Open AccessArticle

Split Common Coincidence Point Problem: A Formulation Applicable to (Bio)Physically-Based Inverse Planning Optimization

Charles E. Chidume

and

Lois C. Okereke

Department of Pure and Applied Mathematics, Mathematics Institute (Emerging Regional Centre of Excellence (ERCE) of the European Mathematical Society (EMS)), African University of Science and Technology, PMB 681, Abuja 900107, Nigeria

Author to whom correspondence should be addressed.

Symmetry 2020, 12(12), 2086; https://doi.org/10.3390/sym12122086

Submission received: 13 October 2020 / Revised: 28 October 2020 / Accepted: 29 October 2020 / Published: 15 December 2020

(This article belongs to the Special Issue Symmetry in Nonlinear Functional Analysis and Optimization Theory)

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Abstract

Inverse planning is a method of radiotherapy treatment planning where the care team begins with the desired dose distribution satisfying prescribed clinical objectives, and then determines the treatment parameters that will achieve it. The variety in symmetry, form, and characteristics of the objective functions describing clinical criteria requires a flexible optimization approach in order to obtain optimized treatment plans. Therefore, we introduce and discuss a nonlinear optimization formulation called the split common coincidence point problem (SCCPP). We show that the SCCPP is a suitable formulation for the inverse planning optimization problem with the flexibility of accommodating several biological and/or physical clinical objectives. Also, we propose an iterative algorithm for approximating the solution of the SCCPP, and using Bregman techniques, we establish that the proposed algorithm converges to a solution of the SCCPP and to an extremum of the inverse planning optimization problem. We end with a note on useful insights on implementing the algorithm in a clinical setting.

Keywords:

coincidence point; inverse planning; convex optimization; fixed point; iterative algorithm; tumor control probability; radiobiological criteria; pseudocontraction

1. Introduction

The grail of modern radiotherapy is to ensure sufficient and conformal irradiation of tumor cells devoid of normal tissue(s) complications. This is achieved through a process known as radiotherapy treatment planning (RTP). A core component of RTP is the fluence map optimization, which involves determining a beam intensity (fluence) profile that would yield a dose distribution closest to a desired clinical outcome. In intensity-modulated radiation therapy (IMRT), for instance, radiation beams from an external source are split into smaller fields called beamlets, whose individual intensities can be regulated (modulated) [1]. To obtain an optimized treatment plan, the inverse problem of finding a set of beamlet intensities that would result in a dose distribution satisfying a set of prescribed clinical objectives is solved. The quality of any treatment plan is usually measured by how well the set of objectives are achieved. Therefore, the objective functions used in describing clinical requirements for treatment plan optimization are an important index in plan quality evaluation.

To this end, several authors have, on the one hand, proposed and analyzed different objective functions for optimizing treatment plans based on physical criteria (such as maximum dose, minimum dose, mean dose, etc.) and/or biological criteria (such as equivalent uniform dose (EUD), tumor control probability (TCP), normal tissue complication probability (NTCP), etc.) [2,3,4,5,6,7,8,9]. On the other hand, other studies have focused on the comparative analysis of objective functions using clinical data [10,11,12,13,14,15,16,17]. From these studies, specific merits and limitations of physically- and biologically-based objectives can be identified. While target coverage enforced by physical criteria is desirable, it has also been reported that better critical organ sparing is achieved using biological criteria. Other results, however, favor hybrid (combination of biological and physical) criteria over just physical or biological criteria. Hence, it is safe to say that no particular objective function can be described as “the best”, that is, better in all respects.

Consequently, inverse planning optimization has been a subject of extensive research. This has led to the development of several optimization formulations based on different objective functions [3,4,9,18], the most widely used being the linear programming approach, largely due to its simplicity and speed. However, this framework has a major drawback: lack of flexibility. Only a relatively limited number of objective functions and constraints can fit into the linear framework. Hence, “with any particular linear programming formulation, it is unlikely that a physician could always achieve an acceptable result” [19]. On the contrary, with nonlinear optimization formulations, a wide range of possible objective functions and constraints including mathematical models for predicting radiobiological responses can be accommodated. This is no doubt advantageous to RTP, even though the uncertainty in determining input parameters of some radiobiological models poses some difficulties.

Therefore, in this paper, we introduce and discuss a nonlinear optimization formulation called the split common coincidence point problem (SCCPP). We propose an iterative algorithm for approximating the solution of the SCCPP, and using Bregman techniques, we establish that the proposed algorithm converges weakly to a solution of the SCCPP. Furthermore, using some biological and/or physical objective criteria recommended for RTP optimization, we show that the SCCPP is a suitable formulation for inverse planning optimization and that the proposed algorithm converges to an extremum of the (bio)physically-based inverse planning optimization problem. It will be interesting to illustrate the performance of the algorithm under conditions obtainable in a clinical setting. Thus, we provide useful computational insights for implementation of the algorithm in such setting.

2. Preliminaries

In this section, we provide some useful definitions and results. We shall denote a reflexive real Banach space and its dual by

E

and

E^{*}

, respectively.

Definition 1.

A function

ψ : E ⟶ \bar{ℝ}

is said to be:

(a): convex if for any $λ \in [0, 1]$ and points $u, v \in E$ ,

$ψ (λ u + (1 - λ) v) \leq λ ψ (u) + (1 - λ) ψ (v)$
(b): strictly convex if the inequality in (a) is strict
(c): totally convex if there exists a function $φ : [0, + \infty) ⟶ [0, + \infty)$ varnishing only at zero such that: $ψ (u) - ψ (v) \geq 〈 v^{*}, u - v 〉 + φ (∥ u - v ∥), v^{*} \in \partial ψ (v)$

The set

d o m (ψ) ≔ {u \in E : ψ (u) < + \infty}

is called the effective domain of

ψ

. When

d o m (ψ)

is nonempty,

ψ

is said to be proper.

Definition 2.

Let

ψ : E ⟶ \bar{ℝ}

be convex, and

u \in d o m (ψ)

. The set

\partial ψ (u) : = {u^{*} \in E^{*} : 〈 u^{*}, v - u 〉 \leq ψ (v) - ψ (u), \forall v \in E},

is called the subdifferential of

ψ

u

. If

\partial ψ (u)

is nonempty, then an element

p \in \partial ψ (u)

is called the subgradient of

ψ

u

Remark 1.

It is well known that if

ψ

is a proper, lower semi-continuous convex function, then for any

u \in d o m (ψ)

\partial ψ (u)

is nonempty. Moreover,

0 \in \partial ψ (u)

iff

u

is a minimizer of

ψ

Definition 3.

A function

ψ : E ⟶ \bar{ℝ}

is said to be Legendre, if

(a): $\partial ψ$ is both locally bounded and single-valued on its domain
(b): $ψ$ is strictly convex on every subset of $d o m (ψ)$ and ${(\partial ψ)}^{- 1}$ is locally bounded in its domain.

Definition 4.

(See [20]) Let

ψ : E ⟶ \bar{ℝ}

be a proper, lower semi-continuous convex function. Then:

(a): the generalized Bregman distance with respect to $ψ$ and a subgradient $p$ is defined as:

$B_{ψ}^{p} (u, v) = ψ (u) - ψ (v) - 〈 p, u - v 〉, p \in \partial ψ (v)$

It is immediate that $B_{ψ}^{p} (u, v) \geq 0 \forall u, v \in d o m (ψ)$ . Furthermore,

$B_{ψ}^{p} (u, w) = B_{ψ}^{q} (u, v) + B_{ψ}^{p} (v, w) + 〈 u - v, q - p 〉, p \in \partial ψ (w), q \in \partial ψ (v)$

when $\partial ψ (u)$ is a singleton, we shall denote $B_{ψ}^{p} (., .)$ simply by $B_{ψ} (., .)$ .
(b): the Bregman projection relative to $ψ$ of a point $u u \in d o m (ψ)$ onto a nonempty, closed, and convex subset $K$ , is defined as the unique vector $\prod_{ψ}^{K} (u)$ satisfying $B_{ψ} (\prod_{ψ}^{K} (u), u) = i n f {B_{ψ} (v, u) : v \in K}$ . If $ψ$ is totally convex and Gateaux differentiable, then $\prod_{ψ}^{K} (u)$ is the unique solution contained in $K$ of the following variational inequalities (see [21]);

(i): $〈 \nabla ψ (u) - \nabla ψ (z), z - w 〉 \geq 0 \forall w \in K$
(ii): $B_{ψ} (w, z) + B_{ψ} (z, u) \leq B_{ψ} (w, u) \forall w \in K$

Definition 5.

A mapping

M : E ⟶ 2^{E^{*}}

is said to be:

(a): monotone, if $〈 m - n, u - v 〉 \geq 0 \forall u, v \in d o m (M), m \in M (u), n \in M (v)$
(b): maximal monotone if $M$ is monotone, and the graph of $M$ is not contained in the graph of any other monotone map
(c): $φ$ -strongly monotone if there exists a non-negative function $φ$ which varnishes only at zero such that:

$m - n, u - v \geq φ (u - v), \forall u, v \in d o m (M), m \in M (u), n \in M (v)$

If $φ$ is the function $φ (x) = k_{0} x^{2}$ , $k_{0} > 0$ , $M$ is called strongly monotone.
Note that $d o m (M) : = {z \in E : M (z) i s n o n e m p t y}$ .

Definition 6.

A mapping

ϕ : E ⟶ X

(

X

is a normed space) is said to be Lipschitz if there exists

L > 0

such that:

∥ ϕ (u) - ϕ (v) ∥_{X} \leq L ∥ u - v ∥_{E} \forall u, v \in E .

Definition 7.

(See [22]) Let

ϕ_{1}, ϕ_{2} : E ⟶ E

be two mappings. A point

u \in E

is called a coincidence point of

ϕ_{1} a n d ϕ_{2}

iff

ϕ_{1} (u) = ϕ_{2} (u)

Lemma 1.

[23] (a) If

ψ

is a lower semi-continuous, proper convex function on

E

, then

\partial ψ

is a maximal monotone operator from

E

E^{*}

. (b) Let

T : E ⟶ E^{*}

be a multivalued mapping. In order that there exists a lower semi-continuous proper convex function on

E

such that

T = \partial ψ

, it is necessary and sufficient that

T

be a maximal cyclically monotone operator. Moreover, in this case,

T

determines

ψ

uniquely up to an additive constant.

Proposition 1.

[24] Let

ψ

be a lower semi-continuous convex function, such that

i n t (d o m (ψ))

is nonempty. Then, the function

ψ

is differentiable at

u \in i n t (d o m (ψ))

iff

\partial ψ (u)

is a singleton. If

ψ

is differentiable on

i n t (d o m (ψ))

, then the derivative is norm to weak* continuous on

i n t (d o m (ψ))

Proposition 2.

[24] Let

ψ

be a convex function such that

i n t (d o m (ψ))

is nonempty. Then, the following are equivalent:

(i): The function $ψ$ is locally bounded from above on $i n t (d o m (ψ))$
(ii): The function $ψ$ is locally bounded on $i n t (d o m (ψ))$
(iii): The function $ψ$ is locally Lipschitz on $i n t (d o m (ψ))$
(iv): The function $ψ$ is continuous on $i n t (d o m (ψ))$ .

Moreover, if

ψ

is lower semi-continuous, then all these conditions are satisfied.

Lemma 2.

[25] Let

ψ

be uniformly convex on every nonempty, bounded and convex subset of

d o m (ψ)

. Then, given two sequences

{u_{k}}_{k = 1}^{\infty}

d o m (ψ)

and

{v_{k}}_{k = 1}^{\infty}

i n t (d o m (ψ))

satisfying

\lim_{k \to \infty} B_{ψ} (u_{k}, v_{k}) = 0,

if one of these sequences is bounded, then, the other is bounded too. Moreover,

\lim_{k \to \infty} ∥ u_{k} - v_{k} ∥ = 0

3. Main Results

3.1. Split Common Coincidence Point Problem (SCCPP)

Definition 8.

(Generalized Coincidence Point) Let

T, S : E_{1} ⟶ E_{2}

be two multivalued mappings. A point

u_{0} \in E_{1}

will be called a coincidence point of

T

and

S

iff there exists

η_{0} \in T (u_{0}) \cap S (u_{0})

.The problem of finding a coincidence point of

T

and

S

is called a generalized coincidence point problem.

Remark 2.

A multivalued version of Definition 7 is obtained immediately by setting

E_{1} = E_{2}

. If in addition, the map

T

S

is the identity operator, then the coincidence point problem reduces to the fixed point problem. It is worthy of mention that a particular case of the generalized coincidence point problem resulting when

E_{1}

is a normed space,

E_{2} = E_{1}^{*}

T

is monotone and

S = J

(the normalized duality map) was introduced and studied by Chidume and Idu [26] as the

J

-fixed point problem. As a tool for solving optimization problems, the

J

-fixed point problem has awaken further research in that direction (See [27,28,29]).

The split common coincidence point problem follows naturally from Definition 8 as Definition 9.

Definition 9.

Let

T, S

be as in Definition 8, let

P, Q : E_{3} ⟶ E_{4}

be two multivalued mappings. Let

A : E_{1} ⟶ E_{3}

be a bounded linear map. Denote by

C_{0} (T, S)

and

C_{0} (P, Q)

, the set of coincidence points of

T

and

S

, and the set of coincidence points of

P

and

Q,

respectively. Then, the SCCPP is

f i n d u_{0} \in E_{1} s u c h t h a t u_{0} \in C_{0} (T, S) a n d A u_{0} \in C_{0} (P, Q) .

It follows that by setting

S = I_{E_{1}}, Q = I_{E_{2}}

(

I

is the identity map), the split common fixed point problem (SCFP) introduced by Censor and Segal [30] and studied extensively (see e.g., [31,32,33,34,35]), is a sub-class of the SCCPP. The SCFP formalism, and its variant introduced and studied by Moudafi [36], has been used in modelling significant real-world inverse problems (see for example [37,38]).

3.2. Optimization by Generalized Coincidence Point Problem

Here, we establish an interconnection between optimization problems and coincidence point problems for a special class of operators,

T

and

S

. For this purpose, we shall assume that

E_{1}

in Definition 8 is a reflexive Banach space, and simply denote it as

E

, and

E_{2} = E^{*}

Definition 10.

(

S

-pseudocontraction) Let

T

and

S

be as in Definition 8. Then,

T

will be called

S

-pseudocontraction (or pseudocontractive with respect to

S

) if for every

u, v \in E

〈 t_{1} - t_{2}, u - v 〉 \leq 〈 s_{1} - s_{2}, u - v 〉 \forall t_{1} \in T u, t_{2} \in T v, s_{1} \in S u, s_{2} \in S v

The concept of

S

-pseudocontraction generalizes that of

J

-pseudocontraction studied in [26].

Lemma 3.

Let

T

and

S

be as in Definition 10; then, the following are equivalent:

(i): $T$ is monotone.
(ii): $S - T$ is $S$ -pseudocontractive.

Proof.

Let

Γ : = S - T

(i) ⟹ (ii)

Let

u, v \in E

be arbitrary; then, for every

t_{u} \in T u, t_{v} \in T v

s_{u} \in S u, s_{v} \in S v

and

γ_{u} \in Γ u, γ_{v} \in Γ v

such that

γ_{u} = s_{u} - t_{u}

and

γ_{v} = s_{v} - t_{v}

, we have that

〈 γ_{u} - γ_{v}, u - v 〉 = 〈 s_{u} - t_{u} - (s_{v} - t_{v}), u - v 〉 = 〈 s_{u} - s_{v}, u - v 〉 - 〈 t_{u} - t_{v}, u - v 〉 \leq 〈 s_{u} - s_{v}, u - v 〉 .

Thus,

Γ : = S - T

S

-pseudocontractive.

(ii) ⟹ (i)

We first note that

T = S - Γ

Now, let

u, v \in E

be arbitrary, let

t_{u} \in T u

and

t_{v} \in T v

, then

t_{u} = s_{u} - γ_{u}

and

t_{v} = s_{v} - γ_{v}

for some

s_{u} \in S u

s_{v} \in S v

γ_{u} \in Γ u

, and

γ_{v} \in Γ v

. Thus,

〈 t_{u} - t_{v}, u - v 〉 = 〈 s_{u} - γ_{u} - (s_{v} - γ_{v}), u - v 〉 = 〈 s_{u} - s_{v}, u - v 〉 - 〈 γ_{u} - γ_{v}, u - v 〉 \geq 0 .

Therefore,

T

is monotone. □

It is easily deducible from Lemma 3 that if

T

S

-pseudocontractive, then a coincidence point of

T

and

S

corresponds to a zero of the monotone map

S - T

. Hence, well known existence results for zeros of monotone operators naturally carry over for coincidence points of operators of this class. The crucial role of zeros of monotone operators in the analysis of solutions to minimization problems (see for example [39,40,41,42]) underscores the interplay between coincidence point problems and optimization problems.

3.3. Examples of S-Pseudocontraction

(1): Let $E = ℍ$ , a real Hilbert space, $S = I$ , the identity map on $ℍ$ . Then, any pseudocontraction on $ℍ$ is $S$ -pseudocontraction.
(2): Every $J$ -pseudocontraction is $S$ -psuedocontraction with $S = J + M$ , where $M$ is any single-valued monotone map.
(3): Let $E$ be a smooth real Banach space, fix $p > 1$ . Define $T_{p} : E ⟶ E^{*}$ and $S_{p} : E ⟶ E^{*}$ by:

$T_{p} (u) = p J (u) S_{p} (u) = (p + μ) J (u) + ∥ u ∥^{p - 2} J (u), μ > 0 .$

Then,

T_{p}

S_{p}

-pseudocontractive and

0 \in C_{0} (T_{p}, S_{p})

We remark that the map,

T_{p}

, failed to be pseudocontractive in the usual sense for

E = ℍ

. Furthermore,

T_{p}

is not

J

-pseudocontractive. In fact, for every mapping

T : E ⟶ E^{*}

, there exists a mapping

S : E ⟶ E^{*}

such that

T

S

-pseudocontractive (Take

S = T + A,

where

A

is any monotone map on

E

Lemma 4.

Let

T : E ⟶ E^{*}

be a single valued map. Let

S : E ⟶ 2^{E^{*}}

be such that

Ψ : = S^{- 1} \circ T

is well defined and single valued. If

T

S

-pseudocontractive, then for every

u, v \in E

T (Ψ (u)) - T (Ψ (v)), Ψ (u) - Ψ (v) \leq T (u) - T (v), Ψ (u) - Ψ (v)

Proof.

Let

u, v \in E

. By definition of

Ψ

, we have that

T (u) \in S (Ψ (u))

and

T (v) \in S (Ψ (v))

. Since

T

S

-pseudocontractive, then

〈 T (Ψ (u)) - T (Ψ (v)), Ψ (u) - Ψ (v) 〉 \leq 〈 T (u) - T (v), Ψ (u) - Ψ (v) 〉

as required. □

3.4. Approximation of Coincidence Points

Theorem 1.

Let

E_{1}

and

E_{2}

be reflexive real Banach spaces, with dual spaces

E_{1}^{*}

and

E_{2}^{*}

respectively. Let

T_{1} : E_{1} ⟶ E_{1}^{*}

and

T_{2} : E_{2} ⟶ E_{2}^{*}

be single-valued cyclically maximal monotone maps. Let

S_{1} : E_{1} ⟶ E_{1}^{*}

and

S_{2} : E_{2} ⟶ E_{2}^{*}

be mappings such that

T_{1}

S_{1}

-pseudocontractive,

T_{2}

S_{2}

-pseudocontractive and

g_{1} : = S_{1}^{- 1} \circ T_{1}

g_{2} : = S_{2}^{- 1} \circ T_{2}

are well defined and single-valued. Let

A : E_{1} ⟶ E_{2}

be a bounded linear map with adjoint

A^{*}

. Assume that

T_{1}^{- 1}

exists and generate inductively the sequence:

{\begin{cases} w_{0} \in E_{1} \\ v_{n} = T_{1}^{- 1} (T_{1} (w_{n}) - r A^{*} (T_{2} (A w_{n}) - T_{2} (g_{2} (A w_{n})))) \\ w_{n + 1} = T_{1}^{- 1} (α T_{1} (v_{n}) + (1 - α) T_{1} (g_{1} (v_{n}))) \\ r > 0, 0 \leq α < 1 . \end{cases}

(1)

Suppose that the set

s o l (S C C P P)

is nonempty, where

s o l (S C C P P) = {z \in E_{1} : z \in C_{0} (T_{1}, S_{1}) a n d A z \in C_{0} (T_{2}, S_{2})} .

Then, given

z \in s o l (S C C P P)

\begin{array}{l} B_{ψ_{1}} (z, w_{n + 1}) \leq B_{ψ_{1}} (z, w_{n}) - B_{ψ_{1}} (v_{n}, w_{n}) - α B_{ψ_{1}} (g_{1} (v_{n}), v_{n}) \\ - r [B_{ψ_{2}} (g_{2} (A w_{n}), A w_{n}) - B_{ψ_{2}} (A v_{n}, A w_{n})] \end{array}

ψ_{1}, ψ_{2}

are convex functions guaranteed by Lemma 1 for

T_{1}, T_{2}

, respectively.

Proof.

Let

z \in s o l (S C C P P)

and

ψ_{1}, ψ_{2}

as in Lemma 1, using

(1)

, definition of

B_{ψ} (., .)

and property of adjoint, we have that

B_{ψ_{1}} (z, v_{n}) \leq B_{ψ_{1}} (z, w_{n}) - B_{ψ_{1}} (v_{n}, w_{n}) + r T_{2} (A w_{n}) - T_{2} (g_{2} (A w_{n})), A z - A v_{n}

Applying the fact that

z \in s o l (S C C P P)

, simple arithmetic, Lemma 4, and second property of

B_{ψ} (., .)

in Definition 4(a), we obtain

\begin{array}{l} B_{ψ_{1}} (z, v_{n}) \leq B_{ψ_{1}} (z, w_{n}) - B_{ψ_{1}} (v_{n}, w_{n}) - r B_{ψ_{2}} (g_{2} (A w_{n}), A w_{n}) - r B_{ψ_{2}} (A v_{n}, g_{2} (A w_{n})) + r B_{ψ_{2}} (A v_{n}, A w_{n}) \\ \leq B_{ψ_{1}} (z, w_{n}) - B_{ψ_{1}} (v_{n}, w_{n}) - r B_{ψ_{2}} (g_{2} (A w_{n}), A w_{n}) + r B_{ψ_{2}} (A v_{n}, A w_{n}) \end{array}

(2)

Next, observe from

(1)

that

w_{n + 1} = T_{1}^{- 1} (T_{1} (v_{n}) - (1 - α) [T_{1} (v_{n}) - T_{1} (g_{1} (v_{n}))])

Then, applying same steps used in the above argument yield

B_{ψ_{1}} (z, w_{n + 1}) \leq B_{ψ_{1}} (z, v_{n}) - α B_{ψ_{1}} (w_{n + 1}, v_{n}) - α B_{ψ_{1}} (g_{1} (v_{n}), v_{n})

(3)

Thus, combining

(2)

and

(3)

gives

B_{ψ_{1}} (z, w_{n + 1}) \leq B_{ψ_{1}} (z, w_{n}) - B_{ψ_{1}} (v_{n}, w_{n}) - α B_{ψ_{1}} (g_{1} (v_{n}), v_{n}) - r B_{ψ_{2}} (g_{2} (A w_{n}), A w_{n}) + r B_{ψ_{2}} (A v_{n}, A w_{n})

(4)

□

Corollary 1.

Under the conditions of Theorem 1, let

T_{1} = T_{2}

S_{1} = S_{2}

A = I

and

α = 0

, then direct substitution gives

B_{ψ} (z, w_{n + 1}) \leq B_{ψ} (z, w_{n}) - (1 - r) B_{ψ} (w_{n + 1}, w_{n}) - B_{ψ} (g (w_{n}), w_{n})

Theorem 2.

Under the conditions of Theorem 1, assume that

T_{1}

is strongly monotone with constant

k_{1}

and weakly sequentially continuous, and

T_{2}

k_{2}

-Lipschitz and

φ

-strongly monotone on bounded sets. Take

r < \frac{k_{1}}{k_{1} + k_{2} ∥ A ∥^{2}}

and

α > 0

, if

I_{E_{1}} - g_{1}

and

I_{E_{2}} - g_{2}

are demi-closed at zero, then, the sequence

{w_{n}}

generated by

(1)

, converges weakly to a point

z_{0} \in s o l (S C C P P)

Proof.

The added properties of

T_{1}

and

T_{2}

(4)

and the fact that

0 < r < \frac{k_{1}}{k_{1} + k_{2} A^{2}}

give the following:

$B_{ψ_{1}} (z, w_{n + 1}) \leq B_{ψ_{1}} (z, w_{n}) - v_{n} - ∥ w_{n} ∥^{2}$
$\lim_{n \to \infty} B_{ψ_{1}} (z, w_{n})$ exists
${w_{n}}$ is bounded
$\lim_{n \to \infty} ∥ v_{n} - w_{n} ∥ = 0$ and ${v_{n}}$ is also bounded.

Also from

(4)

, we deduce that

\lim_{n \to \infty} B_{ψ_{1}} (g_{1} (v_{n}), v_{n}) = \lim_{n \to \infty} B_{ψ_{2}} (g_{2} (A w_{n}), A w_{n}) = 0

Hence, the boundedness of the sequences

{v_{n}}

and

{A w_{n}}

together with Lemma 2 gives the following:

${g_{1} (v_{n})}$ and ${g_{2} (A w_{n})}$ are bounded
$\lim_{n \to \infty} ∥ g_{1} (v_{n}) - v_{n} ∥ = \lim_{n \to \infty} ∥ g_{2} (A w_{n}) - A w_{n} ∥ = 0$

Next, denote the set of weak limit points of

{w_{n}}

W (w_{n})

. The boundedness of

{w_{n}}

guarantees that

W (w_{n})

is nonempty. To conclude our proof, it suffices to show the following:

(i): $W (w_{n}) \subseteq s o l (S C C P P)$
(ii): $W (w_{n})$ is a singleton

Now, for (i), pick

p_{0} \in W (w_{n})

arbitrarily, then there exists

{w_{n_{j}}} \subseteq {w_{n}}

such that

w_{n_{j}} ⇀ p_{0}

. Thus,

v_{n_{j}} ⇀ p_{0}

and

A w_{n_{j}} ⇀ A p_{0}

. Since

∥ g_{1} (v_{n_{j}}) - v_{n_{j}} ∥ ⟶ 0

and

∥ g_{2} (A w_{n_{j}}) - A w_{n_{j}} ∥ ⟶ 0

, we have by demi-closedness that

p_{0} = g_{1} (p_{0})

and

A p_{0} = g_{2} (A p_{0}) .

This gives that

T_{1} (p_{0}) \in S_{1} (p_{0})

and

T_{2} (A p_{0}) \in S_{2} (A p_{0})

. Therefore,

p_{0} \in s o l (S C C P P)

Next, we show that

W (w_{n})

is a singleton. For a contradiction, suppose that there exists

u_{0}, u_{1} \in W (w_{n})

such that

u_{0} \neq u_{1}

, then, there exists

{w_{n_{k}}}

{w_{n_{i}}}

subsequences of

{w_{n}}

such that

w_{n_{k}} ⇀ u_{0}

and

w_{n_{i}} ⇀ u_{1}

. Now we know that

\lim_{n \to \infty} [ψ_{1} (u_{0}) - ψ_{1} (u_{1}) + B_{ψ_{1}} (u_{1}, w_{n}) - B_{ψ_{1}} (u_{0}, w_{n})] = \lim_{n \to \infty} 〈 u_{0} - u_{1}, T_{1} (w_{n}) 〉

exists. Hence,

\lim_{k \to \infty} 〈 u_{0} - u_{1}, T_{1} (w_{n_{k}}) 〈 = \lim_{i \to \infty} 〈 u_{0} - u_{1}, T_{1} (w_{n_{i}}) 〉

By weak sequential continuity of

T_{1}

, we have that

〈 u_{0} - u_{1}, T_{1} (u_{0}) - T_{1} (u_{1}) 〉 = 0 ⟶ u_{0} = u_{1} contradicting that u_{0} \neq u_{1}

Therefore

W (w_{n})

is a singleton. □

Corollary 2.

Let

K_{1}

and

K_{2}

be nonempty, closed, and convex subsets of reflexive spaces

E_{1}

and

E_{2}

respectively. Let

A : E_{1} ⟶ E_{2}

such that

A (K_{1}) \subseteq K_{2}

. Let

T_{1}, T_{2}, S_{1}, S_{2}, g_{1}

g_{2}, r

and

α

satisfy the assumptions of Theorem 2 on

K_{1}

and

K_{2}

. Then, the sequence

{w_{n}}

inductively generated by

{\begin{cases} w_{0} \in K_{1} \\ y_{n} = \prod_{ψ_{1}}^{K_{1}} (T_{1}^{- 1} (T_{1} (w_{n}) - r A^{*} (T_{2} (A w_{n}) - T_{2} (g_{2} (A w_{n}))))) \\ w_{n + 1} = \prod_{ψ_{1}}^{K_{1}} (T_{1}^{- 1} (α T_{1} (v_{n}) + (1 - α) T_{1} (g_{1} (v_{n})))), \end{cases}

(5)

weakly converges to a solution of the

K_{1}, K_{2}

constrained SCCPP.

Proof.

By simple computation using the definition of

B_{ψ_{1}} (., .)

and characterization of

y_{n}

as in Definition 4(b), we have for

z \in s o l (S C C P P)

that

B_{ψ_{1}} (z, y_{n}) \leq B_{ψ_{1}} (z, w_{n}) - B_{ψ_{1}} (y_{n}, w_{n}) + r 〈 T_{2} (A w_{n}) - T_{2} (g_{2} (A w_{n})), A z - A y_{n} 〉

z \in s o l (S C C P P)

, simple arithmetic, Lemma 4, and second property of

B_{ψ_{1}} (., .)

in Definition 4(a), we obtain that

\begin{array}{l} 〈 T_{2} (A w_{n}) - T_{2} (g_{2} (A w_{n})), A z - A y_{n} 〉 \\ \leq B_{ψ_{2}} (A y_{n}, A w_{n}) - B_{ψ_{2}} (g_{2} (A w_{n}), A w_{n}) - B_{ψ_{2}} (A y_{n}, g_{2} (A w_{n})) \end{array}

Hence,

B_{ψ_{1}} (z, y_{n}) \leq B_{ψ_{1}} (z, w_{n}) - B_{ψ_{1}} (y_{n}, w_{n}) + r B_{ψ_{2}} (A y_{n}, A w_{n}) - r B_{ψ_{2}} (g_{2} (A w_{n}), A w_{n})

Similarly,

B_{ψ_{1}} (z, w_{n + 1}) \leq B_{ψ_{1}} (z, y_{n}) - α B_{ψ_{1}} (g_{1} (y_{n}), y_{n}) - α B_{ψ_{1}} (w_{n + 1}, y_{n})

Thus,

B_{ψ_{1}} (z, w_{n + 1}) \leq B_{ψ_{1}} (z, w_{n}) - B_{ψ_{1}} (y_{n}, w_{n}) + r B_{ψ_{2}} (A y_{n}, A w_{n}) - r B_{ψ_{2}} (g_{2} (A w_{n}), A w_{n}) - α B_{ψ_{1}} (g_{1} (y_{n}), y_{n})

(6)

It follows from

(6)

that

$\lim_{n \to \infty} B_{ψ_{1}} (z, w_{n})$ exists
${w_{n}}$ is bounded
$\lim_{n \to \infty} ∥ y_{n} - w_{n} ∥ = 0$ and ${y_{n}}$ is also bounded.
$\lim_{n \to \infty} B_{ψ_{1}} (g_{1} (y_{n}), y_{n}) = \lim_{n \to \infty} B_{ψ_{2}} (g_{2} (A w_{n}), A w_{n}) = 0, thus {g_{1} (y_{n})} and {g_{2} (A w_{n})}$ $are bounded . Moreover, \lim_{n \to \infty} ∥ g_{1} (y_{n}) - y_{n} ∥ = \lim_{n \to \infty} ∥ g_{2} (A w_{n}) - A w_{n} ∥ = 0$

Finally, using a similar argument as in Theorem 2, we deduce that

W (w_{n}) \subseteq s o l (S C C P P) and W (w_{n}) = {u_{0}}

□

4. Application to Inverse Planning Optimization

4.1. An Inverse Planning Optimization Problem (IPOP)

Prior to treatment optimization, the patient’s volume (usually a 3D representation of the patient’s anatomy obtained from medical images such as computed tomography (CT) and magnetic resonance imaging (MRI) scans) is discretized into sub-volumes called voxels. The anatomy is further outlined into various structures grouped as planning target volume (PTV) and organs at risk (OAR). Individual voxels may belong to several structures, but for simplicity, it is usual to associate each to one structure.

Now suppose there are

N

number of voxels indexed by

i = 1, 2, \dots, N

and

M

candidate beams (beamlets) indexed by

j = 1, 2, \dots, M

. Then, the total dose absorbed in the

i

th voxel is given by

d_{i} = \sum_{j = 1}^{M} a_{i j} u_{j}

where

a_{i j}

is the

(i \times j)

th entry of the dose influence matrix

A : ℝ^{M} ⟶ ℝ^{N}

and

u_{j}

is the intensity of the

j

th beam (beamlet) [19].The IPOP is formulated as

find u^{*} \in ℝ_{+}^{M} such that A u^{*} \in \min (f)

where

f : ℝ^{N} ⟶ ℝ

is a dose dependent objective function modeled based on clinical goals, and

\min (f)

denotes the set of minimizers of

f

. For analysis of some convex reformulation of commonly used objective criteria in RTP, we refer the reader to [9,43].

4.2. SCCPP Reformulation of IPOP

We begin with the following assumptions:

(a): $f$ is convex
(b): $f$ is “partly” differentiable, that is, $f$ can be written as a sum of two convex functions $f_{1}$ and $f_{2}$ such that $f_{1}$ or $f_{2}$ is differentiable. Without loss of generality, we shall always assume $f_{2}$ to be differentiable.
(c): There exists a differentiable convex function $h$ and a positive constant $θ$ , such that $θ h - f_{2}$ is Legendre, totally convex, cofinite, and has Lipschitz continuous gradient

Now, let

E_{1} = ℝ^{M}, E_{1} = ℝ^{N}, K_{1} = ⋂_{l = 1}^{ρ} {u \in ℝ_{+}^{M} : σ_{l} (u) \leq ρ_{l}}

for some convex constraint functions

σ_{l}

and scalars

ρ_{l} \in ℝ

and

K_{2} = ℝ_{+}^{N}

. Let

A

be the dose influence matrix, then

A (K_{1}) \subseteq K_{2}

. Let

f : E_{2} ⟶ \bar{ℝ}

, a dose dependent objective function modeled based on clinical goals, satisfy assumptions (a)–(c). Define the following operators

T_{1} : = τ I_{E_{1}} = S_{1}, τ > 0 T_{2} : = κ (\nabla h - \frac{1}{θ} \nabla f_{2}), κ > 0 S_{2} : = κ (\nabla h + \frac{1}{θ} \partial f_{1}), κ > 0

Then, it can easily be shown that

(i): $T_{1}$ is strongly monotone, cofinite, sequentially continuous, and $T_{1}$ inverse exists
(ii): $T_{1}$ is $S_{1}$ -pseudocontractive, and $T_{2}$ is $S_{2}$ -pseudocontractive.
(iii): $T_{2}$ is Lipschitz continuous and $φ$ -strongly monotone
(iv): the mappings $g_{1} : = S_{1}^{- 1} \circ T_{1}$ and $g_{2} : = S_{2}^{- 1} \circ T_{2}$ are well defined and single-valued [44]. Also $I_{E_{1}} - g_{1}$ and $I_{E_{2}} - g_{2}$ are demi-closed at zero.
(v): a solution of the constrained SCCPP associated with $T_{1}, S_{1}$ and $T_{2}, S_{2}$ solves the constrained IPOP (see Lemma 3 and Remark 1).
(i)–(iv) verifies all the assumptions of Corollary 2; hence, by (v), $(5)$ converges to a solution of the IPOP.

4.3. Common Biological and/or Physical Objective Criteria in RTP

We now demonstrate that the reformulation presented in Section 4.2 accommodates several important physical and/or biological objective functions applied in RTP. We consider five examples. In each example, we provide

f_{1}, f_{2}, h

, and

θ

for which assumptions (a)–(c) in Section 4.2 are satisfied. We adopt the following notations:

T : = number of voxels in the target volume (or PTV) O : = number of voxels in Organs at Risk (OAR)

for a total of

N

voxels, we index voxels in OAR by

T + 1, T + 2, \dots, T + O = N

Example 1.

(Physical criteria)

f (w) = \sum_{j = 1}^{T} a_{j} {(w_{j} - D_{p}^{j})}^{q_{1}} + \sum_{j = T + 1}^{N} b_{j} {(w_{j} - D_{m a x}^{j})}_{+}^{q_{2}}

where

{(.)}_{+} : = \max {., 0}

D_{p}^{j}

is the dose prescription for voxel

j

D_{m a x}^{j}

is the maximum dose allowed for voxel

j

a_{j}

and

b_{j}

are positive weighting factors, and the exponents

q_{1}

and

q_{2}

are even. Clearly,

f

is convex.

f_{1} : = 0 f_{2} : = f h : = f + \frac{λ}{2} ∥ . ∥^{2}, λ > 0 θ = 1

A very similar objective function was studied in [4]. Part of the results obtained indicates that with choice of exponents greater than

2

, dose homogeneity inside the target can be effectively improved.

Example 2.

(Biological criteria)

f (w) = - \ln (T C P_{L Q} (w)) - \ln (1 - N T C P_{A N} (w))

where

T C P_{L Q} (w) = \exp [- N_{0} \sum_{j = 1}^{T} v_{j} \exp (- α w_{j} - \frac{β w_{j}^{2}}{n_{f}})] N T C P_{A N} (w) = 1 - \exp [- \sum_{j = T + 1}^{N} v_{j} {(\frac{α w_{j} + \frac{β w_{j}^{2}}{n_{f}}}{α Δ})}^{a}], 1 \leq a < + \infty

N_{0}

is the initial number of clonogenic cells,

v_{j}

is the relative volume of voxel

j

α

and

β

are radiobiologic parameters of the linear quadratic (LQ) cell survival model such that

α^{2} n_{f} > 2 β

, and

n_{f}

is the number of treatment fractions. Clinical

α

β

n_{f}

values in this category for prostate, brain, breast, and liver tumors can be found in [45,46,47,48,49].

Δ

and

a

are structure-dependent scalars.

We remark that

f

is a negative log transform of an uncomplicated tumor control model derived from the LQ-Poisson TCP and a biological effective dose (BED) version of the Alber and Nusslin NTCP model [50]. The convexity of

f

follows from Appendix A and E in [43].

f_{1} : = 0 f_{2} : = f h : = f + \frac{λ}{2} ∥ . ∥^{2}, λ > 0 θ = 1

Example 3.

(Biological criteria)

f (w) = - \ln (1 - {(N T C P_{R S} (w))}^{s}), s > 0

where

N T C P_{R S} (w) = {[1 - \prod_{j = 1}^{N} {(1 - \exp (- s N_{0} \exp (- α w_{j} - \frac{β w_{j}^{2}}{n_{f}})))}^{v_{j}}]}^{\frac{1}{s}}

s

is the relative parameter characterizing the internal organization of a structure (or organ). Other parameters are the same as in Example 2.

f

is a negative log transform of the well known relative Seriality NTCP model. Its convexity was analyzed in [43].

f_{1} : = 0 f_{2} : = f h : = f + \frac{λ}{2} ∥ . ∥^{2}, λ > 0 θ = 1

In [4], a related objective function (weighted sum of NTCPs evaluated with Lyman NTCP model), and subjected to physical constraints specifying the “admissible minimum and maximum dose to targets”, was proposed and implemented.

Example 4.

(Hybrid criteria)

f (w) = \sum_{j = 1}^{T} a_{j} {(w_{j} - D_{p}^{j})}^{q_{1}} - \ln (1 - N T C P_{A N} (w))

where the symbols have the same meaning as in Examples 1 and 2. The convexity of

f

follows easily from [43].

f_{1} : = 0 f_{2} : = f h : = f + \frac{λ}{2} ∥ . ∥^{2}, λ > 0 θ = 1

Example 5.

(Physical criteria)

f (w) = \sum_{j = 1}^{T} a_{j} {(w_{j} - D_{p}^{j})}^{q_{1}} + \sum_{j = T + 1}^{N} b_{j} {(w_{j} - D_{m a x}^{j})}_{+}

where the symbols have the same meaning as in Example 1.

f

is clearly convex.

f_{1} : = \sum_{j = T + 1}^{N} b_{j} {(w_{j} - D_{m a x}^{j})}_{+} f_{2} : = \sum_{j = 1}^{T} a_{j} {(w_{j} - D_{p}^{j})}^{q_{1}} h : = f_{2} + \frac{λ}{2} ∥ . ∥^{2}, λ > 0 θ = 1

4.4. Insights on Algorithm Implementation

The algorithm labeled

(5)

has been carefully designed. Its efficiency in terms of speed and accuracy can easily be demonstrated by a simple numerical example. However, this may not be truly valuable. In this instance, what may truly be of high value and practical usefulness is to demonstrate those, when the algorithm is implemented for the reformulated IPOP, in a clinical setting.

Implementation in a clinical setting involves a series of steps beginning with the selection of a clinical case (for example, a prostate cancer case), and obtaining the relevant clinical datasets. These clinical datasets may include the original Digital Imaging and Communications in Medicine (DICOM) CT scan image, the DICOM radiotherapy structure files containing the contours of targets and organs at risk, and the dose prescriptions as well as treatment schedule. Furthermore, we require optimization data such as size, resolution, and number of voxel grids in each geometrically contoured structure, and beam information such as beam energy, couch, and gantry angles, number of beamlets, and the geometric location of each beamlet, required for the computation of the dose influence matrix, which is then coupled into the optimization algorithm (in this case,

(5)

) to generate the optimized fluence map. These processes are far from trivial. Thus, we refrain from illustrating and analyzing this implementation in this paper. Nevertheless, for the RTP objective functions presented in Section 4.3, we give explicit forms of the operator

g_{2}

appearing in

(5)

(for Examples 1 and 4) and an approximate form of

g_{2}

(for Example 2), to aid in executing

(5)

during clinical implementation.

Forms of

g_{2}

For Example 1, with

κ = 1, q_{1} = 4

and

q_{2} = 2

S_{2}^{- 1} (x) = u \in ℝ^{N}

such that:

i f j \leq T :

u_{j} = D_{p}^{j} + {(\frac{x_{j} - λ a_{j} D_{p}^{j}}{8 a_{j}} + \sqrt{{(\frac{x_{j} - λ a_{j} D_{p}^{j}}{8 a_{j}})}^{2} + {(\frac{λ}{12 a_{j}})}^{3}})}^{\frac{1}{3}} + {(\frac{x_{j} - λ a_{j} D_{p}^{j}}{8 a_{j}} - \sqrt{{(\frac{x_{j} - λ a_{j} D_{p}^{j}}{8 a_{j}})}^{2} + {(\frac{λ}{12 a_{j}})}^{3}})}^{\frac{1}{3}}

e l s e i f x_{j} > λ D_{m a x}^{j} :

u_{j} = \frac{2 b_{j} D_{m a x}^{j} + x_{j}}{2 b_{j} + λ}

e l s e

u_{j} = \frac{x_{j}}{λ}

Hence,

g_{2} (x) = S_{2}^{- 1} (λ x)

For Example 4, with

= 1, q_{1} = 4

a = 2

, and

\nabla^{2} λ > 1

S_{2}^{- 1} (x) = u \in ℝ^{N}

such that:

i f j \leq T :

u_{j} = D_{p}^{j} + {(\frac{x_{j} - λ a_{j} D_{p}^{j}}{8 a_{j}} + \sqrt{{(\frac{x_{j} - λ a_{j} D_{p}^{j}}{8 a_{j}})}^{2} + {(\frac{λ}{12 a_{j}})}^{3}})}^{\frac{1}{3}} + {(\frac{x_{j} - λ a_{j} D_{p}^{j}}{8 a_{j}} - \sqrt{{(\frac{x_{j} - λ a_{j} D_{p}^{j}}{8 a_{j}})}^{2} + {(\frac{λ}{12 a_{j}})}^{3}})}^{\frac{1}{3}}

else

u_{j} = ϕ_{j} = - \frac{α n_{f}}{2 β} + {(\frac{n_{f}^{2} (α^{2} \nabla^{2} x_{j} + \frac{α^{3} \nabla^{2} n_{f} λ}{2 β})}{8 β^{2} v_{j}} + \sqrt{{(\frac{n_{f}^{2} (α^{2} \nabla^{2} x_{j} + \frac{α^{3} \nabla^{2} n_{f} λ}{2 β})}{8 β^{2} v_{j}})}^{2} + {(\frac{n_{f}^{2} (α^{2} \nabla^{2} λ - α^{2} v_{j})}{12 β^{2} v_{j}})}^{3}})}^{\frac{1}{3}} + {(\frac{n_{f}^{2} (α^{2} \nabla^{2} x_{j} + \frac{α^{3} \nabla^{2} n_{f} λ}{2 β})}{8 β^{2} v_{j}} - \sqrt{{(\frac{n_{f}^{2} (α^{2} \nabla^{2} x_{j} + \frac{α^{3} \nabla^{2} n_{f} λ}{2 β})}{8 β^{2} v_{j}})}^{2} + {(\frac{n_{f}^{2} (α^{2} \nabla^{2} λ - α^{2} v_{j})}{12 β^{2} v_{j}})}^{3}})}^{\frac{1}{3}}

Hence,

g_{2} (x) = S_{2}^{- 1} (λ x)

For Example 2, with

κ = 1, a = 2

, and

α^{2} n_{f} > 2 β

S_{2}^{- 1} (x) \approx u \in ℝ^{N}

such that:

i f j \leq T :

{\begin{matrix} u_{j} = \emptyset_{j}, if j \leq T \\ u_{j} = ϕ_{j}, otherwise \end{matrix}

where

\emptyset_{j}

satisfies

a_{j} \emptyset_{j}^{3} + b_{j} \emptyset_{j}^{2} + c_{j} \emptyset_{j} + d_{j} = 0

with

d_{j} = - (N_{0} α v_{j} + x_{j})

a_{j} = \frac{- N_{0} β (α^{2} - \frac{2 β}{n_{f}}) v_{j}}{n_{f}}, b_{j} = N_{0} α (\frac{3 β}{n_{f}} - \frac{α^{2}}{2}) v_{j}, c_{j} = N_{0} v_{j} (α^{2} - \frac{2 β}{n_{f}}) + λ

Hence,

g_{2} (x) \approx S_{2}^{- 1} (λ x)

For other methods of approximation see [51,52,53].

5. Conclusions

We have introduced and discussed a nonlinear optimization formulation called the split common coincidence point problem (SCCPP). We showed that this formulation is applicable to inverse planning optimization with increased flexibility in accommodating several biological and/or physical criteria recommended for RTP optimization. Although the objective functions are required to be convex, equivalent convex reformulations of commonly used non-convex objective criteria in RTP exist in the literature. We developed and proved convergence of an iterative algorithm to a solution of the SCCPP, which in this particular application corresponds to an extremum of some (bio)physically-based inverse planning optimization problem. In addition, we provided useful insights toward the implementation of the algorithm in a clinical setting.

Author Contributions

Conceptualization, C.E.C. and L.C.O.; methodology, C.E.C. and L.C.O.; validation, C.E.C.; formal analysis, L.C.O.; resources, C.E.C. and L.C.O.; writing—original draft preparation, L.C.O.; writing—review and editing, C.E.C. and L.C.O.; supervision, C.E.C.; project administration, C.E.C.; funding acquisition, C.E.C. and L.C.O. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the African Union Commission [HRST/ED/606/1888/09.18], and the APC was funded by the African Development Bank. Lois C. Okereke thanks the African-German Network of Excellence in Science (AGNES) for granting a Mobility Grant in 2019; the grant is generously sponsored by the German Federal Ministry of Education and Research and supported by the Alexander von Humboldt Foundation.

Acknowledgments

The authors thank Bello, A.U. for useful discussions on the computations and proofs of the theorems. The authors also acknowledge the support of Emmanuel A. Onwukwe in generously typesetting this manuscript, and for making critical suggestions that contributed to its structure.

Conflicts of Interest

The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

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Chidume, C.E.; Okereke, L.C. Split Common Coincidence Point Problem: A Formulation Applicable to (Bio)Physically-Based Inverse Planning Optimization. Symmetry 2020, 12, 2086. https://doi.org/10.3390/sym12122086

AMA Style

Chidume CE, Okereke LC. Split Common Coincidence Point Problem: A Formulation Applicable to (Bio)Physically-Based Inverse Planning Optimization. Symmetry. 2020; 12(12):2086. https://doi.org/10.3390/sym12122086

Chicago/Turabian Style

Chidume, Charles E., and Lois C. Okereke. 2020. "Split Common Coincidence Point Problem: A Formulation Applicable to (Bio)Physically-Based Inverse Planning Optimization" Symmetry 12, no. 12: 2086. https://doi.org/10.3390/sym12122086

APA Style

Chidume, C. E., & Okereke, L. C. (2020). Split Common Coincidence Point Problem: A Formulation Applicable to (Bio)Physically-Based Inverse Planning Optimization. Symmetry, 12(12), 2086. https://doi.org/10.3390/sym12122086

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Split Common Coincidence Point Problem: A Formulation Applicable to (Bio)Physically-Based Inverse Planning Optimization

Abstract

1. Introduction

2. Preliminaries

3. Main Results

3.1. Split Common Coincidence Point Problem (SCCPP)

3.2. Optimization by Generalized Coincidence Point Problem

3.3. Examples of S-Pseudocontraction

3.4. Approximation of Coincidence Points

4. Application to Inverse Planning Optimization

4.1. An Inverse Planning Optimization Problem (IPOP)

4.2. SCCPP Reformulation of IPOP

4.3. Common Biological and/or Physical Objective Criteria in RTP

4.4. Insights on Algorithm Implementation

5. Conclusions

Author Contributions

Funding

Acknowledgments

Conflicts of Interest

References

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