Corresponding author: Nassar H. S. Haidar (nhaidar@suffolk.edu)

Academic editor: Yury Korovin

We demonstrate how the therapeutic utility index and the ballistic index for dynamical neutron cancer therapy (^{*} = (^{*}^{*}^{*}^{*}

In both dynamical and/or stationary (B/Gd) neutron cancer therapy (

A standard technique conventionally used for boosting the ballistic index in stationary

It should be underlined, from the outset of this paper, that the diffusion model for neutron transport near boundaries of strongly absorbing

In the therapeutic setup for dynamical (B/Gd) _{a}_{a}

Sketch to illustrate the two opposing neutron beams in a dynamical (B/Gd) _{α}_{α}_{β}_{β}

The first nonlinear inequality constraint _{1}(_{0}) , _{0} is the life time of thermal neutrons in

_{0}.

It is well known that if it happens that the objective functions are conflicting, i.e. if ῆ(

To have a look at the posing problem from a different angle, we note that the results of (

with 6 control variables. Here _{1}, σ_{2}, σ_{3}, σ_{4}) is a vector of slack variables, see e.g. (

Problem (3) is clearly overdefined and may, in general be unsolvable, unless when at least four of the above 10 equations happen to be inactive (or insignificant). Luckily this turns out to take place for the posing therapeutic setup, as to be demonstrated in the result that follows.

In addition to the notation of (_{R}_{;Λ}_{m}_{R}_{;Π}_{m}_{R}_{;Λ} and ℵ_{R}_{;Π} are coupling factors (

we shall also need: (^{+} =

^{*}^{*}^{*}_{0}, ^{*}_{0}/2)^{+}).

in which,

and all the symbols have their usual meaning, as in (70) of (

The basic modal

in which

is a bell-shaped convex function of _{n}^{4}) . A peak height that diminishes with increasing

is _{11} (

Consequently, as a sum (or difference, due to the sign pattern of

should be convex with a minimum at

conceived as a constant independent of

As for the basic modal

the term

is another bell-shaped, flatter than

but concave function of _{n}

i.e. sin[Ω_{mn}

it can be argued that Ņ_{mn}

is Ņ_{11}(

should be concave with a maximum at

conceived as a constant independent of

Hence ῆ(

Second we consider (4), as a function of

, leads to a conclusion that

should be convex with a minimum at

conceived as a constant independent of

As for the basic modal

the term

is another bell-shaped, flatter than

but concave function of _{n}

indicates that _{mn}

is _{11}(

Then we observe that the denominator of (4), with

conceived as a constant independent of

Hence ῆ(

Third, we consider (4) as a function of _{mn}^{–βnε}_{11} (

should be concave with maximum attained at

conceived as a constant independent of

Quite distinctively, the basic modal _{mn}

which is almost periodic in

represents a half-range expansion of a certain periodic function

The denominator of (4) is simply this periodic function added to a constant (independent of

Now ῆ(_{k}

ε_{k}

with

ῆ(ω, _{k}) < ῆ(ω, _{k+1}), ∀

of (74), in (

The basic modal

in which

is a bell-shaped convex function of _{n}

being _{11}(

Consequently, as a sum (or difference, due to the sign pattern (–1)^{n}^{–1}) of equicentral bell shapes,

should be convex with a minimum at

conceived as a constant independent of

Also the basic modal

has the same convex behavior as _{mn}

and

with a minimum at

Then the denominator of (7), with

conceived as a constant independent of

Hence ᾶ(

Second, we consider (7), as a function of

and

leads to the conclusion that ᾶ(

Third, we consider (7) for ᾶ as a function of _{mn}^{–βnε}^{n}^{–1}) of such aligned concave functions, where _{11}(

should be concave with maximum attained at

conceived as a constant independent of

Quite distinctively, the basic modal _{mn}^{–βnε}^{n}^{–1}) of such aligned concave functions, where _{11}(

should be concave with maximum attained at

conceived as a constant independent of

Obviously ᾶ(

_{0}, _{0}, i.e. ^{*}_{0}, Pareto optimal. Also since both ῆ(^{*}^{*}

The situation with _{k}^{*}_{1} ≈ π/ϖ^{*}_{0}/2. However, by structure of _{k}^{*}_{0}/2)^{+}, Pareto optimal. Here the proof ends.

From the previous proof, it is clear how the sensitivities of ῆ and ᾶ to variations either in

In some cancer patients, transport of thermal neutrons by neutron guides or neutron optical fibers through the regions Λ and Π (with respective thicknesses _{Λ} and _{Π} and neutron macroscopic removal cross sections Σ_{Λ} and Σ_{Π}), may turn out to be medically unfeasible. As an approximate substitute to solving the composite Λ ∪

and

As a result, the entire analysis, reported in this note, holds true if we replace each _{m}_{m}

In this note, we have demonstrated that the therapeutic indices in

is proved, for any given pair {

These facts pave the way towards launching the building of the first experimental benchmark for this new kind of ^{*}, can further be investigated.

The author is grateful to two anonymous referees for their critical reading of an original version of this article and for a number of useful comments.