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Imperial-TP-KM-2021-02

# Democratic Lagrangians for Nonlinear Electrodynamics

Zhirayr Avetisyan<EMAIL_ADDRESS>Department of Mathematics,
University of California, Santa Barbara CA 93106-3080, USA  Regional
Mathematical Center of Southern Federal University, Rostov-on-Don 344090,
Russia  Oleg Evnin<EMAIL_ADDRESS>Department of Physics, Faculty of
Science, Chulalongkorn University, Bangkok 10330, Thailand  Theoretische
Natuurkunde, Vrije Universiteit Brussel and International Solvay Institutes,
Brussels 1050, Belgium  Karapet Mkrtchyan<EMAIL_ADDRESS>Theoretical Physics Group, Blackett Laboratory, Imperial College London, SW7
2AZ, UK

###### Abstract

We construct a Lagrangian for general nonlinear electrodynamics that features
electric and magnetic potentials on equal footing. In the language of this
Lagrangian, discrete and continuous electric-magnetic duality symmetries can
be straightforwardly imposed, leading to a simple formulation for theories
with the $SO(2)$ duality invariance. When specialized to the conformally
invariant case, our construction provides a manifestly duality-symmetric
formulation of the recently discovered ModMax theory. We briefly comment on a
natural generalization of this approach to $p$-forms in $2p+2$ dimensions.

Nonlinear electromagnetic theories with actions of the form

$S=\int{\cal L}(s,p)\,d^{4}x,\quad
s\equiv\frac{1}{2}F_{\mu\nu}F^{\mu\nu},\quad
p\equiv\frac{1}{2}F_{\mu\nu}\,{\textstyle\star}F^{\mu\nu}$ (1)

have surfaced historically in relation to topics as diverse as mitigating
classical field divergences due to point charges [1, 2, 3, 4], induced
interactions of photons via coupling to matter [5, 6, 7, 8, 9], effective low-
energy description of open string theory [10, 11] and construction of regular
black holes [13, 14, 15, 16]. Here,
$F_{\mu\nu}\equiv\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ and the Hodge
star is defined by
${\textstyle\star}F_{\mu\nu}=\epsilon_{\mu\nu\sigma\rho}F^{\sigma\rho}/2$,
while most generally we view $\cal L$ as an arbitrary function.

There are many reasons one may want to recast (1) in a language where both the
electric potential $A_{\mu}$ and its dual magnetic potential $B_{\mu}$
(satisfying
${\textstyle\star}F_{\mu\nu}\equiv\partial_{\mu}B_{\nu}-\partial_{\nu}B_{\mu}$
in the free Maxwell theory) appear explicitly. For one thing, one may be
interested in coupling this theory to both electrically and magnetically
charged matter, as done for free theories in [17, 18, 19, 20]. Furthermore in
many cases, such as the free Maxwell theory, or the celebrated Born-Infeld
(BI) theory [1, 10, 11, 12], the equations of motion are invariant under
continuous rotations of $F$ into ${\textstyle\star}F$ [21, 22, 26, 27, 23, 24,
25, 28, 29, 30, 31, 33, 34, 35, 36, 37, 32], and it is desirable to have an
action principle that manifests this symmetry [17, 39, 40, 41, 42, 45, 43, 44,
46, 49, 50, 38, 47, 48].

The purpose of this Letter is to give a democratic formulation in terms of
electric and magnetic potentials to a general theory of the form (1) and to
explore the properties of this formulation for special cases of current
interest. In our construction, we shall rely on the approach of [51, 52] that,
for free fields, provides a polynomial reformulation of the Pasti-Sorokin-
Tonin (PST) theory [40, 41, 42]. It turns out that this approach is well-
suited for including interactions, as we shall now demonstrate.

General formulation: We start by recalling the formulation of [51, 52] for
free fields:

${\cal
L}_{Maxwell}=-\frac{1}{4}H^{b}_{\mu\nu}H^{b}{}^{\mu\nu}+\frac{a(x)}{4}\,\,\epsilon_{bc}\,\varepsilon^{\mu\nu\lambda\rho}\,F^{b}_{\mu\nu}\,Q^{c}_{\lambda\rho}\,,$
(2)

where $H^{b}_{\mu\nu}\equiv F^{b}_{\mu\nu}+a\,Q^{b}_{\mu\nu}$, $b=1,2$, $a(x)$
is an auxiliary scalar familiar from the PST theory, and

$F^{b}_{\mu\nu}=\partial_{\mu}\,A^{b}_{\nu}-\partial_{\nu}\,A^{b}_{\mu}\,,\quad
Q^{b}_{\mu\nu}=\partial_{\mu}\,R^{b}_{\nu}-\partial_{\nu}\,R^{b}_{\mu}\,.$ (3)

Despite the large number of fields ($A^{b}$, $R^{b}$, $a$), this Lagrangian
describes a single propagating Maxwell field whose electric potential is
$A^{1}$ and magnetic potential $A^{2}$. To see this, we observe that (2) is
invariant under the usual gradient shifts of $A^{b}$ and $R^{b}$ (the
Lagrangian depends only on the corresponding field strengths) and has two
additional symmetries:

$\delta a=0,\quad\delta A^{b}_{\mu}=-au^{b}\,\partial_{\mu}a,\quad\delta
R^{b}_{\mu}=u^{b}\,\partial_{\mu}a,$ (4)

where $u^{b}(x)$ is an arbitrary doublet of scalar field parameters, and
another gauge symmetry that shifts $a(x)$ arbitrarily and correspondingly
corrects the other fields. Further details can be found in [52]. As a
consequence of these symmetries, any solution of the equations of motion can
be gauged to

$\displaystyle
R^{b}=0,\qquad{\textstyle\star}F^{a}_{\mu\nu}+\epsilon^{ab}\,F^{b}_{\mu\nu}=0\,,$
(5)

leaving a single propagating Maxwell field.

To include interactions in this formalism, we intend to deform the free action
(2) in a way that maintains its gauge symmetries. To this end, we first note
that $H^{b}_{\mu\nu}$ is invariant by itself under the transformations (4)
while the second term in (2) changes by a total derivative. As a result, if we
replace $H^{b}_{\mu\nu}H^{b}{}^{\mu\nu}$ by an arbitrary scalar function of
$H^{b}_{\mu\nu}$, the resulting interacting theory still automatically
respects both gradient shifts of $A^{b}$ and $R^{b}$, and (4), which is
crucial for the emergence of (5).

There are six functionally independent scalars one can build from
$H^{b}_{\mu\nu}$ ($U^{ab}=U^{ba}$, $V^{ab}=V^{ba}$):

$U^{ab}\equiv\frac{1}{2}\,H_{\mu\nu}^{a}\,H^{b}{}^{\mu\nu}\,,\quad
V^{ab}\equiv\frac{1}{2}\,H_{\mu\nu}^{a}\,{\textstyle\star}{H}^{b}{}^{\mu\nu}\,.$
(6)

Our algorithm to construct a nonlinear generalization of (2) is then to start
with

${\cal L}=a\,\epsilon_{bc}F^{b}\wedge
Q^{c}+f(U,^{11}U,^{12}U,^{22}V,^{11}V,^{12}V^{22})\,,$ (7)

and constrain $f$ by the requirement that there is an extra symmetry shifting
$a(x)$ arbitrarily, so that it is a pure gauge degree of freedom, or
equivalently [52], that the equations of motion for $A^{b}$ and $R^{b}$,

$\displaystyle
d[(f^{U}_{bc}+f^{U}_{cb})\,{\textstyle\star}H^{c}-(f^{V}_{bc}+f^{V}_{cb})\,H^{c}+a\,\epsilon_{bc}\,Q^{c}]=0\,,\hskip
8.53581pt$ (8) $\displaystyle
d[a\\{(f^{U}_{bc}+f^{U}_{cb})\,{\textstyle\star}H^{c}-(f^{V}_{bc}+f^{V}_{cb})\,H^{c}-\epsilon_{bc}\,F^{c}\\}]=0\,,\hskip
8.53581pt$ (9)

where $f^{U}_{ab}\equiv\partial f/\partial U_{ab}$, $f^{V}_{ab}\equiv\partial
f/\partial V_{ab}$ ($f^{U}_{21}\equiv 0\equiv f^{V}_{21}$), imply the equation
of motion for $a(x)$,

$Q^{b}\wedge\,K_{b}=0\,,$ (10)

with

$K_{a}\equiv(f^{U}_{ab}+f^{U}_{ba})\,{\textstyle\star}H^{b}-(f^{V}_{ab}+f^{V}_{ba})\,H^{b}-\epsilon_{ab}\,H^{b}\,.$
(11)

For that, we first note that multiplying (8) by $a(x)$ and subtracting it from
(9), one gets (we use the differential form notation following the derivations
of [52])

$da\wedge K_{b}=0\,.$ (12)

There is a natural way to ensure that (12) implies (10) in a manner analogous
to the free theory [52]. Indeed, if

$K_{a}\pm\epsilon_{ab}\,{\textstyle\star}K_{b}\equiv 0\,,$ (13)

then (12) implies $K_{b}=0$ by elementary differential form algebra. Hence,
(10) is satisfied whenever (8-9) are satisfied 111We do not have a proof that
(13) is the only way to ensure the gauge symmetry that shifts $a$, but it is
satisfactory in that it allows us to construct a sufficiently large class of
theories..

One can translate (13) to the following condition on $f$:

$\pm\,\delta^{ac}\,(f^{U}_{cb}+f^{U}_{bc})-\epsilon^{ac}\,(f^{V}_{cb}+f^{V}_{bc})+\delta^{a}_{b}=0\,.$
(14)

These linear PDEs are solved in full generality by

$\displaystyle f(U,V)=\mp\tfrac{1}{2}U_{aa}+g(\lambda_{1},\lambda_{2})\,,$
(15) $\displaystyle\lambda_{1}=\pm U_{12}-\tfrac{1}{2}\,(V_{11}-V_{22})\,,$
(16) $\displaystyle\lambda_{2}=V_{12}\pm\tfrac{1}{2}\,(U_{11}-U_{22})\,.$ (17)

where $g(\lambda_{1},\lambda_{2})$ is an arbitrary function (we will
henceforth use the upper signs only). The Lagrangian is then

${\cal L}={\cal L}_{Maxwell}+g(\lambda_{1},\lambda_{2})\,,$ (18)

where $\lambda_{1,2}$ can be read off (16-17) and (6). This Lagrangian
respects a gauge symmetry that shifts $a$:

$\delta a=\varphi(x)\,,\;\delta A^{b}_{\mu}=\\!-a\,\delta
R^{b}_{\mu}=\frac{\varphi}{(\partial
a)^{2}}a\partial^{\nu}\\!a(Q^{b}_{\nu\mu}-\epsilon^{bc}\\!\star
Q^{c}_{\nu\mu})\,.$ (19)

The gauge transformation rules do not depend on the function
$g(\lambda_{1},\lambda_{2})$ and, in particular, are the same as in the free
case [52]; $\lambda_{1,2}$ are invariant under (4) and (19). The Lagrangian
(18) contains a single arbitrary function of two variables, as does (1). We
shall proceed to show its relation to the single-field formulation (1) after
discussing the surprisingly simple way additional electric-magnetic symmetries
can be imposed on (18).

Duality symmetry: Under the discrete $Z_{4}$ interchange of electric and
magnetic degrees of freedom, $H^{1}\to H^{2}\,,H^{2}\to-H^{1}$, and hence
$\lambda_{1}\to-\lambda_{1},\lambda_{2}\to-\lambda_{2}$. Therefore, theories
with such discreet duality symmetry are described by
$g(\lambda_{1},\lambda_{2})$ satisfying

$g(-\lambda_{1},-\lambda_{2})=g(\lambda_{1},\lambda_{2})\,.$ (20)

Next, one may ask for the full $SO(2)$ duality symmetry with respect to
rotating $A^{b}$ and $R^{b}$ in the $b$-plane. Under such a rotation by an
angle $\alpha$, the pair $(\lambda_{1},\lambda_{2})$ simply rotates as a
vector by an angle $2\alpha$, therefore only the radial part is invariant.
Hence, the $SO(2)$-invariant theories are encoded in full generality by

$g(\lambda_{1},\lambda_{2})=h(w)\,,\qquad
w=\sqrt{\lambda_{1}^{2}+\lambda_{2}^{2}}\,,$ (21)

and their Lagrangian is given by

${\cal L}={\cal L}_{Maxwell}+h(w)\,.$ (22)

One can show that

$w=\sqrt{-\det{\cal H}}\,,$ (23)

where ${\cal
H}^{ab}\equiv({\textstyle\star}H^{a}_{\mu\nu}-\epsilon^{ac}H^{c}_{\mu\nu})({\textstyle\star}H^{b}{}^{\mu\nu}-\epsilon^{bd}H^{d}{}^{\mu\nu})/2\,$.

Conformal invariance: Another symmetry one may require from (1) is conformal
invariance [34, 35, 36], which can be expressed by a condition of the form

$U^{ab}f^{U}_{ab}+V^{ab}f^{V}_{ab}=f\,.$ (24)

Then, $g(\lambda_{1},\lambda_{2})$ is a homogeneous function of degree one:

$g=\lambda_{1}\,\tilde{g}(\lambda_{1}/\lambda_{2})\,,$ (25)

where $\tilde{g}(x)$ is an arbitrary function.

If we require both $SO(2)$ symmetry and conformal invariance, (22) reduces to

${\cal L}=-\frac{1}{2}\,H^{b}\wedge\star H^{b}+a\,\epsilon_{bc}F^{b}\wedge
Q^{c}+\delta\,w\,,$ (26)

where $\delta$ is an arbitrary real number, $w$ given by (23). As we shall
show below, this construction provides an explicit duality-symmetric
formulation of the ModMax theory recently introduced in [34] (see also [35,
36, 37]).

Analysis of the equations of motion: As explained under (13), equations (8-9)
imply

$K_{b}=0$ (27)

for theories of the form (18). Plugging (27) into (8), we deduce:

$da\wedge dR^{b}=0\,,$ (28)

exactly the same equation as for the free field case [52]. This, in turn,
implies that the fields $R^{b}$ can be gauge-transformed to zero (see [52] for
details):

$R^{b}=0\,,\qquad H^{b}=F^{b}\,.$ (29)

Note that the two equations (27) are Hodge-dual to each other due to (13), and
can hence be expressed as a single equation ($g_{1}\equiv{\partial
g}/{\partial\lambda_{1}},g_{2}\equiv{\partial g}/{\partial\lambda_{2}}$):

${\textstyle\star}F^{1}+F^{2}=\,g_{2}\,({\textstyle\star}F^{1}-F^{2})-\,g_{1}\,{\textstyle\star}({\textstyle\star}F^{1}-F^{2})\,,$
(30)

which includes the free case (5) given by $g(\lambda_{1},\lambda_{2})=0$.
Taking into account that (with $R^{b}=0$)

$\lambda_{1}=\tfrac{1}{2}\,G_{\mu\nu}\,{\textstyle\star}G^{\mu\nu}\,,\quad\lambda_{2}=-\tfrac{1}{2}\,G_{\mu\nu}\,G^{\mu\nu}\,,$
(31)

where $G_{\mu\nu}\equiv{\textstyle\star}F^{1}_{\mu\nu}-F^{2}_{\mu\nu}$,
$g(\lambda_{1},\lambda_{2})$ is expressed through $G$, while the right hand
side of (30) is equal to $-{\partial g}/{\partial G}$. Therefore, (30) is an
expression for ${\textstyle\star}F^{1}+F^{2}$ in terms of
${\textstyle\star}F^{1}-F^{2}$. This means that, similarly to the free field
case, there is only one independent field strength, while all the auxiliary
fields have been gauged away. Hence, we are really dealing with a theory of
one dynamical gauge field, as intended.

Relation to the conventional single-field formulation: The two formulations of
nonlinear electrodynamics, given by (1) and (18), are related by nonlinear
algebraic equations involving derivatives of the corresponding Lagrangians. We
now proceed to establish this relation. If one solves (30) for $F_{1}$, by
Lorentz invariance, the resulting expression must be of the form (see, e.g.,
[33])

$F^{1}=\alpha(s,p)F^{2}+\beta(s,p){\textstyle\star}F^{2}\,,$ (32)

where $s$ and $p$ are the two independent invariants of $F^{2}$,

$s=\frac{1}{2}F^{2}_{\mu\nu}F^{2}{}^{\mu\nu}\,,\quad
p=\frac{1}{2}F^{2}_{\mu\nu}{\textstyle\star}F^{2}{}^{\mu\nu},$ (33)

and $\alpha$ and $\beta$ depend on the specific form of $g$. If we apply the
exterior derivative operator to (32), the left-hand side vanishes, and $F^{1}$
disappears from the equation. What remains is the equation of motion for the
theory (1), where $F$ is identified with $F^{2}$ and

$\alpha(s,p)=-\frac{\partial{\cal L}}{\partial
p}\,,\qquad\beta(s,p)=\frac{\partial{\cal L}}{\partial s}\,.$ (34)

Thus, our democratic formulation reproduces dynamically the theory (1) for a
single electromagnetic field with self-interactions.

One can recast (32) as the following equations in terms of the invariants:

$g_{1}=\frac{2\,\alpha}{\alpha^{2}+(\beta+1)^{2}}\,,\qquad
g_{2}=\frac{\alpha^{2}+\beta^{2}-1}{\alpha^{2}+(\beta+1)^{2}}\,.$ (35)

Here, $g$ is specified as a function of $\lambda_{1}$ and $\lambda_{2}$. The
latter can be extracted from (32) and (16-17) as

$\begin{split}\lambda_{1}=2\,\alpha\,(1+\beta)\,s-[\alpha^{2}-(1+\beta)^{2}]\,p\,,\\\
\lambda_{2}=[\alpha^{2}-(1+\beta)^{2}]\,s+2\,\alpha\,(1+\beta)\,p\,,\end{split}$
(36)

so that

$w\equiv\sqrt{\lambda_{1}^{2}+\lambda_{2}^{2}}=(\alpha^{2}+(\beta+1)^{2})\,\sqrt{s^{2}+p^{2}}\,.$
(37)

For any concrete $g(\lambda_{1},\lambda_{2})$, (35) provides a $2\times 2$
system of nonlinear algebraic equations for $\alpha$ and $\beta$ as functions
of $s$ and $p$. Given a solution of this system, one can reduce the equations
of motion of the democratic theory to the single-field form (32). Conversely,
to recast a given single-field theory of the form (1) in terms of the
democratic formulation (18), one needs to obtain $\alpha$ and $\beta$ from
(34), then $\lambda_{1}$ and $\lambda_{2}$ from (36) and then $g$ from (35).

Single-field formulation for duality-symmetric theories: There is little one
can say in general about solutions to (35-36), but extra structures emerge for
special classes of theories. For the $SO(2)$-invariant case of (22), one gets

$\frac{\lambda_{1}}{w}h^{\prime}=\frac{2\,\alpha}{\alpha^{2}+(\beta+1)^{2}}\,,\quad\frac{\lambda_{2}}{w}h^{\prime}=\frac{\alpha^{2}+\beta^{2}-1}{\alpha^{2}+(\beta+1)^{2}}\,.$
(38)

Eliminating $h^{\prime}$ gives
$\lambda_{1}(\alpha^{2}+\beta^{2}-1)=2\alpha\lambda_{2}$, and then

$\beta^{2}+\frac{2s}{p}\alpha\beta-\alpha^{2}=1,$ (39)

which is, in view of (34), exactly the same as the general $SO(2)$-invariance
condition in the single-field formalism [21, 23, 28, 46, 31, 35, 36]. The
remaining equation can then be written as

$(\alpha
s+(\beta+1)p)\,\,h^{\prime}\Big{|}_{w=\sqrt{s^{2}+p^{2}}(\alpha^{2}+(\beta+1)^{2})}=\alpha\sqrt{s^{2}+p^{2}}.$
(40)

If one is moving from the single-field formalism to the democratic one for
$SO(2)$-invariant theories, (39) is satisfied from the start, and (40) is what
must be solved to reconstruct $h(w)$.

The ModMax theory: For the conformally invariant case (26), equation (40)
becomes linear, and it can be solved together with (39):

$\displaystyle\alpha(s,p)=-\sinh{\gamma}\frac{p}{\sqrt{p^{2}+s^{2}}}\,,$ (41)
$\displaystyle\beta(s,p)=\sinh{\gamma}\frac{s}{\sqrt{p^{2}+s^{2}}}-\cosh{\gamma}\,.$
(42)

With these functions, (32) reproduces the equations of motion of the ModMax
theory introduced in [34] and defined by the Lagrangian

$L(s,p)=-\cosh{\gamma}\,s+\sinh{\gamma}\sqrt{s^{2}+p^{2}}\,.$ (43)

Thus, with the identification

$\delta=\coth{\frac{\gamma}{2}}\,,$ (44)

a democratic description of the ModMax theory is provided by (26). This
solution corresponds to $\delta>1$ ($\gamma>0$) or $\delta<-1$ ($\gamma<0$).
For the region $-1<\delta<1$, which can be given as
$\delta=\tanh{\tfrac{\gamma}{2}}$, we get the same action (43), but with an
overall minus sign. This is what happens if we replace
$F_{\mu\nu}\to{\textstyle\star}F_{\mu\nu}\,,\;\gamma\to-\gamma$ in the
Lagrangian (43), so it is the ModMax theory written in terms of the dual
magnetic potential. The case $\delta=1$ 222For $\delta=1$, one cannot resolve
(30) into (32). Rather, (30) implies that $\mathbf{V}=0$, $\det\mathbf{U}=0$,
where $\mathbf{V}$ and $\mathbf{U}$ are the $2\times 2$ matrices defined by
(6) with $H^{b}\to F^{b}$. These equations manifest enhanced $SL(2,R)$
symmetry. One can then always apply a duality rotation, so that
$U_{11}=0=V_{ab}$. Thus, the electric field strength satisfies $s=0=p$ in the
notation of (1), known as the equations of motion of the Białynicki-Birula
(BB) electrodynamics [53, 54] (see also [36]). This theory does not admit a
single-potential Lagrangian formulation, while it arises naturally from the
democratic Lagrangian we consider here, highlighting the power of our
formalism. corresponds to the Białynicki-Birula electrodynamics [53, 54].

Other duality-invariant theories: While we have outlined the general way to
connect the single-field and democratic formulations, it involves solving
nonlinear algebraic equations, which is only possible explicitly in special
cases. Furthermore, the functions appearing in the democratic Lagrangian may
be complicated for known simple single-field theories, and vice versa.

We demonstrate here the conversion procedure for the generalized BI theory
[34, 36],

$L_{GBI}=\sqrt{UV}-T,\quad U\equiv 2u+e^{\gamma}\,T,\quad
V\equiv-2v+e^{-\gamma}\,T\,,$ (45)

where $u\equiv(s+\sqrt{p^{2}+s^{2}})/2$, $v\equiv(-s+\sqrt{p^{2}+s^{2}})/2$,
and $T$ is an arbitrary constant (related to the string tension in [10]). The
BI theory corresponds to $\gamma=0$. From (38),

$h^{\prime}=\frac{\sqrt{U}-\sqrt{V}}{\sqrt{U}+\sqrt{V}}\,.$ (46)

From (36),

$w=\frac{T}{2}\Big{(}e^{-\gamma}\frac{U}{V}-e^{\gamma}\Big{)}\Big{(}1+\sqrt{\frac{V}{U}}\Big{)}^{2}\,.$
(47)

Introducing $e^{\lambda}\equiv\sqrt{V/U}$, one can multiply (46) by $\partial
w/\partial\lambda$ and integrate to obtain

$\begin{split}&h(\lambda)=4\,T\,\sinh^{2}\frac{\lambda}{2}\,\,\cosh(\lambda+\gamma)\,,\\\
&w(\lambda)=-4\,T\,\cosh^{2}\frac{\lambda}{2}\,\,\sinh(\lambda+\gamma)\,.\end{split}$
(48)

These formulas provide an implicit definition for $h(w)$ corresponding to the
generalized BI theory in the democratic formulation (22).

Conclusions: We have provided a democratic formulation (18) for the nonlinear
electrodynamics (1) that explicitly features both electric and magnetic gauge
potentials. It includes two auxiliary gauge fields and an auxiliary scalar,
all of which are pure gauge degrees of freedom. The propagating degrees of
freedom are those of (1).

The $SO(2)$ duality invariance is expressed in this formalism by the
strikingly simple condition (21) which takes place of the nonlinear PDE given
by (39) and (34) responsible for the same property in the single-field
formalism (this PDE is sometimes referred to [31, 46] as the Courant-Hilbert
equation after the classic treatise [55]). This makes it easy to specify
arbitrary $SO(2)$-invariant interactions in (22), without any need to satisfy
additional constraints. In particular, polynomial interactions can be
straightforwardly introduced by choosing $h(w)$ in (22) as a polynomial in
$w^{2}$. If conformal symmetry is imposed in addition to the $SO(2)$
invariance, the theory further simplifies to (26), which is a democratic
formulation of the ModMax theory introduced in [34].

Our formulation has been developed as a nonlinear extension of the approach
employed for free fields in [51, 52]. In the free field context, this approach
is closely related to the PST formulation [40, 41, 42], which is recovered by
integrating out the auxiliary form fields. The relation is less obvious for
the interacting theories discussed here since the equations of motion are no
longer linear. If the auxiliary gauge fields can be successfully integrated
out, a PST-like formulation of nonlinear electrodynamics will be produced.

Our approach to constructing the Lagrangian (18) has been rather systematic in
that we started with a simple Lagrangian ansatz that automatically respects
all the necessary gauge symmetries except for the symmetry that shifts the
auxiliary scalar. Then, enforcing this last symmetry fixed the form of the
Lagrangian. An advantage of the resulting theory (18) is that all the gauge
symmetries are realized in a universal manner, independent of the form of
interactions 333Even though this approach can access generic theories of the
form (1), it would be interesting to check if there also exist theories that
deform the symmetries (4) and (19) of the free theory..

The approach adopted here naturally lends itself to generalizations to higher
form field interactions, and it would be interesting to explore them. An
educated guess is that for $(2k-1)$-form fields in $4k$ dimensions, the most
general nonlinear democratic Lagrangian is

${\cal L}={\cal L}_{free}+g(\lambda_{i})\,,$ (49)

where the first term is the free Lagrangian from [52] and the second term is
an arbitrary function of all independent Lorentz scalars $\lambda_{i}$ built
out of the tensor ${\textstyle\star}H^{a}-\epsilon^{ab}H^{b}$. The equations
of motion will imply a deformed twisted self-duality relation that expresses,
as in (30), ${\textstyle\star}F^{a}+\epsilon^{ab}F^{b}$ as a function of the
opposite chirality combination ${\textstyle\star}F^{a}-\epsilon^{ab}F^{b}$.
The $SO(2)$ symmetry condition will further constrain $g$ to depend on a
specific set of combinations of $\lambda_{i}$. Similarly, for the chiral
$2k$-forms in $4k+2$ dimensions, a natural guess is of the same form (49),
with the free part given as in [51, 52], while now $\lambda_{i}$ are all
independent Lorentz scalars built out of the tensor ${\textstyle\star}H-H$.
These structures will be explored in more detail in future works.

Acknowledgments: The authors are grateful to Euihun Joung and Arkady Tseytlin
for helpful discussions. OE has been supported by the CUniverse research
promotion project (CUAASC) at Chulalongkorn University. KM is supported by the
European Union’s Horizon 2020 research and innovation programme under the
Marie Skłodowska-Curie grant number 844265.

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A simple mathematical model of cyclic hypoxia and its impact on
hypofractionated radiotherapy
Edward Taylor
Princess Margaret Cancer Centre, University Health Network, Toronto, Canada
Department of Radiation Oncology, University of Toronto, Toronto, Canada
January 10, 2023

<EMAIL_ADDRESS>

###### Abstract

Purpose: There is evidence that the population of cells that experience
fluctuating oxygen levels (“acute”, or, “cyclic” hypoxia) are more
radioresistant than chronically hypoxic ones and hence, this population may
determine radiotherapy (RT) response, in particular for hypofractionated RT,
where reoxygenation may not be as prominent. A considerable effort has been
devoted to examining the impact of hypoxia on hypofractionated RT; however,
much less attention has been paid to cyclic hypoxia specifically and the role
its kinetics may play in determining the efficacy of these treatments. Here, a
simple mathematical model of cyclic hypoxia and fractionation effects was
worked out to quantify this.
Methods: Cancer clonogen survival fraction was estimated using the linear
quadratic model, modified to account for oxygen enhancement effects. An
analytic approximation for oxygen transport away from a random network of
capillaries with fluctuating oxygen levels was used to model inter-fraction
tissue oxygen kinetics. The resulting survival fraction formula was used to
derive an expression for the iso-survival biologically effective dose,
$\mathrm{BED}_{\mathrm{iso-SF}}$. These were computed for some common extra-
cranial hypofractionated radiotherapy regimens.
Results: Using relevant literature parameter values, inter-fraction
fluctuations in oxygenation were found to result in an added 1-2 logs of
clonogen survival fraction in going from five fractions to one for the same
nominal biologically effective dose (i.e., excluding the effects of oxygen
levels on radiosensitivity). $\mathrm{BED}_{\mathrm{iso-SF}}$’s for most
ultra-hypofractionated (five or fewer fractions) regimens in a given tumour
site are similar in magnitude, suggesting iso-efficacy for common
fractionation schedules.
Conclusions: Although significant, the loss of cell-killing with increasing
hypofractionation is not nearly as large as previous estimates based on the
assumption of complete reoxygenation between fractions. Most ultra-
hypofractionated regimens currently in place offer sufficiently high doses to
counter this loss of cell killing, although care should be taken in
implementing single-fraction regimens.

##  I. Introduction

It is well-known that cells at low oxygen tensions (“hypoxia”) are more
radioresistant than those at higher oxygen levels, with as much as three times
more dose required to achieve the same level of cell killing 1, 2 and hence,
the presence of hypoxia in a tumour likely plays a key role in determining the
response to radiation therapy (RT) 3. Several experiments have demonstrated,
either indirectly 4, 5 or directly 6, that this reduction in sensitivity is
transient, however: cells that are chronically ($\gtrsim$ days) hypoxic
exhibit a more modest dose-enhancing factor of $\sim 1.4$ as compared to the
factor of 2.5-3 experienced by cells that are transiently hypoxic, likely the
result of impaired ability to repair radiation-induced DNA damage 6. There is
substantial pre-clinical evidence showing that most tumours harbor a
radiobiologically significant population of such transiently hypoxic cells, an
effect often referred to as acute or cyclic hypoxia 7.

Assuming that this population exists in clinical populations (and there is
only indirect evidence of this 7), we thus hypothesize that it is the
maximally radio-resistant cyclically hypoxic component and its kinetics
through treatment that determine RT response. Historically, the large OER of
the hypoxic compartment was assumed to be mitigated during conventionally
fractionated radiotherapy (typically, 20 or more fractions) by
“reoxygenation”, the process by which cells that are hypoxic early in RT
gradually become oxic at subsequent fractions, rendering them more
radiosensitive 8. Technological advances over the past two decades have led to
the adoption of hypofractionated RT regimens such as stereotactic body
radiotherapy (SBRT), lasting five or fewer fractions. While some reoxygenation
may occur during these abbreviated regimens 9, it is not clear that it is
sufficient to mitigate the impact of hypoxia and cyclic hypoxia may play a
greater role.

In this paper, we develop a probabilistic model of cyclic hypoxia to quantify
its impact on hypofractionated RT. A number of authors have examined the
effect of hypoxia on hypofractionated RT; see e.g. Refs. 10, 11, 12, 13, 14,
15, 16. Comparatively little attention seems to have been paid to modelling
the specific impact of cyclic hypoxia, however. Popple and colleagues 17,
while not explicitly studying hypofractionated RT, quantified the influence of
cyclic and chronic hypoxia by modelling the former as a completely stochastic
population; i.e., by assuming that cells randomly shuttle between oxic and
cyclic hypoxia compartments at each fraction. Ruggieri et al. studied the
affect of cyclic hypoxia on hypofractionated RT by assuming that cells in the
cyclic hypoxia compartment oscillated between oxic and hypoxic states
periodically in time with randomly assigned phases 11. Toma-Daşu and
colleagues calculated cancer clonogen survival by simulating oxygen diffusion
and metabolism from simulated capillary networks; acute hypoxia was simulated
by randomly closing a fraction of simulated capillaries between each fraction
18, 13.

Although differing in some details, the approach taken by us here is
conceptually similar to that of these last two references 18, 13. A key
feature of these works is the spatial aspect of their modelling, via
calculations of oxygen metabolism and diffusion for simulated vascular
architectures. This is important because, in contrast to purely stochastic
approaches, cells that are in well-perfused regions are unlikely to oscillate
between extremes of hypoxia and oxia between fractions while cells that are
far from blood vessels are likewise unlikely to become oxic. The probability
that a cell is oxic or hypoxic at a given fraction is thus not independent of
the oxygenation status at previous fractions, since both are functions of the
local vascular architecture. Mathematically, this lack of complete
stochasticity lessens the impact of decreasing fractionation: for completely
stochastic oxygenation dynamics, the probability that a given cell is
hypoxic–and hence, maximally radioresistant–over the entire treatment
diminishes rapidly with increasing fraction number $N$ as $p^{N}$, where $p$
is the probability that the cell is hypoxic at a single fraction. (This
applies not only to cyclic hypoxia but also assumptions of complete
“stochastic” reoxygenation of a chronically hypoxic compartment between
fractions 12). In contrast, modelling the expected non-stochasticity of tissue
oxygen tension, the probability that a cell is hypoxic for the duration of RT
decays more slowly with $N$.

The non-stochastic nature of hypoxia dynamics may be of relevance for
understanding the apparent success of _ultra_ -hypofractionated regimes
($N\leq 5$): if the spatial-temporal kinetics of hypoxia are completely
stochastic, one would expect a substantial decrease in tumour control in going
e.g., from five fractions to a single one, unless this reduction were
accompanied by a substantial increase in dose. Although ultra-
hypofractionation has largely been accompanied by increases in doses (as
briefly reviewed in Section IV.), these are not sufficient to overcome the
loss of cell killing implied by completely stochastic oxygen kinetics. By
using a simple approximation to the full oxygen metabolism-diffusion problem
18, 13 in the present work, we derive analytic expressions for the clonogen
survival fractions and iso-survival fraction doses under cyclically hypoxic
conditions that we hope will facilitate analysis of fractionation schedules
and give insight into the key parameters that control the magnitude of cyclic
hypoxia’s impact.

We begin by introducing our model of cyclic hypoxia in Section II. Although it
assumes stochastic temporal dynamics in capillary oxygen levels, by
incorporating the spatial dependency of hypoxia with respect to the capillary
architecture in an approximate way, this model captures the non-stochasticity
of tissue oxygen levels described above. In Section III., we combine this
model with the linear quadratic model to obtain an analytic expression for
clonogen survival as a function of the chosen fractionation regimen and the
amplitude of cyclic hypoxia. We close in Section IV. by discussing the
implications of our results for some modern hypofractionated SBRT regimens
currently in use for lung, liver, pancreas, and prostate cancers, and propose
a method to combat cyclic hypoxia.

##  II. Cyclic hypoxia model

Acute or cyclic hypoxia arises from transient fluctuations in red blood cell
flux 19, 20, 21, 22. The impact of a reduced blood cell flux through
capillaries is to reduce the oxygen tension $p_{\mathrm{c}}$ inside the
capillaries 21, the value determined by the interplay between the number of
red blood cells entering a region per unit time (flux) and the rate of oxygen
metabolism in the tissue there. Fluctuations in $p_{\mathrm{c}}$ yield
fluctuations in tissue oxygen levels, yielding cyclic hypoxia. Experiments
have shown that cyclic hypoxia is spatially correlated over domains
approximately 200-300 $\mu$m in extent 22, implicating an “upstream” 21 effect
such as thermoregulation-induced vasomotion 7 that simultaneously impacts
groups of microvessels.

A key quantity entering this work is the hypoxic fraction, denoted by
$\mathrm{HF}_{\Lambda}$, describing the fraction of tissue for which the local
oxygen tension $p(\mathbf{r})$ at a point $\mathbf{r}$ is less than a
radiobiologically relevant threshold oxygen tension $\Lambda$. Given the
spatial extent of cyclic hypoxia correlations, for our purposes, it will be
useful to define $\mathrm{HF}_{\Lambda}$ as the fraction of tissue in a
pseudo-voxel several hundred $\mu$m’s in size. Note that this differs from
standard nomenclature, wherein hypoxic fraction refers to the fraction of
tissue within an entire tumour for which $p(\mathbf{r})<\Lambda$. Amongst
other factors, the oxygen level $p(\mathbf{r})$ in tissue is a function of
that in capillaries, $p_{\mathrm{c}}$. By defining this “microscopic” hypoxic
fraction, we can associate a voxel-specific value $p_{\mathrm{c}}(t)$ of the
time-varying capillary oxygen tension to all tissue in that pseudo-voxel and
hence, express the hypoxic fraction there as a function of this
$p_{\mathrm{c}}(t)$:
$\mathrm{HF}_{\Lambda}=\mathrm{HF}_{\Lambda}(p_{\mathrm{c}})$. Since the
oxygen enhancement ratio rises rapidly below 5 mmHg 23, an appropriate choice
of cutoff $\Lambda$ should be on the order of several mmHg’s.

Now, suppose that $p_{\mathrm{c}}$ cycles (possibly irregularly) between
minimum and maximum values of capillary oxygen tension, denoted by
$p_{\mathrm{c,min}}$ and $p_{\mathrm{c,max}}$, respectively. The fraction
$\mathrm{HF}_{\mathrm{chronic}}$ of tissue that is _chronically_ hypoxic
refers to the fraction of tissue in a pseudo-voxel that is always hypoxic
($p<\Lambda$):

$\mathrm{HF}_{\mathrm{chronic}}\equiv\mathrm{HF}_{\Lambda}(p_{\mathrm{c,max}}).$
(1)

Correspondingly, the _potential_ cyclic hypoxic fraction

$\mathrm{HF}_{\mathrm{pot.}}\equiv\mathrm{HF}_{\Lambda}(p_{\mathrm{c,min}})-\mathrm{HF}_{\Lambda}(p_{\mathrm{c,max}})$
(2)

is defined as the fraction of non-chronically hypoxic tissue that, given
enough time, is guaranteed to cycle between oxic ($p>\Lambda)$ and hypoxic
($p<\Lambda)$ states. This quantity in general differs from the _mean_ cyclic
hypoxic fraction $\langle\mathrm{HF}_{\mathrm{cyclic}}\rangle$, which is the
expectation value of the fraction of non-chronically hypoxic tissue for which
$p(\mathbf{r})<\Lambda$ _at any point in time_. We assume that the
$p_{\mathrm{c}}$ values in each pseudo-voxel and at each time-point (fraction)
fluctuate independently and hence, $p_{\mathrm{c}}$ can be modelled as being
sampled from a distribution $f(p_{\mathrm{c}})$, bounded between
$p_{\mathrm{c,min}}$ and $p_{\mathrm{c,max}}$. The mean cyclic hypoxic
fraction is thus given by

$\langle\mathrm{HF}_{\mathrm{cyclic}}\rangle=\int_{p_{\mathrm{c,min}}}^{p_{\mathrm{c,max}}}f(p_{\mathrm{c}})\mathrm{HF}_{\Lambda}(p_{\mathrm{c}})\;dp_{\mathrm{c}}-\mathrm{HF}_{\Lambda}(p_{\mathrm{c,max}}).$
(3)

Owing to the differing time scales involved, the potential cyclic hypoxic
fraction will play a role in our (brief) discussion of hypoxia binding agents
(e.g., pimonidazole and those used in positron-emission tomography imaging of
hypoxia), while the mean cyclic hypoxic fraction arises in our model of
radiotherapy response.

The above definitions are completely general. As a specific model of the
microscopic hypoxic fraction, consider a randomly distributed vasculature,
shown schematically in Fig. 1(a.). Oxygen diffuses a characteristic distance
$l_{D}\equiv\sqrt{4D_{\mathrm{O}_{2}}p_{\mathrm{c}}/M}$ from each capillary
before being completely consumed 24. Here, $D_{\mathrm{O}_{2}}$ is the
diffusivity of oxygen and $M$ is the oxygen consumption rate (mmHg/s, for
$p_{\mathrm{c}}$ expressed in mmHg). In a “micro-region” of tissue in which
these quantities are very nearly constant, the fraction of tissue that is at
nearly zero oxygen tension is thus approximated by Poisson statistics:

$\mathrm{HF}_{\Lambda}(p_{\mathrm{c}})\sim e^{-\pi n_{c}l_{D}^{2}}\equiv
e^{-\gamma n_{c}},\;\;\gamma\equiv\frac{4\pi
D_{\mathrm{O}_{2}}p_{\mathrm{c}}}{M},$ (4)

where $n_{c}$ is the areal capillary density (number of capillaries per unit
area in a given tissue plane) in the pseudo-voxel in question. Equation (4) is
an approximation to the solution of the oxygen reaction-diffusion equation
with randomly distributed capillaries acting as oxygen sources 25. It is
approximate since it treats the oxygen “fields” [the shaded circles in Fig.
1(a.)] emanating from each capillary as independent of each other. In dealing
with the distance $l_{D}$ from a capillary required for the oxygen tension to
vanish, Eq. (4) also implicitly assumes that $\Lambda=0$; formally, nonzero
$\Lambda$ can be accounted for by correcting the diffusion distance: $l_{D}\to
l_{D}(1-\Lambda/p_{\mathrm{c}})$. We will ignore this small correction and
continue to treat $\Lambda$ as small ($\lesssim$ 5 mmHg), but nonzero.

Figure 1: (a.) Graphical representation of the model shown in Eq. (4). Oxygen
diffuses a characteristic distance $l_{D}$ from capillaries, leaving a
fraction $\mathrm{HF}_{\Lambda}$ of tissue hypoxic (unshaded region). For
randomly distributed capillaries, this fraction is given by Poisson
statistics: $\exp(-\pi n_{c}l_{D}^{2})$. (b.) Potential cyclical and mean
cyclical versus chronic hypoxia, obtained from Eqs. (6-8) for two
representative values of the oxygen fluctuation parameter $y\equiv(\Delta
p_{c}/\bar{p}_{c})=0.3$ and 0.6. Also shown is flow cytometry data for two
cell lines, extracted from Figs. 2 and 5 of Bennewith and Durand 26 (see text
for details). The left and right sides of this plot correspond to the hypo-
($n_{c}\to 0$) and well-perfused ($n_{c}\to\infty$) limits, respectively.

Cyclic and chronic hypoxia are not independent but are connected by the
vascular architecture and capillary oxygen tension kinetics: for a given rate
of oxygen metabolism $M$, both the chronic and cyclic hypoxic fractions are
functions of the capillary density and the amplitude $\Delta
p_{\mathrm{c}}\equiv p_{\mathrm{c,max}}-p_{\mathrm{c,min}}$ of capillary
oxygen tension fluctuations. Thus, chronic and cyclic hypoxia go hand-in-hand,
primarily functions of the underlying tissue capillary density. Choosing for
simplicity a uniform, “top-hat” function

$f(p_{\mathrm{c}})=\begin{cases}1/\Delta
p_{\mathrm{c}}&\mathrm{for}\;\;p_{\mathrm{c,min}}\leq p_{\mathrm{c}}\leq
p_{\mathrm{c,max}}\\\ 0&\mathrm{otherwise}\end{cases}$ (5)

to represent the distribution of capillary oxygen fluctuations in Eqs. (1-3)
gives

$\mathrm{HF}_{\mathrm{chronic}}=e^{-x}e^{-z/2},$ (6)
$\mathrm{HF}_{\mathrm{pot.}}=2e^{-x}\sinh(z/2),$ (7)

and

$\langle\mathrm{HF}_{\mathrm{cyclic}}\rangle=e^{-x}\frac{\sinh(z/2)}{(z/2)}-\mathrm{HF}_{\mathrm{chronic}}.$
(8)

Here, we have defined the dimensionless parameters

$x\equiv\bar{\gamma}n_{c},\;\;y\equiv\left(\frac{\Delta
p_{c}}{\bar{p}_{c}}\right),\;\;\mathrm{and}\;\;z\equiv x\cdot y,$ (9)

with $\bar{p}_{c}$ is the mean capillary oxygen tension, and
$\bar{\gamma}\equiv 4\pi D_{\mathrm{O}_{2}}\bar{p}_{c}/M$ the mean $\gamma$
value. These parameters are the key variables regulating the magnitude of
cyclic hypoxia and its impact on radiotherapy. $x$ describes the efficiency
with which the vasculature delivers oxygen: when $x\lesssim 1$, the mean
distance $l_{D}$ travelled by oxygen from capillaries is shorter than the mean
distance $\sim 1/\sqrt{n_{c}}$ between capillaries, and oxygen does not reach
all tissue, resulting in substantial $\mathrm{HF}_{\mathrm{chronic}}$. In the
opposite regime where, $x\gtrsim 1$, tissue is well-perfused and the chronic
hypoxic fraction is small. $y$ quantifies the amplitude of capillary oxygen
tension fluctuations. In the limit $y\to 0$ that oxygen tension fluctuations
vanish, both terms on the right-hand side of Eq. (8) reduce to $\exp(-x)$ and
hence, $\langle\mathrm{HF}_{\mathrm{cyclic}}\rangle$ vanishes. $z$ is a
compound parameter describing both the amplitude of capillary oxygen
fluctuations as well as the baseline vascular oxygen deliver efficiency and
hence, characterizes the magnitude of tissue oxygen fluctuations that result
from fluctuations in capillary tension.

Any reasonable alternative choice (e.g., log-normal) for the distribution
function shown in Eq. (5) would yield qualitatively similar results to ours
here. We chose the top-hat form since, in addition to yielding analytic
results for the hypoxic fractions and survival fractions following RT, it
minimizes the number of parameters in our model.

###  II.A. Hypoxia model parameter values

Using the classic value $l_{D}\sim 150$ $\mu$m derived by Thomlinson and Gray
24, we estimate $\bar{\gamma}=7\times 10^{-2}$ mm2. In arriving at this value
for $l_{D}$, these authors also assumed $\bar{p}_{c}\sim 40$ mmHg, the value
we adopt here as well. Estimates of the capillary density $n_{c}$ vary widely
27, 28, 29, with typical values spanning $(5-500)$ mm-2. Thus,
$x\equiv\bar{\gamma}n_{c}$ varies from $\sim 0.1$ to $\sim 10$. Note that
vessel quantification studies such as these measure the “microvascular
density” and do not distinguish capillaries from small arterioles and venules.
Thus, these values may overestimate $n_{c}$. Measurements of vascular and
peri-vascular oxygen tension fluctuations reveal maximum amplitudes $\Delta
p_{c}$ on the order of $(5-25)$ mmHg 21, 20 and hence, $y\equiv\Delta
p_{c}/\bar{p}_{c}$ varies from $\sim 0.1$ to $\sim 0.6$. For some context,
using the representative value $y=0.4$, for $x=0.1$ (poorly-perfused tissue),
$\mathrm{HF}_{\mathrm{chronic}}\sim 0.9$ and
$\langle\mathrm{HF}_{\mathrm{cyclic}}\rangle\sim 10^{-2}$, while for $x=10$
(well-perfused), $\mathrm{HF}_{\mathrm{chronic}}\sim 10^{-5}$ and
$\langle\mathrm{HF}_{\mathrm{cyclic}}\rangle\sim 10^{-4}$.

In Figure 1(b.), we plot $\langle\mathrm{HF}_{\mathrm{cyclic}}\rangle$ and
$\mathrm{HF}_{\mathrm{pot.}}$ as functions of
$\mathrm{HF}_{\mathrm{chronic}}$, using $y=0.6$ and $y=0.3$ as illustrative
values. In the same plot, we show the cyclic and chronic hypoxic fractions
estimated from flow cytometry data acquired by Bennewith and Durand 26 from
xenografts exposed to two different hypoxia markers (pimonidazole and
CCI-103F) for different durations to distinguish these fractions. The data
points shown in this plot were extracted from individual bins for each
xenograft–human cervical (SiHa) and colon (WiDr) cancers–, representing
different intensities for the fluorescent perfusion dye Hoechst 33342. This
way, each datapoint shown in Fig. 1(b.) represents a different distance bin
from nearest capillaries. As long as the time $T$ that these tracers spend in
the blood is greater than several periods of cyclic hypoxia fluctuations,
markers such as these serve as “integral” ones, labelling all cells that were
hypoxic at some point during $T$ and hence, the uptake will be proportional to
$\mathrm{HF}_{\mathrm{pot.}}$, and not
$\langle\mathrm{HF}_{\mathrm{cyclic}}\rangle$. We make no attempt to fit this
data to our model, but present it here to give an idea of the magnitudes of
measured transient and chronic hypoxias. However, we note two properties that
our model curves shown in this plot possess that any physiologically valid
model as well as empirical data must satisfy: in the completely avascular
limit, $n_{c}\to 0$, any fluctuations in capillary oxygen tension hypoxia will
not impact tissue oxygen levels, which remain identically zero:
$\mathrm{HF}_{\mathrm{chronic}}\to 1$ and
$\langle\mathrm{HF}_{\mathrm{cyclic}}\rangle\to 0$. In the opposite limit, of
well-vascularized tissue, $n_{c}\to\infty$ (in practice, for our parameter
choices, $n_{c}\gtrsim 100$ mm-2 suffices), fluctuations in $p_{\mathrm{c}}$
likewise have no impact on the completely oxic tissue:
$\mathrm{HF}_{\mathrm{chronic}}\to 0$ and
$\langle\mathrm{HF}_{\mathrm{cyclic}}\rangle\to 0$. Hence, the exact
$\langle\mathrm{HF}_{\mathrm{cyclic}}\rangle$ versus
$\mathrm{HF}_{\mathrm{chronic}}$ relationship must have the general shape
shown in Fig. 1(b.).

##  III. Cyclic hypoxia and clonogen survival fraction during hypofractionated
radiotherapy

Having introduced our model for cyclic hypoxia [Eqs. (6-8)], we now turn to
the problem of working out its impact on RT response. Extending the binary
decomposition of Carlson et al. 12, tumor clonogens are stratified into three
compartments: an oxic, cyclically and chronically hypoxic compartments, each
with their own OER: OER${}_{\mathrm{oxic}}=1$, OERchronic = 1.37, and
OER${}_{\mathrm{cyclical}}=$ 2.5 6. The surviving fraction SF(1) of cancer
clonogens after a single fraction of radiotherapy is approximated as

$\mathrm{SF}^{(1)}\simeq\langle\mathrm{HF}_{\mathrm{cyclic}}\rangle\cdot\mathrm{SF}^{(1)}_{\mathrm{cyclic}}+\mathrm{HF}_{\mathrm{chronic}}\cdot\mathrm{SF}^{(1)}_{\mathrm{chronic}}+\left(1-\langle\mathrm{HF}_{\mathrm{cyclic}}\rangle-\mathrm{HF}_{\mathrm{chronic}}\right)\cdot\mathrm{SF}^{(1)}_{\mathrm{oxic}},$
(10)

where the single-fraction survival fractions in each compartment are given by
the linear-quadratic model expressions

$\mathrm{SF}^{(1)}_{i}\equiv
e^{-\alpha\cdot\mathrm{BED}_{i}(1)},\;\;i=\mathrm{oxic},\mathrm{cyclic},\mathrm{chronic}$
(11)

Here, $\alpha$ is the intrinsic radiosensitivity parameter and

$\mathrm{BED}_{i}(N)\equiv\frac{d\cdot
N}{\mathrm{OER}_{i}}\left(1+\frac{d}{(\alpha/\beta)\mathrm{OER}_{i}}\right)$
(12)

is the compartment-specific biologically effective dose (BED) for $N$
fractions 23, with $\alpha/\beta$ quantifying the intrinsic sensitivity to
fractionation effects and $d$ the per-fraction dose. In reality, the survival
fraction is given by a convolution over a continuous spectrum of oxygen
tensions 30, 16. The rapid variation of the OER with respect to oxygen tension
at low values ($\lesssim$ 5 mmHg) 1 justifies the compartmentalization in Eq.
(10); however, this approximation can lead to quantitative errors in SF 31.

Neglecting proliferation during abbreviated hypofractionated treatments and
assuming that there is no reoxygenation, for us meaning that
$\langle\mathrm{HF}_{\mathrm{cyclic}}\rangle$ and
$\mathrm{HF}_{\mathrm{chronic}}$ remain constant through treatment, the
survival fraction after $N$ fractions is approximated as

$\mathrm{SF}^{(N)}\simeq\sum_{m=0}^{N}P(m,N-m)(\mathrm{SF}^{(1)}_{\mathrm{cyclic}})^{m}(\mathrm{SF}^{(1)}_{\mathrm{oxic}})^{N-m}+\mathrm{HF}_{\mathrm{chronic}}\cdot\left(\mathrm{SF}^{(1)}_{\mathrm{chronic}}\right)^{N}.$
(13)

Here, $P(m,N-m)$ is the probability that a non-chronically hypoxic cell is
hypoxic for $m$ fractions, and oxic for the remaining $N-m$. Because it
excludes the chronically hypoxic population [accounted for in the last term in
Eq. (13)], it satisfies the normalization condition
$\sum_{m=0}^{N}P(m,N-m)=1-\mathrm{HF}_{\mathrm{chronic}}$. For a single
fraction ($N=1$), this probability is simply the mean cyclic hypoxic fraction,

$P(1,0)=\langle\mathrm{HF}_{\mathrm{cyclic}}\rangle$ (14)

and hence, Eq. (13) reduces to Eq. (10) when $N=1$.

As noted in the Introduction, for high dose-per-fraction hypofractionated RT
regimens, clonogen survival is dominated by the cyclically hypoxic
compartment. To see this, compare $d=12$ Gy with a standard conventional
regimen dose-per-fraction, $d=2$ Gy. Using $\alpha=0.4$ Gy-1 and
$\alpha/\beta=10$ Gy, for the hypofractionated regimen,
$\mathrm{SF}^{(1)}_{\mathrm{oxic}}/\mathrm{SF}^{(1)}_{\mathrm{cyclic}}\sim
10^{-4}$ and
$\mathrm{SF}^{(1)}_{\mathrm{chronic}}/\mathrm{SF}^{(1)}_{\mathrm{cyclic}}\sim
10^{-2}$. Conversely, for the conventional regimen,
$\mathrm{SF}^{(1)}_{\mathrm{oxic}}/\mathrm{SF}^{(1)}_{\mathrm{cyclic}}\sim
0.5$ and $\mathrm{SF}^{(1)}_{\mathrm{cyclic}}\sim 0.7$. Thus, for
hypofractionated regimens, the chronically hypoxic and oxic compartments can
be neglected and survival is dominated by cyclically hypoxic clonogens that
remain hypoxic for all $N$ fractions:

$\mathrm{SF}^{(N)}\simeq P(N,0)(\mathrm{SF}^{(1)}_{\mathrm{cyclic}})^{N}.$
(15)

The leading-order correction to this expression arising from the $P(N-1,1)$
contribution remains less than 10% for $d\gtrsim$ 6 Gy and hence, Eq. (15)
well-describes most hypofractionated regimens of five or fewer fractions. For
longer fractionation schedules, in addition to considering all terms in Eq.
(13), one would have to model reoxygenation and proliferation kinetics, which
are beyond the scope of the present work.

It thus remains to derive an expression for $P(N,0)$ to estimate the impact of
fractionation on a tumour with a cyclically hypoxic population. Consider a
pseudo-voxel for which the capillary oxygen tensions through $N$ fractions of
radiotherapy are
$p_{\mathrm{c}}^{(1)},p_{\mathrm{c}}^{(2)},\cdots,p_{\mathrm{c}}^{(N)}$.
Assuming that the vasculature remains constant over the $N$ fractions, the
fraction $p(N,0)$ of non-chronically hypoxic tissue in this pseudo-voxel that
remains hypoxic for all $N$ fractions is

$p(N,0)=\mathrm{HF}_{\Lambda}(\mathrm{max}[p_{\mathrm{c}}])-\mathrm{HF}_{\mathrm{chronic}},$
(16)

where $\mathrm{max}[p_{\mathrm{c}}]$ is the maximum of the
$p_{\mathrm{c}}^{(1)},p_{\mathrm{c}}^{(2)},\cdots,p_{\mathrm{c}}^{(N)}$
values. Note that this is not in general equal to $p_{\mathrm{c,max}}$, but is
bounded from above by it. Although high doses per fraction ($\gtrsim 8$ Gy)
can ablate the vasculature 32, 33, substantial ($\gtrsim 20\%$) losses do not
occur until after 1-2 weeks 33, justifying our approximation of a constant
vasculature over the hypofractionated schedules of interest to us here.

Generalizing the result shown in Eq. (16) to average over all possible
realizations of sequences of $p_{\mathrm{c}}$ values [sampled from the
distribution $f(p_{\mathrm{c}})$] by defining $P(N,0)\equiv\langle
p(N,0)\rangle$, gives

$\displaystyle P(N,0)=N\int_{0}^{\infty}dp_{\mathrm{c}}^{(1)}$ $\displaystyle
f(p_{\mathrm{c}}^{(1)})\mathrm{HF}_{\Lambda}(p_{\mathrm{c}}^{(1)})\Bigg{\\{}\int_{0}^{\infty}dp_{\mathrm{c}}^{(2)}f(p_{\mathrm{c}}^{(2)})\Theta(p_{\mathrm{c}}^{(1)}-p_{\mathrm{c}}^{(2)})\times$
$\displaystyle\cdots\times\int_{0}^{\infty}dp_{\mathrm{c}}^{(N)}f(p_{\mathrm{c}}^{(N)})\Theta(p_{\mathrm{c}}^{(1)}-p_{\mathrm{c}}^{(N)})\Bigg{\\}}-\mathrm{HF}_{\mathrm{chronic}}.$
(17)

The factor $N$ in front of the integrals arises since we have arbitrarily
chosen $p_{\mathrm{c}}^{(1)}$ to be the maximum value of $p_{\mathrm{c}}$, as
enforced by the Heaviside theta functions $\Theta$, and there were $N$ choices
for the fraction with the maximum $p_{\mathrm{c}}$ value.

Using Eq. (5) in Eq. (17), the resulting integral can be evaluated
analytically, with the result

$P(N,0)=\langle\mathrm{HF}_{\mathrm{cyclic}}\rangle\cdot
F(z,N)-\mathrm{HF}_{\mathrm{chronic}}\cdot\left[1-F(z,N)\right],$ (18)

where

$F(z,N)\equiv\frac{N\left[\Gamma(N)-\Gamma(N,z)\right]}{z^{N-1}(1-e^{-z})}.$
(19)

Here, $\Gamma(N)$ and $\Gamma(N,z)$ are the complete and incomplete Euler
gamma functions, respectively. It can be shown that $F(z,1)=1\;\forall z$ and
hence, Eq. (14) is recovered. For nonzero $z$, $F(z,N)$ decreases with
increasing number $N$ of fractions, reflecting the increase in clonogen cell-
killing (decrease in SF) due to inter-fraction oxygen fluctuations. To express
this effect in terms of biologically effective doses, we define an iso-
survival fraction BED as:

$\displaystyle\mathrm{BED}_{\mathrm{iso-
SF}}(N)\equiv-\frac{1}{\alpha}\ln(\mathrm{SF}^{(N)})$
$\displaystyle=\frac{d\cdot
N}{\mathrm{OER}_{\mathrm{cyclic}}}\left(1+\frac{d}{(\alpha/\beta)\mathrm{OER}_{\mathrm{cyclic}}}\right)+\frac{1}{\alpha}\ln[P^{-1}(N,0)]$
$\displaystyle\equiv\mathrm{BED}_{\mathrm{cyclic}}(N)+\frac{1}{\alpha}\ln[P^{-1}(N,0)].$
(20)

When $N=1$, $F(z,1)=1$ and the iso-SF BED for the hypoxic fraction reduces to
$\mathrm{BED}_{\mathrm{cyclic}}$. When $N>1$, the second term on the right-
hand side is non-zero and positive; it represents the added BED arising from
inter-fraction fluctuations in oxygen levels.

Figure 2: Clonogen survival fraction versus (a.) the amplitude $y\equiv(\Delta
p_{\mathrm{c}}/\bar{p}_{c})$ of capillary oxygen partial pressure fluctuations
and (b.), (c.) number of fractions for iso-BED schedules (relative to 60 Gy in
30 fractions, with $\alpha/\beta=10$). The dashed-vertical line in (a.)
indicates $y=0.4$, the representative value used in the remainder of this
manuscript. In (b.) and (c.), SF is shown using this value and $n_{c}=10$ and
$100$ mm-2, respectively. Also shown in these plots are the survival fractions
assuming complete reoxygenation, given by Eq. (21). For all panels,
$\alpha=0.4$ Gy-1 and $\alpha/\beta$ = 10 Gy.

Taken together, Eqs. (15), (18), and (20) constitute the primary results of
this manuscript. They quantify the impact of cyclic hypoxia on
hypofractionated radiotherapy treatments via the estimated fraction of
surviving clonogens and the dose needed to counteract the loss of cell-killing
due to the reduction in oxygen fluctuations with fewer fractions. The key
parameters governing this impact are the fractionation regimen ($d$, $N$) and
the compound parameter $z$, which multiplies the vasculature efficiency
parameter $x$ and the amplitude $y$ of capillary oxygen fluctuations; see Eq.
(9). In Figure 2, using $\alpha=0.4$ Gy-1 and $\alpha/\beta$ = 10 Gy, we plot
SF as a function of (a.) the amplitude $y$ of capillary oxygen fluctuations
and (b.)-(c.), the number of fractions for nominally iso-BED doses. From Fig.
2(a.), it can be seen that, except for very small values of $y$, SF increases
slowly with $y$ and we use $y=0.4$ for the remainder of the results in this
manuscript, noting the weak dependence of our results on this parameter.
Survival fractions in Figs. 2(b.) and 2(c.) are plotted as functions of the
number of fractions, iso-BED with (the arbitrarily chosen conventional RT
schedule 60 Gy in 30 fractions), for two representative values of the
capillary density, $n_{c}=10$ and 100 mm-2. Respectively, these correspond to
$x=0.7$ and 7 (poorly- and well-perfused tumours) and cyclic (chronic) hypoxic
fractions of 0.07(0.43) and 0.001(0.0002). By “iso-BED”, we mean in the usual
sense of iso-$\mathrm{BED}_{\mathrm{oxic}}$; that is, neglecting OER
corrections (by setting OER = 1) in Eq. (12). For $N=1$, for instance, using
$\alpha/\beta=10$, the iso-BED (to 60 Gy in 30) dose is 22.3 Gy. The value
$\alpha\simeq 0.4$ Gy-1 used in these plots is a commonly used value in
theoretical radiobiological studies of e.g., head and neck tumours 30, while
also being consistent with radiobiological studies of lung cancers 14.
$\alpha/\beta=10$ Gy is a standard value for early-responding (non-prostate)
tumour tissues. Also shown in Figure 2(b.) and (c.) are the survival fractions
when complete, _stochastic_ reoxygenation is assumed, corresponding to 12

$\mathrm{SF}^{(N)}=(\mathrm{SF}^{(1)})^{N},\;\;\mathrm{complete}\;\mathrm{reoxygenation}.$
(21)

Figure 2 shows that cyclic hypoxia enhances the efficacy of nominally iso-BED
(iso-$\mathrm{BED}_{\mathrm{oxic}}$) radiotherapy with increasing number of
fractions, a result of tissue that is initially hypoxic becoming oxic at
subsequent fractions due to transient fluctuations. For e.g.,
$\langle\mathrm{HF}_{\mathrm{cyclic}}\rangle$=0.07 [shown in Fig. 2(b.)],
clonogen cell survival increases by approximately a factor 30 in going from
five fractions to a single one. Although SF increases with decreasing fraction
number, the increase is much smaller than arises when complete stochastic
reoxygenation occurs between fractions. In the same figure, the assumption of
complete reoxygenation leads to a _much_ larger increase in SF, by a factor
$\sim 10^{5}$.

The expression for the iso-survival fraction biologically effective dose, Eq.
(20), gives a convenient way of comparing the efficacy of different
fractionation schedules. $\mathrm{BED}_{\mathrm{iso-SF}}$’s for some common
SBRT fractionations schedules are presented in Sec. IV.

##  IV. Discussion

Driven by advances in technology that have allowed for the reliable delivery
of more conformal dose distributions, the use of hypofractionated treatments
such as SBRT has grown considerably over the past two decades. Although SBRT
has clear efficiency benefits, _prima_ _facie_ , it is likely suboptimal
radiobiologically. Setting aside the differential repair of sublethal DNA
damage between early- and late-responding tissues, SBRT challenges two of the
“4 R’s of radiotherapy” 34: reoxygenation and redistribution through the cell
cycle, which enhance radiosensitivity over time. The impact of
hypofractionated radiotherapy regimens on tumours with hypoxia has been widely
discussed, and the general conclusion has been that tumour control probability
will be reduced with fewer fractions because of the reduced extent of
reoxygenation 10, 12. Why has hypofractionated radiotherapy been so successful
then, with no clear evidence of plunging control probabilities even for
single-fraction regimens 35?

Based on our results, we give two potential reasons for this: first, assuming
that reoxygenation does not occur to a substantial degree during
hypofractionated RT, cyclic hypoxia was found to result in a more modest
decrease in clonogen survival with decreasing fractionation, as compared to
models that assume a complete “stochastic” reoxygenation; see Fig. 2. Second,
as we discuss below, in sites such as pancreas and lung, increasing
hypofractionation has been accompanied by increasing BED’s, largely offsetting
the smaller increase in survival predicted here.

“Complete reoxygenation” conventionally refers to the phenomenon wherein the
hypoxic fraction of _viable_ cells (i.e., slated for death but possibly still
intact and metabolising) remains approximately constant at each fraction 36.
Further assuming that cells are randomly re-assigned to this hypoxic
compartment (i.e., stochastic reoxygenation, with probability equal to the
hypoxic fraction) at each fraction leads to the result shown in Eq. (21). As
was noted in the Introduction, this widely-used approximation likely
overestimates the effect of hypofractionation, since it neglects to take into
account that the oxygen level kinetics of a given cell depends on its
proximity to the nearest capillary and is _not_ stochastic. In our model, a
cell in close proximity to a blood vessel [i.e., the population of cells not
described by Eq. (4)] would remain oxic even when the capillary oxygen tension
reaches its minimum value (as long as $p_{\mathrm{c,min}}>0$; i.e., $y<1$).
Mathematically, this lack of stochasticity is manifested as
$P(N,0)\gg\left[P(1,0)\right]^{N}$ being well-satisfied for $N>1$; that is,
the probability that a (non-chronically hypoxic) clonogen will be hypoxic for
all $N$ fractions is much greater than
$\langle\mathrm{HF}_{\mathrm{cyclic}}\rangle^{N}$. Hence, the decay in cell-
killing with decreasing fractionation is much lower than would occur for
completely stochastic dynamics.

Biologically effective doses for some common SBRT regimens
---
Site | $D,N$ | BEDoxic | $\mathrm{BED}_{\mathrm{iso-SF}}$ | $\mathrm{BED}_{\mathrm{iso-SF}}$
|  | ($n_{c}=10$ mm-2) | ($n_{c}=100$ mm-2)
Lung | 50, 5 37 | 100 | 38 | 50
48, 4 38 | 106 | 38 | 49
34, 1 38 | 150 | 39 | 49
27*, 1 39 | 100 | 29 | 40
Pancreas | 40-50, 5 40, 41 | 72-100 | 31-38 | 43-50
36, 3 42 | 79 | 30 | 42
25, 1 43 | 88 | 27 | 37
Liver | 35-40, 5 44, 45 | 60-72 | 29-32 | 43-46
40-48, 3 46, 47 | 93-125 | 34-41 | 48-55
Prostate | 42.7, 7 48 | 130 | 59 | 105
36.25, 5 49 | 124 | 54 | 100
19, 1 50 | 139 | 45 | 84

Table 1: Biologically effective doses for some common SBRT schedules,
described by the total dose $D\equiv d\cdot N$ and number of fractions, $N$.
Lung cancer and liver schedules refer to early-stage non-small-cell and
primary liver cancers, respectively. All BED’s are expressed in units of
Gyα/β. For non-prostate sites, $\alpha=0.4$ Gy-1 and $\alpha/\beta=10$ Gy were
used; for prostate, we used $\alpha=0.15$ Gy-1 and $\alpha/\beta=3$ Gy 51.
Iso-SF BED’s [Eq. (20)] are shown for two representative capillary densities
and were calculated using $y=0.4$. *Estimated dose accounting for tissue
heterogeneity corrections.

Even though our model predicts a much smaller reduction in SF, a 1-2 log
increase in SF [e.g., Fig. 2(b.)] in going from $\sim$five to one fraction
would likely have observable clinical effects except for the fact that, for
the most part, hypofractionation has been accompanied by increases in doses
and are not “iso-BED” (iso-$\mathrm{BED}_{\mathrm{oxic}}$). This is apparent
in Table 1, where we list the biologically effective
doses–$\mathrm{BED}_{\mathrm{oxic}}$ and $\mathrm{BED}_{\mathrm{iso-SF}}$ for
two representative values of the capillary density–for some common SBRT
regimens. As an example, for early-stage non-small cell lung cancers, in going
from 50 Gy in five fractions 37 to 34 Gy in a single fraction (RTOG 0915 38),
$\mathrm{BED}_{\mathrm{oxic}}$ increased from 100 Gy10 to 150 Gy10. At the
same time, the iso-SF BED’s defined in Eq. (20) are nearly identical,
reflecting the fact that the 50 Gy10 increase in
$\mathrm{BED}_{\mathrm{oxic}}$ was just enough to overcome the loss of cell
killing due to the diminished effect of inter-fraction oxygen fluctuations.

Although $\mathrm{BED}_{\mathrm{iso-SF}}$ depends sensitively on the capillary
density $n_{c}$, the relative $\mathrm{BED}_{\mathrm{iso-SF}}$ between
different fractions remains largely unchanged. It is also interesting that
survival fractions were not overly sensitive to the precise value of the
amplitude $y$ of capillary oxygen tension fluctuations [Fig. 2(a.)], meaning
that almost any (non-zero) degree of cyclic hypoxia would produce comparable
$\mathrm{BED}_{\mathrm{iso-SF}}$’s. Together, these suggest that
$\mathrm{BED}_{\mathrm{iso-SF}}$ could be used to compare fractionation
schedules, even without knowing $n_{c}$ or $\Delta p_{c}$ precisely.

For most three-to-five fraction regimens in lung, liver, and pancreatic
cancers, the estimated iso-SF BED’s are nearly the same, providing a possible
explanation for why extreme hypofractionation seems to have largely been a
success 35, 38. Primary liver disease is a particularly interesting case,
given the apparent absence of a strong dose-response relationship 52. Even
though this is undoubtedly due in part to the high rates of local control (LC)
in this disease site 53, it is still surprising that even a factor three
increase in the nominal BED ($\mathrm{BED}_{\mathrm{oxic}}$) has such little
apparent effect on LC. Accounting for cyclic hypoxia, however, the spread in
the iso-SF BED’s is greatly reduced, from three to a factor of $\sim$ 1.4.

Nonetheless, our analysis suggests that care should be taken with single-
fraction regimens, in particular lung 27 Gy 39 and prostate 19 Gy 50 regimens
may be under-dosing tumours when factoring in the impact of cyclic hypoxia.
Local control rates and survival at two years have been reported for the lung
study and are similar to those in comparable few-fraction regimens 39. It is
unlikely, however, that a 1-2 log difference in cell survival would result in
a substantial difference in these metrics at two years post-treatment. As an
example of this, consider a poorly-perfused lung tumour with $x=0.7$
($\langle\mathrm{HF}_{\mathrm{cyclic}}\rangle=0.07$). Using Eq. (15), the SF
is estimated to be $3\times 10^{-9}$ and $1\times 10^{-7}$ for 48 Gy in four
fractions and 27 Gy in a single fraction, respectively. Now, assuming that
some clonogens survived treatment and a tumour doubling time
$\tau_{\mathrm{d}}$ of 100 days 14, it would take $\sim$ 6 and $\sim$ 4 years
($=-\tau_{\mathrm{d}}\cdot\ln(\mathrm{SF})/\ln 2$), respectively, for the
tumours to regrow to their pre-SBRT sizes. Assessments of treatment response
efficacy should thus only be made at least four years after completion of SBRT
in sites with high rates of control. Another example is provided by
monotherapy prostate brachytherapy, where 19 Gy in a single fraction was found
to be inferior to (the iso-$\mathrm{BED}_{\mathrm{oxic}}$ schedule) 27 Gy in
two fractions, a difference that was only clearly apparent _after_ several
years of follow-up 54.

Our estimates of clonogen survival and the resulting iso-SF BED are dependent
on a number of approximations. The accuracy of the linear quadratic model
[Eqs. (11) and (12)] has been questioned for high doses per fraction 55. In
assessing clinical data, however, others have argued that there is no
discernible breakdown of this model 56, in particular when accounting for
heterogeneous oxygen distributions that arise in tumours 57, 14. Also, we have
not considered the impact of cell-cycle kinetics through radiotherapy, which
produce a radio-sensitizing effect with increasing fractionation and which
also effect the OER 36. As we have noted throughout this manuscript, the
spatial relationship between hypoxia and the vasculature plays a key role in
determining the impact of cyclic hypoxia on radiation response since it is
responsible for the non-stochasticity of regional levels of hypoxia going
though treatment. Our simple expression for this spatial relationship–given by
Eq. (4)–as well as the compartmentalization used in Eqs. (13) and (15), are
approximations to the continuum distribution of oxygen, obtained by solving
the oxygen reaction-diffusion equation for simulated capillary architectures
18, 13. We believe, however, that our model captures the essence of cyclic
hypoxia and its impact on hypofractionation and that its simplicity will
facilitate analyses of SBRT response, in particular for sites where functional
imaging is available to assess hypoxia and perfusion.

In addition to solving the full reaction-diffusion equations in their study of
cyclic hypoxia and hypofractionated RT, Lindblom and collaborators modelled
cyclic hypoxia as a transient, localized occlusion of blood vessels 13.
Although there is significant evidence that this is not the case, and that
cyclic hypoxia is caused by transient fluctuations in perfusion 20, 21, 22,
this distinction would not qualitatively change the results of our study or
the work of Lindblom et al. It would still be the case that cells proximal to
capillaries are less likely to find themselves hypoxic at different fractions.
Both studies thus find that the impact of hypofractionation is less than would
be expected based on the assumption of complete, stochastic, reoxygenation.

The final major assumption made by us in this work is that substantial
reoxygenation (stochastic or otherwise) does not take place during
hypofractionated RT. A large body of work has investigated reoxygenation and
two types of reoxygenation can be discerned: 1.) transient reoxygenation,
occurring up to several hours after radiation, due to transient metabolic or
perfusion changes 58, 59 and 2.) “long-term” ($\gtrsim$ 1 week) reoxygenation
kinetics, arising from a reduction in metabolic capacity due to cell death 14
and the lysing kinetics of cells killed by RT resulting in blood vessels being
brought closer to previously hypoxic cells 36. Both these sources of
reoxygenation are unlikely to have a large impact on hypofractionated
treatments lasting $\sim 1$ week, with fractions separated by a day or more.
Consistent with this, recent pre-clinical measurements of hypoxia using
positron-emission tomography in pancreatic tumours revealed only a modest
amount of reoxygenation over five fractions (delivered every second day, with
hypoxia measurements separated by two weeks) 9. Additionally, Kaleda and
colleagues found _increased_ levels of hypoxia after (2-4 days post RT) large-
dose single-fraction radiotherapy in lung tumours 60.

A limitation of the present work is that we have only compared different
ultra-hypofractionated radiotherapy regimens with each other, and not with
conventionally fractionated regimens. To do so would require modelling
reoxygenation 14, 15 and proliferation, as well as using the more general
expression for survival fraction given by Eq. (13) and hence, computation of
$P(m,N-m)$ for $m\neq N$, since these probabilities become non-negligible for
smaller per-fraction doses. These straightforward generalizations will be
considered in future work.

In closing, let us ask: are there any actionable discoveries here? Motivated
by a desire to understand the apparent success of ultra-hypofractionated
radiotherapy regimens, we showed that most of these regimens increase dose
sufficiently with decreasing fraction number to overcome the increase in
clonogen survival arising from the loss of inter-fraction oxygen fluctuations.
Can we offer any recommendations for improvements, then? One possibility
concerns “dose painting”, the delivery of higher doses to more hypoxic
regions, typically quantified by positron-emission tomography (PET) using
specialized hypoxia-sensitive tracers 30. As with pimonidazole (discussed in
Sec. II.A.), these tracers are integral markers of cyclic hypoxia, and tracer
uptake in cyclically hypoxic cells will be correspondingly smaller than that
in chronically hypoxic ones 61. Because chronically hypoxic cells are also
less radioresistant that cyclically hypoxic ones, the survival fraction versus
uptake curve (needed for dose painting 16) is relatively “flat”, restricting
the opportunity to meaningfully dose paint based on PET signal. In contrast,
our model predicts that the SF versus capillary density ($n_{c}$) curve will
be sharply peaked, reflecting the fact that most radioresistant cyclically
hypoxic cells lie in the middle ground between very well-perfused and hypo-
perfused; see Fig. 1(b). This suggests that patients may benefit from dose
painting based on _perfusion_ measurements. For our chosen parameter values,
SF is maximal for $n_{c}\sim 10$ mm-2 and is decreased by a factor $\sim
10^{2}$ in going to $n_{c}\sim 100$ mm-2, encompassing a typical range
measured in solid tumours 28, 29. Thus, delivery of higher doses to regions
with lower $n_{c}$ (but not $\lesssim 10$ mm-2) would decrease clonogen
survival. Apart from the inability of hypoxia-PET imaging to resolve cyclic
and chronic hypoxia, an advantage of this approach over PET imaging is that
qualitative perfusion measurements (e.g., biphasic imaging for liver and
pancreatic cancers) are part of routine clinical practice and moreover, even
qualitative 62 metrics have been shown to correlate with microvascular
density. Although the precise parameter values (e.g., the rate of oxygen
metabolism) required to derive the patient-specific SF versus $n_{c}$ curve
are unknown, the preponderance of perfusion imaging means that it should be
possible to derive empirical population-averaged curves by integrating
perfusion images with dose distribution maps and outcome data. Future work
will explore these directions further.

##  V. Conclusions

Cells that are transiently hypoxic have been found to be approximately twice
as radioresistant as chronically hypoxic cells and between two and three times
more radioresistant than oxic cells, possibly due to a diminished capacity to
repair DNA damage. Hence, the kinetics of this population through
hypofractionated radiotherapy may play a major role in determining treatment
response. Using simple analytic approximations for the spatial-temporal
dynamics of cyclic hypoxia, we estimated that most ultra-hypofractionated
regimens give enough dose to counter losses in clonogen cell-killing with
increasing hypofractionation, although some lung and prostate single-fraction
regimens may be under-dosing tumours. Dose painting based on perfusion imaging
may provide a way to target the cyclically hypoxic population.

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