# A mathematical justification for metronomic chemotherapy in oncology

Luis A. Fernández † Cecilia Pola † Judith Sáinz-Pardo ⋆

###### Abstract

We mathematically justify metronomic chemotherapy as the best strategy to
apply most cytotoxic drugs in oncology for both curative and palliative
approaches, assuming the classical pharmacokinetic model together with the
Emax pharmacodynamic and the Norton-Simon hypothesis.

From the mathematical point of view, we will consider two mixed-integer
nonlinear optimization problems, where the unknowns are the number of the
doses and the quantity of each one, adjusting the administration times a
posteriori.

$\dagger$ Dep. Matemáticas, Estadística y Computación,

Universidad de Cantabria, Avda. de los Castros, s/n, 39005 Santander, Spain

⋆ Instituto de Física de Cantabria (UC-CSIC), Avda. de los Castros, s/n,

39005 Santander, Spain

E-mail address, Luis A. Fernández<EMAIL_ADDRESS>
E-mail address, Cecilia Pola<EMAIL_ADDRESS>
E-mail address, Judith Sáinz-Pardo<EMAIL_ADDRESS>
Mathematics Subject Classiﬁcation: 93C15, 92C50, 90C30

Keywords: Metronomic chemotherapy, mixed-integer nonlinear optimization,
Norton-Simon hypothesis, Emax model.

## Introduction

Since the origins of chemotherapy in oncology, the standard and most widely
used treatment has been the maximum tolerated dose (MTD) one: “the higher the
dose, the better”, whose rationale seems very intuitive. As everyone knows,
the trade-off lies in the side effects due to drug toxicity, which not only
diminishes quality of life for the patient (adding more illness to the
existing one), but also make it necessary to impose rest periods between
cycles of therapy, which compromises a good resolution of the cancer
treatment. For this reason, the design of more effective and less toxic
chemotherapeutic regimens (by choosing the timing and fractionation of the
doses) is a very active area of research in oncology.

A turning point in cancer chemotherapy treatments can be placed in the year
$2000$, when a revolutionary alternative approach called “metronomic
chemotherapy” (MC) was proposed, see [13] and the references therein.
According to National Cancer Institute [7], this is the “treatment in which
low doses of anticancer drugs are given on a continuous or frequent, regular
schedule (such as daily or weekly), usually over a long time. Metronomic
chemotherapy causes less severe side effects than standard chemotherapy.
Giving low doses of chemotherapy may stop the growth of new blood vessels that
tumors need to grow”. In oncology, this last effect is referred to as “anti-
angiogenesis”.

The historical controversy between efficacy in tumor killing and lack of
toxicity has given rise to numerous studies and experimental trials comparing
MTD and MC, the main conclusion of which can be summarized by saying that “the
body of experimental and clinical evidence, coupled with theoretical
considerations […] point to MC as a preferred course of action”, see [5].
Among the theoretical considerations, in addition to the anti-angiogenesis
effect, we can mention that MC can activate anti-tumor immunity and minimize
therapeutic resistance.

In this work, we will justify mathematically that MC is the best strategy in
oncology for most cytotoxic drugs and both curative and palliative treatments,
using simple well-established mathematical models and without taking these
additional benefits into account. To that end, we will study two different
optimization problems that arise when modeling usual chemotherapy treatments
for cancer: to minimize the tumor volume at a fixed final time (curative
approach) or to minimize the total cumulative dose that a patient must take
during a specific period of time for maintaining the tumor volume below a
given threshold (palliative approach). We will assume a Gomperztian type tumor
growth and the Norton-Simon hypothesis to represent the growth-inhibitory
influence due to the cytotoxic chemotherapy effect (see [9, 10]) with the
classical pharmacokinetics and the Emax pharmacodynamic model. As far as we
know, this is the first time that the superiority of MC is mathematically
justified for minimally parameterized models. Up to now, the efficiency of MC
was related with heterogeneous tumor populations (sensitive/resistant cells)
and palliative purposes, see [6] and the references therein.

From the mathematical point of view, the optimization problems under
consideration are nonlinear and of mixed-integer type (where we will have the
number of treatment doses as an integer variable, $N$, and the amount of each
individual dose as a continuous variable, $d_{i}$). In both cases we will give
explicit expressions of the approximated optimal solutions which will lead us
to choose treatments corresponding to MC. A posteriori, the administration
times can be adjusted (in fact, in several ways) completing the description of
the treatment. We will illustrate the utility of our approach with numerical
experiments concerning the treatment of some brain tumors that exhibit
Gompertzian growth (see [14]) using a cytotoxic drug called Temozolomide
(TMZ). This drug is the subject of many clinical and biological recent studies
(see [2, 3, 8, 11] and their references). The origin of this paper was an
academic project [12] submitted by the third author under the supervision of
the other authors.

##  1 Associated optimization problems

The Gompertzian law has often been used to simulate the growth of some
untreated tumors (see for instance [1]). In mathematical terms this can be
described by the following ordinary differential equation (ODE):

$L^{\prime}(t)=\Psi(L(t)),$ (1)

where

$\Psi(L)=\xi L\log\left(\displaystyle\frac{\theta}{L}\right).$

In the above equations, $L(t)$ represents the tumor volume (or tumor size) at
time $t$, $\xi$ its growth rate and $\theta$ the maximum size it can reach
(also called carrying capacity for biological systems). In practice, the
parameters $\xi$ and $\theta$ can also change with time (due to angiogenesis),
but here we will focus on the simpler problem in which they are fixed.

As mentioned before, when using a cytotoxic drug to treat the tumor, we will
consider the Norton-–Simon hypothesis to model its effect; more precisely, we
will assume that the growth-inhibiting effect due to the treatment is
proportional to the growth rate of the untreated tumor (see [9, 10]).
Mathematically, this can be written as

$L^{\prime}(t)=\Psi(L(t))(1-\rho(t)).$ (2)

Here the term $\Psi(L(t))\rho(t)$ represents the growth-inhibitory influence
due to the cytotoxic chemotherapy effect and it reflects both, the level of
the therapy at time $t$, $\rho(t)$, and the tumor’s sensitivity to therapy.

First let us present a general result for the existence and uniqueness of
solution for the Cauchy problem associated with (2) (see [4] for the proof).
As usual, we will denote by $W^{1,\infty}(0,T)$ the Sobolev space of all
functions in $L^{\infty}(0,T)$ having first order weak derivative (in the
distributional sense) also belonging to $L^{\infty}(0,T)$. It is well known
that $W^{1,\infty}(0,T)$ can be identified with $C^{0,1}[0,T]$, the space of
Lipschitz continuous functions in $[0,T]$, after a possible redefinition on a
set of zero measure.

######  Theorem 1.

Let us assume that $\xi$, $\theta$, $L_{0}$ and $T$ are given positive real
numbers, $L_{0}\in(0,\theta)$ and $\rho\in L^{\infty}(0,T)$. Then, there
exists a unique solution $L\in W^{1,\infty}(0,T)$ of the following Cauchy
problem

$L^{\prime}(t)=\xi L(t)\log{\left(\frac{\theta}{L(t)}\right)}(1-\rho(t)),\ \
L(0)=L_{0},$ (3)

given by

$L(t)=\theta\exp{\left(\log{(L_{0}/\theta)}\exp{(-\xi\int_{0}^{t}(1-\rho(s))ds)}\right)},\
\ \forall t\in[0,T].$ (4)

Depending on the expression of $\rho(t)$, different pharmacodynamics (PD) can
be studied. In this work we consider the classic Emax model:

$\rho(t)=\frac{k_{1}c(t)}{k_{2}+c(t)},$ (5)

where $k_{1}$ and $k_{2}$ are fixed positive real numbers and $c(t)$ denotes
the concentration of the drug in the tumor at time $t$. Other choices have
been considered in the literature, as the Skipper model where
$\rho(t)=k_{1}c(t)$. The Emax model seems to be more appropriate from the
clinical point of view because it saturates for high concentration values.

Another crucial term is related with the pharmacokinetics (PK). We use the
following Cauchy problem to describe the PK of the drug:

$\left\\{\begin{array}[]{l}c^{\prime}(t)=-\lambda
c(t)+\displaystyle\sum^{N}_{i=1}\sigma d_{i}\delta(t-t_{i}),\\\ c(0)=0.\\\
\end{array}\right.$ (6)

The coefficient $\lambda$ is the clearance rate and it is related to the half-
life of the drug. The second term on the right hand of the equation depends on
both the specific drug and the way it is administered. We assume that $N$
doses, $\\{d_{i}\\}_{i=1}^{N}$, will be used at $N$ dosage times,
$\\{t_{i}\\}_{i=1}^{N}$, such that $0\leq t_{1}<\ldots<t_{N}$. The coefficient
$\sigma$ is determined on the basis of drug-, patient- and tumor-specific
parameters. For instance, in the case of brain tumors the drug loss during the
transport to the brain has to be taking into account, see [2] and the
references therein. In other cases,
$\displaystyle\sigma=\frac{\alpha}{V_{D}\beta}$, where $\alpha$ is the
patient’s body surface area, $V_{D}$ is the volume of distribution of the drug
and $\beta$ is the patient weight in $kg$ if $V_{D}$ is given in $l/kg$.
Finally, $\delta(t-t_{i})$ is the Dirac delta distribution concentrated at
$t_{i}$.

In principle, it seems reasonable to consider the administration times
$\\{t_{i}\\}_{i=1}^{N}$ as variables of the problem, as well as $N$ and the
doses $\\{d_{i}\\}_{i=1}^{N}$. However, this approach would lead us into
serious difficulties because the criterion for deciding a priori the
characteristics that determine the feasible protocols may be unknown even to
the specialist. In practice, this would make the mathematical approach
unfeasible. Instead, in a first step, we can consider the administration times
as data, fixed according to some pattern. It is clear that this assumption
will implicitly affect the solution obtained: for example, a daily regimen
will inevitably lead to a solution with small doses, because MTD cannot be
administered over a long period due to the serious side effects it produces.
Apparently, it looks impossible to get out of this vicious circle, but we will
see that this can be done for most cytotoxic drugs by following our approach
and readjusting the administration times a posteriori, once $N$ and the doses
$\\{d_{i}\\}_{i=1}^{N}$ have been determined.

Now, let us present our optimization problems: in this paper we are interested
in the solutions of two optimization problems associated with the Cauchy
problem (3) using the Emax model (5) and when the following data are given:
the final time $T$, $T>t_{N}$ and the levels of effectiveness and the toxicity
of each individual dose, $d_{min}$ and $d_{max}$, respectively. We assume
$0<d_{min}<d_{max}$. To simplify some expressions we will use $t_{N+1}=T$. As
we have explained before, the administration times $t_{i}$, $i=1,\ldots,N$,
are fixed a priori following some pattern, but the number $N$ of doses of the
treatment is a variable to be determined.

Firstly, we consider a curative approach with the goal of minimizing the tumor
volume at the final time $T$ with a fixed cumulative dose, $D$, that will be
divided into $N$ smaller doses, $\\{d_{i}\\}_{i=1}^{N}$, to be determined.
Hence we formulate the following problem with $N+1$ variables

$(P_{1})\left\\{\begin{array}[]{lcc}\min\hskip 5.69046ptL(T),\\\
N\in\mathbb{N},d=(d_{1},\ldots,d_{N})\in\mathbb{R}^{N},\\\ \mbox{subject to
}\displaystyle\sum_{i=1}^{N}d_{i}=D,\\\ \hskip 54.06006ptd_{min}\leq d_{i}\leq
d_{max},\hskip 14.22636pti=1,\ldots,N.\end{array}\right.$ (7)

Using (4) and the fact that $L_{0}<\theta$ (where $L_{0}$ is the initial tumor
size), we transform $(P_{1})$ into the following problem:

$\left\\{\begin{array}[]{lcc}\max\hskip
5.69046pt\displaystyle\int_{0}^{T}\rho(s)ds,\\\
N\in\mathbb{N},d=(d_{1},\ldots,d_{N})\in\mathbb{R}^{N},\\\ \mbox{subject to
}\displaystyle\sum_{i=1}^{N}d_{i}=D,\\\ \hskip 54.06006ptd_{min}\leq d_{i}\leq
d_{max},\hskip 14.22636pti=1,\ldots,N.\end{array}\right.$ (8)

Now, let’s formulate the problem related to a palliative approach in which we
want to minimize the total dose (determining $N$ and $d_{i}$, for
$i=1,\ldots,N$) while maintaining the tumor volume at $T$ below a given
threshold $L_{*}>0$ that is not harmful to the patient, allowing him to have
an acceptable quality of life. We consider the same bound constraints on the
individual doses as in the previous optimization problem:

$(P_{2})\left\\{\begin{array}[]{lcc}\min\hskip
5.69046pt\displaystyle\sum_{i=1}^{N}d_{i},\\\
N\in\mathbb{N},d=(d_{1},\ldots,d_{N})\in\mathbb{R}^{N},\\\ \mbox{subject to
}L(T)\leq L_{*},\mbox{ }\\\ \hskip 51.21504ptd_{min}\leq d_{i}\leq
d_{max},\hskip 14.22636pti=1,\ldots,N.\end{array}\right.$ (9)

Using again (4) it is easy to check the equivalence:

$L(T)\leq L_{*}\Longleftrightarrow\int_{0}^{T}\rho(s)ds\geq
T+T_{R},\mbox{with}\ \displaystyle
T_{R}=\frac{1}{\xi}\log\left(\frac{\log\left(L_{*}/\theta\right)}{\log\left(L_{0}/\theta\right)}\right).$
(10)

Therefore, we can reformulate the problem $(P_{2})$ as follows:

$\left\\{\begin{array}[]{lcc}\min\hskip
5.69046pt\displaystyle\sum_{i=1}^{N}d_{i},\\\
N\in\mathbb{N},d=(d_{1},\ldots,d_{N})\in\mathbb{R}^{N},\\\ \mbox{subject to
}\displaystyle\int_{0}^{T}\rho(s)ds\geq T+T_{R},\\\ \hskip
56.9055ptd_{min}\leq d_{i}\leq d_{max},\hskip
14.22636pti=1,\ldots,N.\end{array}\right.$ (11)

##  2 Optimal treatments

In view of our optimization problems it is convenient to have an explicit
formulation for the concentration of the drug.

######  Proposition 1.

The solution of the Cauchy problem (6) is given by

$c(t)=\left\\{\begin{array}[]{ll}0,&\hskip 8.5359pt\mbox{if}\ \
t\in[0,t_{1}),\\\ \sigma e^{-\lambda t}e^{\lambda t_{1}}d_{1},&\hskip
8.5359pt\mbox{if}\ \ t\in[t_{1},t_{2}),\\\ \sigma e^{-\lambda t}(e^{\lambda
t_{1}}d_{1}+e^{\lambda t_{2}}d_{2}),&\hskip 8.5359pt\mbox{if}\ \
t\in[t_{2},t_{3}),\\\ \vdots&\hskip 8.5359pt\vdots\\\ \sigma e^{-\lambda
t}(e^{\lambda t_{1}}d_{1}+\ldots+e^{\lambda t_{N-1}}d_{N-1}),&\hskip
8.5359pt\mbox{if}\ \ t\in[t_{N-1},t_{N}),\\\ \sigma e^{-\lambda t}(e^{\lambda
t_{1}}d_{1}+\ldots+e^{\lambda t_{N}}d_{N}),&\hskip 8.5359pt\mbox{if}\ \
t\in[t_{N},T).\\\ \end{array}\right.$ (12)

###### Proof.

Applying Laplace transform to problem (6) we get

$s{\cal L}(c)(s)=-\lambda{\cal L}(c)(s)+\sum^{N}_{i=1}\sigma
d_{i}e^{-st_{i}},$

and consequently,

${\cal L}(c)(s)=\sum^{N}_{i=1}\sigma
d_{i}\frac{e^{-st_{i}}}{s+\lambda}=\sum^{N}_{i=1}\sigma d_{i}e^{-st_{i}}{\cal
L}(e^{-\lambda t})(s)=\sum^{N}_{i=1}\sigma d_{i}{\cal L}(g_{i})(s),$

where

$g_{i}(t)=\left\\{\begin{array}[]{ll}0,&\hskip 8.5359pt\mbox{if}\ \ 0\leq
t\leq t_{i},\\\ e^{-\lambda(t-t_{i})},&\hskip 8.5359pt\mbox{if}\ \ t\geq
t_{i}.\end{array}\right.$

This implies that

$c(t)=\sum^{N}_{i=1}\sigma d_{i}g_{i}(t),$

which is equivalent to (12). ∎

We continue by obtaining the expression for the integral term appearing in (8)
and (11). In this respect, we present the following result:

######  Lemma 1.

Let us assume that $c(t)$ is given by (12) and $\rho(t)$ by (5). Then, it is
verified that

$\int_{0}^{T}\rho(s)\;ds=\frac{k_{1}}{\lambda}\log\left(\frac{(d_{1}+\tilde{k}_{2}){\displaystyle\prod_{i=2}^{N}}({\displaystyle\sum_{j=1}^{i-1}}d_{j}e^{\lambda(t_{j}-t_{i})}+d_{i}+\tilde{k}_{2})}{{\displaystyle\prod_{i=1}^{N}}({\displaystyle\sum_{j=1}^{i}}d_{j}e^{\lambda(t_{j}-t_{i+1})}+\tilde{k}_{2})}\right),$
(13)

where $\tilde{k}_{2}=k_{2}/\sigma$ and $t_{N+1}=T$.

###### Proof.

Using (12) and the definition of $\rho$, we derive (13) from the following
equalities:

$\int_{t_{1}}^{t_{2}}\displaystyle\frac{c(s)}{k_{2}+c(s)}ds=\frac{1}{\lambda}\log\left(\frac{d_{1}+\tilde{k}_{2}}{d_{1}e^{\lambda(t_{1}-t_{2})}+\tilde{k}_{2}}\right),$

$\int_{t_{i}}^{t_{i+1}}\displaystyle\frac{c(s)}{k_{2}+c(s)}ds=\frac{1}{\lambda}\log\left(\frac{d_{1}e^{\lambda(t_{1}-t_{i})}+\cdots+d_{i-1}e^{\lambda(t_{i-1}-t_{i})}+d_{i}+\tilde{k}_{2}}{d_{1}e^{\lambda(t_{1}-t_{i+1})}+\cdots+d_{i}e^{\lambda(t_{i}-t_{i+1})}+\tilde{k}_{2}}\right),$

for $i=2,\ldots,N-1$ and taking into account that the integral corresponding
to the interval $[t_{N},T]$ is similar to the previous expression changing
$t_{i}$ by $t_{N}$ and $t_{i+1}$ by $T$. ∎

###  2.1 Curative approach

We start by considering the problem $(P_{1})$ defined in Section 1 (in which
we want to minimize the tumor volume at time $T$ with a fixed cumulative dose
$D$). In this case, using (8), (13), the positivity of $k_{1}/\lambda$ and the
monotonicity of the logarithm function, we reformulate the problem $(P_{1})$
as follows:

$\left\\{\begin{array}[]{lcc}Max\hskip
5.69046ptf_{1}(N,d)={\displaystyle\frac{(d_{1}+\tilde{k}_{2}){\displaystyle\prod_{i=2}^{N}}({\displaystyle\sum_{j=1}^{i-1}}d_{j}e^{\lambda(t_{j}-t_{i})}+d_{i}+\tilde{k}_{2})}{{\displaystyle\prod_{i=1}^{N}}({\displaystyle\sum_{j=1}^{i}}d_{j}e^{\lambda(t_{j}-t_{i+1})}+\tilde{k}_{2})}},\\\
N\in\mathbb{N},d=(d_{1},\ldots,d_{N})\in\mathbb{R}^{N},\\\ \mbox{subject to
}\displaystyle\sum_{i=1}^{N}d_{i}=D,\\\ \hskip 54.06006ptd_{min}\leq d_{i}\leq
d_{max},\hskip 14.22636pti=1,\ldots,N.\end{array}\right.$ (14)

Note that it is a mixed-integer nonlinear optimization problem with an integer
variable, $N$, and $N$ continuous variables, $d_{i}$.

In order to simplify the objective function $f_{1}$, along the rest of the
work, we will assume the main hypothesis that we are considering cytotoxic
drugs verifying

$d_{max}e^{-\lambda s}\ll\tilde{k}_{2},$ (15)

where $s$ denotes the minimum of the elapsed time periods; this is

$s=\min\\{t_{i+1}-t_{i}:i=1,\ldots,N\\}.$

Let us stress that this condition holds (at least) when $\lambda$ is
sufficiently large (or equivalently, the half-life of the drug is sufficiently
small) and that most of the usual cytotoxic drugs satisfy this condition, even
when $s$ is small.

Assuming (15), the following approximation for the objective function holds:

$f_{1}(N,d)\approx\displaystyle\prod_{i=1}^{N}\left(\frac{d_{i}}{\tilde{k}_{2}}+1\right),$
(16)

that leads to the mixed-integer nonlinear programming problem:

$(\hat{P}_{1})\left\\{\begin{array}[]{lcc}Max\hskip
5.69046pt\hat{f}_{1}(N,d)=\displaystyle\prod_{i=1}^{N}\left(\frac{d_{i}}{\tilde{k}_{2}}+1\right),\\\
N\in\mathbb{N},d=(d_{1},\ldots,d_{N})\in\mathbb{R}^{N},\\\ \mbox{subject to
}\displaystyle\sum_{i=1}^{N}d_{i}=D,\\\ \hskip 54.06006ptd_{min}\leq d_{i}\leq
d_{max},\hskip 14.22636pti=1,\ldots,N.\end{array}\right.$ (17)

At this point, it is very important to underline that the administration times
${\\{t_{i}\\}}_{i=1}^{N}$ do not appear in the formulation of $(\hat{P}_{1})$.
This is a crucial property that will allow us to break the vicious circle
mentioned in the previous section: for most cytotoxic drugs we will first
determine $N$ and the doses ${\\{d_{i}\\}}_{i=1}^{N}$ as the solution of
$(\hat{P}_{1})$ and then assign appropriate administration times, checking
that (15) holds.

Using the bound constraints and the total dose value, in order to have a non-
empty set of feasible points, we must assume

$[D/d_{max},D/d_{min}]\cap\mathbb{N}\neq\emptyset.$ (18)

In particular, hypothesis (18) implies that only a finite number of values for
variable $N$ have to be taken into account to solve the optimization problem.
Namely,

$N\in\\{\lceil D/d_{max}\rceil,\ldots,\lfloor D/d_{min}\rfloor\\},$ (19)

where $\lfloor x\rfloor$ denotes the greatest integer less than or equal to
$x$ and $\lceil x\rceil$ the least integer greater than or equal to $x$.

Now, for each fixed feasible value of $N$, we can formulate the following
continuous nonlinear programming problem:

$(\hat{P}_{1}^{N})\left\\{\begin{array}[]{lcc}Max\hskip
5.69046pt\hat{f}_{1}^{N}(d)=\displaystyle\prod_{i=1}^{N}\left(\frac{d_{i}}{\tilde{k}_{2}}+1\right),\\\
d=(d_{1},\ldots,d_{N})\in\mathbb{R}^{N},\\\ \mbox{subject to
}\displaystyle\sum_{i=1}^{N}d_{i}=D,\\\ \hskip 54.06006ptd_{min}\leq d_{i}\leq
d_{max},\hskip 14.22636pti=1,\ldots,N.\\\ \end{array}\right.$ (20)

It is not difficult to prove that equal-dosage is the optimal solution of the
previous problem:

######  Theorem 2.

Let us assume that $d_{min}>0$ and (18). Then, for each $N$ verifying (19),
the problem $(\hat{P}_{1}^{N})$ has a unique optimal solution $\hat{d}$ given
by

$\hat{d}_{i}=D/N,\ \ \mbox{for}\ \ i=1,\ldots,N.$

###### Proof.

For any feasible point $d$, using the relation between the geometric and
arithmetic means, we get

$\sqrt[N]{\prod_{i=1}^{N}\left(\frac{d_{i}}{\tilde{k}_{2}}+1\right)}\leq\frac{1}{N}{\displaystyle\sum_{i=1}^{N}\left(\frac{d_{i}}{\tilde{k}_{2}}+1\right)=\frac{D}{N\tilde{k}_{2}}+1},$

and therefore,
$\hat{f}_{1}^{N}(d)\leq\left(\frac{D}{N\tilde{k}_{2}}+1\right)^{N}$. By the
hypotheses, $\hat{d}=(D/N,\ldots,D/N)$ is a feasible point for
$(\hat{P}_{1}^{N})$ such that the corresponding objective function value
reaches the above upper bound, so the conclusion of the theorem follows
straightforwardly. ∎

As a consequence, the optimal solution of the approximate mixed-integer
optimization problem $(\hat{P}_{1})$ leads to the longest feasible treatment:

######  Corollary 1.

Let us assume that $d_{min}>0$ and (18). Then the optimal solution of
$(\hat{P}_{1})$ is given by $(\hat{N},\hat{d})$, with $\hat{N}=\lfloor
D/d_{min}\rfloor$ and $\hat{d}_{i}=D/\hat{N},$ for $i=1,\ldots,\hat{N}.$

###### Proof.

This can be deduced combining Theorem 2 with the fact that the auxiliary
function

$\varphi_{1}(x)=\left(\frac{1}{x}+1\right)^{x}$

is strictly increasing in $(0,+\infty)$ and noticing that

$\hat{f}_{1}(N,\hat{d})=\left(\frac{D}{N\tilde{k}_{2}}+1\right)^{N}=\left(\varphi_{1}\left(\frac{N\tilde{k}_{2}}{D}\right)\right)^{\frac{D}{\tilde{k}_{2}}}.$

∎

Furthermore, we can study the dependence of the objective function optimal
value with respect to $d_{min}$. Using previous expressions, we derive

$\hat{f}_{1}(\hat{N},\hat{d})=\left(\frac{D}{\hat{N}\tilde{k}_{2}}+1\right)^{\hat{N}}\approx\left(\frac{d_{min}}{\tilde{k}_{2}}+1\right)^{\frac{D}{d_{min}}}=\left(\varphi_{1}\left(\frac{\tilde{k}_{2}}{d_{min}}\right)\right)^{\frac{D}{\tilde{k}_{2}}}.$

On the one hand, this implies that to increase $\hat{f}_{1}(\hat{N},\hat{d})$
(and therefore to decrease the tumor size at the fixed final time) is
interesting to take $d_{min}$ as small as possible, in line with the
principles of MC. But, on the other hand, from the clinical point of view, it
is known that for any drug there is a minimum effective dose (MED) (the
smallest dose with a discernible useful effect) that must be taken into
account. In general, the estimation of the MED (and even its definition)
remains controversial and this carries over to the determination of the
optimal $d_{min}$ in the practice.

######  Remark 1.

  * i)

Choosing the Gompertzian law for the untreated tumor growth is not a crucial
aspect for deriving our results. Other typical ODEs with sigmoid-type
solutions may be reasonable choices, such as (1) with

$\Psi(L)=\frac{\xi}{\gamma}L\left(1-\displaystyle{\left(\frac{L}{\theta}\right)^{\gamma}}\right),$

with $\gamma>0$. For $\gamma=1$, we recover the well-known logistic ODE,
meanwhile for small $\gamma>0$ , this is an approximation of the Gompertz ODE,
see [1]. All these options lead to the same results.

  * ii)

Instead, it can be shown that the Norton-Simon hypothesis is crucial for our
argument. There are other classical hypotheses such as the log-kill one. It
states that a given dose of chemotherapy kills the same fraction of tumor
cells regardless of the size of the tumor at the time of treatment, which
leads to the following problem

$L^{\prime}(t)=\xi L(t)\log{\left(\frac{\theta}{L(t)}\right)}-\rho(t)L(t),\ \
L(0)=L_{0},$ (21)

instead of (3). Although the log-kill hypothesis continues to appear
frequently in the literature, it was shown in [9] to be inconsistent with some
clinical experiences.

  * iii)

We have studied a related problem for continuous (non-discrete) drug infusion
in [4] (labeled $(OP_{2})$ with $G=G_{2}$). There, it was proved the
appearance of a constant maintenance infusion rate (during a fairly long time
interval) in the expression of the optimal control. This can be seen as a
piecewise continuous version of our previous Corollary 1.

From a theoretical point of view, we have seen that the optimal curative
treatment (i.e. aiming to minimize the tumor size at a given time with a fixed
cumulative dose using drugs for which (16) holds) is the longest feasible
treatment with equal individual doses. This is the case for drugs with a short
half-life such as Temozolomide (TMZ), an oral alkylating agent with antitumor
activity in high and low grade gliomas (HGG and LGG, respectively). In the
monotherapy phase for adult patients with newly-diagnosed glioblastoma
multiforme, TMZ is usually administered consecutively for $5$ days of every
$28$ days (this is denoted by $5/28d$) for a maximum of $6$ cycles with
individual doses of $150$ $mg/m^{2}$ in first cycle and $200$ $mg/m^{2}$ in
the others. So, taking into account this usual treatment (UT), for our
numerical experiments we have fixed $T=210$ days for the final time and
$D=5{,}750$ $mg/m^{2}$ for the cumulative dose.

The parameter values used for our numerical computations are showed in Table
1. Following the reference cited in [2] we know that TMZ has a mean
elimination half-life of $1.8$ hours (i.e. $t_{1/2}=0.075$ days). The
clearance rate $\lambda$ is estimated using that
$\lambda=log(2)/t_{1/2}=9.242$ $days^{-1}$. The growth rate $\xi$ depends on
the type of tumor; the value $\xi=5.51e$$-3$ could correspond to an HGG with a
volume-doubling time about $126$ days (see [14] for growth dynamics of
glioblastomas). Following [2], we have used $\sigma=1.6(2.5e$$-3)\approx
4e$$-3$ that corresponds to a woman with a body surface of $1.6\;m^{2}$ and it
takes into account the fraction of TMZ getting to her brain interstitium. On
the other hand, it is well known that $k_{1}$ represents the maximum effect of
the drug on the tumor and $k_{2}$ is the effective concentration, i.e
concentration producing $50\%$ of the maximum effect. We have chosen
$k_{2}=0.36\ mg/l$, a value that is consistent with predicted peak
concentrations of TMZ from [2, 11]. Moreover, we have taken $k_{1}=60$ for
illustrative purposes. Let us note that the parameter $k_{1}$ does not play
any role either in the optimization process of this subsection nor in the main
hypothesis (15), but it does influence the evaluation of the tumor size. Let
us stress that (15) holds in this case, because

$d_{max}e^{-\lambda s}\approx 0.01938\ll\tilde{k}_{2}=90,$

with $s=1$. Finally, let us underline that the value of $\theta$ is
unnecessary here, because the initial datum and the results are given rescaled
in the form $L_{0}/\theta$ and $L(t)/\theta$, resp. (see Table 1 and Figure
1).

Table 1: Parameter values used for the numerical experiments. Parameters | Temozolomide (TMZ)
---|---
$\lambda$ ($days^{-1}$) | 9.242
$\xi$ ($days^{-1}$) | 5.51e$-3$
$k_{1}$ | 60
$k_{2}$ ($mg/l$) | 3.6e$-1$
$\sigma$ ($m^{2}/l$) | 4e$-3$
$d_{max}$ ($mg/m^{2}$) | 200
$T$ ($days$) | 210
$L_{0}/\theta$ | 2.5e$-1$
$t_{1}$ | 0

In Table 2 we present some numerical results to illustrate and compare the
optimal treatments corresponding to $(P_{1})$ without using the approximation
(16) (i.e. (14)) with those associated to $(\hat{P}_{1})$ (given by Theorem 2
and Corollary 1) in the case of individual doses between $d_{min}=100$
$mg/m^{2}$ and $d_{max}=200$ $mg/m^{2}$. To solve the problems associated with
$(P_{1})$ when the value for $N$ is fixed, we have used FMINCON (a nonlinear
programming solver provided in MATLAB’s Optimization Toolbox) with the SQP
algorithm and the value $1.e$$-8$ for the optimality tolerance. Here, we
present results only for the feasible values of $N$ taking into account the
length of the treatment interval and the number of dosage times for the 5/28d
schedule: $N\in[29,40]$. This number $N$ appears in the first column; then we
present two columns with the individual doses of the treatment described in
Theorem 2 and the ratio of the final tumor volume to the initial volume,
$L(T)/L_{0}$, for an initial medium size tumor ($25\%$ of carrying capacity);
the next three columns report results given by FMINCON: the minimum and
maximum values reached by the individual doses of each treatment,
$\bar{d}_{min}$ and $\bar{d}_{max}$, and its corresponding ratio $L(T)/L_{0}$.
Let us note at this point that, for each value of $N$, the individual doses
obtained with FMINCON are almost identical and the optimal solution is reached
administering $40$ doses (the longest 5/28d treatment, as Theorem 2 and
Corollary 1 state, using the approximation (16)). The last line of the table
is associated to the usual treatment (UT) described above using $N=30$ doses.
It is noteworthy that with it the final tumor volume is $91\%$ of the initial
volume while our optimal treatment provides $73\%$. At the top of Figure 1 we
see the evolution of the tumor volume, $L(t)/\theta$, for a selection of
treatments of Table 2.

Table 2: Results with $D=5{,}750$ $mg/m^{2}$ and $d_{min}=100$ $mg/m^{2}$ with 5/28d schedule. | $(\hat{P}_{1})$ | $(P_{1})$
---|---|---
$N$ | $D/N$ | $L(T)/L_{0}$ | $\bar{d}_{min}$ | $\bar{d}_{max}$ | $L(T)/L_{0}$
$29$ | $198.28$ | $0.93$ | $198.26$ | $198.32$ | $0.93$
$30$ | $191.67$ | $0.91$ | $191.65$ | $191.71$ | $0.91$
$31$ | $185.48$ | $0.89$ | $185.47$ | $185.54$ | $0.89$
$32$ | $179.69$ | $0.87$ | $179.67$ | $179.72$ | $0.87$
$33$ | $174.24$ | $0.85$ | $174.23$ | $174.28$ | $0.85$
$34$ | $169.12$ | $0.83$ | $169.10$ | $169.15$ | $0.83$
$35$ | $164.29$ | $0.81$ | $164.27$ | $164.32$ | $0.81$
$36$ | $159.72$ | $0.79$ | $159.71$ | $159.77$ | $0.79$
$37$ | $155.41$ | $0.77$ | $155.39$ | $155.43$ | $0.77$
$38$ | $151.32$ | $0.76$ | $151.30$ | $151.34$ | $0.76$
$39$ | $147.44$ | $0.74$ | $147.42$ | $147.46$ | $0.74$
$\mathbf{40}$ | $143.75$ | $\mathbf{0.73}$ | $\mathbf{143.74}$ | $\mathbf{143.78}$ | $\mathbf{0.73}$
30 (UT) |  |  | $150$ | $200$ | $0.91$
Figure 1: Optimization results for TMZ (see Tables 2 and 4).

The most relevant results of this section are given in Table 3 and at the top
of Figure 2, where we illustrate the influence of the level $d_{min}$ and
compare several optimal treatments corresponding to $(P_{1})$ requiring the
same cumulative dose ($D=5{,}750$ $mg/m^{2}$). The rest of the parameters are
given in Table 1. As we have mentioned previously, first we have fixed
$d_{min}$ and solved $(\hat{P}_{1})$; once we have determined the
corresponding solution ($\hat{N}$ and $\\{\hat{d}_{i}\\}_{i=1}^{\hat{N}}$) we
have selected the most appropriate schedules for TMZ used in some studies, see
[8]: the 7/14d with doses of $150$ $mg/m^{2}$, the 21/28d with doses of $100$
$mg/m^{2}$ and $75$ $mg/m^{2}$ and the daily regimen with $50$ $mg/m^{2}$
doses. The results of the penultimate column of Table 3 have been calculated
with the times given by these schedules. Let us underline that the specific
values of the administration times do not affect the optimal solution and the
only relevant factor is the maximum number of doses that can be administered
with the chosen $d_{min}$. We observe that the greatest tumor shrinkage is
achieved with the longest treatment which also has the lowest doses, in line
with MC. Moreover, three of the treatments of Table 3 are more effective than
the optimal treatment of Table 2, all of them improving the usual treatment
UT.

As predicted by the Norton-Simon hypothesis [10], it is apparent that the
optimal solution is more “dose-dense” than other feasible protocols, see the
top of Figure 2 (the period between cycles is shorter). Nevertheless,
concerning the “dose-intensity” (defined as the total dose administered during
a treatment, divided by its duration) it can be seen in the last column of
Table 3 that the optimal protocol has the lowest one.

Table 3: Numerical results for $(\hat{P}_{1})$ with $D=5{,}750$ $mg/m^{2}$ and varying $d_{min}$. $d_{min}$ | $\hat{N}$ | $D/\hat{N}$ | Schedule | $L(T)/L_{0}$ | Dose intensity
---|---|---|---|---|---
$150$ | $38$ | $151.32$ | 7/14d | $0.76$ | 78.77
$100$ | $57$ | $100.88$ | 21/28d | $0.53$ | 80.99
$75$ | $76$ | $75.66$ | 21/28d | $0.41$ | 59.28
$50$ | $115$ | $50.00$ | 28/28d | $0.27$ | 50.00
Figure 2: Optimization results with TMZ (see Tables 3 and 5).

###  2.2 Palliative approach

In this section we are interested in the optimization problem $(P_{2})$ i.e.
we want to minimize the total administered dose with a constraint on the tumor
size at the final time: $L(T)\leq L_{*}$, where $L_{*}$ is a given level
beforehand. Using (10) and Lemma 1, that constraint is equivalent to the
following inequality:

$f_{1}(N,d)\geq e^{\tilde{T}_{R}},$

being $f_{1}$ the objective function of problem (14) and

$\tilde{T}_{R}\stackrel{{\scriptstyle
def}}{{=}}\frac{\lambda}{k_{1}}(T+T_{R}).$ (22)

Let us remember that $T_{R}$ depends on the data $\xi$, $\theta$, $L_{0}$ and
$L_{*}$ (see (10)). Therefore, using (11), the optimization problem $(P_{2})$
can be reformulated as follows:

$\left\\{\begin{array}[]{lcc}\min\hskip
5.69046ptf_{2}(N,d)=\displaystyle\sum_{i=1}^{N}d_{i},\\\
N\in\mathbb{N},d=(d_{1},\ldots,d_{N})\in\mathbb{R}^{N},\\\ \mbox{subject to
}f_{1}(N,d)\geq e^{\tilde{T}_{R}},\\\ \hskip 54.06006ptd_{min}\leq d_{i}\leq
d_{max},\hskip 14.22636pti=1,\ldots,N,\end{array}\right.$ (23)

with $d_{min}>0$ and $\tilde{T}_{R}\in\mathbb{R}$ defined in (22). Using the
approximation (16) in the nonlinear constraint we arrive to the following
nonlinear programming problem

$(\hat{P}_{2})\left\\{\begin{array}[]{lcc}\min\hskip
5.69046ptf_{2}(N,d)=\displaystyle\sum_{i=1}^{N}d_{i},\\\
N\in\mathbb{N},d=(d_{1},\ldots,d_{N})\in\mathbb{R}^{N},\\\ \mbox{subject to
}\hat{f}_{1}(N,d)\geq e^{\tilde{T}_{R}},\\\ \hskip 54.06006ptd_{min}\leq
d_{i}\leq d_{max},\hskip 14.22636pti=1,\ldots,N,\end{array}\right.$ (24)

with $\hat{f}_{1}$ defined in (17). As one can see, $(\hat{P}_{2})$ is also a
mixed-integer nonlinear optimization problem with an integer variable $N$ and
$N$ continuous variables, $d_{i}$.

In this case, let us remark that the set of feasible values for $N$ is
infinite: using the general inequality constraint and the upper bound
constraints, the following inequality should be satisfied

$N\geq\lceil\tilde{T}_{R}/log(d_{max}/\tilde{k}_{2}+1)\rceil\stackrel{{\scriptstyle
def}}{{=}}N_{min}.$

In fact, only a finite subset has practical interest, i.e. those values that
also verify

$N\leq N_{max}\stackrel{{\scriptstyle
def}}{{=}}\lceil\tilde{T}_{R}/log(d_{min}/\tilde{k}_{2}+1)\rceil,$

because at least the pair
$(N_{max},\underbrace{d_{min},\ldots,d_{min}}_{N_{max}})$ is feasible and any
other feasible pair with larger value of $N$ will produce a greater value of
the objective function $f_{2}$. Therefore, to have a non-empty set of feasible
points we will assume

$[N_{min},N_{max}]\cap\mathbb{N}\neq\emptyset.$ (25)

Now, as in the previous subsection, we will focus on the associated continuous
problems generated when the value of the integer variable $N$ is fixed:

$(\hat{P}_{2}^{N})\left\\{\begin{array}[]{lcc}\min\hskip
5.69046ptf_{2}^{N}(d)=\displaystyle\sum_{i=1}^{N}d_{i},\\\
d=(d_{1},\ldots,d_{N})\in\mathbb{R}^{N},\\\ \mbox{subject to }\
\hat{f}_{1}^{N}(d)\geq e^{\tilde{T}_{R}},\\\ \hskip 54.06006ptd_{min}\leq
d_{i}\leq d_{max},\hskip 14.22636pti=1,\ldots,N,\\\ \end{array}\right.$ (26)

with the function $\hat{f}_{1}^{N}$ defined in (20). In the following result
we show that the optimal treatment for $(\hat{P}_{2}^{N})$ has also equal
doses and moreover, we obtain the explicit formula to determine them:

######  Theorem 3.

Let us assume $d_{min}>0$ and

$N\in[N_{min},N_{max}-1].$ (27)

Then, the optimization problem $(\hat{P}_{2}^{N})$ has a unique solution
$\hat{d}=(\hat{d}_{1},\ldots,\hat{d}_{N})$ given by

$\hat{d}_{i}=\tilde{k}_{2}(e^{\tilde{T}_{R}/N}-1),\ \ \ \mbox{for}\ \
i=1,\ldots,N.$ (28)

###### Proof.

First of all, it is immediate to check that the proposed dose vector is
feasible for the problem $(\hat{P}_{2}^{N})$, if $N$ verifies (27). On the
other hand, using once more the relation between the geometric and arithmetic
means, the doses of any feasible treatment for $(\hat{P}_{2}^{N})$ verify the
following inequalities

$\frac{1}{N}\sum_{i=1}^{N}(d_{i}+\tilde{k}_{2})\geq\left(\tilde{k}_{2}^{N}\hat{f}_{1}^{N}(d)\right)^{1/N}\geq\tilde{k}_{2}e^{\tilde{T}_{R}/N},$

and consequently,

$\sum_{i=1}^{N}d_{i}\geq N\tilde{k}_{2}(e^{\tilde{T}_{R}/N}-1).$ (29)

Taking into account that the lower bound of (29) is equal to
$f_{2}^{N}(\hat{d})$, the conclusion follows. ∎

Finally, using (16), we determine the optimal solution of the approximate
mixed-integer optimization problem of the palliative approach.

######  Corollary 2.

Let us assume that $d_{min}>0$ and (25). Then a global optimal solution of
$(\hat{P}_{2})$ problem is given by $(\hat{N},\hat{d})$ with

  * a)

$\hat{N}=N_{max}-1$ and $\hat{d}_{i}=d^{*}$, for $i=1,\ldots,\hat{N}$, if
$(N_{max}-1)d^{*}\leq N_{max}d_{min}$,

  * b)

$\hat{N}=N_{max}$ and $\hat{d}_{i}=d_{min}$, for $i=1,\ldots,\hat{N}$, if
$(N_{max}-1)d^{*}\geq N_{max}d_{min}$,

where $d^{*}=\tilde{k}_{2}(e^{\tilde{T}_{R}/(N_{max}-1)}-1)$.

###### Proof.

This can be deduced combining Theorem 28 with the fact that the auxiliary
function $\varphi_{2}(x)=x(e^{1/x}-1)$ is strictly decreasing in $(0,+\infty)$
and

$f_{2}(N,\hat{d})=N\tilde{k}_{2}(e^{\tilde{T}_{R}/N}-1)=\tilde{T}_{R}\tilde{k}_{2}\varphi_{2}\left(\frac{N}{\tilde{T}_{R}}\right),$
(30)

for $N\leq N_{max}-1$ and using that
$(\underbrace{d_{min},\ldots,d_{min}}_{N_{max}})$ is the solution of
$(\hat{P}_{2}^{N_{max}})$. ∎

######  Remark 2.

  * i)

Let us note that both possibilities presented in Corollary 2 can occur in
practical situations: for example, it can be observed in Table 5 that option
$a)$ is true for $d_{min}\in\\{50,100\\}$, while the option $b)$ holds for
$d_{min}\in\\{75,150\\}$.

  * ii)

For the trivial cases we recover the expected solutions. For instance, when
$L_{*}\rightarrow\theta^{-}$, we derive that $e^{\tilde{T}_{R}}\rightarrow 0$
and the nonlinear constraint is always verified and therefore the solution of
$(\hat{P}_{2})$ is given by $(\hat{N},\hat{d})=(1,d_{min})$, as one can expect
from the beginning.

  * iii)

We can also observe that $N_{max}$ is a decreasing function of $d_{min}$; in
fact, $N_{max}\rightarrow+\infty$ when $d_{min}\rightarrow 0$. Moreover, in
case $a)$, we have

$d_{min}\leq d^{*}\leq\frac{N_{max}}{N_{max}-1}d_{min},$

and $d^{*}\approx d_{min}$ for $N_{max}$ sufficiently large. From here we
deduce that case $a)$ is the prevalent one when $d_{min}$ is small enough.

  * iv)

Once more, we can now study the dependence of the optimal cumulative dose with
respect to $d_{min}$. Taking (30) into account and previous remark, we
conclude that

$f_{2}(\hat{N},\hat{d})=\tilde{T}_{R}\tilde{k}_{2}\varphi_{2}\left(\frac{N_{max}-1}{\tilde{T}_{R}}\right)\approx\tilde{T}_{R}\tilde{k}_{2}\varphi_{2}\left(\frac{1}{\log(d_{min}/\tilde{k}_{2}+1)}-\frac{1}{\tilde{T}_{R}}\right),$

and to decrease $f_{2}(\hat{N},\hat{d})$ we have to take $d_{min}$ as small as
possible. Again, this is in line with the principles of MC.

Table 4 and the plot at the bottom of Figure 1 are devoted to the palliative
approach with TMZ. We have considered the threshold $L_{\star}=0.1813\theta$
which is associated to the size of the tumor at the final time $T$ with the
best treatment of the curative approach (see Table 2). Let us remember that in
this case we are considering treatments with a maximum of $40$ doses
associated to the 5/28d schedule, $d_{min}=100$ $mg/m^{2}$ and the parameter
values of Table 1. Moreover, Table 4 presents two kind of optimal solutions.
For each value of $N$ (number of doses of the treatment) the following three
columns are devoted to problems $(\hat{P}_{2}^{N})$ (that use (16)) and the
last four columns correspond to optimal solutions obtained with FMINCON for
problems $(P_{2}^{N})$, setting the value for $N$ and without using that
approximation. Columns with labels $\bar{d}_{min}$ and $\bar{d}_{max}$ show
the minimum and maximum doses of the treatment given by FMINCON, respectively.
Let us note that both values in each row are almost equal. Moreover, for each
value of $N$, the results for the problems $(\hat{P}_{2}^{N})$ are good
approximations of the results obtained with FMINCON. Third and seventh columns
(with labels $f_{2}^{N}$ and $(f_{2}^{N})_{opt}$) present the total dose
associated to each optimal solution. We check that the longest treatment is
the best from the point of view used in this subsection. Finally, let us note
that the total dose of the approximate solution ($5{,}749.24$ $mg/m^{2}$) is
less than the FMINCON optimal value ($5{,}749.95$ $mg/m^{2}$), but the
constraint on the tumor volume is not satisfied (because the tumor size level
is slightly exceeded). The plot at the bottom of Figure 1 shows the evolution
of the tumor volume with three of these treatments computed with FMINCON.

Last but not least, in Table 5 and at the bottom of Figure 2 we illustrate the
influence of the level $d_{min}$. Now, we compare the optimal treatments
corresponding to the palliative approach. Again, first we have fixed $d_{min}$
and solved $(\hat{P}_{1})$; once we have determined the corresponding solution
($\hat{N}$ and ${\\{\hat{d}_{i}\\}}_{i=1}^{\hat{N}}$) we have selected the
more appropriated schedules used for TMZ. All the treatments in the table keep
the tumor below the given $L_{\star}$ level, and the longest treatment (that
also has the lowest doses) is the best because it has the lowest cumulative
dose. These results are also consistent with MC. The corresponding tumor
volumes, $L(t)/\theta$ , for a medium-sized initial tumor ($25\%$ of carrying
capacity) with the treatments in Table 5 are shown in Figure 2.

The computations were performed on a 3.9 GHz Core i5-8265U machine, with 8 GB
RAM running under the 64-bit version of Windows 10 and MATLAB R2019b.

Table 4: Results for $(P_{2})$ and TMZ with $L_{\star}=0.1813\theta$ and $d_{min}=100$ $mg/m^{2}$ with schedule 5/28d. | $(\hat{P}_{2})$ | $(P_{2})$
---|---|---
$N$ | $\hat{d}_{i}$ | $f_{2}^{N}$ | $L(T)/\theta$ | $\bar{d}_{min}$ | $\bar{d}_{max}$ | $(f_{2}^{N})_{opt}$ | $L(T)/\theta$
$33$ | $196.18$ | $6{,}473.84$ | $0.18134$ | $196.20957$ | $196.21000$ | $6{,}474.92$ | $0.18130$
$34$ | $186.60$ | $6{,}344.54$ | $0.18134$ | $186.63372$ | $186.63413$ | $6{,}345.55$ | $0.18130$
$35$ | $177.87$ | $6{,}225.51$ | $0.18134$ | $177.89874$ | $177.89913$ | $6{,}226.46$ | $0.18130$
$36$ | $169.88$ | $6{,}115.59$ | $0.18134$ | $169.90161$ | $169.90211$ | $6{,}116.46$ | $0.18130$
$37$ | $162.54$ | $6{,}013.81$ | $0.18134$ | $162.55750$ | $162.55786$ | $6{,}014.63$ | $0.18130$
$38$ | $155.77$ | $5{,}919.30$ | $0.18134$ | $155.79148$ | $155.79182$ | $5{,}920.08$ | $0.18130$
$39$ | $149.52$ | $5{,}831.32$ | $0.18133$ | $149.54009$ | $149.54041$ | $5{,}832.07$ | $0.18130$
$\mathbf{40}$ | $143.73$ | $5{,}749.24$ | $0.18133$ | $\mathbf{143.74862}$ | $\mathbf{143.74893}$ | $\mathbf{5{,}749.95}$ | $0.18130$
Table 5: Numerical results for $(\hat{P}_{2})$ with $L_{\star}=0.1813\theta$ varying $d_{min}$. $d_{min}$ | $N$ | $\hat{d}_{i}$ | $f_{2}^{N}$ | Schedule | $L(T)/\theta$ | Dose intensity
---|---|---|---|---|---|---
$150$ | $39$ | $150.00$ | $5{,}850.00$ | 7/14d | $0.18047$ | $79.05$
$100$ | $51$ | $100.25$ | $5{,}112.64$ | 21/28d | $0.18133$ | $78.66$
$75$ | $63$ | $75.00$ | $4{,}725.00$ | 21/28d | $0.18119$ | $61.36$
$50$ | $86$ | $50.29$ | $4{,}324.78$ | 28/28d | $0.18132$ | $50.29$

##  3 Conclusions

Throughout this work we have given a mathematical justification supporting
metronomic chemotherapy (MC) as the best option for most cytotoxic drugs (i.e.
when the main hypothesis (15) holds) and for curative and palliative
approaches in oncology. First, we have obtained explicit expressions of global
optimal solutions of $(\hat{P}_{1})$ and $(\hat{P}_{2})$ problems (given by
Corollary 1 and Corollary 2) that are approximations of $(P_{1})$ and
$(P_{2})$, respectively. Once more, we emphasize that the administration times
do not appear in the formulation of $(\hat{P}_{1})$ and $(\hat{P}_{2})$ and
for both cases the optimal solutions lead to the longest feasible treatments
with equal doses, which are in line with MC. We further deduce mathematically
that it is convenient to take the effectiveness level $d_{min}$ as small as
possible, but above MED (i.e. ensuring a minimum effect!). Numerical
experiments with TMZ support the high quality of the solutions of
$(\hat{P}_{1})$ and $(\hat{P}_{2})$ with respect to those of $(P_{1})$ and
$(P_{2})$, respectively, and illustrate the theoretical results. Moreover, the
optimal treatments calculated for both curative and palliative approaches are
very similar. Finally, with our method the administration times are adjusted a
posteriori (once the number of doses and the individual doses of the treatment
are determined).

In our work we have used a simple mathematical model being crucial the Norton-
Simon hypothesis, the Emax model, the PK of the drug given by (6) and the main
hypothesis (15). In the future, we would like to analyze several
generalizations and extensions from a mathematical point of view, including
some relevant factors such as drug resistance and anti-angiogenesis effect or
the case when (15) fails. We think that both theoretical and numerical results
would be very useful for clinical purposes.

Acknowledgements. The first and second authors were supported by MCIN/
AEI/10.13039/501100011033/ under research projects MTM2017-83185-P and
PID2020-114837GB-I00.

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