A heterotic integrable deformation of the principal chiral model

David Osten

Institute for Theoretical Physics (IFT),
University of Wrocław
pl. Maxa Borna 9, 50-204 Wrocław, Poland

<EMAIL_ADDRESS>
###### Abstract

A novel classically integrable model is proposed. It is a deformation of the
two-dimensional principal chiral model, embedded into a heterotic
$\sigma$-model, by a particular heterotic gauge field. This is inspired by the
bosonic part of the heterotic $\sigma$-model and its recent Hamiltonian
formulation in terms of O$(d,d+n)$-generalised geometry in [1]. Classical
integrability is shown by construction of a Lax pair and a classical
$\mathcal{R}$-matrix. Latter is almost of the canonical form with twist
function and solves the classical Yang-Baxter equation.

###### Contents

  1. 1 Introduction
  2. 2 The classical bosonic heterotic string in the Hamiltonian formalism
  3. 3 The $A$-deformation of principal chiral model
    1. 3.1 Integrability of the undeformed model
    2. 3.2 Introducing the deformation
    3. 3.3 The Lax pair
    4. 3.4 Hamiltonian integrability
  4. 4 Discussion

##  1 Introduction

One of the main impacts of the study of integrable model in high energy
physics is the connection of quantities of integrable Green-Schwarz
superstring $\sigma$-models in supercoset Anti de Sitter backgrounds and those
of dual integrable conformal field theories [2, 3] in context of the AdS/CFT-
duality. In the last years, some effort has been spent to generalise this to
other, less symmetric settings. On the world-sheet side of the duality, new
integrable models were found that are deformations of such (semi)symmetric
(super)cosets, namely Yang-Baxter deformations [4, 5, 6], $\lambda$-[7] and
WZ-term [8] deformations and their interplay [9], or generalisations to non-
symmetric cosets [10, 11, 12]. So far, this survey was focussing on
conventional two-dimensional non-linear $\sigma$-models, i.e. the embedding of
a two-dimensional world-sheet in some $d$-dimensional target space.111See [11,
13] for two recent reviews and more references.

This article aims to slightly step out of this fairly well explored context
and suggest that also the heterotic string [14, 15, 16, 17] might have
interesting integrable models. In context of this paper, the bosonic part of a
heterotic string $\sigma$-model is a $d$-dimensional non-linear
$\sigma$-model, characterised by a metric and a $B$-field, together with $n$
chiral currents, characterised by a gauge field $A$ and a given
$n$-dimensional gauge group.

#### A heterotic $\mathcal{E}$-model.

Recently, progress in understanding the space of two-dimensional classically
integrable $\sigma$-models was made using two quite different constructions:
in terms of as affine Gaudin models [18, 19], coming from a four-dimensional
Chern-Simons theory [20, 21, 22, 23, 24, 25], or as $\mathcal{E}$-models [26,
27, 28, 29, 30, 31, 32]. Latter does not directly clarify the origin of the
integrable structure but its data relates directly to the $\sigma$-model
target space geometry in the generalised flux formulation. Concretely, an
$\mathcal{E}$-model is defined in terms of O$(d,d)$-covariant world-sheet
currents $J_{A}$ with Poisson brackets – the so-called current algebra –

$\\{J_{A}(\sigma),J_{B}(\sigma^{\prime})\\}=\eta_{AB}\delta^{\prime}(\sigma-\sigma^{\prime})-{F^{C}}_{AB}(\sigma)J_{C}(\sigma)\delta(\sigma-\sigma^{\prime}),$
(1.1)

where $A=1,...,2d$, $d$ is the dimension of the target space, $F_{ABC}$ are
the so-called generalised fluxes, which are constant for an
$\mathcal{E}$-model, and $\eta_{AB}$ is the O$(d,d)$-invariant metric,
together with a Hamiltonian $H=\frac{1}{2}\int\mathrm{d}\sigma\
\delta^{AB}J_{A}J_{B}$.

Similar to that, O$(d,d)$ generalised geometry appeared first in the
Hamiltonian formulation and the analysis of the current algebra of the world-
sheet string [33, 34, 35, 36] before it was considered as a tool for
supergravity [37, 38, 39, 40, 41, 42, 43]. For the heterotic string the
progress was reverse. Heterotic supergravity was phrased in terms of
$O(d,d+n)$ generalised geometry [44, 45, 46, 43], but the role of O$(d,d+n)$
has not been illuminated for the heterotic world-sheet model until recently
[1]. This is the latest addition to the program of duality covariant
formulations of world-volume theories either via actions [47, 48, 49, 50, 51,
52] or via Hamiltonian formulations [53, 54, 55, 56, 57, 58, 59, 60].
Following [1], a heterotic $\mathcal{E}$-model is defined in section 2.

#### A new integrable deformation of the principal chiral model.

The new integrable model proposed here is a deformation of the principal
chiral model, embedded into a heterotic string $\sigma$-model. The deformation
is characterised by a heterotic gauge field:

$A^{\alpha}_{a}=\epsilon\mathcal{A}^{\alpha}_{a},\qquad\text{with}\quad\mathcal{A}^{\alpha}_{a}\mathcal{A}^{\beta}_{b}\kappa_{\alpha\beta}=\kappa_{ab},$
(1.2)

which is constant in flat indices $a,b,c$. Here $\epsilon$ is the deformation
parameter and $\kappa_{ab}$, $\kappa_{\alpha\beta}$ are the Killing forms on
$\mathfrak{g}$ (the Lie algebra from the principal model) and a symmetric non-
degenerate bilinear form on $\mathfrak{u}{(1)}^{n}$ (the abelian gauge
algebra) respectively. The Lax pair and the classical $\mathcal{R}$-matrix, to
show Hamiltonian integrability in Maillet’s formalism [61, 62], are computed
in section 3. It also shown that this deformation can be combined with the
introduction of a WZ-term with arbitrary coefficient while maintaining
classical integrability.

##  2 The classical bosonic heterotic string in the Hamiltonian formalism

In a slight stretch of the concept, a (classical) bosonic heterotic string
here will be a $d$-dimensional non-linear $\sigma$-model together with $n$
(abelian) chiral currents. The $d$ coordinate fields $x$, their canonical
duals $p$ and $n$ chiral currents $k$ can be arranged into an
O$(d,d+n)$-vector:

$\mathcal{P}_{\mathcal{M}}=\left(p_{m}(\sigma),k^{\mu}(\sigma),\partial
x^{m}(\sigma)\right).$ (2.1)

Latin indices $k,l,m,...=1,...,d$ will be reserved for the $d$-dimensional
coordinates, Greek letters $\kappa,\lambda,\mu,...=1,...,n$ for the
$n$-dimensional coordinates. Latter can be raised and lowered using
$\kappa_{\mu\nu}$, the Killing metric on $\mathfrak{u}(1)^{n}$. The canonical
Poisson brackets of these fields can (up to topological terms) also be written
in an O$(d,d+n)$-covariant way:

$\\{\mathcal{P}_{\mathcal{M}}(\sigma),\mathcal{P}_{\mathcal{N}}(\sigma^{\prime})\\}=\eta_{\mathcal{M}\mathcal{N}}\delta^{\prime}(\sigma-\sigma^{\prime})$
(2.2)

with the O$(d,d+n)$-invariant metric

$\eta_{\mathcal{M}\mathcal{N}}=\left(\begin{array}[]{ccc}0&0&\delta_{m}^{n}\\\
0&\kappa_{\mu\nu}&0\\\ \delta^{m}_{n}&0&0\end{array}\right)$ (2.3)

which is also used to raise and lower the
$\mathcal{K},\mathcal{L},\mathcal{N},...$-indices. The action on functions of
these currents is
$\\{\mathcal{P}_{\mathcal{M}},f(x)\\}=-\partial_{\mathcal{M}}f(x)=(-\partial_{m}f(x),0,0)$.
The derivative $\partial_{\mathcal{M}}$ is subject to the section condition
$\eta^{\mathcal{M}\mathcal{N}}\partial_{\mathcal{M}}\ast\partial_{\mathcal{N}}\ast=0=\eta^{\mathcal{M}\mathcal{N}}\partial_{\mathcal{M}}\partial_{\mathcal{N}}\ast$,
which is an O$(d,d+n)$-covariant way to express that the functions only depend
on $x$.

The coupling to the bosonic part of heterotic supergravity, namely a
$d$-dimensional metric $G$ and two-form gauge field $B$ and the heterotic
gauge field $A$ can be encoded in a Hamiltonian that is quadratic in the
currents $\mathcal{P}_{\mathcal{M}}$ and contains the couplings in the so
called generalised metric $\mathcal{H}_{\mathcal{M}\mathcal{N}}$:

$\displaystyle H=\frac{1}{2}\int\mathrm{d}\sigma\
\mathcal{H}^{\mathcal{M}\mathcal{N}}(G,B,A)\mathcal{P}_{\mathcal{M}}(\sigma)\mathcal{P}_{\mathcal{N}}(\sigma).$
(2.4)

The spatial diffeomorphism constraint also takes a simple form,
$0=\mathcal{P}_{\mathcal{M}}\mathcal{P}_{\mathcal{N}}\eta^{\mathcal{M}\mathcal{N}}$,
but will not play a role in the further discussion. The explicit generalised
metric $\mathcal{H}(G,B,A)$ and the connection to a Lagrangian picture, and
also the generalisation to non-abelian currents has been discussed in detail
in [1].

For the purpose of this paper the generalised flux formulation is better
suited. To this end, we diagonalise the generalised metric
$\mathcal{H}_{\mathcal{A}\mathcal{B}}={E_{\mathcal{A}}}^{\mathcal{M}}{E_{\mathcal{B}}}^{\mathcal{N}}\mathcal{H}_{\mathcal{M}\mathcal{N}}$
with a generalised vielbein ${E_{\mathcal{A}}}^{\mathcal{M}}$ and some
suitable constant $\mathcal{H}_{\mathcal{A}\mathcal{B}}$. The coupling to the
background is encoded in the Poisson brackets via the so-called generalised
fluxes $F_{\mathcal{A}\mathcal{B}\mathcal{C}}$:

$\\{J_{\mathcal{A}}(\sigma),J_{\mathcal{B}}(\sigma^{\prime})\\}=\eta_{\mathcal{AB}}\delta^{\prime}(\sigma-\sigma^{\prime})-{F^{\mathcal{C}}}_{\mathcal{A}\mathcal{B}}(\sigma)J_{\mathcal{C}}(\sigma)\delta(\sigma-\sigma^{\prime})$
(2.5)

with
$F_{\mathcal{A}\mathcal{B}\mathcal{C}}=3\partial_{\mathcal{[A}}{E_{\mathcal{B}}}^{\mathcal{N}}{E_{\mathcal{C]N}}}$.
In the particular case that these $F_{\mathcal{ABC}}$ can be chosen to be
constant, we would call this construction heterotic $\mathcal{E}$-model, in
analogy to the construction for ordinary $2d$ $\sigma$-models and
O$(d,d)$-generalised geometry. For a generic heterotic supergravity background
these fluxes are not constant.

In the standard (geometric) parametrisation of the background in terms of
metric $G$, two-form gauge field $B$ and heterotic gauge field $A$ the
generalised vielbein is [44]

${E_{\mathcal{A}}}^{\mathcal{M}}=\left(\begin{array}[]{ccc}\delta_{a}^{b}&A_{a}^{\alpha}&B_{ab}-\frac{1}{2}A_{\alpha
a}A^{\alpha}_{b}\\\ 0&\delta_{\alpha}^{\beta}&-A_{\alpha b}\\\
0&0&\delta^{a}_{b}\end{array}\right)\left(\begin{array}[]{ccc}{e_{b}}^{m}&0&0\\\
0&{\delta_{\beta}}^{\mu}&0\\\ 0&0&{e^{b}}_{m}\end{array}\right)$ (2.6)

with ${e_{a}}^{m}{e_{b}}^{n}G_{mn}=\kappa_{ab}$, which will be used to raise
and lower $a,b,...$ indices. Then, the non-vanishing components of
$F_{\mathcal{ABC}}$ are:

$\displaystyle H_{abc}$
$\displaystyle=F_{abc}=\left[\partial_{a}B_{bc}+{f^{d}}_{ab}\left(B_{cd}-\frac{1}{2}A^{\alpha}_{c}A_{d\alpha}\right)+(\partial_{[a}A_{b]}^{\alpha})A_{\alpha
c}\right]+\text{cyclic permutations of }abc$ $\displaystyle{f^{c}}_{ab}$
$\displaystyle={F^{c}}_{ab}=2{e_{n}}^{c}\partial_{[a}{e_{b]}}^{n}$ (2.7)
$\displaystyle{F^{\gamma}}_{ab}$
$\displaystyle=2\partial_{[a}A_{b]}^{\gamma}-{f^{c}}_{ab}A_{c}^{\gamma}.$

In hindsight of what is to come, we write
$J_{\mathcal{A}}=(J_{a},J^{\alpha},J^{a})=(j_{0a},k^{\alpha},j_{1}^{a})$. The
subscripts $0,1$ refer to the world-sheet coordinates $\tau,\sigma$. In this
decomposition the current algebra (2.5) is

$\displaystyle\\{j_{0a}(\sigma),j_{0b}(\sigma^{\prime})\\}$
$\displaystyle=\left(-{f^{c}}_{ab}j_{0c}-{F^{\gamma}}_{ab}k_{\gamma}(\sigma)-H_{abc}j_{1}^{c}\right)\delta(\sigma-\sigma^{\prime})$
$\displaystyle\\{j_{0a}(\sigma),j_{1}^{b}(\sigma^{\prime})\\}$
$\displaystyle=\delta_{a}^{b}\delta^{\prime}(\sigma-\sigma^{\prime})-{f^{b}}_{ca}j_{1}^{c}(\sigma)\delta(\sigma-\sigma^{\prime})$
$\displaystyle\\{j_{0a}(\sigma),k^{\beta}(\sigma^{\prime})\\}$
$\displaystyle=-{F^{\beta}}_{ca}j_{1}^{c}\delta(\sigma-\sigma^{\prime})$ (2.8)
$\displaystyle\\{k^{\alpha}(\sigma),k^{\beta}(\sigma^{\prime})\\}$
$\displaystyle=\kappa^{\alpha\beta}\delta^{\prime}(\sigma-\sigma^{\prime})$
$\displaystyle\\{j_{1}^{a}(\sigma),j_{1}^{b}(\sigma^{\prime})\\}$
$\displaystyle=0=\\{j_{1}^{a}(\sigma),k^{\beta}(\sigma^{\prime})\\}.$

The resulting equations of motion
$\partial_{\tau}\mathcal{O}=\\{\mathcal{O},H\\}$ with the Hamiltonian (2.4) in
the generalised metric frame,

$H=\frac{1}{2}\int\mathrm{d}\sigma\
\left(\kappa^{ab}j_{0a}(\sigma)j_{0b}(\sigma)+\kappa_{ab}j_{1}^{a}(\sigma)j_{1}^{b}(\sigma)+\kappa_{\alpha\beta}k^{\alpha}(\sigma)k^{\beta}(\sigma)\right),$
(2.9)

are

$\displaystyle 0$
$\displaystyle=\partial_{+}j_{-a}+\partial_{-}j_{+a}-2{F^{\beta}}_{ca}k_{\beta}j_{-}^{c}-H_{abc}j_{+}^{b}j_{-}^{c}$
$\displaystyle 0$
$\displaystyle=\partial_{+}j_{-}^{a}-\partial_{-}j_{+}^{a}+{f^{a}}_{bc}j_{+}^{b}j_{-}^{c}$
(2.10) $\displaystyle 0$
$\displaystyle=2\partial_{-}k^{\alpha}-{F^{\alpha}}_{bc}j_{+}^{b}j_{-}^{c}$

in conventions with $V_{\pm}=V_{0}\pm V_{1}$. The generalisation of these
equations of motion to a generic $F_{\mathcal{ABC}}$, corresponding to choices
of frame than (2.6), is straightforward as for O$(d,d)$, e.g. in [29].

##  3 The $A$-deformation of principal chiral model

There is a plethora of integrable deformations of the principal chiral model:
$\lambda$-, $\eta$-, homogeneous Yang-Baxter and WZ-term-deformation and
combinations of those. Here, a new deformation of a $d$-dimensional principal
chiral model by the heterotic gauge field $A$ is proposed.

###  3.1 Integrability of the undeformed model

As starting point we take the principal chiral model on a group $G$ with Lie
algebra $\mathfrak{g}$, embedded into the bosonic heterotic $\sigma$-model. In
the language of the previous section, this will be characterised by the
components of the Maurer-Cartan form on $G$ as vielbein $e$ and $B=A=0$ in the
generalised vielbein (2.6). This corresponds to the generalised fluxes (2.7)

${f^{c}}_{ab},\qquad H_{abc}=0,\qquad{F^{\gamma}}_{ab}=0,$ (3.1)

where ${f^{c}}_{ab}$ are structure constants to the Lie algebra
$\mathfrak{g}$. This corresponds to the following equations of motion

$\partial_{+}j_{-a}+\partial_{-}j_{+a}=0,\qquad\partial_{+}j_{-}^{a}-\partial_{-}j_{+}^{a}+{f^{a}}_{bc}j_{+}^{b}j_{-}^{c}=0,\qquad\partial_{-}k^{\alpha}=0.$
(3.2)

This model is classically integrable, as the currents $k$ and the
$d$-dimensional principal chiral model with the currents $j$ decouple.
Classical integrability is shown by providing a Lax representation and a
classical $\mathcal{R}$-matrix, proving Hamiltonian integrability, i.e.
involution of the charges produced from the Lax representation.

The Lax representation of this model is

$L_{+}(\lambda)=\frac{1}{1-\lambda}\big{(}j_{+}^{a}t_{a}+2k^{\alpha}t_{\alpha}\big{)},\qquad
L_{-}(\lambda)=\frac{1}{1+\lambda}j_{-}^{a}t_{a}\qquad\in\tilde{\mathfrak{g}}=\mathfrak{g}\oplus\mathfrak{u}(1)^{n},$
(3.3)

where $\lambda$ is the spectral parameter. $(t_{a},t_{\alpha})$ are generators
of $\tilde{\mathfrak{g}}=\mathfrak{g}\oplus\mathfrak{u}(1)^{n}$ which is
defined by

$[t_{a},t_{b}]={f^{c}}_{ab}t_{c},\qquad[t_{\alpha},t_{\beta}]=0=[t_{a},t_{\alpha}].$

A method to show involution of the associated tower of conserved charges is
Maillet’s formalism [61, 62]. Central for this is the Lax matrix

$\mathcal{L}(\lambda)=L_{1}(\lambda)=\frac{1}{2}(L_{+}(\lambda)-L_{-}(\lambda))=\frac{\lambda
j_{0}^{a}+j_{1}^{a}}{1-\lambda^{2}}t_{a}+\frac{1}{1-\lambda}k^{\alpha}t_{\alpha}$
(3.4)

The aim is to construct a so-called classical $\mathcal{R}$-matrix, i.e. an
object
$\mathcal{R}(\lambda,\mu)\in\tilde{\mathfrak{g}}\otimes\tilde{\mathfrak{g}}$
such that

$\displaystyle\\{\mathcal{L}_{\mathbf{1}}(\lambda,\sigma),\mathcal{L}_{\mathbf{2}}(\mu,\sigma^{\prime})\\}$
$\displaystyle=\left([\mathcal{R}_{\mathbf{12}}(\lambda,\mu),\mathcal{L}_{\mathbf{1}}(\lambda,\sigma)]-[\mathcal{R}_{\mathbf{21}}(\mu,\lambda),\mathcal{L}_{\mathbf{2}}(\mu,\sigma)]\right)\delta(\sigma-\sigma^{\prime})$
$\displaystyle{}\quad-\left(\mathcal{R}_{\mathbf{12}}(\lambda,\mu)+\mathcal{R}_{\mathbf{21}}(\mu,\lambda)\right)\delta^{\prime}(\sigma-\sigma^{\prime}).$
(3.5)

A bold subscript indicates the place in the tensor product, e.g.
$X_{\mathbf{1}}=X\otimes\mathbb{1}$. A sufficient condition for the involution
of the conserved charges constructed from the Lax pair is the classical Yang-
Baxter equation:

$[\mathcal{R}_{\mathbf{12}}(\lambda_{1},\lambda_{2}),\mathcal{R}_{\mathbf{13}}(\lambda_{1},\lambda_{3})]+[\mathcal{R}_{\mathbf{12}}(\lambda_{1},\lambda_{2}),\mathcal{R}_{\mathbf{23}}(\lambda_{2},\lambda_{3})]+[\mathcal{R}_{\mathbf{32}}(\lambda_{3},\lambda_{2}),\mathcal{R}_{\mathbf{31}}(\lambda_{3},\lambda_{1})]=0.$
(3.6)

The classical $\mathcal{R}$-matrix for the undeformed model (3.2) is

$\mathcal{R}(\lambda,\mu)=\frac{t_{a}\otimes
t^{a}}{\lambda-\mu}\phi^{-1}_{PCM}(\mu)+\frac{t_{\alpha}\otimes
t^{\alpha}}{\lambda-\mu}\phi^{-1}_{chiral}(\mu).$ (3.7)

with the so-called twist functions

$\phi_{PCM}(\lambda)=\frac{1-\lambda^{2}}{\lambda^{2}},\qquad\phi_{chiral}(\lambda)=1-\lambda.$
(3.8)

This is close to the twist function form
$\mathcal{R}(\lambda,\mu)=\mathcal{R}^{(0)}(\lambda,\mu)\phi^{-1}(\lambda)$
with a solution of the classical Yang-Baxter equation $\mathcal{R}^{(0)}$,
like $\mathcal{R}(\lambda,\mu)=\frac{t_{a}\otimes t^{a}}{\lambda-\mu}$ and a
twist function $\phi(\mu)$. This form ensures that this is a solution of the
classical Yang-Baxter equation (3.6). Although not quite of this form, it is
nevertheless easy to see that (3.7) solves the classical Yang-Baxter equation,
as $\mathfrak{t}_{\alpha}\otimes t^{\alpha}$ is a central element of
$\tilde{\mathfrak{g}}\otimes\tilde{\mathfrak{g}}$, and all its contributions
drop out of the classical Yang-Baxter equation (3.6).

###  3.2 Introducing the deformation

The deformation is characterised by a constant (in flat indices) heterotic
gauge field

$A_{a}^{\alpha}=\epsilon\mathcal{A}_{a}^{\alpha},\quad\text{with}\quad\mathcal{A}_{a}^{\alpha}\mathcal{A}_{b}^{\beta}\kappa_{\alpha\beta}=\kappa_{ab},$
(3.9)

so $\mathcal{A}\in\ $O$(d)\subset\ $Mat$(n,d)$. For this one needs that $n>d$
which would be the case for the bosonic part of the heterotic string where
$n=16$.222This is due the fact that U$(1)^{n}$ is the Cartan subgroup of
$E_{8}\times E_{8}$ or SO$(32)$. We use $\kappa_{ab}$ (as the Killing metric
on $\mathfrak{g})$ and $\kappa_{\alpha\beta}$ to raise and lower indices on
$\mathcal{A}$, e.g. to define the object $\mathcal{A}^{a}_{\alpha}$. Care
needs to be taken that when contracting $\mathcal{A}$’s in the $n$-dimensional
indices one will produce $A_{a}^{\alpha}\mathcal{A}_{b\alpha}=\kappa_{ab}$ by
definition (3.9), but contractions of the $d$-dimensional flat indices will
result in a projector on the part of the $n$-dimensional currents, that couple
to the $j_{+}$ (see below in the equations of motions)

$\mathcal{P}^{\alpha\beta}=\mathcal{A}^{\alpha}_{a}\mathcal{A}^{a\beta}.$
(3.10)

In particular, one has that
${\mathcal{P}^{\alpha}}_{\gamma}\mathcal{P}_{\gamma\beta}=\mathcal{P}_{\alpha\beta}$
and
${\mathcal{P}^{\alpha}}_{\beta}\mathcal{A}^{\beta}_{a}=\mathcal{A}^{\alpha}_{b}$.
So contractions like $\mathcal{A}_{a\alpha}k^{\alpha}$ will project out the
decoupled components of $k$ as well. One could, of course, decompose the
$\mathfrak{u}(1)^{n}$-currents into coupled and decoupled currents
immediately, by choosing the generators of $\mathfrak{u}(1)^{n}$ accordingly.
Also, in this sense, there is only one inequivalent $A$-deformation. All
$\mathcal{A}\in\ $O$(d)$ are related by a basis change of
$\mathfrak{u}(1)^{n}$. Keeping in mind possible generalisations, the
$n$-dimensional covariance by introducing $\mathcal{A}$ is kept in the
following.

Resultingly, any choice of gauge field satisfying (3.9), should result in the
same deformation of the fluxes

$\displaystyle H_{abc}$
$\displaystyle=\left(\tilde{h}-\frac{3}{2}\epsilon^{2}\right)f_{abc}=hf_{abc},$
$\displaystyle{f^{c}}_{ab}$ $\displaystyle={f^{c}}_{ab},$ (3.11)
$\displaystyle{F^{\gamma}}_{ab}$
$\displaystyle=-\epsilon\mathcal{A}_{c}^{\gamma}{f^{c}}_{ab},$

where an WZ-term with some free coefficient $\tilde{h}$ was allowed as well,
i.e. a $B$-field such that $H\sim f$. Effectively, this results in a two-
parameter deformation where the WZ-part is, of course, well understood but it
interesting to note that one can combine it with the novel $A$-deformation.
The equations of motion are:

$\displaystyle 0$
$\displaystyle=\partial_{+}j_{-}^{a}+\partial_{-}j_{+}^{a}+2\epsilon\mathcal{A}^{b}_{\beta}{f^{a}}_{bc}k_{+}^{\beta}j_{-}^{c}-h{f^{a}}_{bc}j_{+}^{b}j_{-}^{c},$
$\displaystyle 0$
$\displaystyle=\partial_{+}j_{-}^{a}-\partial_{-}j_{+}^{a}+{f^{a}}_{bc}j_{+}^{b}j_{-}^{c},$
(3.12) $\displaystyle 0$
$\displaystyle=2\partial_{-}k^{\alpha}+\epsilon\mathcal{A}^{\alpha}_{a}{f^{a}}_{bc}j_{+}^{b}j_{-}^{c}.$

It is convenient to decompose the last equation into an $d$-dimensional part
of $k$ that couples to the $j$, and the $n-d$-dimensional part that is still
decoupled:

$\displaystyle 0$
$\displaystyle=\mathcal{A}^{a}_{\alpha}\left(2\partial_{-}k^{\alpha}+\epsilon{f^{a}}_{bc}j_{+}^{b}j_{-}^{c}\right),\qquad
0=(1-P)_{\alpha\beta}\partial_{-}k^{\beta}.$ (3.13)

These decoupled components and the associated generators will be denoted with
a bar in the following:

$\displaystyle{(1-P)^{\alpha}}_{\beta}k^{\beta}=\bar{k}^{\alpha},\qquad{(1-P)_{\alpha}}^{\beta}t_{\beta}=\bar{t}_{\alpha}.$

###  3.3 The Lax pair

In order to show the existence of a Lax pair $L_{\pm}\in\tilde{\mathfrak{g}}$,
we make the ansatz is

$\displaystyle L_{-}(\lambda)=a(\lambda)j_{-}^{a}t_{a},\qquad
L_{+}=\big{(}b(\lambda)j_{+}^{a}+c(\lambda)\epsilon\mathcal{A}^{a}_{\alpha}k^{\alpha}\big{)}t_{a}+\frac{2}{1-\lambda}\bar{k}^{\alpha}\bar{t}_{\alpha}.$
(3.14)

where the coefficient in front of the decoupled currents $\bar{k}$ is taken
from the undeformed case (3.3) and $\lambda$ is the spectral parameter. The
coefficients $a(\lambda),b(\lambda),c(\lambda)$ are determined such that

$0=\partial_{+}L_{-}-\partial_{-}L_{+}+[L_{+},L_{-}]\in\tilde{\mathfrak{g}}$
(3.15)

is valid by virtue of the equations of motion (3.12). This results in the
under-determined system of equations:

$\displaystyle 0=\frac{a+b}{2}-ab-\frac{1}{2}(\epsilon^{2}c+h(a-b)),\qquad
0=-(a-b)+ac.$

A convenient parametrisation of the solution to this is

$a=\frac{1}{1+\lambda},\quad
b=\frac{1-[\epsilon^{2}(1+\lambda)+h]}{f_{\epsilon,h}(\lambda)},\quad
c=-\frac{2\lambda}{f_{\epsilon,h}(\lambda)},$ (3.16)

with
$f_{\epsilon,h}(\lambda)=1-\lambda-(1+\lambda)[\epsilon^{2}(1+\lambda)+h]$. In
that parametrisation one can recognise a typical parametrisation of the Lax
pair of the principal chiral model in the undeformed limit. I.e. for the
$\mathfrak{g}$-valued part of $L(\lambda)$

$L^{a}_{\pm}(\lambda)\longrightarrow\frac{j_{\pm}^{a}}{1\mp\lambda},\quad\text{for}\quad\epsilon,h\rightarrow
0.$ (3.17)

But the full Lax pair $L_{\pm}\in\tilde{\mathfrak{g}}$ in the limit
$\epsilon\rightarrow 0$ does not give the full undeformed Lax pair (3.3) – the
part of the $k^{\alpha}$ that is coupled in the deformed models disappears
from the Lax pair and the equations of motion reconstructed from it.

###  3.4 Hamiltonian integrability

One should check whether the conserved charges from the Lax pair are also in
Poisson involution for arbitrary choices of $\epsilon$ and $h$. The relevant
quantity for this is the Lax matrix

$\displaystyle\mathcal{L}$ $\displaystyle\equiv
L_{1}=\frac{1}{(1+\lambda)f_{\epsilon,h}(\lambda)}\left[\lambda
j_{0}^{a}+\left(f_{\epsilon,h}(\lambda)+\lambda)\right)j_{1}^{a}-\epsilon\
\lambda(1+\lambda)\mathcal{A}_{\alpha}^{a}k_{+}^{\alpha}\right]t_{a}+\frac{1}{1-\lambda}\bar{k}^{\alpha}\bar{t}_{\alpha}.$
(3.18)

A corresponding classical $\mathcal{R}$-matrix is

$\mathcal{R}_{\mathbf{12}}(\lambda,\mu)=\frac{t_{a}\otimes
t^{a}}{\lambda-\mu}\phi^{-1}_{\epsilon,h}(\mu)+\frac{\bar{t}^{\alpha}\otimes\bar{t}_{\alpha}}{\lambda-\mu}\frac{1}{1-\mu},\qquad\text{with}\quad\phi_{\epsilon,h}(\mu)=\frac{(1+\mu)f_{\epsilon,h}(\mu)}{\mu^{2}}.$
(3.19)

Again, $\mathcal{R}$ is almost of the form with twist function and solves the
classical Yang-Baxter equation (3.6) by the same argument as in the undeformed
case. This $\mathcal{R}$-matrix could be guessed relatively easily from
inspiration from the form of the $\mathcal{R}$-matrix of the principal chiral
model. It is easy to see that the $\mathfrak{g}\otimes\mathfrak{g}$-valued
part is reproduced for $\epsilon,h\rightarrow 0$, as then
$f_{\epsilon,h}(\lambda)\rightarrow(1-\lambda)$. But, as also for the Lax
pair, the full $\mathcal{R}$-matrix (3.19) does not reproduce the full
undeformed $\mathcal{R}$-matrix (3.7) in the non-deformation limit.

One checks that (3.19) and (3.18) fit into Maillet’s formalism, by showing
that for the $\mathfrak{g}\otimes\mathfrak{g}$-valued part both the left-hand
and right-hand side of Maillet bracket (3.5) correspond to

$\displaystyle{}\qquad\text{\scriptsize$\frac{t^{a}\otimes
t^{b}}{(1+\lambda)f_{\epsilon,h}(\lambda)(1+\mu)f_{\epsilon,h}(\mu)}$}\Big{(}{f^{c}}_{ab}\
\delta(\sigma-\sigma^{\prime})\
\Big{(}(-\lambda\mu)j_{0c}(\sigma)+(\lambda\mu)\
\epsilon\mathcal{A}^{\gamma}_{c}k_{\gamma}(\sigma)$
$\displaystyle{}\qquad\qquad-\big{[}(\lambda+\mu)-\epsilon^{2}(\lambda+\mu+2\lambda\mu)-h(\lambda+\mu+\lambda\mu)\big{]}j_{1c}(\sigma)\Big{)}$
(3.20)
$\displaystyle{}\qquad\qquad+\kappa_{\alpha\beta}\delta^{\prime}(\sigma-\sigma^{\prime})\
\Big{(}(\lambda+\mu)-\epsilon^{2}(\lambda+\mu+3\mu\lambda-\lambda^{2}\mu^{2})-h(\lambda+\mu+2\mu\lambda)\Big{)}\Big{)}.$

The part valued in $\mathfrak{u}(1)^{n}\otimes\mathfrak{u}(1)^{n}$ remains
unchanged from the discussion in the undeformed case.

##  4 Discussion

In this article a new integrable deformation of the principal chiral model
inspired by the heterotic string was proposed. It builds on the recent
Hamiltonian formulation of the bosonic part of the heterotic string
$\sigma$-model in terms of $O(d,d+n)$-generalised geometry in [1]. Building on
that the analogue to the so-called $\mathcal{E}$-model in O$(d,d)$-generalised
geometry was proposed in section 2. The deformation was introduced in section
3 and is characterised by a certain choice of heterotic gauge field
$A^{\alpha}=\epsilon\mathcal{A}^{\alpha}$. The concrete choice of
$\mathcal{A}\in\ $O$(d)$, can be absorbed into a change of basis for the
$n$-dimensional current $k$. Hence, this deformation of $A$ really only
results in a one-parameter deformation given by the fluxes (3.11). It is also
shown that this deformation can be extended to a two-parameter deformation
together with a WZ-term like deformation, while preserving integrability.
Classical integrability was shown by the explicit construction of a Lax pair
(3.18) and a classical $\mathcal{R}$-matrix (3.19) that solves the classical
Yang-Baxter equation.

To the knowledge of the author this is uncharted territory. From here several
direction should be taken to further explore this space of novel integrable
$\sigma$-models:

  * •

Generalisation to the full (supersymmetric) heterotic string

  * •

Different starting point for the deformation:

    * -

symmetric space coset $\sigma$-models, Yang-Baxter or $\lambda$-deformation,
instead of the principal chiral model for a Lie group $G$

    * -

non-abelian chiral currents $k^{\alpha}$, i.e. some non-trivial gauge algebra
$h$ instead of $\mathfrak{u}(1)^{n}$

  * •

Dual version of the $A$-deformation.333In generalisation of the fact, that one
can have integrable deformations based on the Kalb-Ramond field $B$ (WZ-term)
or their duals $\beta$ ($\eta$\- and homogeneous Yang-Baxter deformations),
there could be also a dual version of $A$-deformation. The natural starting
point would be the dual vielbein to (2.6) with dual fields $\alpha$ and
$\beta$ to $A$ and $B$:

${E_{\mathcal{A}}}^{\mathcal{M}}=\left(\begin{array}[]{ccc}\delta_{a}^{b}&0&0\\\
\alpha_{\alpha}^{b}&\delta_{\alpha}^{\beta}&0\\\
\beta^{ab}-\frac{1}{2}\alpha^{a}_{\alpha}\alpha^{b\alpha}&-\alpha^{a\beta}&\delta^{a}_{b}\end{array}\right)\left(\begin{array}[]{ccc}{e_{b}}^{m}&0&0\\\
0&{\delta_{\beta}}^{\mu}&0\\\ 0&0&{e^{b}}_{m}\end{array}\right).$ (4.1)

Beside this, there are some questions that are not addressed in this article
and left for future research:

#### Non-deformation limit.

As discussed briefly in the main text, the objects of the Lax construction,
i.e. the Lax pair and the classical $\mathcal{R}$-matrix do not have the
correct non-deformation limit. The $\mathfrak{g}$-valued part of the Lax
connection (3.18) correctly reproduces the one of the principal chiral, but
the full $\mathfrak{g}\oplus\mathfrak{u}(n)^{n})$-valued Lax connection does
not reproduce the full undeformed one (3.3). A similar problem also arises for
the poles and zeros of the twist function. Whereas
$\phi_{\epsilon,h}\rightarrow\phi_{PCM}$ and both have double poles at $0$ and
infinity, $\phi_{\epsilon,h}=(1+\lambda)f_{\epsilon,h}(\lambda)\lambda^{-2}$
has three zeros:

$\lambda_{1}=-1,\qquad\lambda_{2,3}=-\frac{1}{2\epsilon^{2}}\left(2\epsilon^{2}+(1+h)\pm\sqrt{8\epsilon^{2}+(h+1)^{2}}\right)$

For an analysis of the $h,\epsilon\rightarrow 0$-limit, a concrete
$\epsilon$-dependence of $h=\tilde{h}-\frac{3}{2}\epsilon^{2}$ resp. also of
$\tilde{h}$ needs to be specified. As an example, let us assume that $h=h_{0}$
has no $\epsilon$-dependence (so $\tilde{h}=h_{0}+\frac{3}{2}\epsilon^{2}$).
Then one of the $\lambda_{2,3}$ has the expected $\epsilon\rightarrow
0$-limit:

$\displaystyle\lambda_{2}=-\frac{1+h_{0}}{\epsilon^{2}}-\frac{3}{1+h_{0}}+\mathcal{O}(\epsilon^{2})\overset{\epsilon,h_{0}\rightarrow
0}{\longrightarrow}-\infty,\qquad\lambda_{3}=\frac{1}{1+h_{0}}+\mathcal{O}(\epsilon^{2})\overset{\epsilon,h_{0}\rightarrow
0}{\longrightarrow}=1$

Latter gives the expected zero in comparison to the twist function of the
principal chiral model, but the role and origin of the additional zero at
infinity is not clear at this stage. Let us summarise these issues:

deformed Lax pair (3.18) $\displaystyle\nrightarrow\text{undeformed Lax pair
\eqref{eq:LPairUndeformed}}$ deformed $\mathcal{R}$-matrix (3.19)
$\displaystyle\nrightarrow\text{undeformed $\mathcal{R}$-matrix
\eqref{eq:RMatrixUndeformed}}\qquad\text{for}\ \epsilon,h\rightarrow 0$ 3
zeros of $\phi_{\epsilon,h}$ $\displaystyle\nrightarrow\text{2 zeros of
$\phi_{PCM}$}.$

It would be interesting to see whether a different construction could be found
which has a correct non-deformation limit.

#### Reduction of a conventional $\sigma$-model.

From the point of view of the systematics of integrable models, the question
is: how new or exotic is this model really? The present Lax representation is
valued in $\mathfrak{g}\oplus\mathfrak{u}(1)^{n}$. A natural candidate for a
more conventional integrable $\sigma$-model, that would reproduce the same
equations of motion (2.10), would be something like a principal chiral model
or a WZW-model with target space $G\times T^{n}$, together with some
constraint that makes $k_{-}$ (which did not appear in the construction here)
vanish.

This is clearly possible for the undeformed model. For the deformed model this
does not seem so simple and deserves further attention.

#### Supergravity and $\alpha^{\prime}$.

The usual question for the integrable deformations and their application in
string theory is, whether the deformed backgrounds are also supergravity
solutions, given that the undeformed one is. A detailed discussion is left for
future research when deformations are known that start at a more realistic
starting point than the principal chiral model.444In comparison to the type II
$\sigma$-models, one could for example not make a principal chiral model to a
string $\sigma$-model by simply choosing a supergroup with vanishing Killing
form.

Another connected issue when comparing to heterotic supergravity is the
following. Matching the present parametrisation (2.6) to the standard one
(i.e. [44, 1]) that would lead to the heterotic supergravity action, one would
need to make the identification $A\rightarrow\sqrt{\alpha^{\prime}}A$ in
(2.6). One notices that deformation parameter enters at exactly the same level
as $\sqrt{\alpha^{\prime}}$. Hence, in cases where the starting point is a
supergravity background, one might identify

$\epsilon^{2}\sim\alpha^{\prime},$ (4.2)

up to dimensionality, and the integrable deformation becomes the
$\alpha^{\prime}$-correction.

## Acknowledgements

The author thanks Falk Hassler and Eric Lescano for comments on the draft.

This research is part of the project No. 2022/45/P/ST2/03995 co-funded by the
National Science Centre and the European Union’s Horizon 2020 research and
innovation programme under the Marie Skłodowska-Curie grant agreement no.
945339.

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# Mixed Supervision of Histopathology Improves Prostate Cancer Classification
from MRI

Abhejit Rajagopal  Antonio C. Westphalen  Nathan Velarde  Tim Ullrich  Jeffry
P. Simko  Hao Nguyen  Thomas A. Hope  Peder E. Z. Larson  Kirti Magudia A.
Rajagopal, N. Velarde, T.A. Hope, and P.E.Z. Larson are with the Department of
Radiology and Biomedical Imaging, University of California, San Francisco,
94158 USA. J.P. Simko is with the Department of Pathology, University of
California, San Francisco, 94158 USA. H. Nguyen is with the Department of
Urology, University of California, San Francisco, 94158 USA. K. Magudia is
with the Department of Radiology, Duke University, Durham 27705 USA. T.
Ullrich is with the Department of Diagnostic and Interventional Radiology,
University of Dusseldorf, Germany. A.C. Westphalen is with the Department of
Radiology, University of Washington, Seattle 98105 USA. Correspondence e-mail:
<EMAIL_ADDRESS>
###### Abstract

Non-invasive prostate cancer detection from MRI has the potential to
revolutionize patient care by providing early detection of clinically-
significant disease (ISUP grade group $\geq 2$), but has thus far shown
limited positive predictive value. To address this, we present an MRI-based
deep learning method for predicting clinically significant prostate cancer
applicable to a patient population with subsequent ground truth biopsy results
ranging from benign pathology to ISUP grade group 5. Specifically, we
demonstrate that mixed supervision via diverse histopathological ground truth
improves classification performance despite the cost of reduced concordance
with image-based segmentation. That is, where prior approaches have utilized
pathology results as ground truth derived from targeted biopsies and whole-
mount prostatectomy to strongly supervise the localization of clinically
significant cancer, our approach also utilizes weak supervision signals
extracted from nontargeted systematic biopsies with regional localization to
improve overall performance. Our key innovation is performing regression by
distribution rather than simply by value, enabling use of additional pathology
findings traditionally ignored by deep learning strategies. We evaluated our
model on a dataset of 973 (testing $n=160$) multi-parametric prostate MRI
exams collected at UCSF from 2015-2018 followed by MRI/ultrasound fusion
(targeted) biopsy and systematic (nontargeted) biopsy of the prostate gland,
demonstrating that deep networks trained with mixed supervision of
histopathology can significantly exceed the performance of the Prostate
Imaging-Reporting and Data System (PI-RADS) clinical standard for prostate MRI
interpretation.

MRI, prostate cancer, deep learning, histopathology, regression, weak spatial
supervision

##  1 Introduction

Although prostate cancer has the highest incidence of any invasive cancer in
American men, survival from localized prostate cancer is 100% [1]. Prostate
Specific Antigen (PSA) blood tests historically used for prostate cancer
screening have high sensitivity but low specificity. Prostate MRI has been
widely incorporated into practice in combination with Prostate Imaging-
Reporting and Data System (PI-RADS), allowing direct MR-guided biopsy of the
prostate gland or more commonly MR/ultrasound fusion biopsy [2, 3]. However,
PI-RADs demonstrates a wide variability in positive predictive value and
overall low positive predictive value for clinically significant prostate
cancer (CS-PCa) [4]. Thus, there is great interest in developing data-driven
deep learning methods to augment prostate MRI interpretation to the benefit of
improved detection of CS-PCa.

Figure 1: Overview of UCNet model for prediction of prostate cancer from MRI
and training via mixed supervision of histopathology ground truth. The 3D
model inputs (blue) are fed to a fully-convolutional neural network (here, a
3D Residual UNet) to produce two primary outputs (purple): a Lesion Risk Map
and a Cancer Grade Map. The primary outputs are compatible with conventional
strong voxel-level ground truth for lesions, but incompatible with biopsy data
from inhomogeneous tissue in the sextants. To handle this, we query the Cancer
Grade Map with geometric masks to produce regional cancer grade histograms,
which are weakly supervised using a final RegionNet classification module and
a novel distribution-based loss for cancer.

Treatment of prostate cancer is driven by biopsying the prostate gland, an
invasive procedure with possible complications. Historically, systematic
biopsy is used to sample sextants of the prostate gland (left apex, left mid
gland, left base, right apex, right mid gland and right base) as estimated by
the urologist. MRI/ultrasound fusion biopsy is able to target MRI lesions
identified as suspicious on prostate MRI. However, prostate biopsies do not
sample the entire gland and may miss small and subtle tumors. Whole-mount
analysis of prostatectomy specimens whereby the entire prostate gland is
surgically removed is better as the entire prostate gland is analyzed.
However, only a small proportion of patients undergo prostatectomy and relying
on this population leads to significant sample bias. Ultimately, biopsy data
is abundant and better represents the distribution of the population being
initially evaluated for prostate cancer by prostate MRI. Thus, an artificial
intelligence (AI) strategy with the aim of detecting CS-PCa in the population
being screened for prostate cancer must incorporate prostate biospy data.

Most deep learning strategies focus on MRI-visible disease, which subsequently
undergo MRI/ultrasound fusion (i.e. targeted) biopsies111However, these prior
approaches often inconsistently report results with missing comparisons to PI-
RADS or the distribution of tumor grades, leading to ultimately poor
generalization performance for developed models [5].. In [6], authors used
lesion biopsy data from and T2-weighted MRI from 312 men with a 2D UNet model,
achieving 62-62.5% balanced accuracy, but found that it was not statistically
better than PI-RADS. In [7], authors focused on clinically significant cancer
in 417 patients receiving prostatectomy, achieving 0.79-0.81 AUC for CS-PCa
detection using multi-parametric MRI, but found it was 1.5-3.5% lower in
accuracy than radiologists. Authors in [8] used the Prostate-X2 challenge
dataset and found a voxel-wise kappa of 0.446$\pm$0.082 using cross-
validation, but a lesion-wise kappa of 0.13$\pm$0.27 on the test set.
Meanwhile, in [9] authors showed with 986 exams that image-level labels can
achieve 0.75 $\pm$0.03 AUC in grade group classification via attention-based
multiple instance learning. Finally, in [10] authors used 1043 in-house and
347 Prostate-X exams, and showed that a 3D UNet could predict CS-PCa in
prostate MRI lesions at nearly 56.1% sensitivity and 62.7% precision, compared
to 30.8% classification accuracy with PI-RADS. Despite differences in the
reported metric and issues balancing model sensitivity and false-positive rate
[11], the consensus here is that advanced deep learning techniques can
feasibly classify cancer inside suspicious MRI-visible lesions.

Most AI models for prostate cancer detection rely on MRI-visible lesions.
However, not all tumors are visible on prostate MRI. For example, a study
utilizing our dataset (973 prostate MRI exams conducted at UCSF from 2015-2018
with subsequent MRI/ultrasound fusion biopsy and full systematic biopsy)
showed that 14.4% of cases had a clinically significant upgrade of prostate
cancer by nontargeted systematic biopsy compared to MRI targeted biopsy (i.e.
MRI visible lesions) [12]. Thus, incorporation of both targeted fusion biopsy
for MRI-visible lesions and non-targeted systematic biopsy of disease not
necessarily perceived prospectively on MRI, may improve automated algorithms
for the detection and risk stratification of prostate cancer.

In this paper, we describe a new approach that can incorporate additional
supervision from all biopsy data to DNNs for identifying clinically
significant prostate cancer (CS-PCa) from MRI. Specifically, we introduce a
unified classification model architecture, 3D UCNet, which is composed of
global, regional, and spatial prediction modules for predicting cancer on an
exam-wise, region-wise, and voxel-wise basis, respectively (Fig. 1). Of
primary importance in our approach is how we encode and exploit histopathology
ground truth. Unlike the cardinal encoding used in [7], we adopt a multi-class
histogram representation that supports regional classification of homogeneous
and inhomogeneous tissue via a classification head, as well as a novel
distribution-based loss function that enables visualization of cancer grade
prediction in each region. The handling of inhomogeneous tissue is essential
for leveraging systematic biopsy data, since the full extent of the cancerous
region sampled by the biopsy core is nonsalient in MRI, and thus the Gleason
pattern or ISUP grade group (GG) extracted from pathology report processing
must be applied delicately to avoid inconsistency during model training.

###  1.1 Contributions

Thus our contributions are as follows:

  * •

We introduce 3D UCNet, which features regional and global prediction modules
that enable gland-level classification of cancer from a semantic segmentation
backbone.

  * •

We introduce a histogram-based representation that enables a new type of
indirect (weak) supervision consistent with how biopsy cores are annotated by
pathologists.

  * •

We demonstrate that mixed supervision techniques expand the type of
histopathology ground truth used in training and improve CNN-based detection
of clinically significant prostate cancer from prostate MRI.

  * •

We open source our models, algorithms, and analysis:
https://gitlab.com/abhe/prostate-mpmri

##  2 Dataset and Assumptions

###  2.1 Multi-parametric Prostate MRI

Multi-parametric prostate MRI (mp-MRI) is composed of T2-weighted imaging
(T2WI), diffusion-weighted imaging (DWI) with an associated apparent diffusion
coefficient (ADC) map, and dynamic contrast-enhanced (DCE) MRI. DWI, which
typically includes both low and high b-value images, and the associated ADC
map represent the restriction of water movement in tissue, which is thought to
complement the anatomic T2WI series for detecting prostate cancer [13]. DCE
provides data on the contrast enhancement kinetics of the imaged soft tissues
and may help to delineate prostate tumors. All together mp-MRI provides a rich
set of information, although variations due to differences in choice of MR
pulse sequence, scanner hardware, and individual patient are commonplace.

Herein, we used bi-parametric (bp-MRI) consisting of just the T2WI and DWI
data [14, 15]. The T2WI, high B-value DWI and ADC map were registered and
resampled uniformly across the dataset to the same spatial resolution of [1.0,
1.0, 2.24]mm for x, y, and z axis respectively and normalized. For analysis,
these imaging volumes are cropped with centering of the prostate gland and
final pixel dimensions of [64, 64, 32].

###  2.2 Histopathology Derived from Prostate Biopsy

Prostate MRI exams are typically performed after the prostate-specific antigen
(PSA) is found to be elevated along with the clinical assessment of a
urologist. For our dataset, fellowship-trained abdominal imaging radiologists
at UCSF provided routine clinical interpretation of prostate MRI exams with
identification of MRI targets, each with an associated PI-RADS scores. These
MRI targets were subsequently underwent MRI/ultrasound fusion guided prostate
biopsy by a urologist, whereby an axial T2WI is fused by software with live
ultrasound to sample MRI targets under ultrasound guidance. In addition to
these targeted biopsies, urologists routinely performed systematic biopsies,
sampling the prostate in 6 sextants (left apex, left mid gland, left base,
right apex, right mid gland, right base). As the exact coordinates of these
systematic biopsies are not recorded, these biopsies are only associated with
the sextant regions of the prostate gland. Thus, for our dataset, we assumed
the location of the systematic biopsies corresponded to a region defined by
the geometric division of the prostate into sextants in the registered MR-
image space. For the targeted biopsies, we assumed the location of the
targeted biopsy corresponded to a radiologist annotated bounding box around
each MRI-identified lesion on each applicable slice 1. The ground truth
histopathology for each biopsy sample was determined by a genitourinary
pathologist with more than 20 years of experience who graded all specimens
using Gleason score [16]. Gleason scores were converted to ISUP grade group
(1-5). We used 0 to represent benign pathology results. ISUP grade group $\geq
2$ indicates clinically significant disease.

###  2.3 Gland Segmentation and Contrast Normalization

As T2 and DWI have arbitrary non-quantitative image amplitudes, we applied
interquartile range (IQR)-based intra-image normalization to control to the MR
image intensity values across research sites and eliminate outlying values
created by imaging artifacts. Specifically, each image was normalized to the
image-level IQR computed within the 3D prostate gland (annotated by a
radiologist or defined by a previously developed neural network segmentation
model) according to [17]:

$\displaystyle
I_{norm}=\frac{I-\text{percentile}(I,1)}{\text{percentile}(I,99)-\text{percentile}(I,1)}$
(1)

We subsequently applied Z-score image normalization to each exam to overcome
the problem of high variability of intensity distribution between different
patients by transforming the intensities to have zero mean and unit variance.

###  2.4 Summary of Datasets

Table 1 summarizes the distribution of ISUP grade groups in the training,
validation, and testing cohorts used in this study. Clinical exam data from
2015-2018 was collected at the University of California, San Francisco (UCSF)
under IRB approval. As evident, the distribution of ISUP grade group is
imbalanced in our dataset.

| Training | Validation | Testing
---|---|---|---
max ISUP 0 | 92 (13.5%) | 17 (17.7%) | 24 (12.1%)
max ISUP 1 | 222 (32.7%) | 27 (28.1%) | 73 (36.9%)
max ISUP 2 | 228 (33.6%) | 30 (31.3%) | 61 (30.8%)
max ISUP 3-5 | 137 (20.2%) | 22 (22.9%) | 40 (20.2%)
Totals | 679 (69.8%) | 96 (9.9%) | 198 (20.3%)

Table 1: Number of exams in each ISUP grade group for each cohort.

All 973 patient exams used in this study included systematic biopsy results
from all 6 sextants and at least one targeted-biopsy result. This results in
7278 biopsy results that were used in training, validation or testing. Each
exam also includes, for each targeted biopsy, a PI-RADS score assigned by a
board-certified radiologist when the case was interpreted clinically.

##  3 UCNet for Mixed Histopathology Supervision

For this problem, we utilized an image-to-image CNN with an additional fully-
connected classification output head, a unified classification architecture we
call “UCNet”. Our UCNet implementation takes 3D T2WI, high b-value DWI and ADC
map as input and predicts lesion segmentation maps, ISUP grade group maps, and
region-wise histograms that are used to determine region-wise and exam-wise
cancer severity. We believe this architecture is applicable to many deep
learning tasks in medical imaging, but it is particularly well-suited to
handling the variety of ground truth histopathology data available for
prostate cancer. Of particular importance here is the dynamically-populated
multi-task objective we used to train UCNet using highly heterogeneous
pathology ground truth collected across the UCSF patient population.

Specifically, we use a fully-convolutional 3D residual UNet [18] as the
backbone for UCNet, which in this case accepts a image tensor
$x\in\mathbb{R}^{3\times X\times Y\times Z}$ as input, and produces voxel-wise
tanh-activated lesion maps $\hat{y}_{\text{seg}}\in\mathbb{R}^{1\times X\times
Y\times Z}$ and softmax-activated grade group membership predictions
$\hat{y}_{\text{gg}}\in\mathbb{R}^{K\times X\times Y\times Z}$, where
$K\in\mathbb{Z}_{[0,5]}$ represents the number of unique grade groups chosen
for regression. For example, $K=2$ in the case of binary detection of CS-PCa
(ISUP grade groups $\geq$ 2). $K$ is an adjustable hyperparameter, but we find
$K=2$ yields slightly higher validation accuracy for detection of CS-PCa
compared to fully multi-class grading approaches.

###  3.1 Voxel-wise Prediction

We used strong supervision for lesion segmentation and lesion grading, when we
had relatively localized information.

We used a combination of dice and weighted binary cross-entropy losses to
supervise the radiological lesion segmentation output, $\hat{y}_{\text{seg}}$,
given the ground truth segmentations $y_{\text{seg}}$, since this has been
shown to achieve higher accuracy [19]:

$\displaystyle\mathcal{L}_{\text{segmentation}}(y_{\text{seg}},\hat{y}_{\text{seg}})=\mathcal{L}_{\text{seg-
dice}}+\mathcal{L}_{\text{seg-BCE}}$ (2) $\displaystyle\mathcal{L}_{\text{seg-
dice}}(y_{\text{seg}},\hat{y}_{\text{seg}})=1-\frac{y_{\text{seg}}\cdot\hat{y}_{\text{seg}}}{y_{\text{seg}}+\hat{y}_{\text{seg}}+\epsilon}$
(3) $\displaystyle\mathcal{L}_{\text{seg-
BCE}}(y_{\text{seg}},\hat{y}_{\text{seg}})=\sum_{z}\sum_{x,y_{\text{seg}}=z}\frac{\text{BCE}(y_{\text{seg}}(x),\hat{y}_{\text{seg}}(x))}{|z|\cdot
n_{z}}$ (4)

where $\epsilon$ is chosen arbitrarily small to prevent overflow, BCE
represents the binary cross-entropy function, $n_{z}$ represents the number of
voxels in class $z\in\\{0,1\\}$ and $|z|$ is the number of classes. Here,
$y_{\text{seg}}$ is determined from as the voxel-wise max of the region masks
$y\in\mathbb{R}^{R\times X\times Y\times Z}$, over regions where the
supervision signal in the first column of the histopathology matrix
$z\in\mathbb{R}^{R\times 2}$ is 1 to represent MRI-annotated lesions.

Similarly, we applied categorical binary cross entropy loss to each voxel of
the predicted grade membership maps,
$\hat{y}_{\text{gg}}\in\mathbb{R}^{K\times X\times Y\times Z}$, given one-hot
encoded ground truth histopathology data $y_{\text{gg}}\in\mathbb{R}^{R\times
K}$ from regions $R^{+}$ where $z$ holds a supervision signal of 1 and a
regression grade group that is not NaN. This is an important subtlety that
allows supervision of segmentation outputs from patient exams even when the
pathological grade group is not known for individual lesions. This strong
voxel-wise grading or classification loss is then defined as:

$\displaystyle\small\mathcal{L}_{\text{GGmap}}(y,z,y_{\text{gg}},\hat{y}_{\text{gg}})=\frac{1}{|R^{+}|}$
$\displaystyle\sum_{r\in
R^{+}}\sum_{k=1}^{K}y_{\text{gg}}[r,k]\log{\hat{y}_{\text{gg}}[\mathbb{1}_{r}]_{k}}$
(5)

where $\mathbb{1}_{r}$ represents an indicator (Kronecker delta) function
selecting the voxels corresponding spatially to region $r$.

###  3.2 Regional Prediction

Unfortunately, strong voxel-wise supervision signals are not available for all
histopathology data types, such as biopsy data. This is because the biopsy
core only targets a small portion of the prostate gland, either a lesion or a
normal-appearing tissue. For targeted lesion biopsies, we assume that the
biopsy sample is representative of a homogeneous cancer profile within a tumor
that is reasonably localized in the MRI coordinate system, enabling the use of
the aforementioned strong classification objective,
$\mathcal{L}_{\text{strong}}$. Systematic biopsies, however, are neither
localized nor representative of the cancer profile in heterogeneous tissue
throughout the gland, preventing the use of such strong voxel-wise objectives.

To this end, we utilized two weak supervision objectives in regions $R^{*}$
where the supervision signal in $z$ is $\geq 1$:

####  3.2.1 Histogram Suppression

Rather than regressing by value with a voxel-wise classification loss, we
regress by distribution by computing and penalizing a histogram of predicted
voxel-wise grade groups for each region of interest. This can be achieved
naturally (maintaining model differentiability) by selecting $K$-dimensional
voxels of $\hat{y}_{\text{gg}}$ using the corresponding region masks in $y$,
and computing their $K$-dimensional average, resulting in a set of approximate
histograms $\hat{h}\in\mathbb{R}^{R^{*}\times K}$.

For regions $R^{\alpha}$ where signal $z=1$ (individual MRI target lesions),
we assume the biopsy core represents a homogeneous cancer profile, so we
penalized non-zero histogram bins that don’t correspond to the ground truth
grade group (Fig. 2a):

$\displaystyle\mathcal{L}_{\text{hist-
strong}}(z,y_{\text{gg}},\hat{h})=\frac{1}{|R^{\alpha}|}$
$\displaystyle\sum_{r\in
R^{\alpha}}\sum_{k=1}^{K}y_{\text{gg}}[r,k]\log{\hat{h}[r,k]}$ (6)

For regions $R^{\beta}$ where signal $z>1$ (sextants, where only the highest
grade is known), we assume the biopsy core represents the highest cancer in a
heterogeneous tissue, so we only penalized non-zero histogram bins that
represented grade groups higher than the ground truth regression grade group
(Fig. 2b):

$\displaystyle\mathcal{L}_{\text{hist-
high}}(z,y_{\text{gg}},\hat{h})=\frac{1}{|R^{\beta}|}$
$\displaystyle\sum_{r\in
R^{\beta}}\quad\sum_{\mathclap{\begin{subarray}{c}k\;>\\\
\underset{k}{\mathrm{argmax}}\;y_{\text{gg}}[r]\end{subarray}}}^{K}y_{\text{gg}}[r,k]\log{\hat{h}[r,k]}$
(7)

The net effect of these losses $\mathcal{L}_{\text{GGmap-
hist}}=\mathcal{L}_{\text{hist-strong}}+\mathcal{L}_{\text{hist-high}}$ is to
suppress the proportion of voxels representing ISUP grade groups not supported
by the histopathology data and its expected uncertainty. Note that when
$\mathcal{L}_{\text{hist-high}}=0$, the histogram bin corresponding to the
ground truth grade group may not necessarily have the highest frequency in the
region, in agreement with the expectation for heterogeneous tissue.

Figure 2: Histogram suppression objectives (Eq. 6-7) penalize predictions of
cancer unequal to the ground truth ISUP grade group in homogeneous lesions
(a), and only above the histopathological support in inhomogeneous tissue (b).
This form of weak supervision helps shape the predicted distribution,
particularly in regions where the most frequent prediction may not represent
the most clinically-significant cancer grade.

####  3.2.2 Regional Grading

To provide a clear indication of ISUP grade group in such cases, without
relying on manual interpretation of the histograms, we fed the regional
histograms $\hat{h}$ into a small dense network with ReLU activation in the
final layer. This results in multi-hot encoded vectors $\hat{z}^{R^{*}\times
K}$, indicating the presence of various ISUP grade groups in regions $R^{*}$.

To maintain the clearest interpretation of $\hat{h}$, we used a single-layer
ReLU-activated network with tunable bias and weight-matrix frozen as identity
$I_{K}$. The bias, therein representing a set of “optimal” thresholds for
declaring clinically-significant prostate cancer in a given region, is
optimized through backpropagation, just as other parameters in the fully-
convolutional portion of UCNet, via the loss:

$\displaystyle\mathcal{L}_{\text{region-
classifier}}(z,y_{\text{gg}},\hat{z})=\frac{1}{|R^{*}|}$
$\displaystyle\sum_{r\in
R^{*}}\sum_{k=1}^{K}y_{\text{gg}}[r,k]\log{\hat{z}[r,k]}$ (8)

###  3.3 Multi-task Objective

The overall multi-task objective is:

$\displaystyle\mathcal{L}(x,y,y_{\text{gg}},z)=\alpha_{1}\lambda_{1}\mathcal{L}_{\text{region-
classifier}}+\alpha_{2}\lambda_{2}\mathcal{L}_{\text{GGmap-hist}}$ (9)
$\displaystyle+\alpha_{3}\lambda_{3}\mathcal{L}_{\text{GGmap}}+\alpha_{4}\lambda_{4}\mathcal{L}_{\text{segmentation}}$

where we empirically choose $\bar{\lambda}=[1.0,0.5,1.0,1.0]$, and
$\bar{\alpha}$ is chosen based on the availability of different annotation
ground truth available for each exam in a minibatch, as well as the type of
supervision signals desired in the experiment. Herein, we choose
$\bar{\alpha}=[1.0,1.0,1.0,1.0]$ to use all described objectives whenever
possible. For training, we used a batch size of 6 MRI exams and the AdamW
optimizer with initial learning rate of $0.0001$, determined empirically by
repeatedly sweeping the initial learning rate.

####  3.3.1 Objective-Aware Balanced-Batching

To achieve the best performance, we curated the distribution of each batch by
round-robin sampling of MRI exams where the highest ISUP grade group for any
region is 0 (negative), 1 (benign), 2 (clinically-significant), 3-5. We find
that this empirically leads to better efficiency in training the multi-task
objective, since there are more representative samples per task per iteration.

####  3.3.2 Global Prediction

The UCNet architecture supports global prediction modules, which can be based
either on the fully-encoded features produced by the encoding branch or the
spatial map that is rendered by the decoding branch of the fully-convolutional
backbone. We selected the later to focus on improving lesion-wise segmentation
and classification accuracy, and to compute a global prediction by maximum
filtering of the individual lesion predictions.

###  3.4 Inference

Inference is performed in a similar fashion to training, except with all
augmentation turned off. Note that, due to the nature of weak histopathology
ground truth, it is not feasible to train UCNet using volumetric patches.
Instead, we feed full (registered and resampled) multi-contrast prostate MRI
exams as input to UCNet, producing a set of predictions corresponding to all
volumetric regions of the prostate gland.

##  4 Experiments

To determine the potential advantage of mixed-supervision deep learning
methods for detection of clinically significant prostate cancer, we designed
experiments that represent variations in the supervision methods used during
training, as well as alternative classification strategies used during
inference or evaluation. Specifically, we choose to withhold or include
various types of annotation ground truth, including lesion segmentation masks,
voxel-wise ISUP grade group in lesions derived from MR-targeted biopsy and
expert histopathological analysis, and maximum ISUP grade group in regions
derived from systematic biopsy and expert histopathological analysis. For the
regions with biopsy ground truth, in particular, we additionally compare two
types of supervision: weak classification-based supervision (Section 3.2.2)
and a novel weak distribution-based supervision (Section 3.2.1).

Each experiment can be identified by a unique experiment ID, represented by a
4-bit vector, corresponding exactly to the
$\bar{\alpha}=\\{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\\}$ in Equation
9. However, there are numerous other differences implied for inference when
changing supervision methods, as elucidated in the following.

###  4.1 Experiment 0001 - lesion segmentation only

$\alpha_{4}=1$, $\alpha_{1,2,3}=0$ represents a baseline lesion segmentation
experiment, where the combined class-balanced BCE-loss and dice loss (Eq. 2)
is the sole loss term in the objective. The network model employed is simply a
3D residual UNet, representing the backbone of the UCNet architecture.

###  4.2 Experiment 0011 - lesion segmentation and classification

$\alpha_{3,4}=1$, $\alpha_{1,2}=0$ represents a baseline lesion segmentation
and classification experiment, utilizing the same 3D residual UNet backbone of
UCNet. The loss terms in the training objective are the segmentation loss (Eq.
2) as well as the strong voxel-wise cross entropy objective (Eq. 5). Here,
classification decisions are based on the most frequent or mean value in each
region based on thresholded voxels of $\hat{y}_{\text{GGmap}}$. Note, for
binary classification the mean and mode inference methods are equivalent. This
experiment serves as an important baseline for subsequent methods that utilize
the same UNet backbone, but with additional supervision signals.

###  4.3 Experiment 0111 - lesion segmentation, lesion classification, and
regional classification with histogram suppression

$\alpha_{2,3,4}=1$, $\alpha_{1}=0$ represents a mixed-supervision objective,
novel for its use of distribution-based representations of histopathology. In
this experiment, the prediction $\hat{y}_{\text{GGmap}}$ is penalized in
regions where the maximum ISUP grade is higher than the ground truth (Eq. 7),
e.g. when only the maximum is known for biopsy cores in sextants that
represent heterogeneous tissue (i.e. a mixture of background prostate tissue
with clinically significant prostate cancer if it were present). When the
biopsy is derived from apparent homogeneous tissue, such as MRI-identified
lesions, voxels are penalized in histogram bins both higher and lower (Eq. 6)
than the ground truth maximum. Specifically, the addition of Equations 6-7
encourages regression by distribution rather than regression by value,
representing the first (to our knowledge) deep learning method to incorporate
the distribution of cancer histopathology. Note that the number of free model
parameters is equal in Experiment 0001, 0011, and 0111. In this setting, and
to maintain a clear comparison to the strongly-supervised case, classification
is made in the same fashion as in Experiment 0011 by computing the mean or
mode value in each region based on thresholded voxels of
$\hat{y}_{\text{GGmap}}$.

###  4.4 Experiment 1111 - lesion segmentation, lesion classification,
regional classification with histogram suppression, and regional
classification with RegionNet

$\alpha_{1,2,3,4}=1$ represents another mixed-supervision objective that
combines the histogram-based penalization of Experiment 0111 together with
traditional weak instance-wise or region-wise classification supervision (Eq.
8). A region-wise classification is produced by passing the histogram
representation of each region to a small neural network, RegionNet. To drive
the clearest interpretation and comparison to Experiment 0111, we choose
RegionNet as a dense ReLU-activated 1-layer neural network of input and output
dimension $K$ and frozen with weight matrix $I_{K}$, so the $K$ free bias
terms represent “optimal” thresholds for declaring clinically-significant
cancer in each region. Classification in this setting is achieved by finding
the most-significant bit (MSB) of the $K$-bit output
$\lceil{\hat{y}_{\text{gg,r}}}\rceil\in\mathbb{Z}^{K}$ for each region $r$,
which represents the highest level of cancer.

This experiment aims to show that accounting for tissue heterogeneity can
dramatically improve classification accuracy of deep learning cancer
classification models. Specifically, the combination of distribution-based
penalization (Experiment 0111), together with optimal thresholding and MSB-
based classification is designed to enable detection of cancer in
heterogeneous tissue where the highest level of cancer is not the most
prevalent (e.g. mean or mode).

Table 2: Comparison of Multi-Task Objectives on withheld test set using as
well as PIRADs predictions. Overall class-balanced accuracy, with specificity
and sensitivity in brackets. Experiment IDs represent varying supervision:
0001 = lesion segmentation only; 0011 = lesion segmentation and
classification; 0111 = lesion segmentation, lesion classification, and
regional classification with histogram suppression; 1111 = lesion
segmentation, lesion classification, regional classification with histogram
suppression, and regional classification with RegionNet.

Experiment ID | 0001 | 0011 | 0111 | 1111 | PI-RADS $\geq 4$ | PI-RADS $\geq 5$
---|---|---|---|---|---|---
Lesion Segmentation IoU | 0.220 | 0.188 | 0.162 | 0.150 | – – |
Lesion Accuracy | – – | 59.5% [94.0%, 25.0%] | 61.1% [93.0%, 29.2%] |  70.3% [76.6%, 63.9%] | 62.4% [31.8%, 93.1%] | 66.7% [82.5%, 52.4%]
Gland Accuracy (via Lesions) | – – | 55.8% [90.7%, 20.8%] | 57.7% [90.7%, 24.8%] |  62.8% [70.1%, 55.4%] | 61.0% [30.9%, 91.1%] | 64.4% [79.4%, 49.5%]

###  4.5 Training Details and Stopping Criteria

For each experiment, UCNet was trained twice: (1) by performing an initial
learning rate sweep and selecting an optimal point based on decrease in the
objective, and (2) by fixing the initial learning rate to 0.0001. Training was
performed for 500 epochs, and the checkpoint with the maximum lesion-wise
validation accuracy was chosen. In all cases, the fixed learning rate run
achieved superior validation performance, and these models were thus chosen
for test-set evaluation and reporting.

###  4.6 Performance Metrics

####  4.6.1 Lesion Segmentation

Lesion segmentation accuracy is measured by the standard intersection-over-
union (IoU) metric, measuring the overlap between the radiologist annotation
(performed as a bounding box) and the predicted binarized lesion segmentation
map $\hat{y}_{\text{seg}}$. Although IoU tends to underestimate the lesion
segmentation performance, we believe this is provides an accurate measure of
model confidence in segmentation on this challenging task. Note that the
assumed radiologist ground truth is not based on pathology, so the IoU metric
is only serving to measure the overlap or consistency between the model and
the radiologist. Furthermore, the bounding box annotations are imprecise which
will also reduce the IoU performance, especially for small non-spherical
lesions.

####  4.6.2 Lesion Classification Accuracy

Unlike many prior works that measure area under the receiver-operating
characteristic curve (AUC) via cross-validation [6, 9, 11], here we measure
overall class-balanced lesion classification accuracy on a withheld test set.
Class-balanced accuracy is computed simply as the arithmetic mean of the true-
positive rate (TPR) and true-negative rate (TNR), or in the multi-class
setting, as the arithmetic mean of the trace of confusion matrix. We use
class-balanced accuracy for two reasons: (1) we believe this is a more
representative measure of model performance because it weights positive and
negative cases equally, and (2) it captures model performance without any
post-training tuning of the final threshold. For clarity, we note that the TPR
and TNR represent sensitivity and specificity, respectively.

####  4.6.3 Gland Classification Accuracy

Gland accuracy is measured in a similar fashion to lesion classification
accuracy using class-balanced accuracy metrics. Specifically in the binary CS-
PCa setting, the maximum prediction for all lesions (or regions, as
appropriate) in an exam is compared to the ground truth maximum to yield a
true-positive or true-negative count. The TPR and TNR are subsequently
averaged to yield the overall class-balanced gland classification accuracy.

##  5 Results

Figure 3: Sample results for a lesion in the right mid posterolateral
peripheral zone that was correctly classified as a true positive CS-PCa in all
experiments. Note that only a single slice from the 3D volume used is shown.
The lesion risk map for all experiments strongly corresponds with the MRI-
identified lesion, and the spatial grade maps for experiments 0011 and 0111
also correctly predict CS-PCa. In Experiment 1111, although the visualized
slice of spatial grade map does not show CS-PCa, RegionNet achieves a correct
classification by picking the highest level of cancer over the entire 3D
region.

###  5.1 Comparison of Multi-Task Objectives

Table 2 displays the withheld test-set performance of the experiments
described in Section 4 with respect to various metrics of success. Experiment
1111, which represents a mixed-supervision objective combining all the
aforementioned strong and weak types of supervision, performs the best with
respect to lesion classification accuracy as well as gland-wise classification
accuracy (based on lesions) compared to the other DNN models, followed by the
model trained in Experiment 0111, followed finally by the model in Experiment
0011. This provides strong evidence that weak supervision methods are
complementary to strong supervision methods.

Moreover, the mixed supervision strategies in Experiment 1111 achieve 70.3%
overall (class-balanced) accuracy in classification of clinically significant
cancer in lesions, which exceeds the 66.7% performance of the best PI-RADS
cutoff (PI-RADS $\geq$5) classification assigned by radiologists for this
dataset. We believe this is achieved not only by learning from hundreds of
datasets, but by incorporation of non-salient cancer histopathology acquired
through systematic biopsy.

Table 2 indicates that although Experiment 1111 performs the best lesion
classification, Experiment 0001 achieves the best overlap with radiologists’
segmentation of lesions, indicating a trade-off between these two metrics for
our dataset. However, high accuracy of the lesion segmentation is not a major
issue in prostate MRI interpretation, as these are relatively readily
visualized by radiologists, but the classification of CS-PCa is the more
challenging task. Yet, the radiologist derived lesion segmentation information
is still valuable for improving classification performance. For example,
Experiment 1110, which includes all mixed-supervision objectives but not
lesion segmentation objectives, does not perform as well as Experiment 1111
with respect to lesion classification accuracy (66.2% vs 70.3%). This suggests
the lesion segmentation provides some extra attention to suspicious areas
during training, improving convergence to desirable minima. We interpret this
lesion segmentation versus classification performance tradeoff in the
following way: while lesion segmentation maps and corresponding IoU metrics
generally correlate with model performance, they do not dictate the lesion
classification performance.

###  5.2 Sample Results

A visual comparison of the experiments described in Section 4 is shown in
Figures 3-5, including the pre-processed input images and the various outputs
which include a lesion risk map, predicted spatial grading map, and results of
the regional classification.

In Figure 3, an example is shown where a PI-RADS 4 lesion is identified by the
model that is T2 hypointense, markedly hypointense on the ADC map and
demonstrating diffusion restriction (high signal intensity) in the right mid
posterolateral peripheral zone. This lesion was correctly predicted as
clinically significant prostate cancer in all experiments, which corresponds
to a biopsy result of ISUP grade group 3.

Figure 4: Sample results for a lesion in the right lateral peripheral zone at
the apex that was correctly classified as a true negative CS-PCa in all
experiments. The lesion risk mask for all for experiments 0001 strongly
corresponds to the MRI-identified lesion. This correlation progressively
weakens with the subsequent experiments, however the spatial grade map and
regional classification correctly predict an absence of CS-PCa.

Figure 5: Sample results for a lesion in the left posterolateral peripheral
zone that was only correctly classified as a true positive CS-PCa with the
addition of weak supervision objectives. The lesion is identified in all
experiments in the Lesion Risk Maps, but was not predicted as CS-PCa until the
RegionNet regional classification was added in Experiment 1111.

In Figure 5, an example is shown where a PI-RADS-4 lesion is identified in the
right lateral peripheral zone at the apex that is T2 hypointense, hypointense
on ADC, and demonstrating restricted diffusion (high signal intensity). This
lesion is predicted as not being CS-PCa on all experiments, which corresponds
to a biopsy result of benign.

In Figure 5, an example is shown where there is a PI-RADS 5 lesion in the left
posterolateral peripheral zone that is subtle to detect on T2 weighted
imaging, hypointense on the ADC map and demonstrating restricted diffusion
(high signal intensity). In experiments 0011 and 0111, the lesion is
accurately segmented but not predicted as clinically significant prostate
cancer. With the addition of weak supervision methods in experiment 1111, the
lesion is predicted to be CS-PCa, corresponding to the biopsy result of an
ISUP grade group 2. Thus, this represents a false negative case that was
converted to a true positive case with the addition of weak supervision
methods.

##  6 Discussion

The results show the benefits of combining strong and weak supervision of
pathology into UCNet deep learning model to predict clinically-significant
prostate cancer, most clearly summarized in Table 2. This is somewhat
intuitive, as we expect improved accuracy by incorporating a much larger set
of training labels. However, taking advantage of both heterogeneous pathology
ground truth in the form of strongly supervised targeted biopsy and weakly
supervised systematic biopsy results required a novel architecture, training,
and evaluation methods. Our results indicate that UCNet was successfully able
to take advantage of these diverse sources of training data. This is
important, as the inclusion of systematic biopsy procedures continues to be
debated clinically [20].

Our results are particularly interesting because the lesion-wise
classification accuracy is improved by utilizing ground truth derived from
poorly localized systematic biopsy data from tissue that did not appear
cancerous on MRI. While this trend seems to occur at the expense of reduced
concordance with radiologist-derived MRI target segmentation, lesion
classification is the key challenge, not segmentation. Furthermore, we note
that the current dataset includes 2D bounding boxes for each lesion over the
applicable axial T2WI MRI slices which systematically overestimates the lesion
size. The absolute IoU metrics must be put in context then, as they will not
be comparable to more finely contoured lesion segmentations, such as those in
[9].

In our experiments we also observed improved performance by classifying
lesions based on the highest grade of cancer rather than the most prevalent
cancer. This is shown by comparing Experiment 0111 and Experiment 1111, where
the latter had improved performance (improved sensitivity) based on the change
to using the RegionNet classifier which applies a bias to threshold the
predictions of ISUP grade present along with a most-significant-bit (MSB)
classification that picks out the highest grade of cancer predicted to be
present. This approach also matches how pathologists interpret biopsy samples,
where scoring is done based on the highest cancer grades observed. We believe
this classification technique allows the deep learning model to preserve the
distribution of cancer. This is achieved mechanistically by providing
flexibility when interpreting the voxel-based representation, which matched
accordingly with the expected uncertainty of the histopathology ground truth.
It is also this approach that expands the number of exams and types of ground
truth that can be used during training via the mixed supervision objective.

Furthermore, through mixed supervision the UCNet lesion classifier
significantly outperforms (70.3%) the MRI-based lesion reporting and
classification used by radiologists based on PI-RADs (PI-RADS$\geq$5 66.7%,
PI-RADS$\geq$4 66.7%). Note that although PI-RADs$\geq$5 achieves better
performance than PI-RADs$\geq$4, this typically targets only the most obvious
MRI-visible lesions, further emphasizing the superior performance of UCNet for
classifying less obvious disease. This suggests that there is additional
information in prostate MRI images apart from those considered by radiologists
using the PI-RADS standardized reporting system that can improve performance.
While UCNet’s improvement over PI-RADS$\geq$4 is minimal for Gland Accuracy
(+1.8%), we note that UCNet with mixed supervision is able to improve the
balance between specificity and sensitivity compared with all other methods
(both human and DNN-based), addressing many of the concerns raised by the
community [5, 11].

This study has limitations. First, this work was conducted with single
institutional data, which typically raises concerns over model
generalizability. However, this study uses one of the largest prostate-MRI
datasets ($\sim$3x larger than Prostate-X [21]) with real world clinical data
variability including varying acquisition protocols and usage of endorectal
coil. Furthermore, our dataset uses the most common approach of combined
fusion and systematic biopsy, while Prostate-X used direct MRI-guided biopsy
only. Overall, we believe our real world dataset will result in improved
generalizability compared to models trained on smaller, more homogeneous
datasets. Conscious of this, our approach is designed to enable scaling to
diverse multi-institutional datasets with diverse histopathology sources,
which can be explored in future work.

Second, in our dataset we had to approximate the systematic biopsy result
locations by geometrically splitting the prostate gland into six sextants to
represent systematic biopsies, and by using retrospective MRI-based
segmentation to represent lesions. This has the potential to cause errors in
the assignment of ISUP grade groups to each region as the exact location of
systematic biopsy within the approximated region was not known. If the
ultrasound-guidance images were available they could potentially be used to
localize the biopsy core by needle position within the gland, as suggested in
[3]. Other datasets, such as Prostate-X, used direct MR-guided biopsy data
that potentially provides more accurate localization, although it is also more
limited given the absence of systematic biopsy data.

Finally, we emphasize here that MRI-based prostate cancer classification is a
difficult task and the major motivating for this project. The current
radiology standard of PI-RADS to classify lesions based on size and signal
characteristics has relatively poor performance [4]. Prior work based on
radiomics, machine learning, and deep learning have also shown promise but
also relatively modest performance improvements [22]. For our dataset, a
sophisticated approach was needed to outperform PI-RADS using both targeted
fusion biopsy and nontargeted systematic biopsy results. We are optimistic
that continued algorithm improvements as well as accumulation of larger, high
quality but diverse datasets will further improve the detection of clinically-
significant prostate cancer and ultimately impact patient management.

##  7 Conclusion

The proposed mixed supervision methods improved the lesion-wise accuracy of
machine learning-based prostate cancer classification algorithms by enabling
use of more, diverse histopathology data. We demonstrated this technique by
augmenting a popular deep neural network architecture, 3D residual UNet, using
regional-query, soft histograms, and a simple threshold-based classification
module that enables simultaneous use of voxel- and region-level histopathology
ground truth, if available. This is particularly useful for improving
classification since systematic biopsy results are typically disregarded in
favor of MRI-targeted lesion biopsy results. The results demonstrated a
significant improvement in lesion-classification accuracy using mixed
supervision objectives, exceeding PI-RADS on a large and diverse dataset of
973 prostate cancer patients.

## Acknowledgements

This work was supported by NIH/NIBIB grant #F32EB030411, NIH/NCI grant
#R01CA229354, the UCSF Benioff Initiative for Prostate Cancer Research, a
Society for Abdominal Radiology Morton A. Bosniak Research Award, a RSNA
Research Resident/Fellow Grant, the Cancer League and Helen Diller Family
Comprehensive Cancer Center at UCSF.

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