Yuting He

Guanyu Yang

Jian Yang

Rongjun Ge

Youyong Kong

Xiaomei Zhu

Shaobo Zhang

Pengfei Shao

Huazhong Shu

Jean-Louis Dillenseger

Jean-Louis Coatrieux

Shuo Li
email: slishuo@gmail.com

Meta Grayscale Adaptive Network for 3D Integrated Renal Structures Segmentation

Keywords: Integrated renal structures, segmentation, Ensemble learning, Meta learning, Automatic hyper-parameter search, Meta grayscale adaptive network, Grayscale interest search, Meta grayscale ensemble

Three-dimensional (3D) integrated renal structures (IRS) segmentation targets segmenting the kidneys, renal tumors, arteries, and veins in one inference. Clinicians will benefit from the 3D IRS visual model for accurate preoperative planning and intraoperative guidance of laparoscopic partial nephrectomy (LPN). However, no success has been reported in 3D IRS segmentation due to the inherent challenges in grayscale distribution: low contrast caused by the narrow task-dependent distribution range of regions of interest (ROIs), and the networks representation preferences caused by the distribution variation inter-images. In this paper, we propose the Meta Greyscale Adaptive Network (MGANet), the first deep learning framework to simultaneously segment the kidney, renal tumors, arteries and veins on CTA images in one inference. It makes innovations in two collaborate aspects: 1) The Grayscale Interest Search (GIS) adaptively focuses segmentation networks on task-dependent grayscale distributions via scaling the window width and center with two cross-correlated coefficients for the first time, thus learning the fine-grained representation for fine segmentation. 2) The Meta Grayscale Adaptive (MGA) learning makes an image-level meta-learning strategy. It represents diverse robust features from multiple distributions, perceives the distribution characteristic, and generates the model parameters to fuse features dynamically according to image's distribution, thus adapting the grayscale distribution variation. This study enrolls 123 patients and the average Dice coefficients of the renal structures are up to 87.9%. Fine selection of the task-dependent grayscale distribution ranges and personalized fusion of multiple representations on different distributions will lead to better 3D IRS segmentation quality. Extensive experiments with promising results on renal structures reveal powerful segmentation accuracy and great clinical significance in renal cancer treatment.

Introduction

Three-dimensional (3D) integrated renal structures (IRS) segmentation on computed tomography angiography (CTA) images is one of the most important tasks for laparoscopic partial nephrectomy (LPN) [START_REF] Shao | Laparoscopic partial nephrectomy with segmental renal artery clamping: technique and clinical outcomes[END_REF][START_REF] Simone | Indications, techniques, outcomes, and limitations for minimally ischemic and off-clamp partial nephrectomy: a systematic review of the literature[END_REF][START_REF] Ljungberg | European association of urology guidelines on renal cell carcinoma: the 2019 update[END_REF]. It targets achieving 3D kidney, renal tumors, arteries, and veins in one inference (Fig. 1(a)), once successful, clinicians will benefit from the 3D visual model of renal structures for accurate preoperative planning [START_REF] Porpiglia | Hyperaccuracy threedimensional reconstruction is able to maximize the efficacy of selective clamping during robot-assisted partial nephrectomy for complex renal masses[END_REF][START_REF] Bianchi | The impact of 3D digital reconstruction on the surgical planning of partial nephrectomy: a case-control study. still time for a novel surgical trend[END_REF]. Preoperatively, the renal arteries will help estimate renal perfusion model [START_REF] Zhang | Application of a functional3-dimensional perfusion model in laparoscopic partial nephrectomy with precise segmental renal artery clamping[END_REF], so that the clinicians will select the tumor-feeding arterial branches and locate the arterial clamping position easily [START_REF] Shao | Precise segmental renal artery clamping under the guidance of dual-source computed tomography angiography during laparoscopic partial nephrectomy[END_REF]. The tumor and kidney models will visually show the lesion regions, thus helping the pre-plan of the tumor resection surface (Fig. 1(b)). Intraoperatively, the preoperative plan will be displayed on screen together with laparoscopic videos to guide the LPN [START_REF] Nicolau | Augmented reality in laparoscopic surgical oncology[END_REF][START_REF] Herrell | Image guidance in robotic-assisted renal surgery[END_REF][START_REF] Hamada | The current status and challenges in augmented-reality navigation system for robot-assisted laparoscopic partial nephrectomy[END_REF]. Renal vessels (veins, arteries) outside hilum will show a clear arterial clamping region visually, thus the clinicians will select arterial clamping branches quickly. The 3D IRS visual model will also guide the clinicians making appropriate decisions (Fig. 1(c)). Therefore, the costs of treatment will be reduced, the quality of LPN will be improved, and the pain of patients will be relieved.

Although the automatic 3D IRS segmentation is clinically urgent, there is no solution yet. Most of the recent studies focus on partial renal structure segmentation (Wang et al., 2020a;[START_REF] Jin | 3D fast automatic segmentation of kidney based on modified AAM and random forest[END_REF][START_REF] He | DPA-DenseBiasNet: semi-supervised 3D fine renal artery segmentation with dense biased network and deep priori anatomy[END_REF] which lacks the countermeasures to segment integrated renal structures. Besides, the separate segmentation of each structure will bring the overlapping problem, especially in the renal hilum where veins, arteries are staggered. Some other studies are dedicated to multiple renal structure segmentation [START_REF] Li | Segmentation of renal structures for image-guided surgery[END_REF][START_REF] Taha | Kid-Net: convolution networks for kidney vessels segmentation from CT-volumes[END_REF], but the Inherent challenges of 3D IRS segmentation are limiting the implementation of automated method: Challenge 1: Narrow task-dependent distribution making low contrast. The CT image has a large grayscale range, but the renal structures are only in a narrow range, making the original CTA images appear unclear soft tissues and boundaries. As shown in Fig. 2, the irrelevant distributions occupy the large grayscale ranges squeezing the renal structures in narrow grayscale ranges, so that the CTA image has coarse-grained distributions and low contrast (a). Therefore, the network will lose the sensitivity to the adjacent structures with close grayscale distribution and weaken the ability to extract detailed features such as the boundaries between the structures. Challenge 2: Distribution variation interimages making representation preferences. The subtypes of the renal tumors have different volumes and abilities to metabolize contrast agents [START_REF] Young | Clear cell renal cell carcinoma: discrimination from other renal cell carcinoma subtypes and oncocytoma at multiphasic multidetector CT[END_REF], so that it will make the differences in the distribution of contrast agent overall, thus making a large grayscale distribution variation inter-images. As shown in Fig. 3, the clear cell carcinomas present bright texture and large size, while the papillary has dark texture and small size. The grey distributions of five tumor subtypes have various centers and peaks. These large variations make it the network difficult to stably learn uniform low-order knowledge (style, pattern, etc.) resulting in representation preferences and large fluctuations of performance inter-images, Solution 1 to focus on the task-dependent grayscale distribution range. Windowing [START_REF] Goldman | Principles of CT and CT technology[END_REF] removes irrelevant grayscale ranges and expands the task-dependent distributions which will augment the contrast for ROIs, making networks focus on a wider distribution of the renal structures to perceive fine-grained patterns. However, it is challenging to find the narrow task-dependent distribution in a large original grayscale range manually according to human experience. Especially for our renal structures segmentation task, it has to be dependent on multiple distributions and has big distribution variation inter-images. As shown in Fig. 2, in the grayscale distribution of the example image, when the grayscale distribution range window (sub-window) covers the renal structures' distributions (c), the sub-window image has fine-grained patterns and the regions of renal structures are significant. When the sub-window is out of the renal structure's distributions (b,d), the renal structures will be lost. Therefore, we propose the Grayscale Interest Search (GIS) strategy which focuses segmentation networks on the task-dependent grayscale distribution in CTA images for the first time. It innovatively [START_REF] Young | Clear cell renal cell carcinoma: discrimination from other renal cell carcinoma subtypes and oncocytoma at multiphasic multidetector CT[END_REF]. f) The grayscale distributions of the tumor subtypes have a large variation. The y-axis is the number of voxels and the x-axis is the grayscale value simulates doctor's observation of different structures in different sub-windows to automatically search the task-dependent distribution via scaling the window width and center with two effective cross-correlated coefficients. Therefore, salient regions with fine-grain patterns are available and networks will learn the fine-grained representation for fine segmentation in details.

Solution 2 to adapt to the grayscale distribution variation. Meta-learning for parameter generation learns cross-tasks knowledge to rapidly generate parameters of models [START_REF] Hospedales | Meta-learning in neural networks: a survey[END_REF], thus dynamically adapting to the variation inter-tasks (tasklevel). However, there is no meta-learning study for the adaptation of grayscale distribution variation inter-images (image-level). In this study, we propose the Meta Grayscale Adaptation (MGA) learning (Fig. 4(c)) which makes each image as a sublearning task to dynamically generate the parameters for each image adapting the distribution variation inter-images. It is an image-level meta-learning strategy which learns the trained kernels via gradient descent for the meta knowledge of IRS segmentation, and generates the meta kernels via prediction for the knowledge of each sub-learning task. It has three key elements: 1) Represent the image robustly with distribution variation. We integrate the feature representation capabilities from multiple distributions covered by different sub-windows (Multiwindowing Group) so that more diverse features will be extracted, reducing the overfitting of representation preferences. 2) Dynamically generate personalized parameters according to image's distribution. We make each image as a sub-learning task and further propose the Meta Perceiver to perceive its distribution characteristic, thus generating personalized feature parameters (meta kernels) to adapt to the distribution variation of each sub-learning task (image). 3) Personalized fusion of features. We propose the Meta Residual Convolution (ResConv) to fuse the meta knowledge of our IRS segmentation task and the personalized knowledge of each sub-learning task. Trained kernels learn meta knowledge on IRS segmentation and the meta kernels generated by our Meta perceiver represent the personalized knowledge of sub-learning tasks. The meta kernels fine-tune the trained kernels for the personalized fusion parameters of each image adapting to the distribution variation inter-images and improving the generalization.

We propose the grayscale distribution optimization from the perspective of the grayscale distribution range adjustment and variation adaptation in our 3D IRS segmentation task, and propose the Meta Greyscale Adaptive Network (MGANet), the first deep learning framework to simultaneously segment the kidney, renal tumors, arteries and veins on CTA images in one inference. The contributions of our study are summarized as the following:

To the best of our knowledge, we achieve the 3D IRS segmentation for the first time which will play an important role in accurate preoperative planning and intraoperative guidance of LPN. We review the relevant technical and clinical value of this task which will provide a valuable reference for follow-up works.

We propose the MGANet to optimize the grayscale distribution of the model representation via grayscale distribution range adjustment and variation adaptation, so that it will obtain fine-grained feature representations and personalized feature fusion for optimal segmentation results.

We propose the MGANet to optimize the grayscale distribution of the model representation via grayscale distribution range adjustment and variation adaptation, so that it will obtain fine-grained feature representations and personalized feature fusion for optimal segmentation results.

We present a novel image-level meta-learning strategy, the MGA learning, which represents diverse robust features from multiple distributions, perceives the distribution characteristic, and generates the model parameters to fuse these features dynamically according to the image's distribution, thus adapting the grayscale distribution variation.

The rest of the paper is organized as follow: We review the related works about renal structures segmentation, ensemble learning in image segmentation, hyper-parameter optimization, and meta-learning for parameters generalization in Sec. 2. Then, we specifically introduce our proposed MGANet in Sec. 3, including the GIS strategy (Sec. 3.1), the MGA learning (Sec. 3.2) and the details of the network structures (Sec. 3.3). Then Sec. 4 describes the materials, experiment setting, and evaluation methods, and Sec. 5 shows and analyzes the results of the experiments. Finally, we discuss and conclude our work in Sec. 6.

Related works

Renal structures segmentation

Renal structure segmentation is significant in accurate preoperative planning and intraoperative guidance of LPN [START_REF] Porpiglia | Hyperaccuracy threedimensional reconstruction is able to maximize the efficacy of selective clamping during robot-assisted partial nephrectomy for complex renal masses[END_REF][START_REF] Bianchi | The impact of 3D digital reconstruction on the surgical planning of partial nephrectomy: a case-control study. still time for a novel surgical trend[END_REF], however, the automatic IRS segmentation has not been reported success. [START_REF] He | DPA-DenseBiasNet: semi-supervised 3D fine renal artery segmentation with dense biased network and deep priori anatomy[END_REF] designed a semi-supervised framework and achieved fine segmentation of renal arteries which is important in the selection of arterial clamping position. Wang et al. (2020) took the graph-cut [START_REF] Rother | Grab-Cut: interactive foreground extraction using iterated graph cuts[END_REF] and proposed a tensor-cut framework for renal artery segmentation. [START_REF] Taha | Kid-Net: convolution networks for kidney vessels segmentation from CT-volumes[END_REF] proposed a Kid-Net for renal vessel segmentation including the arteries, veins, and ureter. The 3D UNet (C ¸içek et al., 2016) was firstly proposed for Xenopus kidney segmentation. [START_REF] Li | Segmentation of renal structures for image-guided surgery[END_REF] proposed a residual U-Net and segmented the renal structures.

However, these works are limited in our task in two aspects: 1) Some works [START_REF] He | DPA-DenseBiasNet: semi-supervised 3D fine renal artery segmentation with dense biased network and deep priori anatomy[END_REF][START_REF] Taha | Kid-Net: convolution networks for kidney vessels segmentation from CT-volumes[END_REF]Wang et al., 2020a;[START_REF] Abdulkadir | 3D U-Net: learning dense volumetric segmentation from sparse annotation[END_REF] only focus on the segmentation of partial renal structures lacking the countermeasures for integrated renal structures in one inference. The separate segmentation of each renal structure via these methods will bring the overlapping problem within segmentation regions, especially in renal hilum where veins, arteries, and ureter are staggered. 2) Some other works [START_REF] Li | Segmentation of renal structures for image-guided surgery[END_REF][START_REF] Taha | Kid-Net: convolution networks for kidney vessels segmentation from CT-volumes[END_REF] lack fine details of the segmented structures such as the interlobar renal arteries which is important to build the renal perfusion model [START_REF] Zhang | Application of a functional3-dimensional perfusion model in laparoscopic partial nephrectomy with precise segmental renal artery clamping[END_REF], limiting the clinical downstream tasks.

Ensemble learning in image segmentation

The ensemble learning [START_REF] Dietterich | Ensemble learning[END_REF] constructs a set of learners and combines their representations for a more accurate discussion. It gives a strategy to embed specific designs for the challenges of tasks, such as error correction, class-imbalanced data, etc. [START_REF] Polikar | Ensemble Machine Learning[END_REF]. The ensemble segmentation models [START_REF] Xia | Bridging the gap between 2D and 3D organ segmentation with volumetric fusion net[END_REF][START_REF] Wang | Hierarchical retinal blood vessel segmentation based on feature and ensemble learning[END_REF][START_REF] Zheng | A new ensemble learning framework for 3D biomedical image segmentation[END_REF] fuse the superiorities from the multiple learners improving the integrated segmentation quality. [START_REF] Xia | Bridging the gap between 2D and 3D organ segmentation with volumetric fusion net[END_REF] proposed the Volumetric Fusion Net (VFN) which train three 2D segmentation network in X, Y, and Z-axis and take a 3D network to fuse the results. [START_REF] Zheng | A new ensemble learning framework for 3D biomedical image segmentation[END_REF] further added a learner in the 3D version and proposed a meta-learner for the ensemble process.

Although the existing works behave promising results on their issues, they are inherently impeded to be applied for our task. They ignore the variation of grayscale distributions caused by the variation of tumor types in our task, making their networks difficult to stably learn uniform low-order knowledge (style, pattern, etc), resulting in representation preferences for common distributions and limiting the generalization for rare distributions. Our MGA learning trains multiple networks on different searched distributions and fuses their feature representations abilities according to the distribution characteristic dynamically. Thereby, their representation for different distributions will be jointed making generalization for distribution variation inter-images.

Hyper-parameter optimization

Hyper-parameter optimization [START_REF] Bergstra | Random search for hyper-parameter optimization[END_REF][START_REF] Tan | EfficientNet: rethinking model scaling for convolutional neural networks[END_REF] is widely used to achieve better accuracy. Numerous prior studies [START_REF] He | Deep residual learning for image recognition[END_REF][START_REF] Huang | Densely connected convolutional networks[END_REF][START_REF] Real | Regularized evolution for image classifier architecture search[END_REF] on network's depth and width proofs that the deeper and wider networks will capture richer and more complex features for more accurate performance. [START_REF] Tan | EfficientNet: rethinking model scaling for convolutional neural networks[END_REF] simultaneously studied the influence of the width, depth, and resolution's compound scaling on network accuracy, and searched the EfficientNet which uses less computing resourcesfor higher accuracy.

In medical image analysis, the grayscale distribution ranges of the medical images are another important hyper-parameter that lacks the attention of researchers. The CT imaging process makes a large grayscale range so that the original image will appear in low contrast and unclear soft tissues. When observing medical images, radiologists use the windowing method [START_REF] Goldman | Principles of CT and CT technology[END_REF] to removes irrelevant grayscale ranges and expands the distributions of the ROIs, thus improving the contrast and bringing clearer details. However, there is no study on the optimization of this hyper-parameter: 1) numerous medical image analysis studies [START_REF] He | DPA-DenseBiasNet: semi-supervised 3D fine renal artery segmentation with dense biased network and deep priori anatomy[END_REF][START_REF] Abdulkadir | 3D U-Net: learning dense volumetric segmentation from sparse annotation[END_REF][START_REF] Li | Segmentation of renal structures for image-guided surgery[END_REF][START_REF] Taha | Kid-Net: convolution networks for kidney vessels segmentation from CT-volumes[END_REF] ignore the optimization of this hyper-parameter and train their models on original grayscale ranges, limiting the segmentation quality on their regions of interest (ROIs); 2) Some other studies [START_REF] Isensee | nnU-Net: self-adapting framework for U-Net-based medical image segmentation[END_REF]Wang et al., 2020b) only set the grayscale range manually based on researchers' experience or clinical criteria directly without further optimization for the better grayscale range in their task. We propose the GIS strategy which adaptively searches the taskdependent grayscale range so that the network will learn more fine-grained feature representation for the interested distributions.

Meta learning for parameters generalization

Meta-learning for parameters generalization directly predict model parameters [START_REF] Hospedales | Meta-learning in neural networks: a survey[END_REF] rather than the gradient in back-propagation, thus generating the personalized parameters and providing dynamic adaptation to the variation. The Hypernetworks [START_REF] Ha | Hypernetworks[END_REF][START_REF] Brock | SMASH: one-shot model architecture search through hypernetworks[END_REF] train a network to generate the parameters of another network for fast learning, which are common applications in model compression or multi-task learning. The Meta networks [START_REF] Munkhdalai | Meta Networks[END_REF] combine the parameters predicted by meta learner and the weights trained cross tasks making rapid generalization. The WeightNet [START_REF] Ma | Weight-Net: revisiting the design space of weight networks[END_REF] unifies the SENet [START_REF] Hu | Squeeze-andexcitation networks[END_REF] and CondConv [START_REF] Yang | CondConv: conditionally parameterized convolutions for efficient inference[END_REF], trains the network in the kernel space and takes the network to generate weights for the classification task.

However, these studies only focus on task-level parameter generation and there is no meta-learning study for the adaptation of grayscale distribution variation inter-images (image-level).

Our MGA learning makes each image as a sub-learning task to dynamically generate parameters for each image, thus making an image-level meta-learning framework to adapt the distribution variation. It represents diverse robust features from multiple distributions and generates the model parameters to fuse these features dynamically according to the input image's distribution in a stable residual structure, thus adapting the grayscale distribution variation inter-images.

Methodology

As shown in Fig. 4, our MGANet adaptively optimizes the grayscale distribution of medical images and achieves the 3D IRS segmentation. It makes innovations in two collaborate components: 1) The Grayscale Interest Search (GIS, Sec. 3.1, Fig. 4 (a)) adaptively optimizes the focused grayscale distribution range via scaling the window width and center with two cross-correlated coefficients, thus adaptively discovering the task-dependent distribution. It further expands the searched distribution via windowing making salient regions with fine-grain patterns, so that segmentation networks will learn fine-grained feature representation for renal structures. 2) The Meta Grayscale Adaptive (MGA, Sec. 3.2, Fig. 4 (b)) learning makes each image as a sub-learning task to adapt the distribution variation via three key elements: the Multi-windowing Group extracts features from multiple grayscale distributions for robust feature representation with distribution variation; the Meta Perceiver perceive the distribution characteristic to generate personalized parameters (meta kernels) to adapt to the distribution variation; the Meta ResConv fuses the meta knowledge of our IRS segmentation task and the personalized knowledge of each sub-learning task, improving the generalization.

Grayscale interest search for task-dependent distributions

Our GIS (Fig. 4 (a)) scales the dimensions of window width and center using two cross-correlated coefficients discovering our task-dependent distributions adaptively and expands the searched distribution via windowing [START_REF] Goldman | Principles of CT and CT technology[END_REF] for fine-grained feature representation of renal structures. Therefore, it focuses segmentation networks on the task-dependent grayscale distribution in CTA images to learn the fine-grained representation for fine segmentation.

Problem Formulation

As shown in Fig. 2, the automatic search for the grayscale range of interest has to scale the dimensions of window width w and center c. The width w determines the size of the grayscale range covered by the window, the wider the window, the richer distribution information will be covered, but a wider window will weaken the discrimination between the distributions of the structures thus reducing the contrast of medical images. The center c determines the position of the window, the closer the window center is to the task-dependent distribution, the clearer the details of these structures will appear in images. We define the images in sub-window as X w,c = W w,c X [gmin,gmax] where the W is the windowing [START_REF] Taha | Metrics for evaluating 3D medical image segmentation: analysis, selection, and tool[END_REF] of all targeted renal structures as the metric to evaluate these segmentation results on the searched sub-window images, thus the searching process of the task-dependent grayscale distribution is as follow:

max w,c DSC (Y(w, c)) s.t. Y(w, c) = N w,c W w,c X [gmin,gmax] c - w 2 ≥ g min c + w 2 ≤ g max (1)
We take the grid search method [START_REF] Bergstra | Random search for hyper-parameter optimization[END_REF] to optimize these two hyper-parameters, and scale an initial window width w 0 = φ and center c 0 = φ in integer multiples using two cross-correlated coefficients, α and β, to reduce the search space:

width : w α = αφ center : c β = βφ s.t. α ∈ Z + β ∈ Z + α + 2β ≤ 2g max φ α + 2β ≥ 2g min φ (2)
where the α and β are the cross-correlated scaling coefficients which have to match the condition

2gmin φ ≤ α + 2β ≤ 2gmax φ
for the width and center. In our experiment, we set φ = 256 to make the grayscale window grid (Fig. 5), so that 32 sub-windows on our grayscale window grid are searched. In particular, we find the best values are α = 3 and β = 5.

Adaptively search the task-dependent distribution range

As shown in Fig. 5, we make a grayscale window grid adaptively via scaling the dimensions of window width and center, and our GIS strategy searches the task-dependent distribution on this grid.

Windowing expends distribution for fine-grained pattern. As shown in Fig. 4(c), a windowing process W removes irrelevant grayscale ranges and expands the distributions of the renal structures thus improving the contrast and bring salient ROIs, so that the network will focus on a wider distribution of the renal structures to perceive their fine-grained patterns. It takes the window width w α and center c β from the grayscale window grid firstly, and then thresholds the original image x on the boundaries, c β -wα 2 and c β + wα 2 , of the sub-window for a thresholded image x thr . Finally, the thresholded image is normalized for sub-window images x w,c to [0, 1] making more salient renal structures. Therefore, the whole windowing process is as follow:

x w,c = W w,c (x) = max(min(x, c β -wα 2 ), c β + wα 2 ) -(c β -wα 2 ) w α (3)
Adaptive sub-window search for task-dependent distribution. Our adaptive sub-window search trains multiple segmentation networks iteratively on our grayscale window grid and evaluates via the DSC metric [START_REF] Taha | Metrics for evaluating 3D medical image segmentation: analysis, selection, and tool[END_REF] for the best window width w and center c which covers the taskdependent distribution. As shown in Fig. 5, multiple sub-window images on the grayscale window grid train their corresponding segmentation networks iteratively to adapt to the segmentation of renal structures on different sub-windows. Therefore, seg- Summary of the advantages. 1) Our GIS strategy adaptively discovers our IRS segmentation taskdependent distribution, so that it will automatically provide the deep learning model with a more appropriate hyper-parameter of the grayscale distribution range. 2) Our GIS strategy removes irrelevant grayscale ranges and expands the taskdependent distribution so that the interruption of task-unconcerned distributions' variations will be weakened, and the contrast of the ROIs will be augmented, thus focusing segmentation networks on a wider distribution of the renal structures to perceive fine-grained patterns.

Meta grayscale adaptive learning for personalized adaptation of distribution variation

Our MGA learning (Fig. 4 (b)) makes an imagelevel meta-learning to extract diverse robust features from multiple distributions via our Multi-windowing Group, perceive the distribution characteristics of each image via our Meta Perceiver, and fuse the meta knowledge (trained kernels) and the personalized knowledge (meta kernels) for a personalized fusion process via our Meta ResConv, thus adapting the distribution variation.

3.2.1. Multi-windowing Group for robust feature representation Our Multi-windowing Group extracts features from multiple distributions covered by different subwindows, thus representing the images in multi-views making the features more diverse to reduce the overfitting of representation preferences to a single distribution. For a CTA image, it obtains M sub-window images {x 0 , ..., x M } with different distributions from the searched M sub-windows via windowing, improving the diversity of image distributions. Independent segmentation networks {N 0 , ..., N M } trained in the searched sub-windows during our GIS process learn the representation of the distributions covered by these sub-windows, and the corresponding sub-window images are put into these networks for the features before the output convolutional layers. These features are concatenated for the features F which integrate the representation of multiple distributions:

F = cat(N 0 (x 0 ), ..., N M (x M )) (4)
where the 'cat' means the concatenation process. Therefore, the features from different distributions will have richer representation capabilities and can be more robustly deal with distribution changes.

Meta perceiver perceives distribution characteristic

Our meta perceiver P perceives the distribution characteristic of each image to dynamically fuse the features from the trained segmentation networks of the M grayscale distributions covered by the searched sub-windows. The features from our Multiwindowing Group are put into our meta perceiver P together with the original image x to generate meta kernels Ŵmeta , bmeta . The meta kernels will fine-tune the trained kernels initialized by the trained weights in the output layers of the segmentation networks for personalized fusion parameters in our Meta ResConv, thus dynamically fusing the features according to the image's distribution:

Ŵmeta , bmeta = P(F, x)

(5)

Meta residual convolution for stable personalized fusion

The personalized fusion parameters from our Meta ResConv has two components: trained kernels W conv , b conv and meta kernels Ŵmeta , bmeta . The trained kernels represent the meta knowledge of our IRS segmentation task and the meta kernels from our Meta Perceiver represent the personalized knowledge of each image. Therefore, the meta kernels dynamically fine-tune the trained kernels via residual [START_REF] He | Deep residual learning for image recognition[END_REF], generating the personalized fusion parameters and improving the generalization for varied distributions inter-images. As shown in Fig. 6, we use the weights of output convolutional layers in the trained segmentation networks to initialize the trained kernels. The meta kernels Ŵmeta , bmeta from our Meta Perceiver are added to the trained kernels W conv , b conv making a dynamic meta Conv Ŵ, b adjust the attention degree of different components in the features F dynamically. Therefore, the representations of different distributions will be dynamically fused according to the distribution characteristic of each image. The formulation of this process is:

Ŵ = Ŵmeta + W conv b = bmeta + b conv ŷ = σ( ŴT × F n + b) (6)
Here, the σ is a softmax activation function for our segmented IRS mask, the n ∈ N is the position of feature vectors in three-dimensional feature maps. The × is the matrix multiplication and the ŴT is the transpose of the matrix Ŵ.

Our Meta ResConv is a dynamic fine-tuning of the output convolutional kernels of the segmentation networks which have been trained in our searched sub-windows. When our meta kernels tend to zero Ŵmeta → 0, bmeta → 0, the Meta Conv degenerates to the original kernels of the trained output convolution layers Ŵ ≈ W conv , b ≈ bconv , so our MGA learning will degenerate to an average ensemble strategy [START_REF] Polikar | Ensemble Machine Learning[END_REF]. Therefore, our MGA learning is a dynamical enhancement of average ensemble which further considers the personalized optimization of distribution variation for each image, so it will bring the generalization of varied distributions inter-images and stable performance.

Summary of the advantages. 1) Our Multwindowing Group integrates feature representation capabilities from multiple distributions, thus extracting more diverse features and reducing the overfitting of representation preferences to a single distribution. 2) Our Meta Perceiver perceives distribution characteristics of each image and our Meta ResConv generates personalized model parameters to fuse the features from different sub-windows, thus adapting the distribution variation in our 3D IRS segmentation task. 3) Our Meta ResConv block uses residuals to generate new fusion parameters based on trained kernels, thus greatly reducing the output space of meta kernels, so that the generated fusion parameters will not make large fluctuations and the personalized fusion for distribution variation will be stable.

Details of the networks

We follow DenseBiasNet structure [START_REF] He | Dense biased networks with deep priori anatomy and hard region adaptation: Semisupervised learning for fine renal artery segmentation[END_REF] as our segmentation network for our IRS segmentation and take continuous convolutions, pooling, ac- tivations, and full connections as our meta perceiver for the perception of discriminative compositions in features.

DenseBiasNet for IRS segmentation

As shown in Fig. 7 (a), we follow DenseBiasNet [START_REF] He | Dense biased networks with deep priori anatomy and hard region adaptation: Semisupervised learning for fine renal artery segmentation[END_REF] as our segmentation network for IRS segmentation, it fuses multi-receptive fields and multi resolution features for the adaptation of scale changes and has achieved success in renal artery segmentation. We take seven resolution stages and each stage has two 3×3×3 convolutional layers followed by a group normalization [START_REF] Wu | Group normalization[END_REF](GN) and a ReLU activation. Max-pooling and up-sampling layers are used to change the resolution. After the final stage, a 1×1×1 convolutional layer followed by a softmax outputs our segmented IRS. The feature maps from each Conv-GN-ReLU layer are compressed via a 1 × 1 × 1 convolutional layer and transmitted to every forward layer thus making a full dense connection network. The number of transmitted feature maps in this network is a hyper-parameter k, and in our experiments, it is 4. To train DenseBias-Net, a crossentropy (ce) loss function L ce [START_REF] Taha | Kid-Net: convolution networks for kidney vessels segmentation from CT-volumes[END_REF], which has good convergence [START_REF] Goodfellow | Deep learning[END_REF] and is widely used in medical image segmentation tasks, is calculated between the IRS labels y and their segmented results ŷ.

Meta Perceiver perceives distribution characteristic

As shown in Fig. 7 (b), our Meta Perceiver takes continuous convolutions, pooling, activations, and full connections for meta kernels which will give a higher response to the discriminative compositions of the feature vectors in feature maps. It has five resolution stages, each stage has one 3 × 3 × 3 convolutional layer followed by a GN and a ReLU activation. Max-pooling follows each stage for down-sampling of the feature maps. Finally, the feature maps are compressed into a one-dimensional feature vector via a global average pooling layer [START_REF] Lin | Network in network[END_REF] and this feature vector is put into two full connections for our meta kernels which will enhance the task-beneficial features. These meta kernels are trained to optimize the trained kernels initialized by the weights of the trained output convolution layers following our Sec. 3.2 via the L ce .

Experiments Configurations

Dataset. Abdominal CTA images from 123 patients who have a tumor in their unilateral kidney were retrospectively selected from the radiology department of Jiangsu Province Hospital in this study. Each image is acquired on a Siemens dual-source 64-slice CT scanner and contrast media was injected during the CT image acquisition. This dataset has five different kidney tumor types including clear renal cell carcinomas, papillary, chromophobe, angiomyolipoma, and eosinophilic adenoma resulting in large heterogeneity and distribution variation. Pixel sizes of these images are between 0.47 mm/pixel and 0.74 mm/pixel, and their slices thicknesses are 0.5 mm/pixel. The cancerous kidney ROIs whose size is 150 × 150 × /200 are pre-extracted automatically via a multi-atlas-based approach following [START_REF] Yang | Automatic kidney segmentation in CT images based on multi-atlas image registration[END_REF]. Kidney, tumor, veins, and arteries are finely labeled by two radiologists with cross-checking.

Data augmentation. To avoid overfitting and improve the generalization of models, the dataset is augmented via batch-generators1 . The combination of random rotation (-20°↔ 20°), cropping (128 × 128 × 128), mirror (in the axis of x, y and z), scaling (0.75 ↔ 1.25 times) and elastic deformation, enlarges the dataset 1,000 times, thus making 123,000 samples. Implementation. Our MGANet has two training stages to search the task-dependent grayscale distributions and learn to fuse the features from multiple distributions dynamically. In the first stage, the initial window width w 0 and center c 0 are 256 and 256, thus 32 sub-windows on our grayscale window grid are searched. The optimization setting of the corresponding segmentation networks (DenseBi-asNet [START_REF] He | Dense biased networks with deep priori anatomy and hard region adaptation: Semisupervised learning for fine renal artery segmentation[END_REF]) trained in these sub-windows is the Adam with the batch size of 1, learning rate of 1 × 10 -4 and iterations of 1.6 × 10 5 . The DSC of their results is ranked for the four best sub-windows. In the second stage, the weights of DenseBiasNets on the searched four sub-windows are fixed to extract features, and the kernels of their output convolutional layers are used to initialize the trained kernels during the training of our Meta ResConv. Same as the first stage, The optimization setting of our Meta perceiver is the Adam with the batch size of 1, learning rate of 1 × 10 -4 , and iterations of 1.6 × 10 5 .

The framework is implemented with PyTorch2 and runs on NVIDIA TITAN Xp GPUs. Five-fold crossvalidation is performed for comprehensive evaluation. All models in our experiments are optimized for 1.6× 10 5 iterations.

Comparison settings. To illustrate the superiority of our framework, we compare the proposed MGANet with V-Net [START_REF] Milletari | V-Net: fully convolutional neural networks for volumetric medical image segmentation[END_REF], 3D U-Net (C ¸içek et al., 2016), Res-U-Net [START_REF] Li | Segmentation of renal structures for image-guided surgery[END_REF], Kid-Net [START_REF] Taha | Kid-Net: convolution networks for kidney vessels segmentation from CT-volumes[END_REF], Dense-BiasNet [START_REF] He | Dense biased networks with deep priori anatomy and hard region adaptation: Semisupervised learning for fine renal artery segmentation[END_REF], and an ensemble model, VFN [START_REF] Xia | Bridging the gap between 2D and 3D organ segmentation with volumetric fusion net[END_REF] on CTA images. To further study the nature of our GIS strategy, we analyze the performance of the segmentation network in different sub-windows, and the variation of their ensemble performance with the number of integrated learners changes. All methods use a same implementation.

Evaluation metric. The segmentation performance of each method is evaluated in two aspects [START_REF] Taha | Metrics for evaluating 3D medical image segmentation: analysis, selection, and tool[END_REF] following [START_REF] He | Dense biased networks with deep priori anatomy and hard region adaptation: Semisupervised learning for fine renal artery segmentation[END_REF]. 1) The area-based metric: we take the Dice coefficient (DSC) to evaluate the area-based overlap index. 2) The distance-based metric: we take the average Hausdorff distance (AVD) to evaluate the coincidence of the surface for stable and less sensitive to outliers. To further compare the segmentation quality of outliers, we also take the Hausdorff distance (HD) which is sensitive to outliers. Average DSC (%), AVD (mm), and HD (mm) of all renal structures are also reported together with their standard deviation (±std).

Results and Analysis

Our MGANet achieves fine-grained feature representation and bridges the representation preferences caused by the distribution variations thus achieving an excellent 3D integrated renal structures segmentation. In this part, we will thoroughly evaluate and analyze the effectiveness of our proposed MGANet: 1) The quantitative evaluation and qualitative evaluation in the comparative study (Sec. 5.1) will show the superiorities of our framework compared with other models.

2) The ablation study (Sec. 5.2) will demonstrate the contribution of our innovations and the influence of different hyper-parameters in our framework.

3) The GIS process, the number of fused subwindows, and the meta kernels will be analyzed in framework analysis (Sec. 5.3)

Comparative study shows our superiority

Quantitative evaluation for metric superiority

As demonstrated in Tab. 1, our MGANet achieves excellent performance compared with other methods. In our MGANet, the artery achieves 89.0% DSC, 22.67 mm HD, and 0.60 mm AVD which will strongly support the perfusion regions estimation and the arterial clamping position selection. The kidney gets 95.1% DSC, 12.28 mm HD, and 0.53 mm AVD, the tumor gets 86.4% DSC, 29.85 mm, and 2.76 mm AVD, and the veins get 81.0% DSC, 18.94 mm HD, and 0.90 mm AVD which will provide the operation guidance. The ensemble model, VFN, fuses 2D information in different perspectives, achieving good performance on tumors (86.3% DSC) and fewer outliers on veins (18.14 mm HD). However, it has poor performance on arteries (81.8% DSC) because of the low contrast in original images and the small arteries in 2D slices. The Kid-Net takes deep supervision thus getting competitive performance with smaller average surface distances on tumors (2.42 mm AVD), but it only gets 78.0% DSC on arteries and 75.4% DSC on veins. Dense-BiasNet which is our basic segmentation network has a decent performance in four structures and achieves 84.6% AVG DSC on the original window. When taking our GIS and MGA, our MGANet achieves a significant average improvement of 3.3% DSC in each renal structure.

Qualitative evaluation for visual superiority

As illustrated in Fig. 8, our proposed MGANet has great visual superiority which will provide visual guidance for renal surgery. In case 1, our MGANet achieved fine IRS segmentation results visually, while DenseBiasNet, Kid-Net, and Res-U-Net have serious mis-segmentations (yellow arrows) between the kidney and the tumor. V-Net and VFN have rough segmentation in tumor regions due to the unclear boundaries caused by the narrow grayscale range and the variable grayscale distribution of tumors. In case 2, the enlarged details show fine segmentation quality on the boundaries of the tumor. The comparison methods segment the tumor regions outside the kidney into the kidney (yellow arrows) owing to the unclear boundaries caused by low contrast in the original window and the representation preferences of the varied tumor distributions. In case 3, the enlarged hilum regions illustrate the excellent segmentation quality of our method in complex regions. The kidneys, veins, and arteries structures are crowded in the narrow renal hilum region, which will make the mis-segmentation of these structures. Our MGANet has fine and complete segmentation of veins, while the other comparison methods have serious mis-segmentation of vein regions that have low contrast in such a complex environment.

Table 1: The quantitative evaluation demonstrates the superiority of our MGANet for our 3D IRS segmentation task. Our MGANet achieves better performance than the comparison methods (V-Net, 3D U-Net, Res-U-Net, Kid-Net, VFN, and Dense-BiasNet) on all four renal structures. The EN means that the model is an ensemble method and the AVG DSC means the average DSC on the four structures.

Method

Ablation study shows improvements of the innovations

As illustrated in Tab. 2, our innovations bring significant enhancements. The DenseBiasNet in the original window achieves 94.2%, 82.6%, 75.3%, and 86.1% DSC on kidneys, tumors, veins, and arteries. When taking our GIS strategy, the best subwindow (w = 768, c = 1280) augments the contrast of the CTA image and make the network focus on a wider distribution of the renal structures to perceive fine-grained patterns, so that the veins and arteries achieve 3.9% and 1.9% DSC improvements. The AVG EN-4 in Tab. 2 is an ensemble models which takes an average ensemble strategy [START_REF] Polikar | Ensemble Machine Learning[END_REF] to calculate the average of the segmentation results from the networks trained in the top four sub-windows. When we take the average ensemble strategy (AVG EN-4), it achieves 95.0%, 85.5%, 79.5%, and 88.1% DSC on kidneys, tumors, veins, and arteries. When using our MGA learning to make the personalized fusion of networks' preferences for different distributions, our MGANet gets great improvements, especially on veins, it achieves the significant 1.5% DSC improvement compared with the average ensemble strategy and finally obtains the 87.9% average DSC for our 3D IRS segmentation task.

Framework analysis

Grayscale interest search analysis

Our GIS strategy achieves multiple sub-windows which will perform better than original CTA images in renal structures. As shown in Fig. 9, we calculate the average DSC of the segmented renal structures is, highlight and select the highest four windows as the sub-windows in our MGANet. 1) The networks' performance present different characteristics of the renal structures when the sub-windows are changed. The kidney has a more stable performance when the sub-window changes than other structures, because it has a relatively large volume, regular shape, and wide grayscale range making the stable learning of feature representation. The vein, tumor, and artery are sensitive to the window center and width, because their volumes are small and grayscale ranges are narrow making the searched sub-windows difficult to cover in some extreme sub-windows. 2) Since the renal 3) The networks trained in the sub-windows which cover the task-dependent distributions will achieve better performance than those trained in the original window. The network trained in the original window achieves 84.6% average DSC which is 1.7% DSC lower than the network trained in the best sub-window.

Fused networks' amount analysis

As shown in Fig. 10, we analyze the DSC change of the models with the amount of the fused networks trained in our GIS increasing. The networks trained during our GIS process are fused start from the highest average DSC. It illustrates the characteristic in two aspects: 1) The fusion performance increases and then decreases. Due to the severe degradation of the networks accuracy in some singular sub-windows (noisy networks), the average DSC first increases and then decreases with the amount of the fused networks increasing. 2) Renal structures have different sensitivity to the increasing of the fused networks. Kidney has little fluctuation due to its large volume and wide grayscale range, making its DSC have less fluctuated. Tumors are sensitive and when the models fuse more than 20 networks, the noisy networks will seriously interrupt the whole accuracy and make severe degradation. Considering model performance and com-

Meta kernels analysis

As shown in Fig. 11, the meta kernels in our MGA learning fine-tune the trained kernels to dynamically adapt the variation of the grayscale distributions. 1) The renal structures have different meta kernel values to fine-tune the trained kernels in different degrees. Because the renal structures have different distributions which present different characteristics, our meta perceiver perceives their characteristics and output the best fine-tuning degrees (meta kernels) for renal structures enhancing the final results. 2) The cases have different meta kernels to fine-tune the trained kernels for the adaptation of the distribution varia- tion. Owing to our MGA learning's significant improvement on veins, we show the meta kernels of the vein in two cases. As the distribution changes interimages, our meta perceiver will perceive the variation and the fine-tuning degree of the trained kernels will also dynamically change.

Discussion and Conclusion

In this paper, we propose and systematically study the grayscale distribution optimization in the 3D IRS segmentation task, and propose the MGANet for 3D IRS segmentation which will provide great technical support for the treatment of renal cancer for the first time. Our MGANet takes the advantage of our GIS strategy and MGA learning achieving powerful generalization ability. Our GIS strategy (fig. 4 (a)) adaptively searches and expands our task-dependent distributions to help segmentation networks learn finegrained feature representation for renal structures. Our MGA learning (fig. 4 (b)) takes image-level meta learning strategy which makes each image as a sub-learning task. It represents diverse robust features from multiple distributions, perceives the distribution characteristic of each sub-learning task, and generates the personalized parameters to fuse these features dynamically according to the sub-learning task's distribution, thus adapting the grayscale distribution variation.

Fine optimization of grayscale distribution significantly improves the performance of medical image segmentation without adding additional model parameters. The experiments of our GIS strategy verified that: 1) The images in suitable sub-window will obtain better performance than in the original window because the sub-window expands the taskdependent distribution making networks focus on a wider distribution of the renal structures to perceive fine-grained patterns. fig. 9 shows that the best sub-window achieves 1.7% average DSC improvement compared with the original window. 2) The segmentation objects have different best sub-windows owing to their different distribution characteristics. For example, the vein achieves the best DSC in (w = 1280, c = 1024), while the tumors best subwindow is (w = 1280, c = 1280). As shown in fig.

12, our GIS strategy automatically searches multiple window centers and widths for windowing to augment the contrasts which adapt to our renal structures, thus bringing clear boundaries, salient regions, and fine-grained textures in different sub-windows. Therefore, the model obtains the discriminative representation of different structures with such stronger regional differences, and finally obtains higher accuracy. Compared with the other methods, our framework achieves higher DSC, lower HD, and AVD in Tab. 1 illustrating our significant improvement.

Our 3D IRS segmentation provides visual assistance for preoperative plans for LPN having great clinical significance. 1) A clinical case (fig. 1 (b)) demonstrates preoperative plans via our 3D IRS segmentation. The 3D IRS model will help doctors estimate the renal perfusion model, locate the tumor resection surface and point the arterial clamping positions visually for preoperative plans. 2) Intraoperatively, the preoperative plan will be displayed on screen together with laparoscopic videos to guide the LPN (fig. 1 (c)). Therefore, 3D IRS visual model will supplement the invisible regions in the laparoscope, so that the difficulties of LPN will be reduced and the quality of LPN will be improved. 3) Thanks to the high computing speed of deep learning frameworks on GPUs, our MGANet has good time efficiency. It only takes 2.305 seconds on each case which will greatly reduce the time cost for doctors to obtain the renal structures 3D model.

Future works. Hyper-parameter optimization for grayscale distribution in medical image analysis is a research direction worthy of further exploration. There are many potential research topics such as the quick search of this hyper-parameter, the grayscale distribution optimization for other medical image tasks, the application of grayscale distribution optimization in cross-domain learning, etc.

Figure 1 :

 1 Figure 1: 3D IRS segmentation will provide accurate preoperative planning and intraoperative guidance for LPN

Figure 2 :

 2 Figure 2: Narrow task-dependent distribution makes original CTA image low contrast and windowing on the distribution makes sub-window image fine-grained pattern. (a) The original CTA image has narrow distribution making low contrast. (b,d) When the window covers the unconcerned distribution, the renal structures will be lost. (c) When the window covers the task-dependent distribution, the sub-window image will have a fine-grained pattern

Figure 3 :

 3 Figure 3: Different renal tumor subtypes make the images have large distribution variation. (a-e) The renal tumor subtypes have different volumes and abilities to metabolize contrast agents making distribution variations[START_REF] Young | Clear cell renal cell carcinoma: discrimination from other renal cell carcinoma subtypes and oncocytoma at multiphasic multidetector CT[END_REF]. f) The grayscale distributions of the tumor subtypes have a large variation. The y-axis is the number of voxels and the x-axis is the grayscale value

Figure 4 :

 4 Figure 4: Our MGANet adaptively optimizes the grayscale distribution of medical images via our GIS strategy and MGA learning, achieving the 3D IRS segmentation

Figure 5 :

 5 Figure 5: Our GIS strategy for our task-dependent grayscale distribution making the networks fine-grained feature representation

  mented results on these windows are available to calculate their average DSCs on the four renal structures. The DSC ranking step ranked these average DSCs. For our following Meta Grayscale Adaptive learning, our framework selects top M windows as the sub-windows. Evaluated by our experiments, we set the M = 4, considering model performance and computational efficiency. We follow DenseBiasNet[START_REF] He | Dense biased networks with deep priori anatomy and hard region adaptation: Semisupervised learning for fine renal artery segmentation[END_REF] as our segmentation networks in our GIS process. The details of DenseBiasNet is introduced in Sec. 3.3.1.

Figure 6 :

 6 Figure 6: Our Meta ResConv takes the trained kernels as basic parameters and fine-tunes them via our meta kernels from our Meta Perceiver for the adaptation of each image.

Figure 7 :

 7 Figure 7: The detailed basic segmentation network structures (DenseBiasNet) for IRS segmentation and meta perceiver for high-order meta kernels.

Figure 8 :

 8 Figure 8: The qualitative evaluation illustrates the visual superiority of our proposed MGANet

Figure 9 :

 9 Figure 9: The heatmaps show the DSC of four renal structures and their average performance from our GIS process

Figure 11 :

 11 Figure 11: The meta kernels in our MGA learning fine-tune the trained kernels to dynamically adapt the variation of the grayscale distributions.

Figure 12 :

 12 Figure 12: Compared with the original images, our GIS strategy automatically searches multiple window centers and widths for windowing to augment the contrasts which adapt to our renal structures, thus bringing clear boundaries, salient regions, and fine-grained details in different sub-windows

https://github.com/MIC-DKFZ/batchgenerators

https://pytorch.org/

Acknowledgements

This research was supported by the National Natural Science Foundation under grants (31571001, 61828101, 31800825), Southeast University-Nanjing Medical University Cooperative Research Project (2242019K3DN08) and Excellence Project Funds of Southeast University. We thank the Big Data Computing Center of Southeast University for providing the facility support on the numerical calculations in this paper.


Pierre Charollois

Matthew Greenberg
email: mgreenbe@ucalgary.ca

RATIONALITY OF SECANT ZETA VALUES

Keywords:

We use the Arakawa-Berndt theory of generalized η-functions to prove a conjecture of Lalín, Rodrigue and Rogers concerning the algebraic nature of special values of the secant zeta function.

Introduction

The cotangent and secant zeta functions attached to an algebraic irrational number α are defined, for Re(s) large enough, by the Dirichlet series

ξ(α, s) = ∞ n=1 cot(πnα) n s and ψ(α, s) = ∞ n=1 sec(πnα) n s ,
respectively. The cotangent zeta function was introduced for s = 2k + 1, k ≥ 1 an integer, by Lerch [START_REF] Lerch | Sur une série analogue aux fonctions modulaires[END_REF], and for general s by Berndt [START_REF] Berndt | Dedekind sums and a paper of G[END_REF] in the course of his study of generalized Dedekind sums. Lerch stated (without proof) the following functional equation valid for algebraic irrational α and sufficiently large k = k(α):

(1)

ξ(α, 2k + 1) + α 2k ξ( 1 α , 2k + 1) = (2π) 2k+1 ϕ(α, 2k + 1)
, where

ϕ(α, n) = n+1 j=0 B j B n+1-j j!(n + 1 -j)! α j-1 ,
and

B k is the k-th Bernoulli number. Suppose α = ±1 is a unit in the quadratic field Q( √ d) of discriminant d. Writing α = a+b √ d 2 and defining ε = ±1 by a 2 -b 2 d = ε, we have 1 α = -ε(α -a).
Using the obvious identities ξ(-α, s) = -ξ(α, s) and ξ(α -1, s) = ξ(α, s) together with (1), we deduce the following rationality result:

(2) ξ(α, 2k + 1) (2π) 2k+1 √ d = 1 √ d ϕ(α, 2k + 1) 1 -εα 2k ∈ Q.
Another proof of this result was given by Berndt [4, Theorem 5.2]. More generally, if α is an arbitrary real quadratic irrationality, then Lerch uses the continued fraction expansion of α to conclude that the left hand side of (2) belongs to Q(α) -see [7, (3)].

Lalín, Rodrigue and Rogers conjecture an analogous result for the secant zeta function:

Conjecture 1.1 ([6, Conjecture 1]). Suppose d and k are positive integers such that d is not a square. Then

ψ( √ d, 2k) π 2k ∈ Q.
They prove a functional equation for ψ(α, s) analogous to [START_REF] Arakawa | Generalized eta-functions and certain ray class invariants of real quadratic fields[END_REF] and use it to deduce many instances of Conjecture 1.1 as above. In §2, we show that their functional equation is actually sufficient to prove in general that ψ(α, 2k) ∈ π 2k Q(α) for all real quadratic irrationalities α, not merely those of the form α = √ d. In §3, we relate ψ(α, s) to the generalized η-functions studied by Arakawa [START_REF] Arakawa | Generalized eta-functions and certain ray class invariants of real quadratic fields[END_REF], fascinating objects in their own right. Using this relationship, we leverage Arakawa's results to give an explicit formula for ψ(α, 2k) for real quadratic irrationalities α, yielding another proof of Conjecture 1.1.

The Lalín-Rodrigue-Rogers functional equation

One can prove Conjecture 1.1 following Lerch's approach for the cotangent zeta function. Most of this argument was given in [START_REF] Lalín | Secant zeta functions[END_REF]; we complete their thought. Let A, B ∈ PSL 2 (Z) be defined by

A = 1 2 0 1 , B = 1 0 2 1 .
The following functional equations are established in [6, (4.1), (4.2)]:

ψ(Aα, 2k) = ψ(α, 2k), ψ(Bα, 2k) = (2α + 1) 1-2k ψ(α, 2k) - π 2k (2k)! 2k m=0 (2 m-1 -1)B m E 2k-m 2k m (α + 1) 2k-m (2α + 1) m-2k -(2α + 1) 1-2k .
Here, B n is the n-th Bernoulli number and E n is the n-th Euler number. It follows that if C ∈ A, B then there is a Q(α)-linear relation between ψ(Cα, 2k), ψ(α, 2k) and π 2k . Thus, if there is a matrix

C ∈ A, B such that Cα = α then ψ(α, 2k) ∈ π 2k Q(α). In [6, §4],
several families of examples of such pairs (α, C) are given and the associated linear relations are worked out explicitly. We merely point out that if α is any real quadratic irrationality, then there is always a C ∈ A, B , C = 1 such that Cα = α.

To see this, let α be a real quadratic irrationality and consider the lattice L = Z + Zα ⊂ Q(α). Let O be the order of L:

O = {x ∈ Q(α) : xL ⊂ L}. Then O is an order in Q(α). Let u ∈ O. Writing u • α = aα + b and u • 1 = cα + d with a, b, c, d ∈ Z, we have u α 1 = a b c d α 1 .
Write j(u) for the matrix (

Generalized η-functions and secant zeta values

Arakawa [START_REF] Arakawa | Dirichlet series ∞ n=1 cot(πnα)/n s , Dedekind sums, and Hecke L-functions for real quadratic fields[END_REF] gave another proof of (2) by relating ξ(α, s) to generalized η-functions, the theory of which he developed in [START_REF] Arakawa | Generalized eta-functions and certain ray class invariants of real quadratic fields[END_REF]. In turn, Arakawa's work has its foundations in papers of Lewittes [START_REF] Lewittes | Analytic continuation of Eisenstein series[END_REF] and Berndt [START_REF] Bruce | Generalized Dedekind eta-functions and generalized Dedekind sums[END_REF]. We show that Arakawa's method can also be used to analyze the secant zeta function.

For x ∈ R, define x (resp., {x}) by 0 < x ≤ 1 (resp. 0 ≤ {x} < 1) and x -x ∈ Z (resp., x -{x} ∈ Z).

We set e(z) = e 2πiz . Following [START_REF] Arakawa | Generalized eta-functions and certain ray class invariants of real quadratic fields[END_REF], let p, q ∈ R and define

η(α, s, p, q) = ∞ n=1 n s-1 e(n(pα + q)) 1 -e(nα)
H(α, s, p, q) = η(α, s, p , q) + e s 2 η(α, s, -p , -q). Theorem 3.1 ([1, Lemma 1 and Theorem 2]). Suppose

α ∈ R ∩ Q and α / ∈ Q.
Then η(α, s, p, q) is absolutely convergent for (s) < 0. If, in addition,

[Q(α) : Q] = 2 and p, q ∈ Q
then H(α, s, p, q) has analytic continuation to C -{0}, and the singularity at s = 0 is at worst a simple pole.

Remark 3.2. The convergence of η(α, s, p, q) relies on the Thue-Siegel-Roth theorem in much the same way that the convergence of ψ(α, s) does -see [START_REF] Lalín | Secant zeta functions[END_REF]Theorem 1].

An elementary computation yields

ξ(α, s) = -2i H(α, 1 -s, 1, 0) 1 + e( s 2 ) + 1 2 ζ(s) ,
so rationality statements for H(α, 1 -s, 1, 0) for even integral s translate to rationality statements for ξ(α, s) at odd integral s. In contrast, ψ(α, s) does not seem to have a simple expression in terms of H(α, s, p, q). Crucially, however, we still have a relation between certain special values of H and ψ relying on the relation 3 4 = -1 4 + 1 :

ψ α 2 , 1 -s = ∞ n=1 n s-1 2 e( nα 4 ) + e(-nα 4

  a b c d ) associated to u above. Then j : O → M 2 (Z) is a ring homomorphism. Since ( α 1 ) is an eigenvector of j(u) for all u ∈ O, we have j(u)α := aα+b cα+d = α when u = 0. By Dirichlet's unit theorem, the group O * + of totally positive units in O is free of rank 1; write O * + = γ . By [9, p. 84], A, B is the principal congruence subgroup Γ(2) ⊂ PSL 2 (Z), this inclusion having index 6. Therefore, C := j(γ 6 ) satisfies C = 1, C ∈ Γ(2) and Cα = α.

Theorem 3 . 4 .

 34 α, s,1 4 , 0) -2η(α, s, -1 4 , 0). If s = 1 -2k, then e( s 2 ) = -1 and we conclude that (3)ψ( α 2 , 2k) = 2H(α, 1 -2k, 1 4 , 0). Suppose α is a real quadratic irrationality and k is a positive integer. Then ψ(α, 2k)π 2k ∈ Q(α). Moreover, if x → x is the nontrivial automorphism of Q(α), then ψ(α, 2k) π 2k = ψ(α , 2k) π 2k .Conjecture 1.1 follows from Theorem 3.4 and the evenness of the secant function.

PC's research is partially supported by the grant REGULATEURS "ANR-12-BS01-0002". MG's research is supported by NSERC of Canada.

For the rest of the paper, suppose that α is a real quadratic irrationality. Formulas of Berndt and Arakawa allow us to evaluate H(α, 1 -2k, 1 4 , 0) rather explicitly. Let

be a matrix such that c > 0 and β := cα + d > 0.

Set (p , q ) = (p, q)V and = {q }c -{p }d.

Theorem 3.3 ([1, Theorem 1 and Eq. (1.19)]). Suppose that p and p are not in Z. Then the following transformation formula holds:

where L(α, s, p , q , c, d) is as in (2) of [START_REF] Bruce | Generalized Dedekind eta-functions and generalized Dedekind sums[END_REF]. If s = -m is a negative integer, then

Here, b is the (normalized) -th Bernoulli polynomial defined by the generating series

By [1, Lemma 4], for any rational numbers p and q there is a totally positive unit β of Q(α) and a matrix V ∈ SL 2 (Z) such that c > 0, (p , q ) := (p, q)V ≡ (p, q) (mod Z 2 ), and

The last condition implies that β = cα + d, consistently with the notation introduced above. Suppose p / ∈ Z . Then p / ∈ Z, too, as p ≡ p (mod Z). Applying Theorem 3.3, observing that H and L only depend on the class of (p, q) ≡ (p , q ) modulo Z 2 , and rearranging terms, we get (β -s -1)H(α, s, p, q) = (2π) -s e(-s 4 )L(α, s, p, q, c, d). By the second part of Theorem 3.3, if s = 1 -2k then ( 4)

Setting (p, q) = ( 1 4 , 0) and using (3), this formula specializes to We conclude :


Spiros Kotopoulis

Ragnhild Haugse

Maja Mujic

Andre Sulen

Stein-Erik Gullaksen

Odd Emmet Mc Cormack

Helge Gilja

Michiel Postema

Tore Bjørn

Gjertsen

Evaluation of the effects of clinical diagnostic ultrasound in combination with ultrasound contrast agents on cell stress: single cell analysis of intracellular phospho-signaling pathways in blood cancer cells and normal blood leukocytes

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Evaluation of the effects of clinical diagnostic ultrasound in combination with ultrasound contrast agents on cell stress: Single cell analysis of intracellular phospho-signaling pathways in blood cancer cells and normal blood leukocytes Spiros Kotopoulis * † , Ragnhild Haugse ‡ § , Maja Mujic ‡ , Andre Sulen ‡ , Stein-Erik Gullaksen ‡ , Emmet Mc Cormack ‡ Odd Helge Gilja * ¶ , Michiel Postema † and Bjørn Tore Gjertsen ‡

With the increased interest in therapeutic ultrasound it is becoming ever more important to understand the effects of ultrasound on cells.

In our work here we investigate the effect of clinical diagnostic ultrasound on several cell signalling proteins (p38 p-Thyr180/p-Thyr182, ERK 1/2 p-Thyr202/p-Thyr204 and p53 ac-Lys382) on leukaemia cells (MOLM-13) and monocytes.

Our results show that leukaemia cells and monocytes react differently to ultrasound and microbubbles. A relatively small increase in p38 signalling was seen in the leukimic cells, and only at higher intensities in combination with microbubbles. In contrast the monocytes showed an increase in p38 signalling at all acoustic intensities with microbubbles and at the high acoustic intensity without microbubbles.

Furthermore, the leukaemic cells showed an overall increase in ERK 1/2 p-Thyr202/p-Thyr204 signalling whereas the monocytes showed a decrease. These results indicate that the leukaemia cells are less sensitive to stress induced by ultrasound and microbubbles when compared to normal monocytes.

In conclusion, our results show that clinical diagnostic ultrasound does have a measurable effect on intracellular signalling but may differ drastically between different cell types. This may affect the conditions necessary for therapeutic ultrasound.

I. INTRODUCTION

Diagnostic ultrasound is widely used in the clinic and is considered a safe imaging modality with little to no side effects. Currently, the only guidelines for the use of ultrasound are based on the thermal and cavitation effects of ultrasound with or without an ultrasound contrast agent (UCA) [START_REF] Piscaglia | The EFSUMB Guidelines and Recommendations on the Clinical Practice of Contrast Enhanced Ultrasound (CEUS): update 2011 on non-hepatic applications[END_REF].

Current state-of-the-art research is looking into the bio-effects possible due to ultrasound and microbubbles. In the past years, 35 it has been made clear that at higher acoustic intensities 36 microbubbles may form pores in cell membranes, aiding in 37 drug delivery, or even resulting in DNA double-strand breaks 38 [START_REF] Furusawa | DNA double-strand breaks induced by cavitational mechanical effects of ultrasound in cancer cell lines[END_REF], [START_REF] Forbes | Examination of Inertial Cavitation of Optison in Producing Sonoporation of Chinese Hamster Ovary Cells[END_REF]. Furthermore, it has been speculated that the cells 39 themselves may be acoustically active, resulting in cell stress 40 due to sonication. Primarily, ultrasound-microbubble research 41 has been done on adherent cells, and focussed on increasing 42 drug uptake, but little to no research has focussed on floating 43 cells and blood cancer (leukaemia) cells.

44

In our work here we evaluate the effect of ultrasound and 45 ultrasound and microbubbles on three specific stress activated 46 intracellular signalling proteins: p38, ERK 1/2 and acetylated 47 p53. The signalling pathway p38 is known to respond to stress 48 and inflammation, and regulate proliferation, differentiation 49 and survival [START_REF] Cuadrado | Mechanisms and functions of p38 MAPK signalling[END_REF]. ERK 1/2 is linked to the regulation of cell 50 growth and differentiation [START_REF] Chambard | ERK implication in cell cycle regulation[END_REF]. The p53 protein acts as a 51 tumour suppressor and transcription factor activated by DNA 52 damage and other cellular stress signals leading to cell-cycle 53 arrest, apoptosis, and DNA repair [START_REF] Blagosklonny | P53: An ubiquitous target of anticancer drugs[END_REF].

54

II. MATERIALS AND METHODS

55

A. Cell types 56

The acute myeloid leukemia cell line, MOLM-13 (DSMZ, 57 Braunschweig, Germany) and peripheral blood mononuclear 58 cells (PBMCs) collected from healthy volunteers were used. 59 PBMCs consist of multiple cell types where approximately 60 20% are monocytes and 80% are T cells; cells that play a 61 critical role to the immune system [START_REF] Roitt | Essential immunology[END_REF].

62

The MOLM-13 cells were cultured as previously described 63 [START_REF] Bredholt | Camptothecin and khat (Catha edulis Forsk.) induced distinct cell death phenotypes involving modulation of c-FLIPL, Mcl-1, procaspase-8 and mitochondrial function in acute myeloid leukemia cell lines[END_REF]. The PBMCs were prepared using density gradient 64 Flow cytometric labelling with CD163 was used to distinguish monocytes from T-cells (c.f., Table II).

B. Ultrasound & microbubble treatment

A custom made setup incorporating a clinical diagnostic ultrasound scanner and disposable cuvette was used to sonicate MOLM-13 and PBMC cells (Fig. 1). An M12L ultrasound probe was used as the ultrasound source and driven by an unmodified Logiq 9 clinical ultrasound scanner (GE Healthcare, Buckinghamshire, United Kingdom). A standard abdominal imaging mode was used, and the focal depth was adjusted to vary the acoustic intensity. The sound field was calibrated in-situ using a 0.2-mm PVDF hydrophone connected to a 3-axis, manual, micrometer stage. The acoustic conditions used are shown in Table I.

During treatment 2.5 ml cells, i.e., 2.5 × 10 6 cells were placed in the cuvette for treatment. SonoVue R ultrasound contrast agent was prepared according to manufacturers recommendation. A total of 40-µL was mixed into the 2.5 ml cell containing cuvette and continuously homogenized using a magnetic stirrer. The total treatment time for each sample was 10 min. Following treatment the cells were placed in an incubator for 0 -24 hours. After the required incubation time the reactions were stopped by fixation in 1.5% paraformaldehyde for 20 min at room temperature and subsequently stored at -80 • C until further analysis by flow cytometry. The control sample consisted of cells of the same culture that had not been treated with neither ultrasound or microbubbles. , Switzerland). Barcoding allows high throughput analysis, 107 reduces antibody consumption -and importantly, reduces 108 analytical variation [START_REF] Krutzik | Fluorescent cell barcoding in flow cytometry allows high-throughput drug screening and signaling profiling[END_REF].

C. Flow cytometry

109

Following treatment, approximately 0.5 × 10 6 cells were 110 stained with a combination of fluorochrome-conjugated 111 antibodies. These anti-bodies stain the cells when a specific 112 cell signalling pathway has been activated. The flow cytometer 113 measured the mean fluorescence intensity (MFI) of specific 114 wavelengths correlating to the signalling pathway. Increased 115 intensity indicates a more active pathway, e.g., a five-fold 116 increase in MFI indicates the path was substantially more 117 active compared to the control sample.

118

The intracellular signalling was evaluated at multiple time 119 points ranging from 0 minutes to 24 hours after ultrasound 120 exposure. This allowed us to evaluate the time-dependant 121 change in intracellular signalling in due to ultrasound and 122 ultrasound and microbubbles. Results are presented as a heat 123 map, allowing for faster comparison in large datasets and a 124 linear graph.

125

D. Cell death and proliferation assays 126

To investigate if the acoustic conditions had any direct 127 affect on cell death, live/dead cells were discriminated by 128 Tryphan blue staining. Trypan blue (Invitrogen, Carlsbad, CA,) 129 is used as a dead/alive discriminator; 10 µl of Trypan blue 130 were added to 10 µl of cell suspension. The dead and alive 131 cells were counted using optical microscopy and a Bürker 132 haemocytometer. Apoptosis was evaluated using DNA specific 133

III. RESULTS AND DISCUSSION

A. Cell signalling

Our results showed that both the monocytic leukemia cells, and the monocytes had a reaction to ultrasound, in terms of phosphorylation of cells. Depending on the pathway and cell type, a different response was elicited at the various stimulation conditions.

In the normal blood monocytes, phosphorylation of p38 (c.f. Fig. 2) was only increased when the cells were exposed to high intensity ultrasound with no microbubbles, and at all intensities when combining with microbubbles with ultrasound. The monocytic leukemic cell line, MOLM-13, showed a smaller fold increase in terms of p38 pathway activity when compared to the monocytes, yet a similar trend was seen. In both cell types the addition of microbubble contrast agent resulted in an 160 increase of p38 stress signalling at medium and low ultrasound 161 intensities. The minor differences were that in the leukemic 162 cell line, ultrasound alone reduced the signalling significantly 163 more when compared to the monocytes. These results indicate 164 that the monocytes were more susceptible to external stresses 165 than the leukemic cell line.

166 ERK 1/2 signalling (c.f. Fig. 3) showed a similar response 167 to that seen in the p38 pathway. In the monocytes we see 168 a decrease in signalling at the same conditions we saw an 169 increase in p38 signalling. This may be explained by the 170 fact that increased activity in the p38 pathway can cause 171 inactivation of the ERK 1/2 pathway in normal cells [START_REF] Cuadrado | Mechanisms and functions of p38 MAPK signalling[END_REF]. This effect is not observed in the leukemic cell line. In the 173 leukemic cells ERK 1/2 phosphorylation is slightly increased 174 after 6 hours, and a further increase is seen at 18-24 hours; 175 both with and without microbubbles. ERK 1/2 activation may 176 contribute to cell cycle arrest or apoptosis in response to 177 different levels of DNA damage [START_REF] Tang | ERK activation mediates cell cycle arrest and apoptosis after DNA damage independently of p53[END_REF], which would be a 178 wanted effect in cancer cells. In the current experiment no 179 significant change proliferation and cell viability was observed 180 (c.f., Fig. 5), and further studies are needed to determine the 181 downstream effects of ultrasound and microbubble induced 182 Figure 4 shows the effect of ultrasound, and ultrasound and microbubbles on Lys384 acetylation of the tumor suppressor p53. Ultrasound alone, showed an increases in both monocytes and MOLM-13 cells. In contrast, when adding microbubbles p53 signalling was reduced. However, the effects were not very pronounced. Acetylation has been found to be important for activation of p53 and regulating downstream processes [START_REF] Tang | Acetylation Is Indispensable for p53 Activation[END_REF].

Negligible change was seen in phosphorylation of p53 (Ser15 and Ser37) (Data not shown) also implying no DNA damage occurred at any acoustic conditions [START_REF] Ashcroft | Stress Signals Utilize Multiple Pathways To Stabilize p53[END_REF]. This was in agreement with the results shown in Fig. 5.

It is currently not known exactly how ultrasound and microbubble ultrasound contrast agents modulate signal transduction in the cells. Cells respond to their surrounding environment though receptors on the cell surface which initiate responses in within the cells. Different kinds of receptors may for instance be implicated in activating the ERK pathway, in response to either ultrasound or fluid shear stress [START_REF] Whitney | Integrin-Mediated Mechanotransduction Pathway of Low-Intensity Continuous Ultrasound in Human Chondrocytes[END_REF], [START_REF] Makino | G protein-coupled receptors serve as mechanosensors for fluid shear stress in neutrophils[END_REF], [START_REF] Rane | Formyl peptide receptors are coupled to multiple mitogen-activated protein kinase cascades by distinct signal transduction pathways: role in activation of reduced nicotinamide adenine dinucleotide oxidase[END_REF]. In addition to the large intracellular signaling proteins, small intracellular signaling molecules like Ca2 + bind to and alter the behavior of signaling proteins or target proteins.

In a study by Kumon et al ultrasound and microbubble induced sonoporation was shown to cause transient changes in Ca2 + concentration in cells [START_REF] Kumon | Spatiotemporal Effects of Sonoporation Measured by Real-Time Calcium Imaging[END_REF], and this may further modulate intracellular signaling. Studies have shown Ca2 + to be involved in activation of both p38 and ERK 1/2 proteins [START_REF] Momberger | Mechanosensitive synoviocytes: A Ca2+-PKCa-MAP kinase pathway contributes to stretch-induced hyaluronan synthesis in vitro[END_REF], [START_REF] Lotem | Suppression or induction of apoptosis 309 by opposing pathways downstream from calcium-activated calcineurin[END_REF].

B. Cell viability and clonogenicity 211

Cell viability and nuclear morphology remained unaltered 212 after sonication with and without microbubbles. Figure 5 213 shows the results of the colony forming assay for the 214 MOLM-13 assay under the various conditions. Whilst a larger 215 standard deviation was seen at the higher acoustic intensities, 216 no statistical significant difference was seen between any of 217 the samples.

218

IV. CONCLUSION 219

Our results indicate that both the cancer cell line and normal 220 blood cells responded 2 to 6 hours after ultrasound exposure 221 with changed phosphorylation of p38 (p-Thr180/p-Tyr182) 222 and ERK 1/2 (p-Thr202/p-Tyr204). Activation was ultrasound 223 contrast agent and acoustic pressure dependent. These 224 signalling proteins are shear stress activated indicating that 225 higher acoustic pressures result in higher shear stresses. 226 Both leukemic and normal cells showed a decrease in p53 227 (ac-Lys382) activation at the higher acoustic pressures or when 228 microbubbles were added. In all experimental conditions, the 229 cell viability remained unaltered, consistent with the safety 230 regulations for clinical diagnostic ultrasound.

Fig. 1 .

 1 Fig. 1. Graphical rendering depicting ultrasound treatment method. An M12L clinical ultrasound probe is clamped in the water bath treating the cells placed in the cuvette. A polycarbonate plate, positioned athe back of the tank at 45 • to reduce reflections.

  Flow cytomtery is a well-established technique that allows single cell analysis and characterisation. Cell samples were labelled with different concentrations of succinimidyl esters of Pacific Blue (Invitrogen, Carlsbad, Ca, USA) and Pacific Orange (Invitrogen, Carlsbad, Ca, USA) in which each sample

Fig. 2 .

 2 Fig. 2. Heatmaps showing modulation of the p38 signalling over time following ultrasound stimulation for leukemic cancer cells (MOLM-13) and normal monocytic (monocytes) cells. In both cell lines an increase in signalling can be seen with the addition of microbubbles.

Fig. 3 .

 3 Fig. 3. Heatmaps showing modulation of the ERK 1/2 signalling over time following ultrasound stimulation for leukemic cancer cells (MOLM-13) and normal monocytic (monocytes) cells. In both cell lines an increase in signalling can be seen with the addition of microbubbles.

Fig. 4 .

 4 Fig. 4. Heatmaps showing modulation of p53 acetylation over time following ultrasound stimulation for leukemic cancer cells (MOLM-13) and normal monocytic (monocytes) cells. In both cell lines an increase in signalling can be seen with the addition of microbubbles.

Fig. 5 .

 5 Fig. 5. Results of colony-forming assay used to asses proliferation in MOLM-13 cells. The number of colonies was measured in sample cultured for 5 days after sonication. No statistical difference was seen between all samples other than a greater variance at higher MI. Error bars indicate standard error of mean.

TABLE I ACOUSTIC

 I CONDITIONS GENERATED FROM M12L DIAGNOSTIC ULTRASOUND PROBE USED TO SONICATE CELLS.

		Frequency	Focal	MI	Intensity	Peak-Negative
		(MHz)	Depth		(mW/cm 2 )	Pressure (MPa)
			(cm)
	Low	6.80	2.22	0.16	0.35	0.419
	Medium 6.40	4.44	0.59	1.17	1.416
	High	5.80	3.89	0.72	1.61	1.813

TABLE II LIST

 II OF MODIFICATION SPECIFIC ANTIBODIES USED TO ASSESS INTRACELLULAR SIGNALING.

	Antigen Epitope	Clone	Conjugate
	p38	p-Thr180 /	36/p38	Alexa Fluor R 488
		p-Tyr182
	ERK 1/2 p-Thr202 /	20A	Alexa Fluor R 647 or
		p-Tyr204		Per-CP-Cy5.5
	p53	p-Ser37	J159-641.79	Alexa Fluor R 647
	p53	p-Ser15	16G8	Alexa Fluor R 488
	p53	ac-Lys382	L82-51	Alexa Fluor R 647
	NFκB	p-Ser529	K10-895.12.50 PE or Alexa Fluor R 488
	CD3	CD3	SK7	PE-Cy7
	CD5	CD5	L17F12	PE-Cy7
	CD20	CD20	H1(FB1)	PerCP-Cy5.5
	CD163 CD163	Mac2-158	PE

can be identified by the specific barcode of fluorescence 103 intensity and emission wavelength. Barcoded cell samples 104 were permeabilized with ice cold methanol, washed and 105 treated with human IgG (Octagam, Octapharma AG, Lachen 106

ACKNOWLEDGEMENTS

232

This study has been supported by MedViz 233 (http://medviz.uib.no/), an interdisciplinary research cluster 234 from Haukeland University Hospital, University of Bergen, 235 and Christian Michelsen Research AS.