# Solving the Wheeler-DeWitt of Small Universe

Shintaro Sawayama<EMAIL_ADDRESS>Sawayama Cram School of Physics
Atsuhara 328, Fuji-city, Shizuoka prefecture 419-0201, Japan

###### Abstract

We can solve the Wheeler-DeWitt equation of the small universe enough to
metric becomes diagonal and take a Gaussian normal coordinate. Our previous
works are concerning to this paper. In this paper, we only write how to solve
the Wheeler-DeWitt equation of such universe. Our motivation is simple, that
is to solve the Wheeler-DeWitt equation. Even if the Wheeler-DeWitt equation
is solved, quantum gravity does not complete yet. However, this work may be
one of the first step to quantum gravity.

###### pacs:

04.60.-m, 04.60.Ds

##  I Introduction

In the quantum gravity, there are many approaches, for example loop quantum
gravityAs Rov Thi  or mini-superspaceHart  approachs or string approachesAL .
However, quantum gravity has not completed yet. In the canonical quantum
gravity, the difficulties comes from the Wheeler-DeWitt equation. At first we
should solve the Wheeler-DeWittDe  equation the quantum gravity does not
start.

In our previous work is concerning the Wheeler-DeWitt equation. Our motivation
is simple to solve the Wheeler-DeWitt equation. However, Wheeler-DeWitt is
difficult to solve. Because it is second order elliptic functional partial
differential equation with non-linear term. The elliptic differential equation
is difficult, but if we choose diagonal metric, this problem is solved. The
partial differential equation is difficult, but if we use additional
constraint equation, this problem is solved. The functional differential
equation is difficult, but we construct how to solve the functional
differential equation. The non-linear term is difficult, but we construct the
method to remove this difficulty.

We quantize diagonal metric universe by using Wheeler-DeWitt without
approximation. We use the fact metric become diagonal by coordinate
transformation and Gaussian normal coordinate. Such universe is small
universe. Or we treat the Wheeler-DeWitt equation locally. Because gravity is
quantize in very small universe. To treat such small universe is theoretical.

In section II, we simplify the Wheeler-DeWitt equation. In section III we
quantize toy model. And in this section we show the step to solve the Wheeler-
DeWitt. In section IV we quantize full quantum gravity, using the obtained
result in section III. In section V we summarize and conclude obtained result.

##  II Simplification of the Wheeler-DeWitt

For simplicity we treat the only diagonal metric universes as,

$\displaystyle\begin{pmatrix}g_{00}&0&0&0\\\ 0&g_{11}&0&0\\\ 0&0&g_{22}&0\\\
0&0&0&g_{33}.\end{pmatrix}$ (1)

And in all the spacetime metrics become diagonal locally. Or we treat small
universe enough to metric become diagonal. We start from decomposition of the
Einstein Hilbert action of the above diagonal metric universe. The action of
the above universe is written by

$\displaystyle S=\int RdM=\int R[g_{\mu\mu}]dSdt.$ (2)

Here $S$ is the hyper-surface with constant time. Because, our method is
different from the usual Wheeler-DeWitt equation formalism, our obtained
Hamiltonian constraint is a different type of the Wheeler-DeWitt equation. If
we decompose this action as 3+1, then we can obtain

$\displaystyle{\cal L}=\dot{q}_{ii}P^{ii}+NH-2\sqrt{q}D^{i}N_{,i}.$ (3)

Here $N$ is the lapse functional and $N_{,i}$ is the sift vectors, and $H$ is
the Hamiltonian constraint such that

$\displaystyle H=\frac{1}{2}q_{ii}q_{jj}P^{ii}P^{jj}+{\cal R}.$ (4)

Here ${\cal R}$ is the three dimensional Ricci scalar and $P^{ii}$ is the
momentum whose commutation relation with $q_{ii}$ is not $i$, it is
$i\sqrt{q}$. In this formulation there are not appear $q_{ij}$ and $P^{ij}$
and sift vectors and momentum constraint. So we can ignore the constraint as
$[P^{ij},H]$ or $[[P^{ij},H],H]$, because we start with metric diagonal
setting. In this simple case we can ignore the diffeomorphism constraints. If
we write the Hamiltonian constraint in the operator representation, we obtain

$\displaystyle
H=\sum_{ij}\frac{1}{2}\frac{\delta^{2}}{\delta\phi_{i}\delta\phi_{j}}+{\cal
R}[q_{11},q_{22},q_{33}]=0$
$\displaystyle=\sum_{ij}\frac{1}{2}\frac{\delta^{2}}{\delta\phi_{i}\delta\phi_{j}}+\sum_{i\not=j}(\hat{\phi}_{i,jj}+\hat{\phi}_{j,i}\hat{\phi}_{i,i})e^{\hat{\phi}_{i}}=0.$
(5)

Here $\phi_{i}=\ln q_{ii}$. If we consider the $\phi$ were only depend
$t,x_{i}$, the Hamiltonian constraint becomes,

$\displaystyle
H=\sum_{ij}\frac{1}{2}\frac{\delta^{2}}{\delta\phi_{i}\delta\phi_{j}}-\Lambda=0.$
(6)

We use this Hamiltonian constraint in section III Because we only treat metric
diagonal universe, the Hamiltonian constraint has different form from the
orthodox Hamiltonian constraint. This setting is similar to mini-superspace
model.

If universe is small enough to apply Gaussian normal coordinate, the
Hamiltonian constraint become

$\displaystyle
H=\sum_{ij}\frac{1}{2}\frac{\delta^{2}}{\delta\phi_{i}\delta\phi_{j}}+\sum_{i\not=j}\hat{\phi}_{i,jj}e^{\phi_{i}}=0$
(7)

We treat this Hamiltonian constraint in section IV

The static restriction deviated from up-to-down methodSa1  is

$\displaystyle\sum_{i\not=j}\frac{\delta}{\delta\phi_{i}\delta\phi_{j}}.$ (8)

##  III Simple example of Wheeler-DeWitt

Before solving Eq.(7), we solve Eq.(6) for simplicity. We consider the
following spacetime

$\displaystyle\begin{pmatrix}-N^{2}&0&0&0\\\ 0&g_{1}(t,x)&0&0\\\
0&0&g_{2}(t,y)&0\\\ 0&0&0&g_{3}(t,z)\end{pmatrix}.$ (9)

To quantize this spacetime we take steps. The step 1 is using the static
restriction we obtain special solution. The step 2 is to remove the static
restriction we obtain general solution. The we carry out step 1. The Eq.(6)
and Eq.(8) is consistent and simultaneously quantized. And the solution is

$\displaystyle
f[\phi_{1},\phi_{2},\phi_{3}]=\prod_{i}\exp(a_{i}\Lambda^{1/2}\int\delta\phi_{i})$
(10)

where

$\displaystyle a_{1}a_{2}+a_{1}a_{3}+a_{2}a_{3}=1$ (11) $\displaystyle
a_{1}^{2}+a_{2}^{2}+a_{3}^{2}=1$ (12)

Then we carry out step 2. Using the above solution as a special solution, we
assume the state is a form

$\displaystyle|\Psi\rangle=f[\phi_{i}]g[\phi_{i}].$ (13)

And we remove the static restriction, then $g[\phi_{i}]$ should satisfy

$\displaystyle\nabla(\nabla+a)g[\phi_{i}]=0$ (14)

Here, $\nabla$ is defined by

$\displaystyle\nabla=\sum_{i}\frac{\delta}{\delta\phi_{i}}.$ (15)

And $a$ is defined by $a=\Lambda^{1/2}\sum_{i}a_{i}$ This equation can be
solved easily and solution is

$\displaystyle
e^{-a\int\delta\phi_{1}}(-2\int\delta\phi_{1}+\int\delta\phi_{2}+\int\delta\phi_{3})+e^{-a\int\delta\phi_{2}}(-2\int\delta\phi_{2}+\int\delta\phi_{1}+\int\delta\phi_{3})$
$\displaystyle+e^{-a\int\delta\phi_{3}}(-2\int\delta\phi_{3}+\int\delta\phi_{2}+\int\delta\phi_{1})$
(16)

So the quantum state of (9) is

$\displaystyle|\Psi(\phi_{i})\rangle=\prod_{i}\exp(a_{i}\Lambda^{1/2}\int\delta\phi_{i})\bigg{[}e^{-a\int\delta\phi_{1}}(-2\int\delta\phi_{1}+\int\delta\phi_{2}+\int\delta\phi_{3})$
$\displaystyle+e^{-a\int\delta\phi_{2}}(-2\int\delta\phi_{2}+\int\delta\phi_{1}+\int\delta\phi_{3})+e^{-a\int\delta\phi_{3}}(-2\int\delta\phi_{3}+\int\delta\phi_{1}+\int\delta\phi_{2})\bigg{]}$
(17)

##  IV Solving the Wheeler-DeWitt of small universe

By the same method we can quantize following universe

$\displaystyle\begin{pmatrix}-N^{2}&0&0&0\\\ 0&g_{1}(t,x,y,z)&0&0\\\
0&0&g_{2}(t,x,y,z)&0\\\ 0&0&0&g_{3}(t,x,y,z)\end{pmatrix}$ (18)

Here $x,y,z$ are Gaussian normal coordinates. The step 1 is to use a static
restriction to the Eq.(7), then we obtain

$\displaystyle\sum_{i}\frac{\delta^{2}}{\delta\phi_{i}^{2}}+2\phi_{i,jj}e^{\phi_{i}}=0.$
(19)

Then we use parameter separation, i.e. the solution of the above equation is
assumed to be written as

$\displaystyle f[\phi_{i}]=f_{1}[\phi_{1}]f_{2}[\phi_{2}]f_{3}[\phi_{3}]$ (20)

Then Eq. (19) becomes

$\displaystyle\frac{\delta^{2}}{\delta\phi_{i}^{2}}+2\phi_{i,jj}e^{\phi_{i}}=0$
(21)

Or,

$\displaystyle\frac{\delta^{2}}{\delta a_{i}^{2}}+8\partial_{j}\partial^{j}\ln
a_{i}=0.$ (22)

Here $a_{i}=g_{i}^{1/2}$. Then we use a following equation

$\displaystyle-i(\ln a_{i,jj})^{1/2}\frac{\delta}{\delta
a_{i}}+i\frac{\delta}{\delta a_{i}}(\ln a_{i,jj})^{1/2}=i\delta\frac{1}{2}(\ln
a_{i,jj})^{-1/2}a_{i,jj}^{-1}$ (23)

Now we briefly write $\partial_{j}\partial^{j}\ln a_{i}=\ln a_{i,jj}$. Off
course it is diferent, we simplisitily use the latter. Then Eq.(22) becomes

$\displaystyle\frac{\delta^{2}}{\delta a_{i}^{2}}+2\sqrt{2}i(\ln
a_{i,jj})^{1/2}\frac{\delta}{\delta a_{i}}-2\sqrt{2}i\frac{\delta}{\delta
a_{i}}(\ln a_{i,jj})^{1/2}+8\ln a_{i,jj}=i\sqrt{2}\delta(\ln
a_{i,jj})^{-1/2}a_{i,jj}^{-1}$ (24)

Or

$\displaystyle\bigg{(}\frac{\delta}{\delta a_{i}}+2\sqrt{2}i(\ln
a_{i,jj})^{1/2}\bigg{)}\bigg{(}\frac{\delta}{\delta a_{i}}-2\sqrt{2}i(\ln
a_{i,jj})^{1/2}\bigg{)}=\sqrt{2}i\delta(\ln a_{i,jj})^{-1/2}a_{i,jj}^{-1}.$
(25)

Then we take following assumptions

$\displaystyle\frac{\delta}{\delta a_{i}}+2\sqrt{2}i(\ln
a_{i,jj})^{1/2}=g_{1}[a_{i}]$ (26) $\displaystyle\frac{\delta}{\delta
a_{i}}-2\sqrt{2}i(\ln a_{i,jj})^{1/2}=g_{2}[a_{i}].$ (27)

Here,

$\displaystyle g_{1}[a_{i}]g_{2}[a_{i}]=i\delta\sqrt{2}(\ln
a_{i,jj})^{-1/2}a_{i,jj}^{-1}$ (28)

Then second partial functional derivative become ordinal functional derivative
and the equation become following

$\displaystyle\frac{\delta f^{1/2}[a_{i}]}{\delta a_{i}}+2\sqrt{2}i(\ln
a_{i,jj})^{1/2}f^{1/2}[a_{i}]=g_{1}[a_{i}]f^{1/2}[a_{i}]$ (29)
$\displaystyle\frac{\delta f^{1/2}[a_{i}]}{\delta a_{i}}-2\sqrt{2}i(\ln
a_{i,jj})^{1/2}f^{1/2}[a_{i}]=g_{2}[a_{i}]f^{1/2}[a_{i}].$ (30)

From this equation we obtain

$\displaystyle\frac{1}{2}\ln f[a_{i}]=\int g_{1}[a_{i}]+2\sqrt{2}i(\ln
a_{i,jj})^{1/2}\delta a_{i}$ (31) $\displaystyle\frac{1}{2}\ln f[a_{i}]=\int
g_{2}[a_{i}]-2\sqrt{2}i(\ln a_{i,jj})^{1/2}\delta a_{i}$ (32)

We can obtain the solution

$\displaystyle f[a_{i}]=\exp\bigg{(}2\int g_{1}[a_{i}]+2\sqrt{2}i(\ln
a_{i,jj})^{1/2}\delta a_{i}\bigg{)}$ (33) $\displaystyle
f[a_{i}]=\exp\bigg{(}2\int g_{2}[a_{i}]-2\sqrt{2}i(\ln a_{i,jj})^{1/2}\delta
a_{i}\bigg{)}$ (34)

Because $f[a_{i}]$ is same

$\displaystyle g_{1}[a_{i}]=g_{2}[a_{i}]-4\sqrt{2}(\ln a_{i,jj})^{1/2}.$ (35)

Inserting this equation to Eq.(28), we obtain

$\displaystyle g_{2}[a_{i}]^{2}-4\sqrt{2}(\ln
a_{i,jj})^{1/2}g_{2}[a_{i}]=i\delta\sqrt{2}(\ln
a_{i,jj})^{-1/2}a_{i,jj}^{-1}.$ (36)

Exchangeng this equation, we obtain

$\displaystyle g_{2}[a_{i}]^{2}-4\sqrt{2}(\ln
a_{i,jj})^{1/2}g_{2}[a_{i}]-i\delta\sqrt{2}(\ln
a_{i,jj})^{-1/2}a_{i,jj}^{-1}=0.$ (37)

By solving this second order function equation, we obtain

$\displaystyle g_{2}[a_{i}]=2\sqrt{2}(\ln a_{i,jj})^{1/2}\pm\sqrt{8\ln
a_{i,jj}+i\delta\sqrt{2}(\ln a_{i,jj})^{-1/2}a_{i,jj}^{-1}}.$ (38)

Inserting this equation to Eq.(34), we obtain

$\displaystyle f[a_{i}]=\exp\bigg{(}\int\pm 2\sqrt{8\ln
a_{i,jj}+i\delta\sqrt{2}(\ln a_{i,jj})^{-1/2}a_{i,jj}^{-1}}\delta
a_{i}\bigg{)}.$ (39)

So the quantization of the spacetime as (18) is

$\displaystyle f[a_{1},a_{2},a_{3}]=\prod_{i}\exp\bigg{(}\int\pm 2\sqrt{8\ln
a_{i,jj}+i\delta\sqrt{2}(\ln a_{i,jj})^{1/2}a_{i,jj}^{-1}}\delta
a_{i}\bigg{)}.$ (40)

This is the special solution of quantization of the spacetime (18).

Then we carry out step 2, we remove the static restriction and we assume the
state is the form of the

$\displaystyle|\Psi\rangle=f[a_{i}]h[a_{i}]$ (41)

Then we obtain

$\displaystyle\nabla(\nabla+h^{\prime}[a_{i}])h[a_{i}]=0.$ (42)

Here,

$\displaystyle h^{\prime}[a_{i}]=\sum_{i}\pm\sqrt{8\ln
a_{i,jj}+i\delta\sqrt{2}(\ln a_{i,jj})^{1/2}a_{i,jj}^{-1}}$ (43)

The functional partial derivative

$\displaystyle(\nabla+h^{\prime}[a_{i}])h[a_{i}]=0$ (44)

is solved analytically. The solution is

$\displaystyle
h[a_{i}]=\exp\bigg{(}\int^{a_{1}}h^{\prime}[a_{1}^{\prime},-a_{1}+a_{2}+a_{1}^{\prime},-a_{1}+a_{3}+a_{1}^{\prime}]\delta
a_{1}^{\prime}\bigg{)}(-2\int\delta a_{1}+\int\delta a_{2}+\int\delta a_{3})$
$\displaystyle+\exp\bigg{(}\int^{a_{2}}h^{\prime}[a_{1}-a_{2}+a_{2}^{\prime},a_{2}^{\prime},-a_{2}+a_{3}+a_{2}^{\prime}]\delta
a_{2}^{\prime}\bigg{)}(-2\int\delta a_{2}+\int\delta a_{1}+\int\delta a_{3})$
$\displaystyle+\exp\bigg{(}\int^{a_{3}}h^{\prime}[-a_{3}+a_{1}+a_{3}^{\prime},-a_{3}+a_{2}+a_{3}^{\prime},a_{3}^{\prime}]\delta
a_{3}^{\prime}\bigg{)}(-2\int\delta a_{3}+\int\delta a_{1}+\int\delta a_{2})$
(45)

So the quantum state of spacetime (18) is

$\displaystyle|\Psi\rangle=\prod_{i}\exp\bigg{(}\int\pm 2\sqrt{8\ln
a_{i,jj}+i\delta\sqrt{2}(\ln a_{i,jj})^{-1/2}a_{i,jj}^{-1}}\delta
a_{i}\bigg{)}$
$\displaystyle\times\bigg{[}\exp\bigg{(}\int^{a_{1}}h^{\prime}[a_{1}^{\prime},-a_{1}+a_{2}+a_{1}^{\prime},-a_{1}+a_{3}+a_{1}^{\prime}]\delta
a_{1}^{\prime}\bigg{)}(-2\int\delta a_{1}+\int\delta a_{2}+\int\delta a_{3})$
$\displaystyle+\exp\bigg{(}\int^{a_{2}}h^{\prime}[a_{1}-a_{2}+a_{2}^{\prime},a_{2}^{\prime},-a_{2}+a_{3}+a_{2}^{\prime}]\delta
a_{2}^{\prime}\bigg{)}(-2\int\delta a_{2}+\int\delta a_{1}+\int\delta a_{3})$
$\displaystyle+\exp\bigg{(}\int^{a_{3}}h^{\prime}[-a_{3}+a_{1}+a_{3}^{\prime},-a_{3}+a_{2}+a_{3}^{\prime},a_{3}^{\prime}]\delta
a_{3}^{\prime}\bigg{)}(-2\int\delta a_{3}+\int\delta a_{1}+\int\delta
a_{2})\bigg{]}$ (46)

This is the main result of our paper.

##  V Conclusion and Discussions

We quantized diagonal metric space with Gaussian normal coordinate. By solving
the Wheeler-DeWitt equation, we know the form of the solution. We quantize two
universe. The common feature is similarity of step 2. The step 2 is all ways
solved. So the important point is to find the special solution.

There are many further work. Once we should calculate the inner product and
norm. If the normalization is end the state is used to calculate the averaged
value. Or we should discuss problem of the norm. This open issue is very
important one. And we should search the initial singularity or the black hole
singularity. Our work does not end.

## References

  * (1) C.J.Isham and A.Ashtekar Class. Quant. Grav. 9 1433 (1992)
  * (2) C.Rovelli Quantum Gravity; Cambridge monographs on mathematical physics (2004)
  * (3) T.Thiemann gr-qc/0110034 (2001)
  * (4) A.Linde hep-th/0503195 (2005)
  * (5) S.Sawayama gr-qc/0604007 (2006)
  * (6) B.S.DeWitt Phys. Rev. 160 1113 (1967)
  * (7) J.J.Halliwell and J.B.Hartle Phys. Rev. D 43 1170 (1991)


# SAM on Medical Images: A Comprehensive Study on Three Prompt Modes
††thanks: This study was supported by National Key Research and Development
Program of China (2020YFB1711500, 2020YFB1711503), and the 1·3·5 project for
disciplines of excellence, West China Hospital, Sichuan University
(ZYYC21004).

Dongjie Cheng2, Ziyuan Qin2, Zekun Jiang2, Shaoting Zhang3, Qicheng Lao47,
Kang Li27  2West China Biomedical Big Data Center,West China Hospital, Sichuan
University, Chengdu, China  3Shanghai Artificial Intelligence Laboratory,
Shanghai, China  4School of Artificial Intelligence, BUPT, Beijing, China
7Corresponding Author: Qicheng Lao, Kang Li Email<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The Segment Anything Model (SAM) made an eye-catching debut recently and
inspired many researchers to explore its potential and limitation in terms of
zero-shot generalization capability. As the first promptable foundation model
for segmentation tasks, it was trained on a large dataset with an
unprecedented number of images and annotations. This large-scale dataset and
its promptable nature endow the model with strong zero-shot generalization.
Although the SAM has shown competitive performance on several datasets, we
still want to investigate its zero-shot generalization on medical images. As
we know, the acquisition of medical image annotation usually requires a lot of
effort from professional practitioners. Therefore, if there exists a
foundation model that can give high-quality mask prediction simply based on a
few point prompts, this model will undoubtedly become the game changer for
medical image analysis. To evaluate whether SAM has the potential to become
the foundation model for medical image segmentation tasks, we collected more
than 12 public medical image datasets that cover various organs and
modalities. We also explore what kind of prompt can lead to the best zero-shot
performance with different modalities. Furthermore, we find that a pattern
shows that the perturbation of the box size will significantly change the
prediction accuracy. Finally, Extensive experiments show that the predicted
mask quality varied a lot among different datasets. And providing proper
prompts, such as bounding boxes, to the SAM will significantly increase its
performance.

##  I Introduction

Recently, large language models pre-trained with billions or even trillions of
parameters have become the most influential trends in AI. Large language
models (LLM) such as ChatGPT or LLaMA can be easily extended to different
tasks and scenarios without any further training or tuning. All you need is to
prompt the LLMs to activate the relevant knowledge. These amazing zero-shot
generalizations make the name ”foundation model” become well-known. However,
most of the breakthroughs in foundation models happened in either NLP fields
or cross-modality tasks such as image text retrieval. However, for many
computer vision tasks, such as segmentation, abundant training data does not
exist. Thus, the first segmentation foundation model, SAM [15], was born and
showed impressive zero-shot inference performance in different domains. On the
other hand, segmentation has long been an important topic in medical image
analysis, involving the segmentation of organs, abnormalities, tumors, and
others. Nevertheless, since domain discrepancy and modality differences exist
in medical image tasks, current solutions usually rely on designing a
specialized model architecture and training with significant data. Therefore,
we need a foundation model that only needs a few prompts to generalize to a
new domain or task in medical image analysis. However, acquiring web-scale
images and annotations is not realistic. So, in this work, we want to
investigate whether the SAM can be the foundation model for medical image
analysis segmentation tasks.

For this purpose, we split our experiments into 3 parts. Firstly, we evaluate
the SAM with auto-prompt mode, and conclude that SAM’s zero-shot
generalization in auto-prompt mode is not enough to compete with the baseline
models. We further explore the box-prompt mode and point-prompt mode of the
SAM model to investigate their zero-shot generalization. We conduct extensive
experiments to show that box-prompt mode achieves higher dice accuracy than
other prompt modes. We also observe a pattern that adding jitters to the boxes
can significantly affect prediction accuracy. For point-prompt mode, we
evaluate with single point, 3 point, and 10 points setting. We conclude that
with the increasing of prompt points, the zero-shot performance is approaching
that of the box-prompt mode.

Though SAM achieves competitive zero-shot generalization on many domains and
modalities when we give the model proper prompts, the overall performance is
still not as good as the trained model with the supervised method. Afterall,
SAM provides a reference for the development of medical foundation models,
which is worth exploring in more depth in future research.

Figure 1: Segmentation results with different prompt modes

##  II Related Works

It has been over three weeks since the publication of ”Segment Anything” [15],
and a search of SAM-related research on arVix now yields more than 20 preprint
articles. We anticipate that the number of research studies using SAM will
continue to grow in the coming period, as it has successfully captured the
attention of scholars in AI, computer vision (CV), image processing, and
various cross-disciplinary application fields.

###  II-A About Segment Anything

The emergence of LLM has led to a shift in the research goals of many AI
researchers toward the development of large or foundation models. This trend
has gained immense popularity and is currently one of the hottest topics in AI
research. One viewpoint in the current academic community is that emergence
capability can be achieved when the model parameters reach a sufficiently
large level, resulting in impressive intelligent processing capabilities. As a
result, many AI researchers are considering whether the same is true in the CV
domain and looking forward to developing CV foundation models. In this
context, segmenting anything becomes an imperative foundation model. Recently,
two major parallel works were opened, one is SAM [15] and the other is
”Segment Everything Everywhere All at Once” named SEEM [35]. During the past
two weeks, SAM had a greater degree of impact, but SEEM showed that it has
many aspects of performance that outperform SAM and also deserve our
attention, measurement, and secondary development.

###  II-B SAM in Medical Imaging

Overall, SAM has opened up the foundation models to facilitate zero-shot
processing and secondary development for various downstream tasks. In the
field of medical image processing, there are currently more than 10 preprint
papers. Seven of these works were focused solely on segmentation in specific
medical areas, including digital pathology images [7], multi-phase liver
tumors in CT [11], abdominal CT [25], brain MRI [22], brain tumors in BraTS
[23], polyps [34], and ophthalmology [24]. Meanwhile, three other works,
mainly from Harvard Medical School, Duke University, and the University of
Toronto, respectively, conducted more comprehensive evaluations of
segmentation comparisons across multiple organ sites and medical imaging
multi-modalities [10, 21, 19]. Of these, Ma et al [19] performed the most
extensively evaluated and provided a tutorial for fine-tuning SAM. Unlike
these three work, we focus on which prompt=mode is suitable for the SAM model
in medical image analysis. There is also a study that integrates SAM into 3D
Slicer [18], a widely-used open-source medical image processing software.
Additionally, a blog called ”SAM in Medical Imaging” provides a program to
directly process medical image formats using SAM.

###  II-C Other SAM Works

In other areas, there also has been a considerable amount of research work
based on SAM. There are approximately more than 10 studies, of which 2 have
evaluated SAM on a broader range of natural images [14, 30], while the other
works have improved upon SAM to varying degrees or used it for other tasks.
SAM-Adapter was developed for better segmentation [6]. Moreover, SAM was also
used for inpainting anything[33], 3D reconstruction[26], style
transformer[17], counting anything[20], and segmenting 3D anything[5], etc.

Figure 2: We show the pattern of dice changing with different level of jitter
over most of the datasets. TABLE I: Segmentation performance comparison in
different bbox jitters

Dataset | OCT | FUSC | BUSI | TN3k | TG3k | PolypBenchmark | X-ray | ADNI | LiTs-Liver | LiTs-Tumor
---|---|---|---|---|---|---|---|---|---|---
SAM bbox | 0.7812 | 0.8486 | 0.8421 | 0.7843 | 0.2179 | 0.9086 | 0.7813 | 0.2885 | 0.8667 | 0.7402
SAM bbox jitter=0.01 | 0.7807 | 0.8489 | 0.8424 | 0.7838 | 0.2177 | 0.9080 | 0.7815 | 0.2887 | 0.8665 | 0.7398
SAM bbox jitter=0.05 | 0.7804 | 0.8503 | 0.8401 | 0.7821 | 0.2220 | 0.9028 | 0.7653 | 0.2883 | 0.8642 | 0.7393
SAM bbox jitter=0.1 | 0.7808 | 0.8447 | 0.8391 | 0.7705 | 0.2364 | 0.9022 | 0.7181 | 0.2865 | 0.8465 | 0.7377
SAM bbox jitter=0.25 | 0.6941 | 0.7896 | 0.7585 | 0.6747 | 0.2446 | 0.8339 | 0.5228 | 0.2684 | 0.7255 | 0.6803
SAM bbox jitter=0.5 | 0.4255 | 0.5567 | 0.4700 | 0.4558 | 0.1328 | 0.5438 | 0.2924 | 0.2221 | 0.4378 | 0.4816
SOTA | 0.8447 | 0.8923 | 0.8577 | 0.8126 | / | 0.8946 | 0.9840 | / | 0.9630 | 0.6740

TABLE II: Segmentation performance comparison in different point prompts

Dataset | OCT | FUSC | BUSI | TN3k | TG3k | PolypBenchmark | X-ray | ADNI | LiTs-Liver | LiTs-Tumor
---|---|---|---|---|---|---|---|---|---|---
SAM Auto | 0.4318 | 0.7731 | 0.5312 | 0.4324 | 0.2318 | 0.6540 | 0.4749 | 0.0941 | 0.4641 | 0.1288
SAM midpt prompted | 0.4416 | 0.6852 | 0.6447 | 0.4884 | 0.1989 | 0.6572 | 0.3686 | 0.0320 | 0.4354 | 0.1442
SAM 3pts prompted | 0.4728 | 0.6708 | 0.6180 | 0.6008 | 0.2541 | 0.7686 | 0.5288 | 0.1014 | 0.6362 | 0.2613
SAM 10pts prompted | 0.6425 | 0.8453 | 0.8044 | 0.7745 | 0.4889 | 0.8560 | 0.7890 | 0.1303 | 0.8336 | 0.4231
SOTA | 0.8447 | 0.8923 | 0.8577 | 0.8126 | / | 0.8946 | 0.9840 | / | 0.9630 | 0.6740

Figure 3: Segmentation results with different prompt modes

##  III Methods

###  III-A Dataset

We collected 12 different public available datasets covering CT, X-ray, MRI,
Endoscopy, Ultrasound, and OCT. (1) BUSI: [1] collected at baseline include
breast ultrasound images among women in ages between 25 and 75 years old. (2)
CVC-300 [31], CVC-ClinicDB [2], CVC-ColonDB [29], Kvasir [13], and ETIS [27]:
These five datasets are part of the polyp region detection benchmark that was
used in PraNet [8]. (3) FUSC is from Foot Ulcer Segmentation Challenge 2021
[32], with the purpose to segment ulcer on the feet. (4) COVID-Lung is one of
the largest benchmark dataset designed for COVID-19 detection and
segmentation, with 33,920 CXR images, including 11,956 COVID-19 samples [28],
(5) ADNI is a MRI dataset initiated for Alzheimer’s Disease classification
[4], but it is also used for hippocampus segmentation purpose. (6) TN3K and
TG3K are from the same dataset but with different segmentation annotations.
This dataset containing more than three thousand ultrasound images with high-
quality annotation of thyroid nodule and thyroid gland respectively [9]. (7)
LiTs is a data set for the Liver and Liver Tumor Segmentation Benchmark (LiTS)
[3]. (8) OCT dataset is from a well-known Chinese AI challenge platform,
initiated by Chinese enterprise [12].

In the study, we used SAM to perform segmentation and dice computation
directly on the test sets of different datasets. Then, the segmentation
results were compared with the state-of-the-art (SOTA) results of supervised
learning from the related dataset studies. The relevant SOTA results can be
found in [32, 12, 16], etc.

###  III-B Applying SAM with different prompt modes

In this work, we test three different prompt modes and thoroughly evaluate
their performance on medical image segmentation tasks. As mentioned, the SAM
can run in three different modes: auto-prompt, point-prompt, and box-prompt.
For the auto-prompt mode, SAM will be automatically prompted with a regular
grid of points and predicate a set of masks for each point prompt. For box-
prompt mode, we first generate a bounding box according to the ground truth
for each object and input the curated bounding boxes as prompts to the SAM
with different scales of jitter. Finally, we also tried different settings for
point-prompt mode. Under the single-point prompt setting, we chose the center
point of the curated bounding box as our prompt. Unlike most of the evaluation
papers for the SAM model, we didn’t use the center point of ground truth mask
as a prompt because sometimes the center point of an irregular shape will drop
outside of the circumscribed area, and we think that will give the SAM model a
lousy prompt. However, for some non-contiguous masks, that center point can
still fall on the background are that may mislead the SAM model. This
conjecture also aligns with the observation mentioned in [21]. We also test
the point-prompt mode with 3 and 10 points settings.

No matter under which prompt setting, the zero-shot prediction accuracy of the
SAM model is still lower than that of the commonly used segmentation models
trained with fully supervised methods. Another topic worth diving into is how
to generate high-quality prompts, as prompts curated from ground truth mask is
no longer available in real-world usage.

###  III-C How to properly prompt the SAM with boxes and points

One interesting pattern we observed in the prediction accuracy is decreasing
with the increase of the jitter scale. We conjecture that the bounding box
scopes more background or unrelated areas after we add more perturbation and
jitter. Thus, we can also conclude that the SAM model’s prediction quality is
sensitive to the box size and its reliability. How to produce the reliable and
suitable size of box-prompts will be a possible research topic for applying
the SAM model to different domains. For the point-prompt mode, We found that
the predicted mask quality is rising and competing with the box-prompt
performance with the increase of given prompt points. This pattern is indeed
consistent with our intuition. The reason that the box-prompt mode beat the
few-point-prompt mode in our experiments could be complicated. One of the
contributing factors could be the uneven boundary of some abnormalities. For
those not well-circumscribed areas, several points sampled from the mask area
can not represent the irregular shape well.

##  IV Results

As shown in Table I, II, and Fig 1, the performance of SAM varies among
different datasets and prompt modes. Specifically, the box-prompt mode with
zero jitters is the best way to utilize the SAM model on medical image tasks.
The box-prompt mode achieves the highest average dice compared with the other
two modes. This prompt method also occasionally gives results that are close
to the SOTA methods. Thus we argue that box-prompt mode might be the most
suitable prompt method to leverage the SAM model in medical image segmentation
tasks. In General, the Dice of the SAM model with box-prompt mode is 0.

In Fig 2, we show how the prediction accuracy would change after more jitters
are applied in all datasets .one can tell that the jitters added to the boxes
can significantly affect the prediction mask accuracy. We discussed this
observation before. We also give a visualization of some segmentation results
in Fig 3.

Overall, the SAM zero-shot results are generally lower than SOTA results, but
there are still several datasets that obtain performance exceeding SOTA. There
is variability in the results of SAM segmentation on different medical image
datasets, and the reasons for this variability and how to make improvements
will be further explored in our future studies.

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# Clonal dynamics of surface-driven growing tissues

Ruslan I. Mukhamadiarov<EMAIL_ADDRESS>Arnold Sommerfeld Center
for Theoretical Physics and Center for NanoScience, Department of Physics,
Ludwig-Maximilians-Universität München, Theresienstrasse 37, D-80333 Munich,
Germany  Max Planck Institute for the Physics of Complex Systems, Nöthnitzer
Strasse 38, Dresden, D-01138, Germany  Matteo Ciarchi  Arnold Sommerfeld
Center for Theoretical Physics and Center for NanoScience, Department of
Physics, Ludwig-Maximilians-Universität München, Theresienstrasse 37, D-80333
Munich, Germany  Max Planck Institute for the Physics of Complex Systems,
Nöthnitzer Strasse 38, Dresden, D-01138, Germany  Fabrizio Olmeda  Max Planck
Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, Dresden,
D-01138, Germany  Steffen Rulands<EMAIL_ADDRESS>Arnold Sommerfeld Center for
Theoretical Physics and Center for NanoScience, Department of Physics, Ludwig-
Maximilians-Universität München, Theresienstrasse 37, D-80333 Munich, Germany
Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse
38, Dresden, D-01138, Germany

###### Abstract

The self-organization of cells into complex tissues relies on a tight
coordination of cell behavior. Identifying the cellular processes driving
tissue growth is key to understanding the emergence of tissue forms and
devising targeted therapies for aberrant growth, such as in cancer. Inferring
the mode of tissue growth, whether it is driven by cells on the surface or
cells in the bulk, is possible in cell culture experiments, but difficult in
most tissues in living organisms (_in vivo_). Genetic tracing experiments,
where a subset of cells is labeled with inheritable markers have become
important experimental tools to study cell fate _in vivo_. Here, we show that
the mode of tissue growth is reflected in the size distribution of the progeny
of marked cells. To this end, we derive the clone-size distributions using
analytical calculations and an agent-based stochastic sampling technique in
the limit of negligible cell migration and cell death. We show that for
surface-driven growth the clone-size distribution takes a characteristic
power-law form with an exponent determined by fluctuations of the tissue
surface. Our results allow for the inference of the mode of tissue growth from
genetic tracing experiments.

The self-organization of cells into tissue relies on the coordination of cell
proliferation and differentiation in space and time. Broadly, tissue growth
can be driven by the spatially homogeneous proliferation of cells (bulk
growth). This mode of growth is characteristic of softer tissues like
tendroins, arteries, or brain [1]. Alternatively, tissues may grow by the
preferential proliferation of cells on the surface, for example, because these
cells have access to signaling molecules or vasculature. Surface-driven growth
is often found in shells, horns, some bones [1], or tumors, where cells on the
tumor surface have better access to nutrients [2]. As a specific example of
surface-driven growth, in some fish and amphibians the eyecup forms by cell
division in the outer part of the eye, the ciliary marginal zone [3].
Understanding whether a given tissue grows by cell proliferation on its
surface or in its bulk is important for targeting treatments during aberrant
growth, such as cancer, it can form a template for developing synthetic
tissues, and for understanding pathological development scenarios. In the
example of the eyecup, cell divisions outside of the ciliary marginal zone, in
the bulk, lead to the formation of additional blood vessels and scar tissue,
and eventually to a disorder called proliferative retinopathy and to a
complete loss of the eye’s functionality [4].

The regulation of cell proliferation and the ensuing spatial distribution of
proliferation patterns is governed on the one hand by complex biochemical
signaling networks and cell-to-cell communication [5]. On the other hand, it
relies on how microscopic mechanical parameters, such as stresses, translate
to a macroscopic scale. The connection between both is not well understood
[6], such that an a priori inference of the mode of tissue growth is
infeasible from a tissue mechanics perspective [7, 5]. Live imaging gives
access to spatio-temporally resolved cell kinetics and allows for the
estimation of tissue mechanical parameters [8, 9, 10, 11]. However, live
imaging is highly challenging _in vivo_ and it is usually limited to specific
cases of embryonic development [12] or to studies of epithelium or other
surface tissues [13, 14].

The recent advent of genetic tracing experiments allows studying cell-fate
behavior _in vivo_. In these experiments, a subset of cells is genetically
labeled with fluorescent markers or genetic barcodes [15]. As cells divide,
these labels are passed on to all progeny of a labeled cell, termed a _clone_.
The probability density of the sizes of such clones provides indirect
information about the history of cell division, differentiation, and cell
death events between labeling and the time point of analysis [16, 17]. For
example, the first moment of the clone size distribution, the average clone
size, reflects the rate of proliferation and whether both daughter cells
remain proliferative or not. The functional form of the clone size
distribution reflects how the fate of individual cells is decided [9, 10, 18],
and the presence of mechanical forces [19] leading to clone fragmentation and
merging[11, 20, 8]. Therefore, the combination of genetic tracing experiments
and tools from statistical physics has become a standard method for unveiling
cell-fate behavior _in vivo_ [21, 22, 23, 24].

Here, we derive a theory that allows identifying the mode of tissue growth
from genetic tracing experiments. By studying the stochastic dynamics of clone
boundaries and employing ideas from the range expansion process [25, 26], we
show that, for surface-driven tissue growth, the clone size distribution
follows a characteristic power-law decay. The decay exponent depends on the
roughness of the surface, which in turn is determined by the mechanisms
regulating the tissue interface. We confirm our theoretical predictions with
Monte Carlo stochastic lattice simulations with forward labeling.

We consider a tissue in $d$ spatial dimensions which are separated from other
tissues by a $d-1$ dimensional boundary (Fig. 1). At a time $t=0$ a random
subset of cells is labeled and, when cells divide, this label is passed on to
all progeny of a labeled cell. We are interested in how the size distribution
of the number of cells that carry a given label at time $t$ relates to the
growth mode of the tissue. To begin our analysis of the clonal dynamics in
surface-driven growing tissues, we make a simplifying assumption that cells
labeled with the same marker remain spatially segregated. For this to hold
true, the rate of cell death needs to be small compared to the rate of cell
proliferation, which is generally the case for growing tissues. Moreover, the
typical length scale associated with cell migration also needs to be small
compared to the spatial extension of clones that result during the time course
of the experiment. Under these conditions, cells that share the same marker
form spatially-segregated clones such that boundaries that separate clones
with different markers are well-defined. Such spatially segregated domains
have indeed been observed in experiments of the growing retina of medaka fish
[27]. Under these assumptions, we show that the clone size distribution can be
obtained from stochastic and geometric arguments alone, without making further
assumptions about tissue mechanics.

As the tissue grows, the boundaries of clones are subject to stochastic
fluctuations which originate from the random nature of cell divisions at the
tissue surface [25, 26]. In the following, we will first derive expressions
for the clone-size distribution in two spatial dimensions and then extend
these results to three spatial dimensions. To this end, we will consider the
stochastic wandering dynamics of clone boundaries an approach that was first
applied in the context of the random genetic drift of the range expansion
process [25, 26].

In two dimensions, the clone boundary dynamics stemming from the stochasticity
of cell divisions can be defined by a stochastic process, $X(t)$. The
difference in distance between two adjacent clone boundaries, $\Delta X$, has,
as the label does not influence proliferation, a vanishing mean,
$\langle\Delta X\rangle=0$, and the time evolution of its variance is
described by a wandering exponent $\zeta$,

$\langle(\Delta X)^{2}\rangle\sim t^{2\zeta}.$ (1)

As an example, a wandering exponent $\zeta=1/2$ describes Brownian Brownian
motion of the distance between clone boundaries. The additional presence of
surface fluctuations alters the wandering exponent, such that for surfaces
belonging to the Kardar-Parisi-Zhang universality class the wandering exponent
takes a value $\zeta=2/3$ [28, 25].

Figure 1: Schematic illustration of expected clonal dynamics in surface-driven
growing tissues – tissues where the cell divisions occur predominantly at
tissue’s surface. Cells that share the same color belong to the same clone.
The arrow indicates the growth direction and $\Delta X$ indicates the distance
between two clone boundaries.

As the tissue grows, adjacent clone boundaries are subject to stochastic
coalescence events. Thereby, a clone that is enclosed by two merging
boundaries loses its access to the growing tissue surface (see Fig. 1). As a
result of this merging event of the domain boundaries, the number of
persisting clones, $N_{p}$, i.e. number of clones that have access to the
front and that continue to forward their label and grow in size, decreases
with time as

$N_{p}\sim 1/\sqrt{\langle(\Delta X)^{2}\rangle}\sim t^{-\zeta}\,.$ (2)

In order to derive the size of persisting clones we note that, in contrast to
non-growing or bulk-driven tissues, the clonal dynamics in surface-driven
tissue growth depend on the proximity of the clone to the surface of the
growing tissue. Only clones containing cells at the tissue’s surface can
continue growing and contributing to the asymptotic shape of the clone-size
distribution. If the linear extension of the tissue surface stays constant at
a value $L$, we get an approximate expression for the average size of
persistent clones, $\langle n_{p}\rangle$, by dividing the total tissue area
by the number of persistent clones at time $t$,

$\langle n_{p}(t)\rangle\sim Lv\,t/N_{p}\sim t^{1+\zeta}\,.$ (3)

Here, $v$ is the growth rate of the tissue that we assume to stay constant for
a given cell division rate [29, 30].

Asymptotically, the fraction of clones with access to the surface vanishes.
Therefore, the clone size distribution, $P(n)$, is well approximated by the
distribution of clones that have lost access to the surface. To calculate
$P(n)$, we therefore first calculate the number of clones that have lost their
access to the moving front in a time interval $\text{d}t$,

$\displaystyle N_{lost}(t)\text{d}t=$
$\displaystyle-\left[N_{p}(t+\text{d}t)-N_{p}(t)\right]\text{d}t$ (4)
$\displaystyle\sim-(\text{d}N_{p}(t)/\text{d}t)\,\text{d}t$ $\displaystyle\sim
t^{-\zeta-1}\text{d}t\,.$

Then, using the mean-field argument that all persisting clones grow with the
same average rate $n(t)=\langle n_{p}(t)\rangle$ in Eq. (3), we obtain the
clone-size distribution for surface-driven growing tissues,

$\displaystyle P(n)\text{d}n$
$\displaystyle=N_{lost}(\text{d}t/\text{d}n)\text{d}n\sim
n^{-1}n^{-\zeta/(1+\zeta)}\text{d}n$ (5)
$\displaystyle=n^{-(1+2\zeta)/(1+\zeta)}\text{d}n\,.$

The clonal size distribution has a unique, previously unobserved, power-law
form, with an exponent that only depends on the wandering exponent $\zeta$
that describes the stochastic motion of clone boundaries. This result is in
contrast to log-normal distribution and exponential distributions, observed
for bulk-driven growing tissues and in homeostatic tissues, respectively [11,
9, 10]. As a remark, one obtains the same results by formally identifying
coalescence events of clone boundaries with first passage events of a Brownian
walker that hit the origin (see Supplement).

In three-dimensional tissues, clone boundaries are defined by stochastic
surfaces. For a given clone, consider a slice along the direction of the
growth. Within this slice, we expect the distance between the clone
boundaries, $\langle(\Delta X)^{2}\rangle$ to scale like $t^{2\zeta}$. In the
absence of anisotropies, this scaling holds for all $d-1$ directions
perpendicular to the growth direction. We now consider the number of cells
that share the same marker in a given slice perpendicular to the growth
direction, $A$. Its deviation from the average, $\Delta A$, fluctuates as

$\langle(\Delta A)^{2}\rangle\sim(\langle\Delta X^{2}\rangle)^{2}\sim
t^{4\zeta}\,.$ (6)

If the number of cells in a given slice remains constant, the number of clones
that have access to the surface decreases as (cf. Eq. (2))

$N_{p}\sim 1/\sqrt{\langle(\Delta A)^{2}\rangle}\sim t^{-2\zeta}\,.$ (7)

For growth with a constant growth rate $v$, the average size of persistent
clones increases as

$\langle n_{p}(t)\rangle\sim L^{2}v\,t/N_{p}\sim t^{1+2\zeta}\,.$ (8)

Finally, utilizing the same line of argument as for two-dimensional tissues,
the clone size distribution in $d=3$ reads

$P(n)\text{d}n\sim n^{-(1+4\zeta)/(1+2\zeta)}\text{d}n\,.$ (9)

Figure 2: Snapshots of Monte Carlo lattice simulation of the birth process
with label forwarding for (a) two-dimensional $1000\times 500$ lattice and (b)
three-dimensional $400\times 200\times 200$ lattice. In both cases, the
simulation begins at $X=0$ with only the first perpendicular layer being
filled with agents, each having a unique label that they can forward when they
reproduce. The snapshots are taken right before the front reaches the other
end of the lattice $X=L_{x}$. For both (a) and (b) we have kept periodic
boundary conditions in directions perpendicular to the growth. Figure 3: (a)
Time evolution of the number of persisting clones $N_{p}$ and their average
size $\langle n_{p}\rangle$ (inlay), obtained from two- and three-dimensional
Monte Carlo lattice simulations of a simple birth process with label
forwarding. (b) The clonal size distribution and its local decay exponent. The
inlay shows the local exponent. Error bars depict $\pm$ standard deviation.
The data in (a) was obtained from simulations on $1000\times 500$ and
$1000\times 100\times 100$ latices, and were averaged over $10^{4}$
independent realizations. The data for the clonal size distribution in (b) was
obtained from simulations on $500\times 200$ and $100\times 50\times 50$
lattices and were averaged over $10^{6}$ independent simulation runs.

Taken together, these scaling arguments predict that the clone size
distributions follows characteristic power laws. The associated exponents
depend on the spatial dimension and the wandering exponent of clone
boundaries, which again is influenced by the roughness of the tissue surface.
For flat surfaces, where $\zeta=1/2$, the clone size distribution decays with
an exponent of $4/3$ for planar tissues and $3/2$ for volumnar tissues. For a
large class of fluctuating surfaces belonging to the Kardar-Parisi-Zang
universality class ($\zeta=2/3$) the clone size distribution decays with
exponents $7/5=1.4$ and $11/7\approx 1.57$, respectively.

We derived these results in the limit of negligible curvature. For curved
tissue surfaces, clone boundary coalescence halts asymptotically if the mean
squared displacement of clone boundaries increases slower than the expansion
of the tissue surface [26, 31, 27]. Therefore, the results are strictly valid
if $2\,\zeta>d-1$ for surfaces with constant curvature. Even if this is not
the case, our results are applicable if the linear extension of clones,
$\Delta X$, is much smaller than the length scale associated with the
curvature. This is generally the case not too long after labeling. Since
genetic tracing experiments typically trace clones over several rounds of cell
divisions we expect our results to be broadly applicable.

To test the validity of our analytical predictions, we performed numerical
simulations of surface-driven growth in $d=2$ and $d=3$. For these simulations
to verify the predicted power-law exponents they need to generate clones
spanning orders of magnitude in size. Simulations of such extent are
impossible when considering tissue mechanics and the details of biochemical
processes underlying cell fate regulation. However, for surface-driven growth,
if the rate of cell motility and loss are significantly smaller than the rate
of cell division, the clone-size distribution is expected to depend only on
the wandering and coalescence statistics of clone domain boundaries, and not
on the underlying tissue mechanics or regulatory biochemical signaling
network. Therefore, we use computationally efficient lattice simulations that
capture the stochastic dynamics of clone domain boundaries and their relation
to the clone-size distribution without necessarily being accurate microscopic
representations of the tissue mechanics and regulatory processes.

Specifically, we employ a modified version of the Eden cluster growth model,
which is a minimal agent-based model that produces surface-driven cluster
growth [32, 29]. In addition to the diffusion-limited branching process
$A\xrightarrow{\lambda}A+A$ that increases the size of clusters by 1 with a
rate $\lambda$, we randomly label agents $A$ at the beginning of the
simulation and allow them to pass their label upon replication (see Fig. 2).
To produce and sample clone statistics, we employ Monte Carlo simulations of
the described diffusion-limited birth process with label forwarding on two-
and three-dimensional lattices. All of our simulations are initialized with a
fully-occupied line ($d=2$) or plane ($d=3$) at $x=0$, while the rest of the
lattice is empty. Initially, each agent is endowed with a unique label. We
update the system state using the Monte Carlo random sequential updating
scheme. For a randomly chosen agent, we select at random an empty nearest
neighbor lattice site and generate a new agent with the same label with a rate
$\lambda$.

We simulated systems, where clones have boundaries that follow Brownian
dynamics, i.e., $\zeta=1/2$ in Eq. (1) and Eq. (6), respectively, and a class
of systems where fluctuations in the surface lead to super-diffusive motion of
domain boundaries $\zeta=2/3$ (see Supplement). In the former case, the
standard deviation of fluctuations stemming from the tissue surface needs to
be constant at all times. We achieve that by choosing a space-dependent growth
rate, $\lambda=(1-\tanh\left[\alpha(x-x_{0})\right])/2$, where the coefficient
$\alpha$ sets the surface sharpness, $x_{0}=vt$ determines the surface
position, and $v$ sets the growth velocity. Choosing $\lambda$ to vary only
along the growth direction prohibits the development of surface fluctuations
in directions perpendicular to the growth. As such, the cluster interface
stays flat at all times, and clone boundaries To simulate clones with super-
diffusive boundaries, we set $\lambda$ to be spatially homogeneous, which
gives rise to surfaces belonging to the Kardar-Parisi-Zang universality class
and$\zeta=3/2$ [33, 34, 35].

We collect the number of different labels $N_{p}$ that have access to the
front and measure the size of these clones to compute $\langle n_{p}\rangle$.
To obtain the clonal size distribution, we collected the sizes of the clones
that have lost their access to the advancing surface. In Figs. 2 and 3 we show
results for the case $\zeta=1/2$. We give the results for $\zeta=2/3$ in the
supplement. In both cases, our simulations reflect the predictions made by the
scaling arguments given above. Specifically, for two-dimensional growth all
results follow the predicted values. In three spatial dimensions, our
simulations deviate slightly from our mean-field analysis (Fig. 3, inlay). It
is plausible that the slower decay in the number of persistent clones $N_{p}$
for $d=3$ stems from the fragmentation and merging clones, which can occur in
$d=3$ and is not considered in the mean-field theory.

In summary, we have studied the dynamics of clones for both $d=2$ and $d=3$
surface-driven growing tissues. We found that the clone-size distribution
takes a power-law form with exponents depending on the tissue dimension and
the nature of fluctuations in the surface. The power laws in the clone size
distribution translate to associated power laws in the time evolution of the
average clone size and the number of clones with access to the surface. These
results allow identify the mode of tissue growth using genetic tracing
experiments. While genetic tracing experiments using fluorescent markers
typically do not produce a sufficiently high number of clones to confidently
identify such power laws, recently developed technologies using genetic
barcodes produce thousands of unique clones in a tissue and can therefore be
used to infer the mode of tissue growth as well as distinguish different kinds
of surface fluctuations.

We thank Martin Lenz, Ricard Alert, and Frank Jülicher for useful discussions.
This project has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation program (grant
agreement no. 950349).

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# Light speed variation from GRB 221009A

Jie Zhu<EMAIL_ADDRESS>Bo-Qiang Ma<EMAIL_ADDRESS>
###### Abstract

It is postulated in Einstein’s relativity that the speed of light in vacuum is
a constant for all observers. However, the effect of quantum gravity could
bring an energy dependence of light speed, and a series of studies on high-
energy photon events from gamma-ray bursts (GRBs) and active galactic nuclei
(AGNs) suggest a light speed variation
$v(E)=c\left(1-E/E_{\mathrm{LV}}\right)$ with $E_{\mathrm{LV}}=3.6\times
10^{17}~{}\mathrm{GeV}$ or a bound $E_{\mathrm{LV}}\geq 3.6\times
10^{17}~{}\text{GeV}$. From the newly observed gamma-ray burst GRB 221009A, we
find that a 99.3 GeV photon detected by Fermi-LAT is coincident with the sharp
spike in the light curves detected by both Fermi-GBM and HEBS under the above
scenario of light speed variation, suggesting an option that this high-energy
photon was emitted at the same time as a sharp spike of low-energy photon
emission at the GRB source. Thus this highest energy photon event detected by
Fermi-LAT during the prompt emission of gamma-ray bursts might be considered
as an optional signal for the linear form modification of light speed in
cosmological space.

###### keywords:

gamma-ray burst, high-energy photon, light speed variation, Lorentz violation

††journal: arXiv:2210.11376; Published in J. Phys. G: Nucl. Part. Phys. 50
(2023) 06LT01

[aff1] organization = School of Physics, Peking University, city = Beijing
100871, country = China

[aff2] organization = Center for High Energy Physics, Peking University, city
= Beijing 100871, country = China

[aff3] organization = Collaborative Innovation Center of Quantum Matter, city=
Beijing, country = China

On 9 October 2022, the Gamma-ray Burst Monitor (GBM) [1] onboard the Fermi
Gamma-ray Space Telescope (FGST) triggered a special gamma-ray burst (GRB),
numbered as GRB 221009A, at $T_{0}=13:16:59.000\text{ UT}$ [2]. The prompt
emission of this extraordinarily bright GRB 221009A was detected not only by
Fermi [2, 3, 4, 5, 6], but also by other space observatories such as Swift [7,
8], AGILE [9, 10], INTEGRAL [11], Solar Orbiter [12], SRG [13], Konus [14],
GRBAlpha [15], STPSat-6 [16], HEBS [17], and by ground observatories such as
LHAASO [18] with striking very-high-energy features [19]. This burst, located
at around RA = 288.282 and Dec = 19.495 [5], is a long burst but with a very
small redshift $z=0.1505$ [20, 21] compared to most other long bursts. It is
estimated that there is only a 10% probability to observe an event like GRB
221009A about 50 years after the discovery of the first GRB [22].

The Fermi Large Area Telescope (LAT) [23] also detected this GRB. The highest
energy photon observed by LAT reaches 99.3 GeV (with a probability of 99.2%)
[5], and this photon event represents the highest GRB photon energy ever
detected by Fermi-LAT during the prompt emission phase of gamma-ray bursts. We
show in this Letter that such a high-energy photon event of Fermi-LAT offers a
precious opportunity to explore the light speed variation related with Lorentz
violation (LV) [24].

It is speculated from quantum gravity that the Lorentz invariance might be
broken at the Planck scale ($E_{\mathrm{Pl}}\simeq 1.22\times
10^{19}~{}\mathrm{GeV}$), and that the light speed may have a variation with
the energy of the photon. Amelino-Camelia et al. [25, 26] first suggested
testing Lorentz violation by comparing the arrival times between high-energy
and low-energy photons from GRBs. For energy $E\ll E_{\mathrm{Pl}}$, the
modified dispersion relation of the photon can be expressed in a model-
independent way as

$E^{2}=p^{2}c^{2}\left[1-s_{n}\left(\frac{pc}{E_{\mathrm{LV},n}}\right)^{n}\right],$
(1)

from which we have the following speed relation

$v(E)=c\left[1-s_{n}\frac{n+1}{2}\left(\frac{pc}{E_{\mathrm{LV},n}}\right)^{n}\right],$
(2)

where $n=1$ or $n=2$ as usually assumed, $s_{n}=\pm 1$ indicates whether high-
energy photons travel faster ($s_{n}=-1$) or slower ($s_{n}=+1$) than low-
energy photons, and $E_{\mathrm{LV},n}$ represents the nth-order Lorentz
violation scale. Jacob and Piran [27] suggested a formula of arrival-time
difference between astroparticles with different energies as

$\Delta
t_{\mathrm{LV}}=s_{n}\frac{1+n}{2H_{0}}\frac{E^{n}_{\mathrm{h}}-E^{n}_{\mathrm{l}}}{E^{n}_{\mathrm{LV,}n}}\int_{0}^{z}\frac{(1+z^{\prime})^{n}\mathrm{d}z^{\prime}}{\sqrt{\Omega_{\mathrm{m}}(1+z^{\prime})^{3}+\Omega_{\Lambda}}},$
(3)

where $E_{\mathrm{h}}$ and $E_{\mathrm{l}}$ correspond to the energies of the
observed high-energy and low-energy photons, $z$ is the redshift of the source
GRB, $H_{0}$, $\Omega_{\mathrm{m}}$ and $\Omega_{\mathrm{\Lambda}}$ are
cosmological constants. Eq. (3) was derived in the standard model of
cosmology, and later was found the same as the time-difference formula in
expanding Finsler spacetime [28]. In this work, we adopt the present day
Hubble constant $H_{0}=67.3\pm 1.2~{}\mathrm{km~{}s}^{-1}\mathrm{Mpc}^{-1}$
[29], the pressureless matter density
$\Omega_{\mathrm{m}}=\mathrm{0.315^{+0.016}_{-0.017}}$ [29] and the dark
energy density $\Omega_{\Lambda}=\mathrm{0.685^{+0.017}_{-0.016}}$ [29].
Earlier researches on cosmic photon data of observed GRBs set robust bounds on
Lorentz violation [30, 31, 32, 33]. A series of researches [34, 35, 36, 37,
38, 39, 40, 41, 42, 43] on time flights of high-energy photon events from GRBs
detected by Fermi Gamma-ray Space Telescope reveal a regularity of high-energy
photon events, leading to a light speed variation with $s=+1$, $n=1$, and
$E_{\mathrm{LV}}=3.6\times 10^{17}~{}\text{GeV}$. This result suggests a
scenario that high-energy photons travel slower than low-energy photons. It is
also shown that several phenomena related to the light curves of three active
galactic nuclei (AGNs), namely Markarian 421 (Mrk 421), Markarian 501 (Mrk
501) and PKS 2155-304, can serve as the supports for the light speed variation
determined from GRBs at a same scale of $E_{\mathrm{LV}}=3.6\times
10^{17}~{}\text{GeV}$ [44]. Such a value is compatible with constraints from
analyses on variety GRB data [45, 46, 47, 48], including recent analyses using
eight Fermi-LAT gamma-ray burst data [49] and using spectral lag data from the
BATSE satellite [50]. It is also found to be consistent with predictions in a
string theory model for space-time foam [51, 52, 53, 54, 55, 56]. From a
conservative viewpoint, we may consider these results as a suggestion for a
lower bound of the Lorentz violation scale $E_{\mathrm{LV}}\geq 3.6\times
10^{17}~{}\text{GeV}$.

As the newly detected GRB 221009A is the brightest one among all detected GRBs
and also is the one with a very short distance from the Earth, the light
curves of this rare GRB should contain rich information from which we can
reveal novel features of our Universe. To simplify the discussion, here we
treat the GBM trigger time $T_{0}$ as 0. Detailed information of the highest
GRB photon energy ever detected by Fermi-LAT during the prompt phase of GRBs
can be found at the LAT data server [57]. The observed energy of this photon
is $E=99.3254~{}\text{GeV}$, the position of this photon in J2000 is (287.533,
19.875), and the arrival time is $T_{1}=T_{0}+241.326~{}\text{s}$. Also, the
information of low-energy photons can be found at the Fermi server [58], and
the light curve of GRB 221009A from two brightest GBM trigger detectors
combined (GBM NaI-n7 and NaI-n8, 7keV-1MeV) are shown in Fig. (1). The Lorentz
violation scenario provided above suggests that the low-energy photons (less
than 1 MeV) emitted at the same time as the 99.3 GeV photon at the source
travel faster than the latter one, and the arrival time difference can be
calculated by Eq. (3) as

$\Delta
t=\frac{E}{E_{\mathrm{LV}}H_{0}}\int_{0}^{z}\frac{(1+z^{\prime})\mathrm{d}z^{\prime}}{\sqrt{\Omega_{\mathrm{m}}(1+z^{\prime})^{3}+\Omega_{\Lambda}}}=19.8\text{
s}.$ (4)

Thus the low-energy photons emitted at the same time as the 99.3 GeV photon at
the source would be detected at $T_{1}^{\prime}=T_{1}-\Delta
t=221.5~{}\text{s}$. Turn to the light curve of GBM, amazingly we find that at
$T_{1}^{\prime}=221.5~{}\text{s}$ (the red vertical line in Fig. (1)) there is
a significant sharp peak of low-energy photons. That may imply that at the
source the 99.3 GeV photon was emitted at the same time as the low-energy
photons detected by GBM as a sharp spike centered at around
$T_{1}^{\prime}=221.5~{}\text{s}$. Since the energy resolution of LAT is
within 10% uncertainty and the time resolution for GBM and LAT is smaller than
10 $\mu s$, the error-bars of the Fermi data can be safely neglected in our
analyses. Also, the energy value of the low-energy photons is taken as zero in
the time-lag analysis Eq. (4) compared to the 99.3 GeV photon event, so the
energy values of the low-energy photons can be neglected.

For the low-energy photon events of GRB 221009A, the HEBS collaboration also
published the light curve [17] (Fig. (2)). Different from GBM, the energy
range of HEBS is from 400 keV to 5 MeV. We can also see that there is a tiny
sharp peak of low-energy photon events at $T_{1}^{\prime}=221.5~{}\text{s}$
(the red vertical line in Fig. (2)). Due to the long duration of the GRB, the
99.3 GeV event locates in the emission phase of low-energy photons detected by
the Fermi detectors at the Earth, but the actually observed position (the blue
dashed line in Fig. (2)) of this Fermi-LAT event is outside the main peak
ranges of the HEBS light curves, showing the difficulty to associate it with
the accompanying low-energy photons in the HEBS light curves without
considering the Lorenz violation effect. We can see that the estimated
position of this Fermi-LAT event just corresponds to the sharp spike of low-
energy photons in both GBM and HEBS light curves, as shown with a red vertical
line in both Fig. (1) and Fig. (2). The coincidence of this fact supports the
Lorentz violation scenario of $v(E)=c\left(1-E/E_{\mathrm{LV}}\right)$ with
$E_{\mathrm{LV}}=3.6\times 10^{17}~{}\text{GeV}$ or a bound
$E_{\mathrm{LV}}\geq 3.6\times 10^{17}~{}\text{GeV}$.

Figure 1: Light curves of Fermi-GBM with the two brightest trigger detectors
combined (GBM NaI-n7 and NaI-n8, 7keV-1MeV) for GRB 221009A, with events
binned in 2-second intervals. The red vertical line represents
$T_{1}^{\prime}=221.5~{}\text{s}$, the estimated position of the 99.3 GeV
photon event detected by Fermi-LAT if there is no light speed variation, and
the blue dashed vertical line represents $T_{1}=241.3~{}\text{s}$, the
observed position of this photon event. The red line very well matches a peak
of low-energy photons. Figure 2: Light curves of HEBS for GRB 221009A from GCN
Cir. 32751 [17]. The red vertical line represents
$T_{1}^{\prime}=221.5~{}\text{s}$, the estimated position of the 99.3 GeV
photon event detected by Fermi-LAT if there is no light speed variation, and
the blue dashed vertical line represents $T_{1}=241.3~{}\text{s}$, the
observed position of this photon event. The red line also very well matches a
peak of low-energy photons.

In conclusion, we analyze the $99.3$ GeV photon event detected by Fermi-LAT at
$T_{1}=241.3$ s after the GRB 221009A trigger time $T_{0}=0$, and find that in
the Lorentz violation scenario of $v(E)=c\left(1-E/E_{\mathrm{LV}}\right)$
with $E_{\mathrm{LV}}=3.6\times 10^{17}~{}\text{GeV}$, at the GRB source this
photon was emitted at the same time as the low-energy photon emission observed
as a sharp spike centered at around 221.5 s in the light curves reported by
Fermi-GBM and HEBS. This remarkable coincidence between the highest energy
photon event observed by Fermi-LAT during the prompt phase and the sharp spike
in the low-energy light curves of brightest gamma-ray burst GRB 221009A might
be considered as an optional signal for the light speed variation with
$E_{\mathrm{LV}}=3.6\times 10^{17}~{}\text{GeV}$ or the bound
$E_{\mathrm{LV}}\geq 3.6\times 10^{17}~{}\text{GeV}$ suggested in previous
studies [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44]. Of course, more scrutiny
is necessary to test the scenario suggested in this Letter when more data are
available.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## Data availability statement

All data that support the findings of this study are referenced within the
article (and any supplementary files).

## Acknowledgements

This work is supported by National Natural Science Foundation of China (Grant
No. 12075003).

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