# Hodge-theoretic variants of the Hopf and Singer Conjectures

Donu Arapura  Department of Mathematics, Purdue University, West Lafayette, IN
47907, USA<EMAIL_ADDRESS>,  Laurentiu Maxim  Department of Mathematics,
University of Wisconsin-Madison, 480 Lincoln Drive, Madison WI 53706-1388,
USA<EMAIL_ADDRESS>and  Botong Wang  Department of Mathematics,
University of Wisconsin-Madison, 480 Lincoln Drive, Madison WI 53706-1388,
USA<EMAIL_ADDRESS>
###### Abstract.

We propose several Hodge theoretic analogues of the conjectures of Hopf and
Singer, and prove them in some special cases.

###### Key words and phrases:

Singer-Hopf conjecture, aspherical manifold, Euler characteristic, sectional
curvature, nef bundle, Fano variety

###### 2020 Mathematics Subject Classification:

14F35, 14F45, 14C30, 14J45, 32S60

First author partially supported by a grant from the Simons Foundation

Second author partially supported by a grant from the Simons Foundation

##  1\. Introduction

###  1.1. Singer-Hopf conjecture.

In 1931, Hopf made the following conjecture on the sign of the Euler
characteristic of a Riemannian manifold, formulated here as strengthened by
Singer in the context of aspherical manifolds, e.g., see [17, Conjecture
25.1]:

######  Conjecture 1.1 (Singer-Hopf).

If $X$ is a closed aspherical manifold of real dimension $2n$, then

(1.1) $(-1)^{n}\cdot\chi(X)\geq 0.$

Recall that a connected CW complex is said to be aspherical if its universal
cover is contractible, so the above conjecture applies more generally to
topological manifolds. Conjecture 1.1 is clearly true if $n=1$. Examples of
aspherical manifolds include closed Riemannian manifolds with non-positive
sectional curvature, for which the conjecture is true if $n=2$ since the
Gauss-Bonnet integrand has the desired sign, cf. [9, Theorem 5]. However,
Conjecture 1.1 is not yet known for all closed aspherical $4$-manifolds, and
it is open for $n\geq 3$.

Gromov [15] introduced the notion of Kähler hyperbolicity, which includes
compact Kähler manifolds with negative sectional curvature as a special case,
and he verified Conjecture 1.1 for Kähler hyperbolic manifolds. Cao-Xavier [6]
and Jost-Zuo [19] introduced the concept of Kähler nonellipticity, including
compact Kähler manifolds with non-positive sectional curvature, and proved the
Singer-Hopf conjecture in this case. See also [12]. These works proved the
desired sign of the Euler characteristic by means of vanishing
$L^{2}$-cohomology. More recently, Liu-Maxim-Wang [26] proved the complex
projective version of Conjecture 1.1 under the assumption that the
(holomorphic) cotangent bundle $T^{*}X$ of $X$ is numerically effective (nef),
e.g., globally generated. Moreover, it was conjectured in [26, Section 6] that
aspherical complex projective manifolds have nef cotangent bundles. Building
on ideas of [11] and [26], Arapura-Wang [1] gave a new proof of Conjecture 1.1
for compact Kähler manifolds with non-positive sectional curvature, using the
fact that such manifolds have nef cotangent bundles. Several generalizations
of the Singer-Hopf conjecture in the singular context have been recently
proposed in [27, Conjectures 1.2 and 1.3] by using constructible functions.

###  1.2. Hodge-theoretic variants of the Singer-Hopf conjecture.

One of the aims of this note is to formulate Hodge-theoretic analogues of
Conjecture 1.1. For this purpose, we only consider compact Kähler (e.g.,
complex projective) manifolds.

For a compact Kähler manifold $X$ of complex dimension $n$, the Euler
characteristic can be computed as

(1.2) $\chi(X)=\sum_{p\geq 0}(-1)^{p}\cdot\chi^{p}(X),$

with $\chi^{p}(X):=\chi(X,\Omega_{X}^{p})$, and
$\chi^{p}(X)=(-1)^{n}\chi^{n-p}(X)$ by Serre duality. Here, $\Omega_{X}^{p}$
denotes the sheaf of holomorphic $p$-forms on $X$, and note that $\chi^{0}(X)$
is just the Todd genus of $X$.

We propose the following natural Hodge theoretic enhancement of the Singer-
Hopf Conjecture 1.1:

######  Conjecture 1.2.

If $X$ is a compact Kähler manifold of dimension $n$ which is aspherical or
has a nef cotangent bundle, then for any $0\leq p\leq n$ one has:

(1.3) $(-1)^{n-p}\cdot\chi^{p}(X)\geq 0.$

Via the Riemann-Roch theorem, (1.3) yields inequalities for the Chern numbers
of $X$ (of weighted degree $2n$), usually referred to as Arakelov type
inequalities. Moreover, if $p=n$, (1.3) reduces to the non-negativity of the
holomorphic Euler characteristic
$\chi(X,K_{X})=(-1)^{n}\cdot\chi(X,\mathcal{O}_{X})$ of the canonical bundle
$K_{X}$, which Kollár [22] conjectured for complex projective manifolds with
generically large fundamental groups (recall here that if $X$ is an aspherical
projective manifold, then $\pi_{1}(X)$ is large, e.g., see [26, Proposition
6.7]). Let us also mention that is was shown in [44, Theorem 4] that if $X$ is
projective with $\Omega^{1}_{X}$ nef and $\chi(X,K_{X})>0$ then $K_{X}$ is
ample, with the converse statement being conjectured in [44] and proved in the
special case when the cotangent bundle is globally generated (cf. [44, Theorem
3]) or if $\dim X\leq 4$.

Conjecture 1.2 is implied by work of Gromov [15] when $X$ is Kähler
hyperbolic, and by Jost-Zuo [19] in the Kähler nonelliptic case, see also
[12]. The works [15, 12, 19] prove such inequalities via vanishing theorems
for $L^{2}$-cohomology.

More recently, Popa-Schnell [35] proved (1.3) for a complex projective
manifold $X$ with a semi-small Albanese map, by showing a Nakano-type generic
vanishing theorem for the sheaves $\Omega^{p}_{X}$ of holomorphic $p$-forms.
Note that if the cotangent bundle is globally generated, then the Albanese map
of $X$ is an immersion, and (1.3) follows in this case from work of Popa-
Schnell [35].

In Proposition 2.4 we show that Conjecture 1.2 holds for complex projective
curves and surfaces with nef cotangent bundles. A similar argument can be used
to show the non-negativity of $\chi(X,K_{X})$ for $X$ a complex projective
manifold with nef cotangent bundle and complex dimension $n\leq 4$ (see
Theorem 2.5).

One can further generalize the statement of Conjecture 1.2 as follows:

######  Conjecture 1.3.

If $X$ is a compact Kähler (or complex projective) manifold which is
aspherical or has a nef cotangent bundle, and $\mathcal{M}$ is a mixed Hodge
module on $X$, then for any integer $p$ one has:

(1.4) $\chi(X,Gr_{F}^{p}DR(\mathcal{M}))\geq 0,$

where $Gr_{F}^{p}DR(\mathcal{M})$ are the graded pieces, with respect to the
Hodge filtration111Here we use the notation $Gr_{F}^{p}$ in place of
$Gr^{F}_{-p}$ corresponding to the identification $F^{p}=F_{-p}$, which makes
the transition between Saito’s increasing filtration $F_{-p}$ appearing in the
$D$-module language and the classical Hodge-theoretic situation of a
decreasing filtration $F^{p}$., of the de Rham complex associated to
$\mathcal{M}$.

For instance, if $n=\dim X$ and $\mathcal{M}=\mathbb{Q}^{H}_{X}[n]$ is the
constant Hodge module, then $Gr_{F}^{p}DR(\mathcal{M})\cong\Omega^{p}[n-p]$,
so Conjecture 1.3 reduces in this case to Conjecture 1.2. Work of Popa-Schnell
[35] on generic vanishing via mixed Hodge modules can be used to prove the
inequality (1.4) in the case when $X$ is a smooth subvariety of, or more
generally, admits a finite morphism to, an abelian variety (see Theorem 2.6.).

###  1.3. Hopf conjecture for positive sectional curvature and
generalizations.

We also propose a Hodge theoretic variant of another conjecture of Hopf, which
can be stated as follows (see, e.g., [43]).

######  Conjecture 1.4 (Hopf).

A compact, even-dimensional Riemannian manifold with positive sectional
curvature has positive Euler characteristic. A compact, even-dimensional
Riemannian manifold with non-negative sectional curvature has non-negative
Euler characteristic.

If $X$ is a compact Kähler manifold with non-negative (resp., positive)
sectional curvature, then the holomorphic tangent bundle $TX$ is nef (resp.,
ample). So the following result from [11], which is derived from
(semi-)positivity properties of nef/ample vector bundles (cf. [11, 13, 3]),
can be seen as a generalization in the compact Kähler context of Conjecture
1.4.

######  Theorem 1.5.

If $X$ is a compact Kähler (resp., complex projective) manifold with a nef
(resp., ample) tangent bundle $TX$ (e.g., $X$ has a non-negative (resp.,
positive) sectional curvature), then $\chi(X)\geq 0$ (resp., $>0$).

Let us remark that, in the case when $X$ has positive bisectional curvature
(e.g., if $X$ has positive sectional curvature), Siu-Yau's proof [38] of
Frankel's conjecture shows that $X$ is biholomorphic to a complex projective
space. The same conclusion was proved by Mori [30] for complex projective
manifolds with ample tangent bundles. So the positive version of Theorem 1.5
can also be deduced from these deep results. Furthermore, a classification of
all compact Kähler manifolds with non-negative bisectional curvature was
obtained by Mok [29] (see also [16]), and this can be used as in Theorem 2.8
to prove Theorem 1.5 in this situation. The general nef case follows, e.g.,
from [11, Proposition 2.1] and the Gauss-Bonnet theorem.

In this note, we propose the following conjectural statement, which provides a
Hodge-theoretic analogue of Theorem 1.5, and which holds, e.g., in the case of
compact Kähler manifolds with non-negative bisectional curvature (e.g., non-
negative sectional curvature), see Theorem 2.8 for a proof in this case.

######  Conjecture 1.6.

If $X$ is a compact Kähler manifold of dimension $n$ with a nef tangent
bundle, then for $0\leq p\leq n$ one has

(1.5) $(-1)^{p}\cdot\chi^{p}(X)\geq 0,$

with $\chi^{p}(X):=\chi(X,\Omega_{X}^{p})$.

As we discuss in Subsection 2.3, by results of [11] it suffices to prove
Conjecture 1.6 in the case when $X$ is a Fano manifold (i.e., a projective
algebraic manifold with $K_{X}^{-1}$ ample). In Proposition 2.7, we prove this
remaining case of Conjecture 1.6 under the additional assumption that $X$ has
an (algebraic) cellular decomposition, in the sense that there is a chain of
Zariski closed subsets $X_{i}$ such that $X_{i}\setminus X_{i+1}$ is a union
of affine spaces (see, e.g., [14, Example 1.9.1]). Examples of such varieties
include rational homogeneous projective manifolds, i.e., projective quotients
$G/P$, with $G$ a semi-simple Lie group and $P\subset G$ is a parabolic
subgroup (see [4, Section 3]). In fact, one has the following conjectural
statement of Campana-Peternell [5], which, in view of Proposition 2.7, would
complete the proof of Conjecture 1.6.

######  Conjecture 1.7 (Campana-Peternell).

Any Fano manifold with nef tangent bundle is rational homogeneous.

This is a natural generalization to the nef case of the above-mentioned result
of Mori [30] (formerly known as the Hartshorne conjecture), and it is
currently proved up to dimension $5$ and in several more cases; see, e.g.,
[10, 25, 40, 20, 21, 41, 32, 34] for recent progress, as well as [31] for a
survey on this conjecture. Furthermore, in the unpublished preprint [10],
Conjecture 1.7 is reduced to proving the non-vanishing of the top Segre class
of $X$ (cf. [10, Theorem 1.2, Conjecture 1.3]).

Given that in order to prove Conjecture 1.6 we do not need the full strength
of Conjecture 1.7, we propose here the following weaker statements which imply
Conjecture 1.6.

######  Conjecture 1.8.

Any Fano manifold $X$ with nef tangent bundle has an (algebraic) cellular
decomposition.

######  Conjecture 1.9.

If $X$ is a Fano manifold with nef tangent bundle, then its Hodge numbers are
concentrated on the diagonal, i.e., $h^{p,q}(X)=0$ if $p\neq q$.

For example, Conjecture 1.8 is true if $X$ has non-negative bisectional
curvature. This is an application of Mok's proof of the generalized Frankel
conjecture [29], using the fact that a Fano manifold is simply-connected. This
observation already leads to the proof of Conjecture 1.6 when $X$ has non-
negative bisectional curvature, see Theorem 2.8. Moreover, the proof of
Proposition 2.7 shows that Conjecture 1.8 implies Conjecture 1.9.

Let us also note that, by results of Carrell-Liebermann [8], in order to prove
Conjecture 1.6, it would also suffice to show that $X$ admits a holomorphic
vector field with only isolated zeros, see Remark 2.9. This latter fact was
conjectured by Carrell [7] to be equivalent to $X$ being rational (see [23,
18] for progress on this conjecture).

We conclude this introduction with yet another conjecture, which may be of
independent interest, and which is easily implied by either Conjecture 1.8 or
Conjecture 1.9, so in particular it is true whenever any of these conjectures
hold (e.g., if $X$ has non-negative bisectional curvature).

######  Conjecture 1.10.

If $X$ is a Fano manifold with nef tangent bundle, then its odd Betti numbers
vanish.

##  2\. Proofs of results

In this section, we discuss various aspects of the Hodge theoretic variants of
the Singer and Hopf conjectures, namely Conjectures 1.2, 1.3 and 1.6 mentioned
in the introduction.

We begin by recalling the following definition.

######  Definition 2.1.

If $E$ is a vector bundle on a projective (or compact complex) manifold $X$,
denote by $\mathbf{P}(E)$ the projective bundle of hyperplanes in the fibers
of $E$. A vector bundle $E$ on $X$ is called ample (resp. nef) if the line
bundle $\mathcal{O}_{E}(1)$ on $\mathbf{P}(E)$ is ample (resp. nef).

In the complex manifold case, the nefness of a line bundle is considered in
the sense of [11, Definition 1.2], and this coincides with the usual
definition when $X$ is projective. Note that globally generated bundles are
nef. Properties of nef bundles are studied, e.g., in [11, 24].

###  2.1. On Conjecture 1.2

In this section we restrict our attention to the projective context. We will
make use of positivity properties for nef vector bundles, which we now recall.

######  Definition 2.2.

The Schur polynomial $P_{\underline{a}}\in\mathbb{Z}[c_{1},\ldots,c_{n}]$ of
weighted degree $2n$ (with $\deg c_{i}=2i$) corresponding to a partition
$\underline{a}=(a_{1},\ldots,a_{n})$ of $n$, i.e.,

$n\geq a_{1}\geq a_{2}\geq\cdots\geq a_{n}\geq 0$ , with $\sum_{j}a_{j}=n$,

is defined as the $n\times n$ determinant

$P_{\underline{a}}(c)=\det\left(c_{a_{i}-i+j}\right)_{1\leq i,j\leq n}.$

Here, $c_{i}=0$ if $i\notin[0,n]$ and $c_{0}=1$.222The Schur polynomial
corresponding to the partition $\underline{1}^{n}=(1,\ldots,1)$ is the Segre
class $s_{n}$.

Then, as a special case of [11, Theorem 2.5], one has the following.

######  Theorem 2.3.

Let $X$ be a compact Kähler manifold of complex dimension $n$. Then the Schur
polynomials $P_{\underline{a}}(c(E))$ of weighted degree $2n$ in the Chern
classes of a nef vector bundle $E$ of rank $n$ on $X$ are non-negative, that
is,

(2.1) $\int_{X}P_{\underline{a}}(c(E))\geq 0.$

For simplicity, in what follows, we will drop the integral symbol.

The Riemann-Roch theorem, positivity results from [11] as recalled in Theorem
2.3, and the Bogomolov-Miyaoka-Yau inequality333Almost all results concerning
the Bogomolov-Miyaoka-Yau inequality are proved under the projective
assumption. For a recent extension to compact Kähler manifolds under a
suitable positivity condition on the canonical bundle, see [33]. [28, 42, 39]
yield the following:

######  Proposition 2.4.

Conjecture 1.2 is true for smooth projective curves and surfaces with nef
cotangent bundles.

###### Proof.

In the calculations below, we use the notation

$c_{i}:=c_{i}(\Omega^{1}_{X})=(-1)^{i}c_{i}(TX).$

Assume $\dim X=1$. The Riemann-Roch theorem and the non-negativity result of
Theorem 2.3 yield:

$\chi(X,\Omega^{1}_{X})=-\chi(X,\mathcal{O}_{X})=\frac{c_{1}}{2}\geq 0.$

Assume $\dim X=2$. The non-negativity of Schur polynomials of $T^{*}X$ implies
that $c_{2}\geq 0$ and $c_{1}^{2}-c_{2}\geq 0$. Together with the Riemann-Roch
theorem, this yields:

$\chi(X,\Omega^{2}_{X})=\chi(X,\mathcal{O}_{X})=\frac{c^{2}_{1}+c_{2}}{12}\geq
0.$

Also, one gets by Riemann-Roch that

$\chi(X,\Omega^{1}_{X})=\frac{c_{1}^{2}-5c_{2}}{6}.$

The non-positivity of the above expression follows from $c_{2}\geq 0$, by
using the Bogomolov-Miyaoka-Yau inequality (e.g., in the sense of [39] since
$K_{X}$ is nef), which in our notations can be written as:

$c_{1}^{2}\leq 3c_{2}.$

∎

We next show that the above arguments can be extended to dimensions $n=3$ and
$n=4$, to show the non-negativity of $\chi(X,K_{X})$ in these dimensions.

######  Theorem 2.5.

Let $X$ be a complex projective manifold of dimension $n\leq 4$ with a nef
cotangent bundle. Then

(2.2) $\chi(X,K_{X})=(-1)^{n}\cdot\chi(X,\mathcal{O}_{X})\geq 0.$

###### Proof.

The case of curves and surfaces was already considered in Proposition 2.4.

Assume $\dim X=3$. The non-negativity of Schur polynomials of $\Omega^{1}_{X}$
implies:

$P_{(3,0,0)}(c)=c_{3}\geq 0,\ P_{(2,1,0)}(c)=c_{1}c_{2}-c_{3}\geq 0,\
P_{(1,1,1)}(c)=c_{1}^{3}-2c_{1}c_{2}+c_{3}\geq 0.$

Then, by using Riemann-Roch, we have

$\chi(X,\Omega^{3}_{X})=-\chi(X,\mathcal{O}_{X})=\frac{c_{1}c_{2}}{24}\geq 0.$

Assume next that $\dim X=4$. The non-negativity of Schur polynomials of
$\Omega^{1}_{X}$ implies:

$P_{(4,0,0,0)}(c)=c_{4}\geq 0,\ P_{(3,1,0,0)}(c)=c_{1}c_{3}-c_{4}\geq 0,\
P_{(2,2,0,0)}(c)=c_{2}^{2}-c_{1}c_{3}\geq 0,$
$P_{(2,1,1,0)}(c)=c_{1}^{2}c_{2}-c_{1}c_{3}-c_{2}^{2}+c_{4}\geq 0,$
$P_{(1,1,1,1)}(c)=c_{1}^{4}-3c_{1}^{2}c_{2}+2c_{1}c_{3}+c_{2}^{2}-c_{4}\geq
0.$

A calculation via the Riemann-Roch theorem yields the following:

$\chi(X,\Omega^{4}_{X})=\chi(X,\mathcal{O}_{X})=\frac{-c_{1}^{4}+4c_{1}^{2}c_{2}+c_{1}c_{3}+3c_{2}^{2}-c_{4}}{720}.$

Since $c_{1}^{4}$ has a negative coefficient in $\chi(X,\Omega^{4}_{X})$, the
non-negativity of the Schur polynomials is not sufficient to guarantee that
$\chi(X,\Omega^{4}_{X})\geq 0$. However, since $\Omega^{1}_{X}$ is nef by our
assumptions, it follows that $K_{X}$ is nef (hence $X$ is minimal), so the
corresponding Miyaoka-Yau type inequality [39, 45] yields:

(2.3) $c_{1}^{4}\leq\frac{5}{2}c_{1}^{2}c_{2}.$

We then have:

$\displaystyle-c_{1}^{4}+4c_{1}^{2}c_{2}+c_{1}c_{3}+3c_{2}^{2}-c_{4}$
$\displaystyle\geq-
c_{1}^{4}+4c_{1}^{2}c_{2}+c_{1}c_{3}+3c_{2}^{2}-c_{4}-P_{(1,1,1,1)}(c)$
$\displaystyle=-2c_{1}^{4}+7c_{1}^{2}c_{2}-c_{1}c_{3}+c_{2}^{2}$
$\displaystyle\geq-2c_{1}^{4}+7c_{1}^{2}c_{2}-c_{1}c_{3}+c_{2}^{2}-P_{(2,2,0,0)}(c)$
$\displaystyle=-2c_{1}^{4}+7c_{1}^{2}c_{2}$
$\displaystyle\geq\frac{4}{5}c_{1}^{4},$

where the last inequality follows by (2.3). Finally, we have by [11, Corollary
2.6] that $c_{1}^{4}\geq 0$. Altogether, we get that
$\chi(X,\Omega^{4}_{X})=\chi(X,\mathcal{O}_{X})\geq 0$. ∎

###  2.2. On Conjecture 1.3

In this section, we prove the following variant of Conjecture 1.3. We refer to
[36, 37] for background material on mixed Hodge modules.

######  Theorem 2.6.

Let $X$ be a complex projective manifold with a finite morphism $f\colon X\to
A$ to an abelian variety, and let $\mathcal{M}$ be a mixed Hodge module on
$X$. Then for any integer $p$ one has:

(2.4) $\chi(X,Gr_{F}^{p}DR(\mathcal{M}))\geq 0,$

where $Gr_{F}^{p}DR(\mathcal{M})$ are the graded pieces, with respect to the
Hodge filtration, of the de Rham complex associated to $\mathcal{M}$.

###### Proof.

Since $f$ is a finite morphism,
$\mathcal{N}:=f_{*}\mathcal{M}=f_{!}\mathcal{M}$ is a mixed Hodge module on
the abelian variety $A$. And since $f$ is proper and the functor
$Gr_{F}^{p}DR$ commutes with proper pushforward, we have:

$\chi(X,Gr_{F}^{p}DR(\mathcal{M}))=\chi(A,Gr_{F}^{p}DR(\mathcal{N})).$

Finally, by [35, Corollary 2.7], there exists $L\in Pic^{0}(A)$ such that

$H^{i}(A,Gr_{F}^{p}DR(\mathcal{N})\otimes L)=0$, for all $i\neq 0$.

Therefore,

$\chi(A,Gr_{F}^{p}DR(\mathcal{N}))=\chi(A,Gr_{F}^{p}DR(\mathcal{N})\otimes
L)\geq 0,$

which proves the assertion. ∎

###  2.3. On Conjecture 1.6

In this section we discuss various aspects of Conjecture 1.6.

First, it was shown in [11, Proposition 3.10] that, if $TX$ is nef, then one
of the two situations occurs:

$(i)$ $c_{1}(X)^{n}=c_{1}(TX)^{n}=0$, in which case it follows that all Chern
polynomials of $X$ of weighted degree $2n$ vanish as well (cf. [11, Corollary
2.7]). In particular, by Riemann-Roch, $\chi(X,\Omega^{p})=0$ for all $p$, and
Conjecture 1.6 is true in this case.

$(ii)$ $c_{1}(X)^{n}>0$, in which case $X$ is a Fano manifold (i.e., a complex
projective manifold with $K_{X}^{-1}$ ample). Then the Kodaira vanishing
theorem implies that

$H^{q}(X,\mathcal{O}_{X})=H^{q}(X,K_{X}\otimes K_{X}^{-1})=0,\ \ \ {\rm for}\
q\geq 1.$

Hence, $\chi^{0}(X)=\chi(X,\mathcal{O}_{X})=\dim H^{0}(X,\mathcal{O}_{X})=1$.
By Serre duality, we further get that $(-1)^{n}\chi^{n}(X)>0$.

It therefore remains to prove Conjecture 1.6 in the Fano case for $0<p<n$. In
what follows, we prove this case of Conjecture 1.6 under the additional
assumption that $X$ has a cellular decomposition, in the sense that there is a
chain of Zariski closed subsets $X_{i}\subseteq X$ such that $X_{i}\setminus
X_{i+1}$ is a union of affine spaces. More precisely, we have the following.

######  Proposition 2.7.

If the complex projective manifold $X$ has a cellular decomposition (e.g., $X$
is rational homogenous), then $(-1)^{p}\cdot\chi^{p}(X)>0$ for all $p\leq\dim
X$.

###### Proof.

Since $X$ has a cellular decomposition, the cycle map $cl_{X}:A^{*}(X)\to
H^{*}(X)$ is an isomorphism, see [14, Example 19.1.11(b)]. In particular, the
cohomology of $X$ is generated by algebraic cycles. Therefore the Hodge
numbers of $X$ are concentrated on the diagonal, i.e., $h^{p,q}=0$ unless
$p=q$. Furthermore, it follows by Hard Lefschetz and the fact that $h^{0,0}=1$
that $h^{p,p}>0$ for all $p\leq\dim X$. (An alternative proof of this fact
follows along the lines of [2, Theorem 3.1], using induction over the number
of cells together with the fact that $H^{i}_{c}(\mathbb{C}^{k};\mathbb{Q})=0$
for $i\neq 2k$ and $H^{2k}_{c}(\mathbb{C}^{k};\mathbb{Q})=\mathbb{C}$ is pure
of weight $2k$.) Therefore,

$(-1)^{p}\cdot\chi(X,\Omega^{p})=h^{p,p}>0.$

∎

Altogether, we obtain the following.

######  Theorem 2.8.

If $X$ is a compact Kähler manifold of dimension $n$ with non-negative
bisectional curvature (e.g., non-negative sectional curvature), then for
$0\leq p\leq n$ one has

(2.5) $(-1)^{p}\cdot\chi^{p}(X)\geq 0,$

with $\chi^{p}(X):=\chi(X,\Omega_{X}^{p})$.

###### Proof.

First recall that the bisectional curvature can be written as a positive
linear combination of two sectional curvatures (e.g., see [46]). So if $X$ has
non-negative sectional curvature, then it also has a non-negative bisectional
curvature.

Secondly, our assumptions imply that the tangent bundle $TX$ is nef. Indeed, a
compact Kähler manifold $X$ with non-negative bisectional curvature has (by
definition) a tangent bundle $TX$ which is Griffiths semipositive. It is also
known that Griffiths semipositive bundles are nef, cf. [11].

As above, it suffices to further restrict our attention to the case when $X$
is a Fano manifold. Then $X$ is simply-connected, and hence, by Mok's theorem
[29, Main Theorem], $X$ is a product of complex projective spaces and compact
Hermitian symmetric spaces (of rank $\geq 2$), all of which are known to admit
a cellular decomposition. Hence $X$ admits a cellular decomposition, and the
claim follows now by Proposition 2.7. ∎

######  Remark 2.9.

By results of Carrell-Liebermann [8, Theorem 1], an alternative approach to
proving the diagonal concentration of Hodge numbers, and thus to prove
Conjectures 1.9 and 1.6, is to show that $X$ admits a holomorphic vector field
with only isolated zeros. Note that, if $X$ is as in $(ii)$ above, then Bott's
residue formula implies that any vector field on $X$ has a non-empty vanishing
locus. Moreover, if $TX$ is globally generated, Bertini's theorem tells us
that the zero locus of a general vector field on $X$ consists of only isolated
points.

######  Remark 2.10.

As already mentioned in the introduction, Proposition 2.7 can be seen as
reducing the proof of Conjecture 1.6 to the Campana-Peternell Conjecture 1.7.
This also applies to the situation described in the previous remark, since
rational homogeneous spaces are known to carry holomorphic vector fields with
only isolated zeros. Moreover, while the nefness of the tangent bundle implies
by (2.1) the non-negativity of the top Segre class $s_{n}(X):=s_{n}(TX)$, it
is known that rational homogeneous spaces have positive top Segre class [10,
Theorem 4.1]. This motivates the reduction in [10] of Conjecture 1.7 to the
non-vanishing of the top Segre class.

## References

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Study Duration Prediction for Clinical Trials with Time-to-Event Endpoints
Using Mixture Distributions Accounting for Heterogeneous Population

Hong Zhang1∗, Jie Pu1, Shibing Deng1, Satrajit Roychoudhury1, Haitao Chu1,2,
Douglas Robinson1

1 Global Biometrics and Data Management, Pfizer Inc., New York, New York, USA

2 Division of Biostatistics, School of Public Health, University of Minnesota,
Minneapolis, Minnesota, USA

$*$ Corresponding author<EMAIL_ADDRESS>
Abstract

In the era of precision medicine, more and more clinical trials are now driven
or guided by biomarkers, which are patient characteristics objectively
measured and evaluated as indicators of normal biological processes,
pathogenic processes, or pharmacologic responses to therapeutic interventions.
With the overarching objective to optimize and personalize disease management,
biomarker-guided clinical trials increase the efficiency by appropriately
utilizing prognostic or predictive biomarkers in the design. However, the
efficiency gain is often not quantitatively compared to the traditional all-
comers design, in which a faster enrollment rate is expected (e.g. due to no
restriction to biomarker positive patients) potentially leading to a shorter
duration. To accurately predict biomarker-guided trial duration, we propose a
general framework using mixture distributions accounting for heterogeneous
population. Extensive simulations are performed to evaluate the impact of
heterogeneous population and the dynamics of biomarker characteristics and
disease on the study duration. Several influential parameters including median
survival time, enrollment rate, biomarker prevalence and effect size are
identitied. Re-assessments of two publicly available trials are conducted to
empirically validate the prediction accuracy and to demonstrate the practical
utility. The R package _detest_ is developed to implement the proposed method
and is publicly available on CRAN.

Keywords: Study duration, Clinical trials, Biomarkers, Mixture distribution,
Time-to-event endpoints

##  1 Introduction

In therapeutic areas where the primary outcome is often time to an event of
interest, e.g., time to disease progression or death, the statistical power
and sample size calculation of a clinical trial is mostly driven by the number
of events observed and the statistical analyses are typically carried out
after the pre-specified event milestones have occurred. Given the considerable
amount of operational efforts and expenses for conducting a clinical trial, it
is of great value to accurately predict the study duration, i.e. when the
milestone(s) will be reached, for both resource and strategic planning
purposes.

Several parametric and non-parametric models have been proposed to predict
trial duration based on factors such as patient enrollment rate and event
rate. For example, Rubinstein et al. [1] used the Poisson accrual process and
exponential survival model to predict trial duration needed for achieving the
desired number of events. Bagiella and Heitjan [2] developed a Bayesian
parametric model to estimate both the point and the interval predictions of
milestone time based on the accumulating data from the trial while Ying et al.
[3] later proposed a non-parametric approach to make the point and interval
prediction. In Anisimov’s work [4], patient enrollment was modeled using a
delayed Poisson process to handle different enrollment rates across centers.
To facilitate an easier examination of the relationship between sample size
(planned number of participants) and the expected study duration and to assess
the variation in study duration for a given sample size, Machida et al. [5]
developed a graphical approach and a probability density function of the study
duration. We refer the interested readers to [6, 5] for more comprehensive
literature review of study duration prediction methodology.

Under the presence of biomarker subgroups, however, the study duration is
dependent on the characteristics of the biomarker. Without accounting for this
information, most of the previous work may lead to inaccurate study duration
prediction, as evidenced by our empirical results in Figure 3. The objective
of this article is to introduce a statistical model for predicting study
duration under patient heterogeneity. The proposed model is an extension of
the work by Machida et al. [5] with a newly added component to incorporate
different patient risk subgroups. This generalization is motivated by the
practical importance to inform the design of a prognostic enrichment study
that aims at increasing trial efficiency for drug development. For event-
driven trials, inclusion of patients with lower risk of developing an outcome
related event may limit the chance to detect a clinically meaningful treatment
effect and may impact the trial efficiency due to increased sample size or
prolonged study duration to achieve the targeted event milestone(s) [7] [8]
[9]. By increasing the proportion of patients with higher risk, a prognostic
enrichment study design may potentially help improve the trial efficiency,
e.g., reducing sample size or shortening study duration, compared to an all-
comer design. Given the complexity of predicting study duration described
earlier, a systematic comparison between an enrichment vs. an all-comer design
in trial efficiency is needed to inform decision making during the planning
stage.

The remainder of this article is structured as follows: In Section 2, we
present the model framework for calculating trial duration and conduct a
theoretical analysis of how various parameters affect the duration. Section 3
is dedicated to evaluating the performance of our proposed calculation methods
and showcasing some intriguing applications. In Section 4, we apply our
calculation method to actual trial data to validate its accuracy and
demonstrate its practical usefulness in predicting trial duration. Finally, we
conclude this paper with some discussions and possible future research
directions in Section 5.

##  2 Theoretical Framework for Study Duration

Consider a clinical trial of $K$ treatment arms on a patient population with
$L$ subgroups. Assume the proportion of patients who are in subgroup $l$ and
receive treatment $k$ is $r_{kl}\geq 0$,
$\sum_{k=1}^{K}\sum_{l=1}^{L}r_{kl}=1$. In the randomized setting where the
treatment assignment is independent from the subgroups, we may write
$r_{kl}=p_{k}q_{l}$, where $p_{k}$ is the proportion of treatment arm $k$, and
$q_{l}$ is the prevalence of subgroup $l$.

Let $T_{kl}$ be the random variable of the time from study initiation to event
observed for patients who are in subgroup $l$ and receive treatment $k$,

$\displaystyle T_{kl}=\left\\{\\!\begin{aligned} &+\infty,&&V_{kl}>W_{kl}\\\
&U_{kl}+V_{kl},&&V_{kl}\leq W_{kl}\end{aligned}\right.$ (1)

where $U_{kl}$ is the time from study initiation to enrollment, $0<U_{kl}<a$
for some enrollment finishing at a given time $a$, $V_{kl}$ is the time from
enrollment to an event, $W_{kl}$ is the time from enrollment to drop-out.
$U_{kl}$, $V_{kl}$ and $W_{kl}$ are assumed to be independent. The above
definition says when an event is observed, the time to the observed event is
the summation of time from study initiation to enrollment and the time from
enrollment until that patient experiences an event. However, if a patient is
censored, the event is never observed and thus the time to the observed event
is defined as infinity.

To derive the cumulative distribution function (CDF) of $T_{kl}$, we use the
total probability theorem to obtain the following integral expression (see
Appendix A1).

$\displaystyle F_{T_{kl}}(t)=P(T_{kl}\leq
t)=\int_{0}^{\min\\{t,a\\}}\int_{0}^{t-u}\int_{v}^{+\infty}f_{W_{kl}}(w)f_{V_{kl}}(v)f_{U_{kl}}(u)dwdvdu,\quad{}0<t<+\infty,$
(2)

where $f_{U_{kl}}$, $f_{V_{kl}}$, and $f_{W_{kl}}$ are the probability density
functions (PDF) of the enrollment time, event time and drop-out time,
respectively. It is straightforward to check that, as $t$ goes to infinity,
the probability of observing an event before time $t$ converges to the
probability of no censoring. That is,
$\lim_{t\to+\infty}F_{T_{kl}}(t)=P(V_{kl}\leq W_{kl})$. We further define
$F_{T_{kl}}(+\infty)=1$ to make it a valid CDF.

Denote $T$ the time from study initiation to an observed event for all
patients in the study. $T$ is the mixture of $T_{kl}$, $k=1,...,K$,
$l=1,...,L$, with mixing parameters $r_{kl}$. The CDF of $T$ may be written as

$\displaystyle F_{T}(t)=\sum_{k=1}^{K}\sum_{l=1}^{L}r_{kl}F_{T_{kl}}(t).$ (3)

We can also check that
$\lim_{t\to+\infty}F_{T}(t)=\sum_{k=1}^{K}\sum_{l=1}^{L}r_{kl}P(V_{kl}\leq
W_{kl})$, and similarly define $F_{T}(+\infty)=1$.

### 2.1. Study Duration of Clinical Trials with Time-to-Event Endpoints

In clinical trials with time-to-event endpoints, the study planning is often
driven by a target number of events, $d$. There are different methods to
calculate $d$, such as Schoenfeld formula [10] and Freedman formula [11].
Here, we assume such $d$ has been determined. Therefore, the study duration
may be defined as the time from study initiation to observing the $d$th event,

$\displaystyle\text{Study Duration}=T_{(d)},$ (4)

the $d$th order statistics from $T_{(1)}\leq T_{(2)}\leq...\leq T_{(n)}$,
where $n$ is the sample size.

It is important to note that the study duration $T_{(d)}$ is a random
variable. In the case of heavy censoring, the $d$th event may never be
observed, i.e., $T_{(d)}$ has a non-zero probability of being $+\infty$.
However, recognizing that the study duration is random, we may still need a
point estimate to guide clinical trial design. One approach is by noticing
that, according to the strong law of large numbers for order statistics,

$\displaystyle\lim_{\begin{subarray}{c}n\to\infty,\\\ d/n\to
s\end{subarray}}T_{(d)}\to F_{T}^{-1}(s)\text{ in probability},$ (5)

where $0<s<+\infty$ is the convergent value for $d/n$. Under large samples,
$s\approx d/n$, $F_{T}^{-1}(d/n)$ may be a good estimate of the study
duration. When $F_{T}(t)$ is known, the percentiles may be efficiently found
using the bisection method.

The above asymptotics may no longer hold, however, under moderate or small
sample sizes. In such cases, we may use the median, $med(T_{(d)})$, as our
study duration estimate. The distribution of $T_{(d)}$ is theoretically known
by the classic order statistics theory [12] when $F_{T}(t)$ is available. To
obtain the median, a direct calculation approach is viable following the
similar idea in Machida et al [5]. Alternatively, we propose a simulation
based approach (See Algorithm 1), which is computationally easier and more
stable than the direct calculation approach. It is worth mentioning that both
the percentile approach and the median approach may cover any continuous
survival and enrollment distributions in 1.

Input: Sample size $n$, number of events $d$, number of treatments $K$,
proportion of treatment arm $k$, $p_{k}$, number of biomarker groups $L$,
prevalence of biomarker subgroup $l$, $q_{l}$, CDF of time from study
initiation to event observed $F_{T_{kl}}$, number of repetitions $R$ and
confidence level $\alpha$.

Output: Study duration estimates.

1 Calculate $P_{k}=\sum_{i=1}^{k}p_{i},Q_{l}=\sum_{i=1}^{l}q_{i}$,
$k=1,...,K$, $l=1,...,L$.

2 for  _i = 1 to R_ do

3 for  _j = 1 to n_ do

4 $u\leftarrow$ runif(0, 1).

5 $v\leftarrow$ runif(0, 1).

6 if  _$P_{k-1} <u\leq P_{k}$ $\And$ $Q_{l-1}<v\leq Q_{l}$_ then

7 generate a random number $x_{j}$ from $F_{T_{kl}}$, e.g., by inverse
transform sampling.

8

9 Find the $d$th order statistic, $x_{(d)}$, from $x_{1},...,x_{j},...,x_{n}$.

Output the median, $\alpha/2$ th and $1-\alpha/2$ th percentiles of
$x_{(d),1},...,x_{(d),i},...,x_{(d),R}$.

Algorithm 1 Pseudo algorithm for study duration calculation

### 2.2. Exponential Event Time and Potentially Non-uniform Enrollment

For some parametric family of survival distributions and enrollment time
distribution, we may develop an analytical expression for $F_{kl}$ in equation
2. Here, we drop the subscript $kl$ to simplify the notations. Assume
exponential survival $V\sim Exp(\lambda_{V})$, exponential drop-out $W\sim
Exp(\lambda_{W})$ and enrollment follows a scaled Beta distribution $U/a\sim
Beta(\alpha=1,\beta)$, where $a$ is the enrollment completion time. Its PDF is

$\displaystyle f_{U}(u)$
$\displaystyle=\frac{\beta}{a}\left(1-\frac{u}{a}\right)^{\beta-1},0<u<a.$ (6)

This family covers a wide variety of situations for enrollment. When
$\beta=1$, the enrollment is uniform over $(0,a)$. When $\beta<1$, the
enrollment curve is convex such that the enrollment is slower at the beginning
and then increasingly faster later on. When $\beta>1$, the enrollment curve is
concave meaning the enrollment is faster at the beginning and then slows down
later.

Under these parametric assumptions, we may evaluate (Appendix A2) the integral
in equation (2) to get the CDF

$\displaystyle\begin{split}F_{T_{kl}}(t)=\frac{\lambda_{V}}{\lambda_{V}+\lambda_{W}}\left(1-\max\\{0,\frac{a-t}{a}\\}^{\beta}-\frac{\beta\Gamma(\beta)e^{-(\lambda_{V}+\lambda_{W})(t-a)}}{(a(\lambda_{V}+\lambda_{W}))^{\beta}}\left(F_{G}(a)-F_{G}(\max\\{0,a-t\\})\right)\right),\end{split}$
(7)

where $F_{G}$ is the CDF of the Gamma$(\beta,\lambda_{V}+\lambda_{W})$
distribution. By Leibniz integral rule, we may show that , for any given
$t>0$, the CDF is an increasing function in $\beta$,

$\displaystyle\frac{\partial}{\partial\beta}F_{T_{kl}}(t)>0,\quad{}0<t<+\infty.$

That is, when two trials have the same enrollment period, the one with larger
$\beta$ will have larger probability of observing an event at or before any
given time point $t$.

Another observation from equation (7) is that, if we assume uniform
enrollment, namely $\beta=1$, we may simplify

$\displaystyle
F_{T_{kl}}(t)=\frac{\lambda_{V}}{\lambda_{V}+\lambda_{W}}\left(1-\max\\{0,\frac{a-t}{a}\\}-\frac{e^{-(\lambda_{V}+\lambda_{W})t}}{a(\lambda_{V}+\lambda_{W})}\left(e^{(\lambda_{V}+\lambda_{W})(\min(a,t))}-1\right)\right).$
(8)

One may verify that, when $t>a$, this formula is the same as Machida et al’s
formula [5] for exponential survival distribution. Formula (8), however, is
more general to include the situation where $t<a$, i.e., when the event of
interest occurred before the enrollment period ends.

### 2.3. Study Duration Comparison between All-comer and Enrichment Designs

One important application of the theoretical framework is to compare the all-
comer and enrichment design in terms of their study duration.

Consider a population of two subgroups $k=1,2$, e.g., biomarker positive vs
biomarker negative. Assume a single-arm ($L=1$) study. Let $r_{1}$, $r_{2}$ be
the prevalence of the biomarker positive and biomarker negative, respectively.
The $\lambda_{1}$, $\lambda_{2}$ are the hazard rates of these two subgroups.
Further assume no drop-out and uniform enrollment. The enrollment rates are
$m_{1}=r_{1}m$, $m_{2}=r_{2}m$, when $m$ is the enrollment rate of the entire
population. Under uniform enrollment, the full study sample size $n=ma$. If we
want to enroll the same number of patients in only a subset, e.g., biomarker
positive, of the population, the enrollment rate will decrease to $m*r_{1}$
and therefore, the enrollment period has to extend to $a/r_{1}$.

For the all-comer design on these two subgroups, the time from study
initiation to the observed event is a random variable whose CDF

$\displaystyle\begin{split}F_{T_{A}}(t)&=r_{1}F_{T_{1}}(t)+r_{2}F_{T_{2}}(t)\\\
&=1-\max\\{0,1-t/a\\}-\frac{r_{1}e^{-\lambda_{1}t}}{a\lambda_{1}}\left(e^{\lambda_{1}\min(a,t)}-1\right)-\frac{r_{2}e^{-\lambda_{2}t}}{a\lambda_{2}}\left(e^{\lambda_{2}\min(a,t)}-1\right).\end{split}$
(9)

For the enrichment design of the same sample size on the biomarker positive
patients, its CDF of time from study initiation to observed event is

$\displaystyle
F_{T_{E}}(t)=1-\max\\{0,1-r_{1}t/a\\}-\frac{r_{1}e^{-\lambda_{1}t}}{a\lambda_{1}}\left(e^{\lambda_{1}\min(a/r_{1},t)}-1\right),$
(10)

where the enrollment period is extended to $a/r_{1}>a$ to account for the fact
that only a fraction $r_{1}$ of the entire population is eligible for the
enrichment trial.

Recall in formula 5, we show that the study duration is approximately
$F_{T}^{-1}(d/n)$. Thus, we may compare the study duration of different
designs by examining the CDFs $F_{T_{A}}$ and $F_{T_{E}}$. If, for example, we
may show that for some $t_{E}^{*}>0$,

$\displaystyle\frac{d}{n}=F_{T_{E}}(t_{E}^{*})>F_{T_{A}}(t_{E}^{*}),$

then,
$t_{E}^{*}=F_{T_{E}}^{-1}(\frac{d}{n})<F_{T_{A}}^{-1}(\frac{d}{n})=t_{A}^{*}$.
That is, the enrichment design has shorter duration if its CDF at the time of
completion is larger than the CDF of the all-comers design.

Generally, we may derive the following formula for the difference between
these two CDFs (Appendix A3).

$\displaystyle\begin{split}F_{T_{E}}(t)-F_{T_{A}}(t)=\begin{cases}\frac{1-r_{1}}{a}\left(\frac{1-e^{-\lambda_{2}t}}{\lambda_{2}}-t\right),&0<t<a\\\
-(1-r_{1})+\frac{r_{1}}{a}\left((t-a)-\frac{1-e^{-\lambda_{1}(t-a)}}{\lambda_{1}}\right)+\frac{1-r_{1}}{a}\left(\frac{e^{-\lambda_{2}(t-a)}-e^{-\lambda_{2}t}}{\lambda_{2}}\right),&a<t<a/r_{1}\\\
\frac{r_{1}}{a}\left(\frac{e^{-\lambda_{1}(t-a)}-e^{-\lambda_{1}(t-a/r_{1})}}{\lambda_{1}}\right)+\frac{1-r_{1}}{a}\left(\frac{e^{-\lambda_{2}(t-a)}-e^{-\lambda_{2}t}}{\lambda_{2}}\right),&t>a/r_{1}\end{cases}\end{split}$
(11)

There are a few remarks may be made regarding the above formula.

######  Remark 1.

In the first period, $0<t<a$, the enrollment is ongoing in both trials. Using
known inequalities in Appendix A4, we may prove that
$F_{T_{E}}(t)-F_{T_{A}}(t)<0$.. That means, if the all-comer trial completes
before the enrollment, i.e., $0<t_{A}^{*}<a$, then the enrichment design
always has longer duration. Intuitively speaking, if the enrollment is slow
such that the target number of events of the all-comer trial can be reached
before $a$, we should not expect any enrichment design to be faster.

######  Remark 2.

In the second period, $a<t<a/r_{1}$, the all-comer enrollment is completed but
the enrichment enrollment is still ongoing. The first term $-(1-r_{1})<0$. The
second term
$\frac{r_{1}}{a}\left((t-a)-\frac{1-e^{-\lambda_{1}(t-a)}}{\lambda_{1}}\right)>0$
is monotone increasing in $r_{1}$ and $\lambda_{1}$. The third term
$\frac{1-r_{1}}{a}\left(\frac{e^{-\lambda_{2}(t-a)}-e^{-\lambda_{2}t}}{\lambda_{2}}\right)$
is also larger than $0$ but monotone decreasing in $r_{1}$ and $\lambda_{2}$.
If everything else is equal but the biomarker positive prevalence $r_{1}$ is
higher, then it is possible that $F_{T_{E}}(t)>F_{T_{A}}(t)$ for some $t>a$.
Thus, the enrichment design has a shorter duration than the all-comers design.
Similarly, if $\lambda_{1}$ is large and $\lambda_{2}$ is small, i.e., hazard
ratio of the biomarker positive, $\frac{\lambda_{1}}{\lambda_{2}}$ is large,
it is possible for the second term to become larger than the first so that the
enrichment design saves time.

######  Remark 3.

In the third period, $t>a/r_{1}$, both trials have completed the enrollment.
Notice that the first term
$\frac{r_{1}}{a}\left(\frac{e^{-\lambda_{1}(t-a)}-e^{-\lambda_{1}(t-a/r_{1})}}{\lambda_{1}}\right)$
is less than $0$ and monotone increasing in $r_{1}$ and $\lambda_{1}$.
Following the same argument in Remark 2, we need a large hazard ratio within
the biomarker positive subgroup, $\frac{\lambda_{1}}{\lambda_{2}}$, such that
the enrichment design is more time-efficient.

The derivation for multi-arm studies, $L\geq 2$, and/or more biomarker
subgroups, $K>2$, is similar but cumbersome. The comparison between all-comer
and the enrichment design will be carried out numerically in the Section 3.4.

##  3 Numerical Results

### 3.1. The Distributions of $T$ and $T_{(d)}$

In this section, we will focus on the improvement of the proposed model
compared to Machida et al. [5] by demonstrating that formula 7 is more general
and that there may be situations where this generalness is needed for accurate
duration calculation. Here, we define the target number of events $d=88$,
sample size $n=140$, no biomarker subgroups, and 1-1 treatment allocation
ratio. Two scenarios were considered:

  1. 1.

Scenario 1: The enrollment rate is 10 patients per month, or a 14 months
enrollment period under uniform enrollment assumption. The survival
distributions are assumed to be exponential, and the median survival is 10 and
20 months for the placebo group and the treatment group, respectively.

  2. 2.

Scenario 2: Similar to Scenario 1, however the enrollment rate is modified to
be 3.88 patients per month (for a 36 months enrollment period). The median
survival changes to 5 and 10 months for the placebo group and the treatment
group, respectively.

Figure 1: The distributions of $T$ (left) and $T_{(d)}$ (right). Scenario 1:
enrollment rate $=10$ pts/month, median survival $=10,20$ months for the
placebo and treatment groups, respectively. Scenario 2: enrollment rate
$=3.88$ pts/month, median survival $=5,10$ months for the placebo and
treatment groups, respectively.

Figure 1 shows that, in terms of the CDF of $T$, the proposed method matches
with the Machida et al method when the time of interest is longer than $a$,
the enrollment period in both scenarios. However, when $T<a$, the CDF
calculated by Machida et al method is no longer a valid CDF. The observation
is consistent with the remark made in Machida et al (2021) [5] that they ‘do
not consider the situation in which the study is completed during the
enrollment period’.

When comparing these two methods in terms of the distribution of $T_{(d)}$,
Figure 1 shows that, in scenario 1, both methods yield almost identical
results while, in scenario 2, the two methods differ significantly. The reason
is that in scenario 2, there is a non-negligible probability for the study to
complete before the enrollment ends due to the slow enrollment and fast
disease progression. Therefore, the $(T<a)$ section of the CDF of $T$ is vital
for the accurate calculation of the duration $T_{(d)}$.

### 3.2. The Impacts of Non-uniform Enrollment

To examine the impact of non-uniform enrollment on study duration, we consider
the following variations of the two scenarios in 3.1:

  1. 1.

Scenario $1^{\prime}$: similar to Scenario 1 but the enrollment follows a
Beta(1, 0.45) as in formula 6.

  2. 2.

Scenario $2^{\prime}$: similar to Scenario 2 but the enrollment follows a
Beta(1, 1.25) as in formula 6.

As we discussed in Section 2.2 from a theoretical point of view, Figure 2
shows that a Beta(1,$\beta$) distribution with $\beta<1$ means the enrollment
starts slower than the average rate. Its enrollment curve (black dotted-
dashed) is below the corresponding uniform enrollment curve (black solid),
even though these two have the same sample size and enrollment period. On the
other hand, if $\beta>1$, the enrollment rate is higher in the beginning than
the average. The resulting enrollment curve (red dotted-dashed) is above the
corresponding uniform enrollment curve (red solid).

From the right panel of Figure 2, we see that, under uniform enrollment,
scenario 1 is about 8 months faster than scenario 2. However, under non-
uniform enrollment, these two scenarios could have about the same predicted
study duration. Specifically, this happens when the enrollment is slower than
average ($\beta<1$) in the beginning for scenario 1 and the opposite
($\beta>1$) for scenario 2.

Figure 2: The enrollment distributions and their impacts on study duration
distributions. Scenario 1: enrollment period = $14$ months, median survival
$=10,20$ months for the placebo, treatment groups, respectively. Scenario 2:
enrollment period = $36$ months, median survival $=5,10$ months for the
placebo, treatment groups, respectively.

### 3.3. The Impacts of Patient Heterogeneity

In this section, as in Scenario 1, we assume the sample size $n=140$ with 1-1
treatment-placebo allocation, target number of events $d=88$, uniform
enrollment and exponential survival distribution. The hazard ratio of the
treatment is assumed to be $0.5$, i.e., if the overall median survival time of
the placebo group is $MST_{pbo}$, then the overall median survival time in the
treatment group $MST_{trt}=2*MST_{pbo}$.

We introduce patient heterogeneity by assuming the population consists of two
groups of patients: the biomarker positive and the biomarker negative. Then,
$MST_{trt}$ and $MST_{pbo}$ are from mixtures of biomarker positive and
negative patients. Next, we shall briefly describe how we define the median
survival time for each biomarker subgroup.

We may derive, for example, the relationship between $MST_{pbo}$ and
$MST_{pbo,neg}$ (the median survival of biomarker negative patients that
received placebo) as the following

$\displaystyle
qe^{-log(2)HR_{pos}\frac{MST_{pbo}}{MST_{pbo,neg}}}+(1-q)e^{-log(2)\frac{MST_{pbo}}{MST_{pbo,neg}}}=0.5,$

where $q$, $HR_{pos}$ are the prevalence and hazard ratio of the biomarker
positive patients, respectively. The biomarker effect is assumed to be
prognostic, i.e., $HR_{pos}$ is the same in the treatment and placebo groups.
Using the above equation, for given $q$ and $HR_{pos}$, we may solve for
$MST_{pbo,neg}$ numerically. Subsequently, we may calculate
$MST_{pbo,pos}=MST_{pbo,neg}/HR_{pos}$. Then, $MST_{trt,neg}$ and
$MST_{trt,pos}$ may be obtained in a similar fashion. To model the extent of
heterogeneity, we set the prevalence $q$ ranges from $10\%$ to $90\%$ and the
hazard ratio $HR_{pos}=1,2,...,5$.

The study duration for each pair of the prevalence and hazard ratio of the
biomarker is summarized in Figure 3. The black solid line is the benchmark
where the hazard ratio is $1$, i.e., no biomarker effect. Thus, the line is
perfectly horizontal regardless of the prevalence. When the biomarker hazard
ratio increases, the colored curves begin to show a concaved down behavior,
showing that the study duration increases when the biomarker effect is present
even if the median survival times are the same for every point along each
curve. Ignoring the biomarker subgroups will result in underestimating the
study duration of the traditional placebo-controlled, all-comers design by up
to 10% across different median survival times and enrollment rates that we
examined. The maximum discrepancy occurred when the prevalence is about $45\%$
and the hazard ratio, as expected, is the largest at $5$.

Figure 3: The impacts of patient heterogeneity on study duration. The
enrollment is assumed to be uniform. Left: median survival $MST_{pbo}=7.5$
months, enrollment rate $=7$ patients/month. Middle: median survival
$MST_{pbo}=10$ months, enrollment rate $=10$ patients/month. Right: median
survival $MST_{pbo}=15$ months, enrollment rate $=20$ patients/month.

### 3.4. Comparison between the All-comers Design and the Enrichment Design

One of the important applications of the proposed calculation method is to
compare the study duration of the all-comers design and the biomarker
enrichment design. Here, we keep the settings of the all-comers’ design the
same as in Section 3.3. For the enrichment design, the difference is that only
biomarker positive patients are enrolled over a longer period, $a/q>a$, where
$a$ is the enrollment period of the all-comers design and $q$ is the
prevalence of the biomarker positive. For each pair of prevalence and hazard
ratio of the biomarker, we calculate the difference of the study duration of
the all-comers design and the enrichment design. The results are summarized in
the heatmaps in Figure 4.

There are multiple interesting observations. First, there seems to exist a
decreasing function in the prevalence - hazard ratio space as the boundary
that separates the region (in red) where the enrichment is more time-efficient
and the opposite region (in blue). Second, the enrichment-favoring region is
larger when the enrollment rate is quicker and/or the median survival is
longer. For example, when the median survival is $15$ months and $20$ patients
may be enrolled per month, then an enrichment design based on a mild biomarker
effect, such as a hazard ratio of $2$ with $30\%$ prevalence, already has a
shorter duration than the all-comers design. Lastly, the white boundary seems
to have steep slope from the beginning, which means a decrease in the
prevalence has to be compensated by a larger increase in the hazard ratio.
Therefore, the prevalence seems to be a more important factor than the hazard
ratio when we are choosing the biomarker for enrichment.

Figure 4: The heatmap of the study duration difference between the all-comers
design and the enrichment design. Columns: the enrollment rate $=7,10,20$
patients per month. Rows: median survival time of the placebo group
$=7.5,10,15$ months. within each panel, x-axis: the prevalence of the
biomarker positive; y-axis: hazard ratio of the biomarker positive.

Additionally, we also considered the comparison in the space of enrollment
rate and median survival time. Figure A1 summarized the results in a similar
format, where we varied the enrollment rate from $7$ to $20$ patients per
month and the median survival time from $7.5$ to $15$ months in a finer grid
of values. The observations are similar: 1) There exists a boundary in terms
of the enrollment rate and median survival, where the enrichment is favorable
in the region above the boundary. For example, if the enrollment is fast and
median survival is long, then the enrichment design can save a significant
amount of time. 2) A biomarker with higher prevalence and stronger effect
makes the enrichment-favorable region larger.

##  4 Real Data Analysis

To illustrate the accuracy and practical utility of the proposed study
duration calculation method, we applied it to re-assess two publicly available
clinical trial datasets. In a hypothetical randomized, double-blind, placebo-
controlled trial setting, we consider two designs: one is a biomarker
enrichment design where patients are included only if they belong to a certain
biomarker positive group, while the traditional all-comers design does not
impose this inclusion criteria. Next, for a given sample size $n$ and, a given
number, $d$, of events, we briefly describe the procedures of the re-
assessment.

  1. 1.

To extract the _actual_ study duration, we choose, from the real clinical
data, the first $n$ patients in the biomarker positive population (or all-
comers population) as our hypothetical trial dataset in the enrichment (all-
comers) design. Then, we record the $d$th event date of the hypothetical trial
dataset. The difference between the first patient enrollment date and the
$d$th event date is the trial duration.

  2. 2.

To _calculate_ the study duration, we estimate the survival curves of each
subgroups (biomarker-by-treatment) using the hypothetical trial dataset by
assuming a Weibull distribution. The enrollment distribution is estimated
using the same data by assuming a beta distribution in formula 6.

The purpose of this application is to show that under reasonable
distributional assumptions and the availability of representative historical
data, the proposed model may indeed calculate the study duration accurately.

### 4.1. Example 1: _adtte_ Data from the R _visR_ Package [13]

The _adtte_ data is a legacy data used in the CDISC SDTM/ADaM Pilot Project
[14] that was provided by Eli Lilly and Company from a phase II clinical trial
of the safety and efficacy of the Xanomeline Transdermal Therapeutic System
(TTS) in patients with mild to moderate Alzheimer’s disease.

For the purpose of this paper, we are interested in the time to first
dermatologic adverse event. We selected the lower Xanomeline dose for the
treatment group, resulting in a dataset of 170 patients (88 in the treatment
group, 82 in the placebo group). Although not statistically significant, the
hazard ratio for male patients compared to females is 1.39. For illustrative
purposes, we selected male as the biomarker for our enrichment design, taking
into account the prevalence of $39.4\%$. The hypothetical trial dataset will
have a sample size of 67, which is the number of male patients in the _adtte_
after filtering the high dose treatment.

The results are summarized in the left panel of Figure 5. The black solid
curve is universally higher than the red solid curve, meaning for any given
number of events, the all-comer design will always have shorter duration than
the enrichment design. On the other hand, given any desired duration, the all-
comer design is always able to achieve a higher number of events, which could
translate to higher power. The biomarker enrichment result should not be
surprising since the biomarker effect (HR=1.39) is not strong enough to
compensate the difficulty of recruiting biomarker positive (male only)
patients. Another important observation is that the calculated duration
(dashed) from the proposed model is close to the actual duration, which shows
the accuracy of the calculation algorithm.

### 4.2. Example 2: _udca_ Data from the R _survival_ Package [15]

The _udca_ data came from a double-blind, placebo-controlled, randomized trial
of ursodeoxycholic acid (UDCA) in patients with primary biliary cirrohosis
[16, 17]. The primary endpoint is the time to treatment failure, which is
defined as death, liver transplantation, histological progression or other
conditions listed in [16]. The baseline bilirubin level was found to be
strongly associated with the time to treatment failure. In this example, we
define the biomarker positive patients as those with baseline bilirubin > 1
mg/dL. The resulting biomarker positive subgroup has a hazard ratio of $2.59$
and prevalence $50\%$. Similar to Example 1, we limit the sample size to 84,
which is the total number of biomarker positive patients that are available in
_udca_.

From the right panel of Figure 5, we may see that the all-comers design has a
relatively shorter duration when the required number of events is low, e.g.,
$d<30$. If the trial targets more than 30 events, however, the enrichment will
save as much as 10 months. The calculated durations reflect the transition,
starting from $d\approx 20$, of the enrichment design being more time-
efficient as the required number of events increases. The overlapping
calculated curves and actual curves also show the potential utility of the
proposed algorithm to inform study planning.

Figure 5: The study duration of the hypothetical all-comers and enrichment
designs. Left: _adtte_ data from R _visR_ package, sample size $n=67$. Right:
_udca_ data from R _survival_ package, sample size $n=84$.

##  5 Discussion

The proposed model is a generalization of the order statistic model in [5]. We
provided a theoretically rigorous foundation for the derivation of the CDF of
study duration $T_{(d)}$. Our model is more general by considering the patient
heterogeneity induced by the biomarker effect. We also proposed an algorithm
based on simulation that covers a a wide variety of survival distribution and
enrollment assumptions. The empirical results show a more robust performance
compared to the reference. Several interesting examples and real datasets are
also examined to show the broad application potential of the proposed method,
such as the impact of non-uniform enrollment on the study duration and the
comparison between the enrichment design and traditional all-comers design.

Although we mainly focused on calculating study duration, the proposed
algorithm may be used to answer different questions, such as, given a target
duration, how many samples do we need; what is the minimum enrollment rate; or
what are the desirable biomarker characteristics to achieve a certain amount
of duration reduction by adopting an enrichment design? This is because our
proposed model connects the study duration with other parameters of the
disease, trial and patient population. Numerically, the proposed algorithm may
be used to search for the parameter values, e.g., via bisection search, that
achieve desirable study duration conditions.

One of the major motivations of this research is to quantify the benefits of
biomarker guided designs to improve trial efficiency. By integrating
innovative biomarker-guided strategies, clinical trials can become more
efficient, reducing costs and timelines, and ultimately accelerating the
availability of new treatments for patients in need. As mentioned in the
Introduction, many such innovative designs have a component of biomarker
enrichment. By accurately predicting the (dis)advantages of this enrichment
component relative to the traditional all-comers component, we are able to
select the most appropriate design under reasonable assumptions and historical
data.

An example of the innovative biomarker strategies is the utilization of
circulating tumor DNA (ctDNA), which holds significant promise in expediting
clinical trials across various cancer types. CtDNA, originating from tumor
cells, provides a non-invasive means to monitor the molecular changes within a
patient’s tumor. Pre-treatment ctDNA variant allele frequency (VAF) has been
found to be a prognostic biomarker across many cancer types [18, 19]. A higher
VAF often suggests higher tumor burden, therefore a higher chance of
progression or death. The enrichment of patients with high baseline ctDNA may
allow for faster readout of trials and make the efficacious treatment options
available quicker. As the field of ctDNA analysis continues to advance, its
potential to guide clinical trial design is becoming increasingly evident. Our
proposed method will help with this process by providing statistically
credible and quantifiable evidence to support innovative trial designs.

Lastly, although the proposed model may accommodate both prognostic and
predictive biomarkers, most of our discussions focus on prognostic ones where
the treatment effect are similar between the patient subgroups. Many
innovative biomarker-guided trial designs, however, focused on predictive
biomarker where a differential treatment effect is assumed for the biomarker
positive and negative groups. In reality, a biomarker may be both prognostic
and predictive. For example, high risk patients defined by a prognostic marker
may have better or worse response to treatment than low risk patients. A
future research direction is to explore the impact of the additional
parameters of the effect of predictive biomarkers on the study duration
comparison under the proposed framework.

##  6 Acknowledgements

The authors would like to thank Jean-Francois Martini and Douglas Laird for
helpful discussions.

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## Appendix

### A1. Derivation of Equation (2)

$\displaystyle F_{T_{kl}}(t)=P(T_{kl}\leq t)$ $\displaystyle=P(T_{kl}\leq
t,V_{kl}\leq W_{kl})+P(T_{kl}\leq t,V_{kl}>W_{kl})$
$\displaystyle=P(U_{kl}+V_{kl}\leq t,V_{kl}\leq W_{kl})+P(+\infty\leq
t,V_{kl}>W_{kl})$ $\displaystyle=P(U_{kl}+V_{kl}\leq t,V_{kl}\leq W_{kl})$
$\displaystyle=\int_{0}^{\min\\{t,a\\}}P(V_{kl}\leq t-u,V_{kl}\leq
W_{kl})f_{U_{kl}}(u)du$
$\displaystyle=\int_{0}^{\min\\{t,a\\}}\int_{0}^{t-u}\int_{v}^{+\infty}f_{W_{kl}}(w)f_{V_{kl}}(v)f_{U_{kl}}(u)dwdvdu$

### A2. Derivation of Equation (7)

$\displaystyle F_{T_{kl}}(t)=$
$\displaystyle\int_{0}^{\min\\{t,a\\}}\int_{0}^{t-u}\int_{v}^{+\infty}f_{W_{kl}}(w)f_{V_{kl}}(v)f_{U_{kl}}(u)dwdvdu$
$\displaystyle=$
$\displaystyle\int_{0}^{\min\\{t,a\\}}\left(\int_{0}^{t-u}e^{-\lambda_{W}v}\lambda_{V}e^{-\lambda_{V}v}dv\right)\frac{\beta}{a}\left(1-\frac{u}{a}\right)^{\beta-1}du$
$\displaystyle=$
$\displaystyle\int_{0}^{\min\\{t,a\\}}\frac{\lambda_{V}}{\lambda_{V}+\lambda_{W}}\left(1-e^{-(\lambda_{V}+\lambda_{W})(t-u)}\right)\frac{\beta}{a}\left(1-\frac{u}{a}\right)^{\beta-1}du$
$\displaystyle=$
$\displaystyle\frac{\lambda_{V}}{a^{\beta}(\lambda_{V}+\lambda_{W})}\int_{\max\\{0,a-t\\}}^{a}\left(1-e^{-(\lambda_{V}+\lambda_{W})(x+t-a)}\right)\beta
x^{\beta-1}dx$ $\displaystyle=$
$\displaystyle\frac{\lambda_{V}}{\lambda_{V}+\lambda_{W}}\left(1-\max\\{0,1-t/a\\}^{\beta}-e^{-(\lambda_{V}+\lambda_{W})(t-a)}\frac{\beta\Gamma(\beta)}{(a(\lambda_{V}+\lambda_{W}))^{\beta}}\left(F_{G}(a)-F_{G}(\max(0,a-t))\right)\right)$

### A3. Derivation of Equation (11)

$\displaystyle F_{T_{E}}(t)-F_{T_{A}}(t)$ $\displaystyle=$
$\displaystyle\max\\{0,1-\frac{t}{a}\\}-\max\\{0,1-r_{11}\frac{t}{a}\\}+\frac{r_{11}e^{-\lambda_{11}t}}{a\lambda_{11}}(e^{\lambda_{11}\min(a,t)}-e^{\lambda_{11}\min(\frac{a}{r_{11}},t)})+\frac{r_{21}e^{-\lambda_{21}t}}{a\lambda_{21}}\left(e^{\lambda_{21}\min(a,t)}-1\right)$
$\displaystyle=$
$\displaystyle\begin{cases}-(1-r_{11})\frac{t}{a}+0+\frac{r_{21}e^{-\lambda_{21}t}}{a\lambda_{21}}\left(e^{\lambda_{21}t}-1\right),&0<t<a\\\
-(1-r_{11}\frac{t}{a})+\frac{r_{11}e^{-\lambda_{11}t}}{a\lambda_{11}}\left(e^{\lambda_{11}a}-e^{\lambda_{11}t}\right)+\frac{r_{21}e^{-\lambda_{21}t}}{a\lambda_{21}}\left(e^{\lambda_{21}a}-1\right),&a<t<a/r_{11}\\\
0+\frac{r_{11}e^{-\lambda_{11}t}}{a\lambda_{11}}\left(e^{\lambda_{11}a}-e^{\lambda_{11}a/r_{11}}\right)+\frac{r_{21}e^{-\lambda_{21}t}}{a\lambda_{21}}\left(e^{\lambda_{21}a}-1\right),&t>a/r_{11}\end{cases}$
$\displaystyle=$
$\displaystyle\begin{cases}\frac{1-r_{11}}{a}\left(\frac{1-e^{-\lambda_{21}t}}{\lambda_{21}}-t\right),&0<t<a\\\
-(1-r_{11})-\frac{r_{11}}{a}\left(\frac{1-e^{-\lambda_{11}(t-a)}}{\lambda_{11}}-(t-a)\right)+\frac{1-r_{11}}{a}\left(\frac{e^{-\lambda_{21}(t-a)}-e^{-\lambda_{21}t}}{\lambda_{21}}\right),&a<t<a/r_{11}\\\
\frac{r_{11}}{a}\left(\frac{e^{-\lambda_{11}(t-a)}-e^{-\lambda_{11}(t-a/r_{11})}}{\lambda_{11}}\right)+\frac{1-r_{11}}{a}\left(\frac{e^{-\lambda_{21}(t-a)}-e^{-\lambda_{21}t}}{\lambda_{21}}\right),&t>a/r_{11}\end{cases}$

### A4. Some Inequalities

  1. 1.

If $b>0$, $x\geq 0$, then $\frac{1-e^{-bx}}{b}-x<0$.

  2. 2.

Define $f(x)=\frac{e^{-bx}-e^{-cx}}{x}$, $x\geq 0$.
If $b>c>0$, then $f(x)$ is a monotone increasing function, $c-b<f(x)<0$.
If $c>b>0$, then $f(x)$ is a monotone decreasing function, $0<f(x)<c-b$.

### A5. Supplement Figures

Figure A1: The heatmap of the study duration difference between the all-comers
design and the enrichment design. Columns: the hazard ratio of the biomarker
positive $=2,3,4$. Rows: the prevalence of the biomarker positive
$=0.25,0.33,0.5$. within each panel, x-axis: the enrollment rate of the all-
comers trial; y-axis: median survival time of the placebo group.