Shan Dong Provincial Natural Science Foundation of China (No.ZR2015PA001).
The authors declare they have no conflict of interest.
[1] |
M. L. Goodenberger, R. B. Jenkins, Genetics of adult glioma, Cancer Genet., 205 (2012), 613–621. https://doi.org/10.1016/j.cancergen.2012.10.009 doi: 10.1016/j.cancergen.2012.10.009
![]() |
[2] |
D. N. Louis, A. Perry, G. Reifenberger, A. Deimling, D. Figarella-Branger, W. K. Cavenee, et al., The 2016 World Health Organization Classification of Tumors of the Central Nervous System: A summary, Acta Neuropathol., 131 (2016), 803–820. https://doi.org/10.1007/s00401-016-1545-1 doi: 10.1007/s00401-016-1545-1
![]() |
[3] |
D. N. Louis, A. Perry, P. Wesseling, D. J. Brat, I. A. Cree, D. Figarella-Branger, et al., The 2021 WHO Classification of Tumors of the Central Nervous System: A summary, Neuro-Oncol., 23 (2021), 1231–1251.https://doi.org/10.1093/neuonc/noab106 doi: 10.1093/neuonc/noab106
![]() |
[4] |
J. S. Barnholtz-Sloan, Q. T. Ostrom, D. Cote, Epidemiology of brain tumors, Neurol. Clin., 36 (2018), 395–419. https://doi.org/10.1016/j.ncl.2018.04.001 doi: 10.1016/j.ncl.2018.04.001
![]() |
[5] | M. Decuyper, R.V. Holen, Fully automatic binary glioma grading based on Pre-therapy MRI using 3D Convolutional Neural Networks, preprint, arXiv: 1908.01506 |
[6] |
A. Patra, A. Janu, A. Sahu, MR Imaging in neurocritical care, Indian J. Crit. Care Med., 23 (2019), 104–114. https://doi.org/10.5005/jp-journals-10071-23186 doi: 10.5005/jp-journals-10071-23186
![]() |
[7] |
Ö. Polat, C. Güngen, Classification of brain tumors from MR images using deep transfer learning, J. Supercomput., 77 (2021), 7236–7252.https://doi.org/10.1007/s11227-020-03572-9 doi: 10.1007/s11227-020-03572-9
![]() |
[8] |
S. Gore, T. Chougule, J. Jagtap, J. Saini, M. Ingalhalikar, et al., A review of radiomics and deep predictive modeling in glioma characterization, Acad. Radiol., 28 (2021), 1599–1621. https://doi.org/10.1016/j.acra.2020.06.016 doi: 10.1016/j.acra.2020.06.016
![]() |
[9] |
H. Jiang, Z. Diao, Y. Yao, DL techniques for tumor segmentation: A review, J. Supercomput., 78 (2022), 1807–1851. https://doi.org/10.1007/s11227-021-03901-6 doi: 10.1007/s11227-021-03901-6
![]() |
[10] |
S. Waite, J.Scott, B. Gale, T. Fuchs, S. Kolla, D. Reede, Interpretive error in radiology, Am. J. Roentgenol., 208 (2017), 739–749. https://doi.org/10.2214/ajr.16.16963 doi: 10.2214/ajr.16.16963
![]() |
[11] |
R. Ranjbarzadeh, A. B. Kasgari, S. J. Ghoushchi, S. Anari, M. Naseri, M. Bendechache, Brain tumor segmentation based on DL and an attention mechanism using MRI multi-modalities brain images, Sci. Rep., 11 (2021), 10930. https://doi.org/10.1038/s41598-021-90428-8 doi: 10.1038/s41598-021-90428-8
![]() |
[12] |
M.-A. Schulz, B. T. Thomas Yeo, J. T. Vogelstein, J. Mourao-Miranada, J. N. Kather, K. Kording, Different scaling of linear models and deep learning in UKBiobank brain images versus machine-learning datasets, Nat. Commun., 11 (2020). https://doi.org/10.1038/s41467-020-18037-z doi: 10.1038/s41467-020-18037-z
![]() |
[13] |
K. Yasaka, H. Akai, A. Kunimatsu, S. Kiryu, O. Abe, Deep learning with convolutional neural network in radiology, Jpn. J. Radiol., 36 (2018), 257–272. https://doi.org/10.1007/s11604-018-0726-3 doi: 10.1007/s11604-018-0726-3
![]() |
[14] |
S. Fathi, M. Ahmadi, A. Dehnad, Early diagnosis of Alzheimer, Comput. Biol. Med., 146 (2022), 105634. https://doi.org/10.1016/j.compbiomed.2022.105634 doi: 10.1016/j.compbiomed.2022.105634
![]() |
[15] |
H. Özcan, B. G. Emiroglu, H. Sabuncuoğlu, S. Özdoğan, A. Soyer, T. Saygı, A comparative study for glioma classification using deep convolutional neural networks, Math. Biosci. Eng., 18 (2021), 1550–1572. https://doi.org/10.3934/mbe.2021080 doi: 10.3934/mbe.2021080
![]() |
[16] |
A. Krizhevsky, I. Sutskever, G. E. Hinton, ImageNet classification with deep convolutional neural networks, Commun. ACM, 60 (2017), 84–90. https://doi.org/10.1145/3065386 doi: 10.1145/3065386
![]() |
[17] | K. Simonyan, A. Zisserman, Very deep convolutional networks for large-scale image recognition, (2014), preprint, arXiv: 1409.1556. |
[18] | H. Dong, G. Yang, F. Liu, Y. Mo, Y. Guo, Automatic brain tumor detection and segmentation using U-Net based fully Convolutional Networks, preprint, arXiv: 1705.03820 |
[19] |
S. Khawaldeh, U. Pervaiz, A. Rafiq, R. S. Alkhwaldeh, Noninvasive grading of glioma tumor using magnetic resonance imaging with Convolutional Neural Networks, Appl. Sci., 8 (2017), 27. https://doi.org/10.3390/app8010027 doi: 10.3390/app8010027
![]() |
[20] |
A. K. Anaraki, M. Ayati, F. Kazemi, Magnetic resonance imaging-based brain tumor grades classification and grading via convolutional neural networks and genetic algorithms, Biocybern. Biomed. Eng., 39 (2019), 63–74. https://doi.org/10.1016/j.bbe.2018.10.004 doi: 10.1016/j.bbe.2018.10.004
![]() |
[21] |
H. Mzoughi, I. Njeh, A. Wali, M. B. Slima, A. B. Hamida, C. Mhiri, et al., Deep multi-scale 3D Convolutional Neural Network (CNN) for MRI Gliomas brain tumor classification, J. Digit. Imaging, 33 (2020), 903–915. https://doi.org/10.1007/s10278-020-00347-9 doi: 10.1007/s10278-020-00347-9
![]() |
[22] |
Y. Zhuge, H. Ning, P. Mathen, J. Y. Cheng, A. V. Krauze, K. Camphausen, et al., Automated glioma grading on conventional MRI images using deep convolutional neural networks, Med. Phys., 47 (2020), 3044–3053. https://doi.org/10.1002/mp.14168 doi: 10.1002/mp.14168
![]() |
[23] |
S. Gutta, J. Acharya, M. S. Shiroishi, D. Hwang, K. S. Nayak, Improved Glioma grading using Deep Convolutional Neural Networks, AJNR Am. J. Neuroradiol., 42 (2020), 233–239. https://doi.org/10.3174/ajnr.a6882 doi: 10.3174/ajnr.a6882
![]() |
[24] |
Z. Lu, Y. Bai, Y. Chen, C. Su, S. Lu, T. Zhan, et al., The classification of gliomas based on a Pyramid dilated convolution resnet model, Pattern Recognit. Lett., 133 (2020), 173–179.https://doi.org/10.1016/j.patrec.2020.03.007 doi: 10.1016/j.patrec.2020.03.007
![]() |
[25] |
M. A. Naser, M. J. Deen, Brain tumor segmentation and grading of lower-grade glioma using deep learning in MRI images, Comput. Biol. Med., 121 (2020), 103758. https://doi.org/10.1016/j.compbiomed.2020.103758 doi: 10.1016/j.compbiomed.2020.103758
![]() |
[26] |
M. Decuyper, S. Bonte, K. Deblaere, R. Van Holen, Automated MRI based pipeline for segmentation and prediction of grade, IDH mutation and 1p19q co-deletion in glioma, Comput. Med. Imaging Graph., 88 (2021), 101831. https://doi.org/10.1016/j.compmedimag.2020.101831 doi: 10.1016/j.compmedimag.2020.101831
![]() |
[27] |
G. S. Tandel, A. Tiwari, O. Kakde, Performance optimisation of deep learning models using majority voting algorithm for brain tumour classification, Comput. Biol. Med., 135 (2021), 104564. https://doi.org/10.1016/j.compbiomed.2021.104564 doi: 10.1016/j.compbiomed.2021.104564
![]() |
[28] |
G. S. Tandel, A. Tiwari, O. G. Kakde, Performance enhancement of MRI based brain tumor classification using suitable segmentation method and deep learning-based ensemble algorithm, Biomed. Signal Process. Control., 78 (2022). https://doi.org/10.1016/j.bspc.2022.104018 doi: 10.1016/j.bspc.2022.104018
![]() |
[29] |
S. E. Nassar, I. Yasser, H. M. Amer, M. A. Mohamed, A robust MRI-based brain tumor classification via a hybrid deep learning technique, J. Supercomput., 80 (2023). https://doi.org/10.1007/s11227-023-05549-w doi: 10.1007/s11227-023-05549-w
![]() |
[30] | T.-Y. Hsiao, Y.-C. Chang, C.-T. Chiu, Filter-based deep-compression with global average pooling for Convolutional Networks, in IEEE International Workshop on Signal Processing Systems (SiPS), (2018). https://doi.org/10.1109/sips.2018.8598453 |
[31] |
T. G. Dietterich, Ensemble methods in machine learning, in multiple classifier systems, MCS 2000. Lecture Notes Computer Sci., 1857 (2020). https://doi.org/10.1007/3-540-45014-9_1 doi: 10.1007/3-540-45014-9_1
![]() |
[32] |
B. Menze, A. Jakab, S. Bauer, J. Kalpathy-Cramer, K. Farahani, J. Kirby, et al., The Multimodal Brain Tumor Image Segmentation Benchmark (BRATS), IEEE Trans. Med. Imaging, 34 (2015), 1993–2024. https://doi.org/10.1109/TMI.2014.2377694 doi: 10.1109/TMI.2014.2377694
![]() |
[33] | S. Bakas, M. Reyes, A. Jakab, S. Bauer, M. Rempfler, A. Crimi, et al., Identifying the best machine learning algorithms for brain tumor segmentation, progression assessment, and overall survival prediction in the BRATS challenge, (2018), preprint, arXiv: 1811.02629. |
[34] |
S. Bakas, H. Akbari, A. Sotiras, M. Bilello, M. Rozycki, J. S. Kirby, et al., Advancing the Cancer genome atlas glioma MRI collections with expert segmentation labels and radiomic features, Sci. Data, 4 (2017). https://doi.org/10.1038/sdata.2017.117 doi: 10.1038/sdata.2017.117
![]() |
[35] |
A. Man, S. Anand, Method of multi-region tumour segmentation in brain MRI images using grid-based segmentation and weighted bee swarm optimisation, IET Image Process., 14 (2020), 2901–2910. https://doi.org/10.1049/iet-ipr.2019.1234 doi: 10.1049/iet-ipr.2019.1234
![]() |
[36] |
K. Maharana, S. Mondal B. Nemade, A review: Data pre-processing and data augmentation techniques, Glob. Transit., 3 (2022), 91–99. https://doi.org/10.1016/j.gltp.2022.04.020 doi: 10.1016/j.gltp.2022.04.020
![]() |
[37] |
H. Moradmand, S. M. R. Aghamiri, R. Ghaderi, Impact of image preprocessing methods on reproducibility of radiomic features in multimodal magnetic resonance imaging in glioblastoma, J. Appl. Clin. Med. Phys., 21 (2019), 179–190. https://doi.org/10.1002/acm2.12795 doi: 10.1002/acm2.12795
![]() |
[38] | O. Ronneberger, Invited Talk: U-Net Convolutional Networks for Biomedical Image Segmentation, in Bildverarbeitung für die Medizin 2017, Informatik aktuell, (2017), 3. https://doi.org/10.1007/978-3-662-54345-0_3 |
[39] |
S. Das, M. K. Swain, G. K. Nayak, S. Saxena, S. C. Satpathy, Effect of learning parameters on the performance of U-Net Model in segmentation of Brain tumor, Multimed. Tools Appl., 81 (2021), 34717–34735. https://doi.org/10.1007/s11042-021-11273-5 doi: 10.1007/s11042-021-11273-5
![]() |
[40] |
A. Rusiecki, Trimmed categorical cross-entropy for deep learning with label noise, Electron. Lett., 55 (2019), 319–320. https://doi.org/10.1049/el.2018.7980 doi: 10.1049/el.2018.7980
![]() |
[41] |
H. Seo, M. Bassenne, L. Xing, Closing the gap between deep neural network modeling and biomedical decision-making metrics in segmentation via adaptive loss functions, IEEE Trans. Med. Imaging, 40 (2021), 585–593. https://doi.org/10.1109/tmi.2020.3031913 doi: 10.1109/tmi.2020.3031913
![]() |
[42] |
A. Taha, A. Hanbury, Metrics for evaluating 3D medical image segmentation: Analysis, selection, and tool, BMC Med. Imag., 15 (2015). https://doi.org/10.1186/s12880-015-0068-x doi: 10.1186/s12880-015-0068-x
![]() |
[43] |
A. Tharwat, Classification assessment methods, Appl. Comput. Inform., 17 (2020), 168–192. https://doi.org/10.1016/j.aci.2018.08.003 doi: 10.1016/j.aci.2018.08.003
![]() |
[44] |
C. Huan, M. Wan, Automated segmentation of brain tumor based on improved U-Net with residual units, Multimed. Tools Appl., 81 (2022), 12543–12566. https://doi.org/10.1007/s11042-022-12335-y doi: 10.1007/s11042-022-12335-y
![]() |
[45] | M. Noori, A. Bahri, K. Mohammadi, Attention-guided version of 2D UNet for automatic brain tumor segmentation, in 9th International Conference on Computer and Knowledge Engineering (ICCKE), (2019). https://doi.org/10.1109/iccke48569.2019.8964956 |
[46] | F. Isensee, P. F. Jager, P. M. Full, P. Vollmuth, K. H. Maier-Hein, NnU-Net for brain tumor segmentation, in Brainlesion: Glioma, Multiple Sclerosis, Stroke and Traumatic Brain Injuries, BrainLes 2020, Lecture Notes in Computer Science, 12659. https://doi.org/10.1007/978-3-030-72087-2_11 |
[47] |
W. Ayadi, W. Elhamzi, M. Atri, A deep conventional neural network model for glioma tumor segmentation, Int. J. Imaging Syst., 33 (2023), 1593–1605. https://doi.org/10.1002/ima.22892 doi: 10.1002/ima.22892
![]() |
[48] | Y. Zhang, Y. Han, J. Zhang, MAU-Net: Mixed attention U-Net for MRI brain tumor segmentation, Math Biosci. Eng., 20 (2023), 20510–20527. https://10.3934/mbe.2023907 |
[49] |
M. U. Rehman, S. Cho, J. H. Kim, K. T. Chong, BU-Net: Brain tumor segmentation using modified U-Net architecture, Electronics, 9 (2020), 2203. https://doi.org/10.3390/electronics9122203 doi: 10.3390/electronics9122203
![]() |
[50] |
M. U. Rehman, J. Ryu, I. F. Nizami, K. T. Chong, RAAGR2-Net: A brain tumor segmentation network using parallel processing of multiple spatial frames, Comput. Biol. Med., 152 (2023), 106426. https://doi.org/10.1016/j.compbiomed.2022.106426 doi: 10.1016/j.compbiomed.2022.106426
![]() |
[51] |
J. Linqi, N. Chunyu, L. Jingyang, Glioma classification framework based on SE-ResNeXt network and its optimization, IET Image Process., 16 (2021), 596–605. https://doi.org/10.1049/ipr2.12374 doi: 10.1049/ipr2.12374
![]() |
[52] |
Y. Yang, L-F. Yan, X. Zhang, Y. Han, H-Y. Nan, Y-C. Hu, et al., Glioma grading on Conventional MR Images: A deep learning study with transfer learning, Front. Neurosci., 12 (2018), 804. https://doi.org/10.3389/fnins.2018.00804 doi: 10.3389/fnins.2018.00804
![]() |
[53] |
S. V. Rubio, M. T. Garcia-Ordas, O. García-Olalla Olivera, H. Alaiz-Moretón, M. González-Alonso, J. A. Benítez-Andrades, Survival and grade of the glioma prediction using transfer learning, PeerJ Comput. Sci., 9 (2023). https://doi.org/10.7717/peerj-cs.1723 doi: 10.7717/peerj-cs.1723
![]() |
[54] |
H. E. Hamdaoui, A. Benfares, S. Boujraf, N. E. H. Chaoui, B. Alami, M. Maaroufi, et al., High precision brain tumor classification model based on deep transfer learning and stacking concepts, Indones. J. Electr., 24 (2021), 167–177. https://doi.org/10.11591/ijeecs.v24.i1.pp167-177 doi: 10.11591/ijeecs.v24.i1.pp167-177
![]() |
[55] |
Z. Khazaee, M. Langarizadeh, and M. E. Shiri Ahmadabadi, Developing an artificial intelligence model for tumor grading and classification, based on MRI sequences of human brain gliomas, Int. J. Cancer Manag., 15 (2022). https://doi.org/10.5812/ijcm.120638 doi: 10.5812/ijcm.120638
![]() |
[56] |
K. Dang, T. Vo, L. Ngo, H. Ha, A deep learning framework integrating MRI image preprocessing methods for brain tumor segmentation and classification, IBRO Neurosci. Rep., 13 (2022), 523–532. https://doi.org/10.1016/j.ibneur.2022.10.014 doi: 10.1016/j.ibneur.2022.10.014
![]() |
[57] |
P. C. Tripathi, S. Bag, A computer-aided grading of glioma tumor using deep residual networks fusion, Comput. Methods Programs Biomed., 215 (2022), 106597. https://doi.org/10.1016/j.cmpb.2021.106597 doi: 10.1016/j.cmpb.2021.106597
![]() |
[58] |
A. B. Slama, H. Sahli, Y. Amri, H. Trabelsi, Res-Net-VGG19: Improved tumor segmentation using MR images based on Res-Net architecture and efficient VGG gliomas grading, Appl. Eng. Sci., 16 (2023), 100153. https://doi.org/10.1016/j.apples.2023.100153 doi: 10.1016/j.apples.2023.100153
![]() |
[59] | J. Sivakumar, S. R. Kannan, K. S. Manic, Automated classification of brain tumors into LGG/HGG using concatenated deep and handcrafted features, in Frontiers of Artificial Intelligence in Medical Imaging, (2022). https://doi.org/10.1088/978-0-7503-4012-0ch7 |
[60] | M. M. Mahasin, A. Naba, C. S. Widodo, Y. Yueniwati, Development of a modified UNet-based image segmentation architecture for brain tumor MRI segmentation, in Proceedings of the International Conference of Medical and Life Science (ICoMELISA 2021), (2023), 37–43. https://doi.org/10.2991/978-94-6463-208-8_7 |
[61] |
S. Ambesange, B. Annappa, S. G. Koolagudi, Simulating federated transfer learning for lung segmentation using modified UNet model, Procedia Comput. Sci., 218 (2023), 1485–1496. https://doi.org/10.1016/j.procs.2023.01.127 doi: 10.1016/j.procs.2023.01.127
![]() |
[62] |
J. Ryu, M. U. Rehman, I. F. Nizami, K. T. Chong, SegR-Net: A deep learning framework with multi-scale feature fusion for robust retinal vessel segmentation, Comput. Biol. Med., 163 (2023), 107132. https://doi.org/10.1016/j.compbiomed.2023.107132 doi: 10.1016/j.compbiomed.2023.107132
![]() |
[63] |
T. Tiwari, M. Saraswat, A new modified-unet deep learning model for semantic segmentation, Multimed. Tools Appl., 82 (2023), 3605–3625. https://doi.org/10.1007/s11042-022-13230-2 doi: 10.1007/s11042-022-13230-2
![]() |
[64] |
A. K. Upadhyay, A. K. Bhandari, Semi-supervised modified-UNet for lung infection image segmentation, IEEE Trans. Radiat. Plasma Med. Sci., 7 (2023), 638–649. https://doi.org/10.1109/trpms.2023.3272209 doi: 10.1109/trpms.2023.3272209
![]() |
[65] |
R. Ranjbarzadeh, P. Zarbakhsh, A. Caputo, E. B. Tirkolaee, M. Bendechache, Brain tumor segmentation based on optimized convolutional neural network and improved chimp optimization algorithm, Comput. Biol. Med., 168 (2024), 107723. https://doi.org/10.1016/j.compbiomed.2023.107723 doi: 10.1016/j.compbiomed.2023.107723
![]() |
[66] |
R. Ranjbarzadeh, S. J. Ghoushchi, N. T. Sarshar, E. B. Tirkolaee, S. S. Ali, T. Kumar, et al., ME-CCNN: Multi-encoded images and a cascade convolutional neural network for breast tumor segmentation and recognition, Artif. Intell. Rev., 56 (2023), 10099–10136. https://doi.org/10.1007/s10462-023-10426-2 doi: 10.1007/s10462-023-10426-2
![]() |
[67] | A, B. Kasgari, R. Ranjbarzadeh, A. Caputo, S. B. Saadi, M. Bendechache, Brain tumor segmentation based on zernike moments, enhanced ant lion optimization, and convolutional neural network in MRI images, metaheuristics and optimization, in Computer and Electrical Engineering, Lecture Notes in Electrical Engineering, 1077 (2023). Springer, Cham. https://doi.org/10.1007/978-3-031-42685-8_10 |
[68] |
S. Anari, N. S. Tataei, N. Mahjoori, S. Dorosti, A. Rezaie, Review of deep learning approaches for Thyroid Cancer Diagnosis, Math. Probl. Eng., (2022), 1–8. https://doi.org/10.1155/2022/5052435 doi: 10.1155/2022/5052435
![]() |
[69] |
Z. Zhu, X. He, G. Qui, Y. Li, B. Cong, Y. Liu, Brain tumor segmentation based on the fusion of deep semantics and edge information in multimodal MRI, Inf. Fusion, 91 (2023), 376–387. https://doi.org/10.1016/j.inffus.2022.10.022 doi: 10.1016/j.inffus.2022.10.022
![]() |
[70] |
Y. Li, Z. Wang, L. Yin, Z. Zhu, G. Qi, Y. Liu, X-Net: A dual encoding–decoding method in medical image segmentation, Vis. Comput., 39 (2021), 2223–2233. https://doi.org/10.1007/s00371-021-02328-7 doi: 10.1007/s00371-021-02328-7
![]() |
[71] |
X. He, G. Qi, Z. Zhu, Y. Li, B. Cong, L. Bai, Medical image segmentation method based on multi-feature interaction and fusion over cloud computing, Simul. Model Pract. Theory, 126 (2023), 102769. https://doi.org/10.1016/j.simpat.2023.102769 doi: 10.1016/j.simpat.2023.102769
![]() |
[72] |
Y. Xu, X. He, G. Xu, G. Qi, K. Yu, Li. Yin, et al., A medical image segmentation method based on multi-dimensional statistical features, Front. Neurosci., 16 (2022). https://doi.org/10.3389/fnins.2022.1009581 doi: 10.3389/fnins.2022.1009581
![]() |
1. | Zhen Zhang, Shance Wang, Relative subcategories with respect to a rigid subcategory, 2025, 0092-7872, 1, 10.1080/00927872.2025.2509823 |