This work introduces a novel capacitive‐sensing technology capable of detecting respiratory motion with high temporal frequency (200 Hz). The system does not require contact with the patient and has the capacity to sense motion through clothing or plastic immobilization devices.This work presents and evaluates a novel capacitive monitoring system (CMS) technology for continuous detection of respiratory motion during radiation therapy. This modular system provides real‐time motion monitoring without any contact with the patient, ionizing radiation, or surrogates such as reflective markers on the skin.The novel prototype features an array of capacitive detectors that are sensitive to the position of the body and capable of high temporal frequency readout. Performance of this system was investigated in comparison to the RPM infrared (IR) monitoring system . The prototype included three (5 cm × 10 cm) capacitive copper sensors in one plane, located at a distance of 8–10 cm from the volunteer. Capacitive measurements were acquired for central and lateral‐to‐central locations during chest free‐breathing and abdominal breathing. The RPM IR data were acquired with the reflector block at corresponding positions simultaneously. The system was also tested during deep inspiration and expiration breath‐hold maneuvers.Capacitive monitoring system data demonstrate close agreement with the RPM status quo at all locations examined. Cross‐correlation analysis on RPM and CMS data showed an average absolute lag of 0.07 s (range: 0.03–0.23 s) for DIBH and DEBH data and 0.15 s (range: 0–0.43 s) for free‐breathing. Amplitude difference between the normalized CMS and RPM signal during chest and abdominal breathing was within 0.15 for 94.3% of the data points after synchronization. CMS performance was not affected when the subject was clothed.This novel technology permits sensing of both free‐breathing and breath‐hold respiratory motion. It provides data comparable to the RPM system but without the need for an IR tracking camera in the treatment room or use of reflective markers on the patient. For many sites, special attention to motion management is required to ensure accurate delivery. One of the most prevalent sources of motion is respiration, and has a prominent effect when treating breast, lung, or abdominal indications. Management of respiratory motion can result in improved targeting accuracy and reduced normal tissue toxicity, and also ameliorates imaging motion artifacts enabling improved tumor visualization and alignment.Implanted radiofrequency transponders are used for motion management but involve surgical intervention with a chance of major or minor complications, transponder migration, and introduce imaging artifacts.In this work, we present the first report of a capacitive‐sensing technology capable of detecting respiratory motion. The technology extends the application of capacitive sensors described previously22.Ad), the area of the capacitor plates (A) and the material between the plates which is introduced as permittivity (ε). Therefore, capacitance will change as the distance between the sensor and patient changes due to breathing.The respiratory monitoring system works by tracking the position of the area of interest, for example, chest wall or abdomen. It can detect the motion of the region of interest (ROI) and provide information used for gating, or determining breath‐hold amplitude. The system is comprised of thin (0.0254 mm) copper conductive sensors mounted on an acrylic horizontal plate above the patient's chest or abdominal area. The human body is naturally electrically conductive,Constant monitoring of the system capacitance allows tracking of the motion of the region of interest. Multiple sensors can be used to track the motion of different regions of the body depending on the clinical needs. Capacitive monitoring of the system can be accomplished by using a capacitive proximity sensor such as MPR121 . This system has been shown to be stable in linac conditions and under high dose rate photon irradiation.The prototype shown in Fig. 2.BData collection was performed with the help of a volunteer. The volunteer was positioned supine on the couch, comfortably level, with both arms raised and supported on an indexed breast board , and made comfortable using a knee wedge. The breast board was centrally indexed to the treatment couch using a standard carbon fiber locating bar . Experiments were conducted in the CT simulator  equipped with standard flat top carbon fiber couch top and real‐time position management system (RPM) . The CMS sensors have been shown to be unaffected by the environment of the linear accelerator, and a similar setup and measurement could be achieved there.The RPM block was setup in contact with the anatomy of interest, and the CMS sensor array was located anterior to the same region. The volunteer was asked to take a deep breath that was used to synchronize both systems in postprocessing. The CMS system was setup to acquire data for 150 s at 200 Hz using in‐house software. Once the CMS data acquisition was concluded, the RPM system was turned off. The CMS data were processed using an exponential weighting methodThe data were synchronized and normalized to compare the behavior of the two systems. Cross‐correlation analysis was performed to determine any relative time lag (phase shift) between the two datasets. Since the CMS data were acquired at 200 Hz it was downsampled to 30 Hz to match the RPM data and allow cross‐correlation analysis. Amplitude comparison was performed on synchronized and normalized CMS and RPM data during free‐breathing after applying the cross‐correlation shifts.To investigate the similarity of the measurements from the two systems under different possible clinical conditions, experiments were repeated during chest breathing, belly breathing, deep inhalation, and expiration breath‐hold.2.B.1Breathing amplitude traces were acquired during chest free‐breathing using CMS and RPM simultaneously, for three ROIs: central, where the central sensor was place on the xiphoid process, and left/right lateral where the sensors were placed 5 cm lateral to the central sensor. An additional set of measurements was acquired during abdominal free‐breathing for central and lateral ROIs which were 10 cm inferior to the chest breathing ROIs.In each case, the RPM block was affixed to the volunteer's skin and the CMS sensor was positioned above the ROI with a distance of about 8–10 cm from the skin surface. When placing the RPM block in lateral positions, gauze was placed under the block to compensate for the natural curvature of the body and keep the RPM block level and in view of the IR camera.The volunteer was instructed by a radiation therapist to perform chest breathing by concentrating on sternal rise and fall during inhale/exhale, and belly breathing by concentrating on abdominal rise and fall during inhale/exhale. The volunteer was instructed to maintain even breathing by counting to four each for inhale and exhale. A total of six chest breathing and seven abdominal breathing experiments were performed.2.B.2Breathing traces were acquired during DIBH and DEBH using CMS and RPM simultaneously, for the central region of interest (xiphoid process). In each case the RPM block was affixed to the volunteer’s skin and the CMS sensor was above the ROI with a distance of 8–10 cm from the volunteer.The volunteer was coached by a radiation therapist to take a slow deep breath to maximally inflate the chest, and to maintain this maneuver for approximately 20–25 s. Additionally, the volunteer was instructed not to arch their back during DIBH/DEBH maneuvers. The volunteer was given time to practice maneuvers and breathing instructions prior to data capture.2.B.3Performance of the CMS system was tested with no direct view of the chest. The volunteer was clothed, and the CMS sensor was placed above the xiphoid, again at an 8–10 cm distance from the skin, and data were gathered for three DIBH instances. The RPM system was inoperable in this scenario due to the fact that the reflective block could not be stably positioned on clothing.2.CAnother set of volunteer experiments was performed in a Varian Clinac EX accelerator  treatment room to determine the effect of different sensor‐body distance on the signal as well as ensure that no additional effects are present as a result of migration to the linac environment. The volunteer was positioned supine on the couch, comfortably level, with both arms raised and supported on an indexed breast board , and made comfortable using a knee wedge. These experiments monitored abdominal respiration with the sensor placed 10 cm inferior to the xiphoid process in the absence of the RPM block at sensor‐body distances of 3.5, 5.5, and 7.5 cm.A denotes the amplitude of signal or noise as specified by the subscript.Signal to noise ratio was calculated for the raw data using Eq.  below whThe amplitude of the signal was defined as the average change in acquired signal from exhale to inhale point (breathing amplitude) over the 120 s of data acquisition. The noise was estimated on a 0.005 s rolling window by calculating the signal change between two adjacent data points, averaged over the time series of acquired data (120 s) and rounded. This provides an understanding of the ratio between the range of signal and the amplitude of noise which can be helpful for clinical comparison and decision making.33.AFigure 3.BData for the xiphoid sensor trace during chest and abdominal breathing are shown in Fig. 3.CBreathing traces for DIBH and DEBH are shown in Fig. 3.DFigure 3.EFigure 4This manuscript presents a novel method for noncontact respiratory motion detection. The system provides near real‐time (200 Hz) respiratory motion information during treatment. The system is modular, low cost in comparison to IR or surface monitoring cameras, requires no contact with the patient, and offers the flexibility of using different regions of interest. Additionally, the system does not require unobstructed view of the patient, as it can detect motion through clothes and, was previously shown to detect human motion even within a full thermoplastic mask.The modular nature of this device presents several advantages relative to IR or surface imaging camera‐based systems. The device may be coupled reproducibly to a couch top and therefore may be relocated easily between treatment units, CT, PET/CT or angiography platforms. It may also be used outside of these rooms, for example, on a mock‐up of a treatment couch, thereby providing an offline platform for patient education and coaching without tying up expensive capital equipment resources. The device could be produced at comparatively low cost, which is amenable to equipping multiple treatment and imaging rooms. This option is increasingly important, as techniques such as DIBH for treatment of left‐sided breast cancer become more common.Additionally, while the RPM system detects motion from a single plane, the CMS sensor detects one single capacitance value which is related to the average distance between the sensor and the body over the area of the sensor. As the breathing occurs, the distance between the sensor and body surface varies. The distance averaging resulting from a strip sensor geometry (rather than a point sensor) helps detect the global respiratory motion despite the curvature of the individual’s body.The system charge/discharge time is 2 µs and draws a low current of 24 µA with the charge/discharge process, which occurs 200 times per second. Considering these variables and the fact that the dielectric material used in this case is air, no significant capacitive leakage or stability issues are observed or expected.The capacitive system signal relies on the electrical conductivity of the body. This precludes conducting a phantom study, specifically in the chest/abdominal area. In this study, the CMS detection system was used on two volunteers to acquire proof of concept data and to investigate the viability of the design. However, our next step would be a clinical study with a cohort of volunteers in different clinical setup positions. The presented CMS data shows good agreement with the RPM infrared monitoring system  that has been used for both gating and breath‐hold monitoring in our institution. However, increased noise in signal was observed in case of deep expiration breath‐hold Fig. . This waThe cross‐correlation analysis shows a small lag between the RPM and CMS systems. The average absolute lag values of 0.07 and 0.15 s were observed for DIBH/DEBH, and free‐breathing, respectively. A contributing factor to the lag is the synchronization process of the RPM and CMS positioning data. That, combined with the mandatory down‐sampling of the high frequency MPR data from 200 to 30 Hz in order to perform the cross‐correlation analysis could explain the sub‐second lag values. However, the respiratory motion with larger amplitude (DIBH\DEBH) shows almost twice the average lag value compared to the respiratory motion with smaller amplitude (free‐breathing). We believe parameters such as the nature of postprocessing of data (our protocol vs what might be used in the RPM software) and the source of the signal acquired (conductive tissue as opposed to the skin motion) could be responsible for the difference.Amplitude comparison analysis for RPM and CMS central chest and abdominal experiments, showed that on average, 98.2% and 99.8% of all data points were within 15% of one another for chest and abdominal breathing, respectively. The value decreased to 94.3% when averaged over all chest and abdominal respiration experiments for central and lateral sensors. This is possibly due to the fact that, in order to detect the lateral motion using the RPM block, the block is taped in place on a curved section of the body surface and gauze is placed under the block to help compensate for the curvature of the body. This may introduce additional uncertainties in the RPM data due to the reduced stability of the RPM block during respiratory motion.A comparison of the raw and processed signals shows a slight temporal shift in the processed data. The phase shift was measured at the maxima  for the cases presented in Fig. While this comparison was made to show the similarity between the operation of this technology compared to a clinical status quo (RPM), clinical use of the CMS device would require the acquisition of the breathing trace at the time of CT simulation and during treatment with the CMS device. This would reduce any effects of inter‐device variations in respiratory amplitude detection.The current prototype provides one dimensional positional information regarding respiratory motion. In concept, the system can be extended to include a larger array of sensors in different axial planes to provide a three‐dimensional mapping of respiratory motion during treatment. The prototype described herein included an acrylic frame for ease of machining and construction. A clinical version will be made of carbon fiber, which is the status quo for clinical accessories in radiation therapy due to its radiotransmissive properties. Additionally, the current design only requires an area of 10 cm by 5 cm for the conductive sensor. This allows for reducing the size of the sensor platform (Figure 5This work presents a novel noncontact and modular technology for real‐time monitoring of respiratory motion. The current prototype can detect respiratory motion at different regions, providing positional data at 200 Hz readout frequency. The system is minimally intrusive as it does not require unobstructed view of the chest and can provide motion detection for extracranial lesions through fabric, or thermoplastic immobilization material.Ms. Sadeghi reports grants from Atlantic Canada Opportunities Agency‐ACOA, non‐financial support, and other from Brainlab AG, during the conduct of the study. In addition, Ms. Sadeghi has a provisional patent application pending and a licensing agreement with Brainlab AG. Ms. Moran reports grants from Atlantic Canada Opportunities Agency‐ACOA, non‐financial support, and other from Brainlab AG, during the conduct of the study. Dr. Robar reports grants from ACOA, non‐financial support and other from Brainlab AG, during the conduct of the study, and outside the submitted work. In addition, Dr. Robar has a provisional patent application pending and a licensing agreement with Brainlab AG.


Proteins are characterized by their structures and functions, and these two fundamental aspects of proteins are assumed to be related. To model such a relationship, a single representation to model both protein structure and function would be convenient, yet so far, the most effective models for protein structure or function classification do not rely on the same protein representation. Here we provide a computationally efficient implementation for large datasets to calculate residue cluster classes (RCCs) from protein three-dimensional structures and show that such representations enable a random forest algorithm to effectively learn the structural and functional classifications of proteins, according to the CATH and Gene Ontology criteria, respectively. RCCs are derived from residue contact maps built from different distance criteria, and we show that 7 or 8 Å with or without amino acid side-chain atoms rendered the best classification models. The potential use of a unified representation of proteins is discussed and possible future areas for improvement and exploration are presented. Proteins are molecules found in living organisms and participate in many diverse cellular and molecular functions. It is generally recognized that the three-dimensional (3D) structures of proteins are related to their functions, yet the relationship remains to be elucidated, since many attempts to predict the correct functions of proteins based on their 3D structures still result in false predictions .Different approaches have been described to account for such a relationship, including those based on physical and chemical forces ,3, proteWe have previously described a representation of 3D structure that is learnable ; such a representation allowed us to identify structural neighbors and classify the protein’s 3D structure with the best performance and reliability reported so far . The repAlgorithm 1: RCC calculation  Input: Cartesian coordinates of protein atoms Output: RCC coordinates 1: Generate graph with the contacts of all residues. Two residues are in contact if they are within a distance threshold. 2: Calculate maximal cliques using Tomita algorithm (see below) 3: Calculate RCC from maximal cliques.The pseudo code to obtain the RCC from PDB entries is depicted in Algorithm 1:d with the use of several geometric restrictions:We accelerated the calculation of all contacts given a distance threshold d that corresponded with the contact threshold distance; thus, given a residue we can identify the cube that contains it by using a hashing function; every cube in the grid has a list of all the residues that are inside of it. Given a residue r and the cube in the grid where it is found, rG, we know that any other residue inside rG must be in contact with r. Additionally, if a residue s is in contact with r, it must be that the cubes rG and sG are neighbors in the grid . This way we reduced the search space dramatically (see n (n is the number of amino acids in a protein) in the linear portions of the protein and 16*n on average (given that at 5 Å a residue is in contact with four other residues on average).(i) The hashing approach using a 3D grid over the space. We generated a grid over the 3D space of width ally see , which tr we enveloped it in a sphere of radius dr and center cr. For two residues r and s, let D be the distance between rc and sc,. They must be in contact if (ii) Rules for quick evaluation see . For a rd and center cr overlaps half the volume of the sphere enveloping s or vice versa . Any pair of residues that do not fulfill one of these rules needs the explicit computation of the distance between its pair of atoms to be performed (with an early stop if a contact is found).Finally, they must be in contact if a sphere of radius Algorithm 2: Contact map calculation  Input: Cartesian coordinates of protein atoms  Output: Contact graphr in the protein:  For each residue rG using the hash function.   Calculate its cube in the grid L inside the cubes neighboring rG.   Get the list of residues s in L:   For each residue D distance between cr and cs.    Calculate D < d and continue.    Quick inclusion if D–dr, D–ds} > d and continue.    Quick exclusion if minimum {r and s.     Perform contact check for each pair of atoms from In the scenario in which the side chains are ignored, the spheres are calculated with the remaining atoms of the residue. The geometric restrictions work in the same way as if the spheres were calculated with all the atoms in the residue, and similarly, when a pair of residues does not meet the criteria for quick evaluation, the explicit computation of the distances between their pair of atoms is performed exclusively with the atoms in the backbone. The pseudo-code for this calculation is detailed in Algorithm 2:Calculating the maximal cliques from a graph is a classical problem in computer science presented by Tomita , and theWe only considered maximal cliques containing at least three residues and at most six residues, which are grouped in 26 cluster classes. This grouping for a given protein produces different frequencies for each of these 26 clusters, wherein the first 3 frequencies correspond with maximal cliques with 3 residues ; the next 5 frequencies are maximal cliques with 4 residues ; the next 7 frequencies are maximal cliques containing 5 residues ; and the last 11 frequencies include maximal cliques with 6 residues . The number of residues that are adjacent in the protein sequence define the class of a residue cluster. For instance, an RCC with 3 residues in which all residues are not adjacent in the protein sequence is referred to as ; an RCC with 4 residues in which 2 residues are adjacent in the protein sequence, and other 2 residues are also adjacent in the protein sequence  is represented as  see . The regFor the 3D structural classification, we obtained the RCC for every protein domain reported in CATH (version 4.2.0) using 7 different distance cut-off values  and including or not the atoms of the amino acid side chain; hence, for each protein domain, 16 RCC representations were obtained. Our RCC dataset includes 354,079 different proteins . A total of 2,831,584 different RCCs corresponding to those representations with amino acid side-chains or without side see .For the functional classification, we used the Gene Ontology (GO) database  and calculated the RCCs for every protein in the PDB database . SimilarFor identifying a model to classify protein structures, the full set of RCCs was labeled according to the annotated CATH classification in each of its 2 levels: 4 Classes  and 41 Architectures. This full set was used to train models explored using the AutoWeka plugin , which pThe statistical parameters  reported by Weka and AutoWeka were used to evaluate the learning performance of the classifiers.The code implementing an efficient computation of residue cluster classes (RCCs) is freely see ; the comThis computational efficiency allowed us to compute RCC values for all protein structures in the PDB database  in lessWe built different RCC representations of protein structures to identify the best distance criterion to classify 3D protein structures. These included distances at 5, 6, 7, 8, 9, 10 and 15 Å; we originally used only a distance of 5 Å ; hence, We observed that the best models were obtained using RCC representations with a contact distance threshold of 7 or 8 Å without side chains see . Table 2We used the same protein structure representations described for structural classification for protein function classification; to include or not the side-chain atoms here represent an exploration of the relevance of side-chain contacts for protein function. For the functional annotation, we used the Gene Ontology (GO) annotations, consisting of three main classes: molecular function (F), biological process (P) and cellular localization (C). As described before, AutoWeka and cross-validation were used to identify the best models to learn GO functional annotation from RCC representations.The best models were again observed with 7 or 8 Å of distance between the atoms of residues with or without side-chain atoms see . Table 3An important aspect in functional annotation is the biased compositions of different classes; such a bias may cause predictors to classify proteins according to the most abundant functional classification. To rule out the possibility that such a bias may have affected the reliability of our predictions, we conduced a set of calculations using the ZeroR classifier in Weka; this classifier only predicts the most frequent class; hence, any machine-learning algorithm must achieve a performance better than this value to be reliable. In Including the efficient implementation of Tomita’s algorithm into the RCC calculation rendered a constant time performance independent of the protein lengths analyzed; we also noted that the time required to compute the contacting distances and get the RCCs was less than reading the input file. In terms of computing time, assuming these calculations would be done on a single central processing unit or CPU, the global time required to compute a given RCC is divided in the time required to load the protein’s 3D structure or PDB file (elapsed time) and the time to obtain the RCC (CPU time), which includes constructing the contact map of residues and identifying the maximal cliques. Our results indicate that most of the time is taken by reading the PDB file; this may be improved by implementing a non-serial reading function . This inThe latest Critical Assessment for Structure Prediction (CASP) competition for the first time showed that ML (AlphaFold) improved on previous methods using protein sequence features , yet theOur screening to build RCCs reveals that the backbone contains enough information to represent both 3D structure and function. This does not imply that side chains are not relevant for 3D protein structure or function; after all the backbone conformation depends on the side chains. The relevance of building RCCs without side chains is that on the one hand, such a representation does not need high-resolution structures to build a useful model; this would be relevant to further exploration: what is the range of protein structure resolution that renders a reliable model for protein structural and functional classification? On the other hand, having a model that concerns only on the protein backbone may facilitate the development of methods to predict 3D protein structure based on RCCs.Having the same representation of proteins to model 3D structure and function would allow one to analyze the possible co-localization of structural and functional classes in the 26-dimensional space of RCCs, for instance; this would eventually lead to a better understanding of the structure–function relationship of proteins. RCCs would allow exploring for regions in this 26-dimensional space where no examples of 3D protein structure or function are known, and potentially, designing new proteins.In summary, in this work we distribute a computationally efficient implementation with which to compute RCCs from a 3D protein structure—a representation of proteins that allows for effective modeling of both 3D protein structure and functional classification.


In this paper, we explore the advantages of a fractional calculus based watermarking system for detecting Gaussian watermarks. To reach this goal, we selected a typical watermarking scheme and replaced the detection equation set by another set of equations derived from fractional calculus principles; then, we carried out a statistical assessment of the performance of both schemes by analyzing the Receiver Operating Characteristic (ROC) curve and the False Positive Percentage (FPP) when they are used to detect Gaussian watermarks. The results show that the ROC of a fractional equation based scheme has 48.3% more Area Under the Curve (AUC) and a False Positives Percentage median of 0.2% whilst the selected typical watermarking scheme has 3%. In addition, the experimental results suggest that the target applications of fractional schemes for detecting Gaussian watermarks are as a semi-fragile image watermarking systems robust to Gaussian noise. Digital watermarking has gained popularity in the past few decades as a copyright enforcement tool. It is an active research field that includes applications such as data authentication and data indexing among other practical applications ,2,3. TheA watermarking system embeds a signal, called the watermark, into another signal known as the cover; a cover might be digital media such as an image, audio, video, or other digital media. Most of the proposed watermarking systems generate a pseudo-random signal (the watermark) using a user’s key and then embeds this watermark into the cover; conversely, the watermarking system is able to detect the watermark or even retrieve it from the watermarked cover. If watermark samples are in the set {−1,1}, then the watermark is called binary; sometimes, designers let the watermark be a pseudo-random sequence with Gaussian distribution, this kind or watermark is called Gaussian Watermark.Error type I:The system failed to find a watermark; this is called False Negative (FN).Error type II:The system found a given watermark even when either no watermark or another watermark was embedded; this is called False Positive (FP).A watermarking system can have two types of errors during its attempt to detect a watermark:A FP is considered flawed that must be avoided because this might lead to a legal dispute on the copyrights of the digital media. For this reason, systems that exhibit a high FPP are impractical and thus excluded from literature. Usually, a watermarking system has a negligible FPP for detecting binary watermarks; conversely, some systems might have a high FPP when detecting Gaussian watermarks.To clarify this issue, consider the following example: Assessed statistically meaningful results by extending the data set up to 10,000 images.Carried out a statistical analysis to compare the FPP of the original watermarking scheme versus the corresponding version with detection equations derived from fractional calculus.Evaluated the quality of both schemes as a watermark detector by comparing their ROC curves.Examined the successful detection rate after performing a number of signal processing operations on the watermarked images to define robustness of the system and recommend target applications of fractional detector equations.Complemented our previous study about binary watermarks with this study about Gaussian watermarks.To help to mitigate this issue, we proposed in a previous paper to replace the detection equations of watermarking systems to reduce the FPP. Although results were interesting and seem promising, our tests were not conclusive due to the low number of images in the database used in the experiments; thus, the purpose of this paper is to fill the remaining gaps in our previous proposal by analyzing the cases we left unexplored using a bigger image database. In this paper, we put our early proposal on a firmer basis, we:With this results as a basis, we expect designers of watermarking systems to take advantage of Gaussian watermarks when appropriate to meet his design goals. At the moment, it is difficult to detect watermarks using simple equations, so we look forward to provide an alternative to reuse previously proposed schemes by using fractional calculus based equations.The system designer wants to enhance the discriminative power of a system already proposed.The watermark is some information that closely holds the Gaussian distribution.The complexity of the watermarking system has to be low.The usage scenarios for such schemes include:Another scenario will be discussed later.The rest of the paper is organized as follows: in Before continuing with the background fundamentals, let us define the terminology used in the remainder of this paper. One often refers to different watermarks, so we will call the set of different watermarks A simple model approach to watermarking is to make analogies to the field of the theory of communications. In this context, we assume that the watermark is transmitted through a communications channel as pictured in There are two basic rules for embedding watermarks: the additive rule and the multiplicative rule. We will focus on the additive embedding rule since it is widely used in most related works.g is the embedding gain.The watermarking system embeds the watermark A typical watermark system assesses the presence of the watermark by computing two statistics: a decision variable which is a measurement of the presence of the watermark within the watermarked image, and a threshold that helps to decide if the watermark is present or absent. If the decision variable is greater than or equal to the threshold, then the watermark was detected; otherwise, the watermark is absent as shown in Most watermarking systems have detected watermarks using the cross-correlation formula since the early works on watermarking; an example is the highly influential paper by Cox et al. . The watequation :(3)ThI for  for 3) fquations –(3) is sThe decision of the system for the integer equation set is computed as:Many works use  and 2)  2) to emOn the other hand, Fractional Calculus (FC) has gained attention in recent years; for example, Refs. ,6,7,8 arSome authors use a fractional derivative for watermarking since there is a relationship between the order of the derivative and the resulting function; this relationship is difficult to establish. For example, the authors of  use the The work  is similThe work . The maiThe work  use the Other authors use the Fractional Fourier Transform (FrFT) for watermarking since there is a strong dependency between the orders and the resulting coefficient set of the FrFT, a dependency that seems random. The algorithm proposed in  uses theThe Random Fractional Fourier Transform (RFrFT) is a variation of the FrFT; it has the same properties of FrFT but has the advantage that the spectrum is random and exhibits a high embedding capacity and robustness for watermarking applications. An RFrFT application to watermarking is presented in . This syAnother fractional calculus based transform, the Discrete Fractional Random Transform (DFRNT), was used in . This woOne more fractional based transform is the Fractional Dual-Tree Complex Wavelet Transform (FrDT-WT); the FrDT-WT is used to find the wavelet transform in the Fourier domain resulting in a mathematical description of the multiresolution properties. The work presented in  and explAnother work is  that is The authors of  present All of these works have in common the use of  and 2)  2) to emOn the other hand, the authors of  analyzedThe authors of  exploredThe case of the fractional scheme proposed in  for the The detection variable derived from FC principles proposed in  is:5)dFdF(5)dF(wshold is:ThF(wl)=kshold is:ThF(wl)=kThe decision of the system for the fractional equation set is:If we use  and 6)  6) for dTo carry out experiments, we used 10,000 images of the BOWS database as the set We used the embedding scheme shown in The goal of the tests was to assess the capacity of watermarking schemes shown in For the first test, we computed the FP of each image. To achieve this, each image in the set ed using  and 8)..X was waFor the sake of simplicity, and without loss of generality, we indicated We selected the Receiver Operating Characteristic (ROC) since it is regarded as an objective measure to evaluate performance of a decision technique. Thus, as a second test, we computed the ROC as follows: each image in the set With a last test, we examined the robustness of the fractional scheme. To achieve this goal, each image in the set Procedure 1 Procedure to record measures for False positives.Require: Image set  1: Open log file for writing.for Each image do 2:  3: Select randomly a watermark  4: Partition set  5: Embed the watermark for Each watermark do 6: for Each watermarking scheme do 7:   8:   Compute ifthen 9:   10:    end if11:   end for12.  end for13: 14: Record end for15: return Log file.16: Procedure 2 Procedure to record measures for getting the ROC curve.Require: Image set  1: Open log file for writing.for Each image do 2.  3: Select a random watermark  4: Embed the watermark  5: Compute  6: Record  7: Get the set of indexes of true negatives, these indexes form a set m from set  8: Draw at random an index  9: Record 10: Get the set of indexes false positives, these indexes form a set for Each index k in do11: 12:  Record the values of end for13: end for14: 15: Close log file.return Log file.16: Procedure 3 Procedure to record Detection Rate.Require: Image set  1: Open log file for writing.for Each image do 2:  3: Select randomly a watermark  4: Embed the watermark  5: Perform an attack on the watermarked image. 6: Compute ifthen 7:  8:  end if 9: end for10: 11: Compute detection rate (12: Record return Log file.13: As a first test, we computed the false positive percentages for both the integer and the fractional schemes and build a boxplot. In our second test, we evaluated the quality of both schemes; we used the data we collected to draw the ROC curve shown in The last test consisted of examining the successful detection rate after attacks; this was accomplished by attacking each watermarked image in the set A bar plot of the percentage of detected watermarks after the image set We performed a similar attack; this time, a median filter was used to filter the set The next test is comprised of adding Gaussian noise to each watermarked image in the set The following test consists of adding salt and pepper noise and then detecting the watermark. Results depicted in We carried out the next test by adding speckle noise before trying to detect the watermark. Results in A very common scenario is to compress images using the JPEG standard, so the next test comprised watermarking the image and compressing the image with the JPEG standard, and then detecting the watermark. Another common signal processing operation is the cropping attack. t rows and t columns at random, and then removes these rows and columns from the watermarked image; the resulting image is smaller than the original watermarked image, so the image is then scaled to match the size of the original watermarked image. The watermark was then detected, and the results are shown in The next test selects t rows and t columns at random, and then substitutes the selected rows and columns with the adjacent row or column of the same image. The watermark was detected, and results are shown in Another test, similar to the previous one, selects Finally, we carried out a scaling attack; the watermarked image was scaled to make it smaller and then the image was restored back to its original size. The results are shown in In this study, we compared the FPP of the original watermarking scheme versus the corresponding version with detection equations derived from fractional calculus; evaluated the quality of both schemes as a watermark detector by comparing their ROC curves, and examined the successful detection rate after attacking the watermarked images to define robustness of the system. We performed several tests that allowed us to conclude the following facts:The False Positives percentage is much lower for the fractional scheme than the corresponding percentages of the integer scheme. According to Results show that the fractional scheme is a reliable method for detecting Gaussian watermarks according to In addition, the experimental results in Since the target application might be authenticating information, it will be convenient to propose another value of The results provide designers of watermarking systems with an alternative to take advantage of Gaussian watermarks when appropriate to meet their design goals. Thus, the proposed strategy is an alternative to reuse previously proposed schemes by using fractional calculus based equations.The results obtained in this study complement the study in  since thThe characteristics to discriminate between patterns with Gaussian statistical distribution suggest that the fractional equations might be used in pattern recognition applications where samples have a Gaussian distribution.


Medical products are an integral and pivotal part of health care delivery. They need to be available, affordable and quality-assured. The SADC region is prone to threats arising from the availability and use of substandard and falsified (SF) medical products. This is something that needs to be actively addressed.A survey, constructed around four themes, was carried out between September 2018 and January 2019. The National Medicines Regulatory Authorities (NMRAs) of the 16 Member States within the SADC region were asked to respond to the survey questionnaire. The objective was to map existing fameworks, mechanisms and approaches to prevention, detection and response (PDR) to SF medical products.Responses were received from twelve out of the sixteen NMRAs. Only three of the twelve respondents had included elements for PDR for SF medical products in their national medicine policies. Regardless of the status in terms of policies, legislation is however in place for the majority of NMRAs. The mandate for regular sampling, an important detection mechanism, was enshrined in the legislation of nine of the twelve respondents. In terms of response mechanisms, six of the respondents had both inter-agency and intra-agency co-ordination for responding to SF products.Though findings point to some deficiencies in terms of policies and implementation plans, the majority of countries have the mandate and legislation to deal with substandard and falsified medical products. Effective enforcement requires more investments into human resources, infrastructure, stakeholder coordination and public outreach. WHO has an important source of actionable information about incidience of substandard and falsified medical products. It needs to be leveraged to improve outreach to stakeholders and to raise awareness about SF problem and mechanisms available to address it. The extent, to which mechanisms and procedures are in place, varies. Some elements of the desired approach exist in the region; however, they will benefit from targeted strengthening to ensure a holistic approach across 12 action areas recommended by WHO. Access to essential medicines and health products holds significant importance in addressing health problems . HoweverThe WHO study on the public health and socioeconomic impact of SF medical products articulates the problem and impact of SF products. The study establishes that these products threaten the health of those who take them, as they may be of poor quality, unsafe or ineffective. The study highlights the impact of SF products used in the first-line treatment of uncomplicated malaria: incremental deaths in sub-Saharan Africa due to SF antimalarials make up approximately 2.1 to 4.9% of total malaria deaths; and approximately 3.8 to 8.9% of malaria deaths relating to cases seeking treatment . The thrThe WHO Global Surveillance and Monitoring System for substandard and falsified medicines, vaccines and in vitro diagnostics, based on data gathered in its first 4 years of operation, highlights the possible impact of SFs on individuals, families, national health systems and the economy and presents actual data from the surveillance system and other sources . The repThe risk posed by SF medical products is very real in the African region, which is currently leading in terms of the number of reports to the WHO rapid alert system. Forty-two percent of all reports to the WHO Global Surveillance and Monitoring System on SF medical products reported between 2013 and 2017 came from Africa, highlighting the magnitude of the problem . HoweverA 2009 WHO baseline survey provided a general overview of the pharmaceutical situation in terms of demographic and economic aspects, health services data, medicines policy and access to medicines, including procurement and supply, intellectual property aspects and medicine regulation in the SADC region . InformaThe PDR approach recommended by WHO guides countries in protecting their health systems from the detrimental impact of substandard and falsified products. Their implementation requires engagement of a wide range of national stakeholders and cross-border collaborations. In August 2016, the Regional Committee for Africa at its sixty-sixth session adopted the Regional Strategy for Regulation of Medical Products, 2016–2025. Addressing the burden of substandard and falsified medical products was highlighted as one of the key elements for the African region in the strategy. The present survey was undertaken to monitor the progress in the implementation of the regional targets in the area of SF counteraction.The study was conducted as part of an exercise to map the national approaches to counteract SF medical products in the WHO AFRO region. This article analyses the data describing national mechanisms to prevent, detect or respond to the incidence of SF medical products in the SADC region. The 2017 WHO GSN study report on SF medical products makes a case for collecting evidence. Strong evidence is needed to help prevent, detect and respond to SF products and the threat they represent. Therefore, the aim of the survey was to gather information that could form a basis to describe the status of national systems for counteracting SF medical products in the SADC region.A survey was conducted between September 2018 and January 2019 to map the capacity and approaches for Prevention, Detection and Response to the incidence of substandard and falsified medical products by the National Medicines Regulatory Authorities (NMRAs) in the 16 Member States within SADC.The survey was conducted using a structured, self-administered questionnaire developed in English, French and Portuguese. The development of the questionnaire was guided by the ‘Prevention, Detection and Response’ approach captured by the WHO Global Surveillance Network and approved by the Member State Mechanism; questions were structured around four themes, namely the presence of policies and frameworks, mandate and legislation, and coordination and response.The tool underwent internal review in WHO and was amended upon the receipt of feedback from the inhouse experts.The survey tool was distributed to the National Medicines Regulatory Authority (NMRA), which is either an autonomous or semi-autonomous regulatory authority or a department within the ministry of health, in each country. Return email addresses were provided on the questionnaire.Results were received back from the NMRAs and compiled for use as part of the WHO mandate to generate evidence for effective health interventions and bring people to enjoy the highest attainable level of health and well-being. Obtained results were compiled on an Excel spreadsheet. For the purposes of data validation, the results were presented to the SADC NMRA heads present and the findings discussed at the 6th African Medicines Regulators’ Conference, which took place from 02–04 October 2019 in Victoria Falls, Zimbabwe.The target group of the survey were NMRAs in the sixteen SADC countries. TwelveAs the pool of respondents included all the countries in the SADC region and this being a small number, results are presented as absolute numbers and not in percentages. The survey results are outlined below and grouped into six thematic areas.The number of NMRAs within SADC which had included elements for PDR to SF within their national medicines policies or strategy documents was very low, as shown in Fig. Most indicated that they had no policy frameworks to underpin prevention, detection or response to SF. Only three of the twelve respondent countries had this framework in place, with the same 3 being the only ones with implementation plans of action. It was noted that only one respondent that had a policy on SF had gone further and aligned their policies to the African Regional Strategy for Regulation of Medical Products, 2016–2025 endorsed by the 66th session of the WHO Regional Committee for Africa.Nine out of twelve respondents indicated that they criminalised the manufacture, supply or distribution of SF products. Financial penalties and imprisonment were the most common sanctions provided for in national legislations. Jail terms ranging from not less than 6 months up to 15 years can be imposed. Out of the twelve respondents, five had legal provisions for seizure and confiscation of property other than the offending SF products.In the case of administrative liability or sanctions, they are defined as seizure and confiscation of SF products, suspension of the right to discharge official or professional functions, ban to practice or hold administrative or managerial positions and financial penalties. It was noted that they were applicable in nine of the twelve respondents. Of interest was that two of the responding NMRAs specified that they had provisions for criminal liability or administrative liability, and not both.The mandate for regular sampling was enshrined in the legislation of nine of the twelve respondent countries. As illustrated in Fig. A risk-based approach to sampling was applied by five of the respondents. Sampling plans were indicated to be regularly updated in six of the respondents that have a mandate to sample. The testing of the samples is carried out to some degree in the region. However, only three of the laboratories used were either ISO-certified or WHO-prequalified.The survey captured information on other activities that are related to sampling. Half of the respondents, that is, six confirmed that documented procedures were in place to allow the implementation of decisions based on sample testing outcomes. Where SF products are detected, seven of the responding NMRAs had the authority and power to seize and dispose of them without having to resort to law enforcement agencies. However, only five of the respondents had the required infrastructure and documented procedures to ensure timely collection, handling, storage and in appropriate cases, disposal of samples or seized medical products that are unfit for use. Notably, one NMRA had the infrastructure and documented procedures for the handling of SF medical products but is legally required to involve law enforcers.Alleviating the SF burden requires engagement of all stakeholders and existence of coordination mechanisms. From the NMRA perspective, there are two types of stakeholders—internal and external. Internal stakeholders are NMRA departments, units and services responsible for various regulatory functions. External stakeholders include other health sector players, i.e., public health programmes, medical product procurement agencies, health professionals and their associations; representatives of other government authorities, such as law enforcers, investigators, judiciary and customs; and general public including mass media and patient communities. NMRAs were requested to indicate the existence of national interagency or interdisciplinary collaborative mechanisms such as national taskforces as to well as explain, which agencies were represented in these taskforces. Six respondents had both inter-agency and intra-agency co-ordination for SF products. They provided collaboration platforms for involving stakeholders to ensure coordination of the fight against SF trafficants. In six countries, there were also linkages between nationally coordinated mechanisms and national healthcare programmes.WHO has a global surveillance programme which produces medical product alerts to timely inform the healthcare stakeholders of the confirmed SF cases. The survey looked into the impact of this alert programme at the national health system level. Eleven of the twelve respondents reported regularly receiving the WHO alerts for SF medical products. Eight also reported having standard operating procedures to take action on the basis of the WHO alerts for SF medical products. One NMRA indicated that though a formal, written procedure was not in place, there were structured mechanisms to immediately alert professionals. Whilst the procedures were in place in almost all the NMRAs, the focus areas differed somewhat. All responding NMRAs that had procedures in place focussed on cascading the information to and on actions involving health professionals. Five of these also focussed on disseminating information to the general public, the extent to which this was done varying amongst the respondents.The study was the first attempt to map national frameworks established for implementation of priority interventions for preventing, detecting and addressing substandard and falsified medical products in the SADC region.There are two major streams of work in combating SF products. The first is the judicial approach, which is the preserve of the law enforcement agencies such as the police or the office of drug and crime. The focus in this approach is to introduce liability and in so doing, dissuade people from dealing with SF medical products. The second approach, taken by WHO and the focus of this article, puts forward the public health perspective to combine it with the judicial measures. SF medical products are a threat to public health, and NMRAs are the focal point for protecting public health with regard to medical products. In that regard, the survey targeted the NMRAs and looked in detail at their role in the fight against SF.The study responses are the result of the review undertaken by the NMRAs with the help of the questionnaire. At present, there is no consensus on a global tool for assessing national systems and approaches to PDR for SF medical products. The survey could be the first step towards developing a substantive tool for evaluating PDR for SF products.The study shows that policies and implementation plans are lacking for the majority of NMRAs. This largely affects guidance and direction in terms of the approach and strategy for the PDR for SF medical products, leaving the SADC region susceptible to such products being in their market. Notably, some NMRAs did include comments indicating that the need for formalising PDR policies had been identified in their strategic planning but was yet to be translated into policy documents and implementation plans.Regionally, there is strength in terms of legislation. Nevertheless, limiting the penalties and sanctions to the seizure of detected samples or non-conforming medical products does not completely deal with the ill-gained wealth from criminal activities associated with their trafficking. Similarly, those NMRAs countries that do not include administrative liability or sanctions in their legislation must be encouraged to do so in order to put in place significant deterrents to engaging in trade in SF medical products.Monitoring the supply chain, particularly through sampling and testing of products, is one of the measures for detecting SF products. A limitation noted was that there was a slew of countries without regular funding allocated collection of product samples for quality control testing which could affect the number of SF detection efficiency. At the time of the study, only three NMRAs had access to ISO-certified or WHO-prequalified laboratories. This has a negative impact on the confidence in and credibility of test results from the part of other NMRAs.No NMRA has limitless resources for testing. In any case, it is not always feasible to test all products on the market. By the same token, there are significant costs associated with handling and storing detected SF products. This indicates the need to embrace the risk-based approach to sampling. NMRAs should also be encouraged to regularly update their sampling plans based on risk assessments in order to focus on susceptible products and have a relevant post-marketing surveillance system.The trade in SF medical products is quite complex. Multi-pronged strategies are needed. Intra- and inter-agency co-ordination for responding to SF products, which was in existence in 50% of the countries, should be further strengthened and taken up by the other countries. A coordinated approach provides a platform for involving key stakeholders to ensure effectiveness of SF counteraction interventions.The ability for NMRAs to seize and dispose of SF medical products without involving law enforcers balances out stakeholder involvement and allows medicine regulators to act swiftly, should the need arise. Issues of bureaucracy when involving other agencies could pose a risk for NMRAs without this mandate and could hamper the efficiency in responding to SF medical products.Testing is essentially a means to an end. The true endpoint is enforcement of sound regulatory decisions based on the results of testing. NMRAs need to be assisted to utilise test results, starting with documenting procedures to do so.Cognisance of and contributing to the WHO Global SF alerts, confirmed in eleven of the twelve respondents, are commendable. Equally commendable was the relatively high number of respondents who translated this to action, by putting in place putting in place and abiding by standard operating procedures. The study results did however give an indication of the need to broaden the scope and focus of actions. Public awareness and knowledge are powerful weapons in combating SF medical products, especially given the increasing complexity of global supply chains and the spread of e-commerce.The study was an initial step in documenting the evidence of existing SF-counteraction frameworks in the SADC region of Africa. A limitation of the study lies in the methodology—a self-administered tool without external validation of the presented information. This calls for further efforts for comprehensive development of tools and devising acceptable mechanisms for data validation.Constrained access to affordable, safe and quality medical products can lead to the proliferation of SF medical products , 12. It There is a need to support NMRAs in the SADC region to strengthen PDR frameworks Access to ISO-certified or WHO prequalified testing facilities has to be increased.Detection activities need to be further strengthened to stem the prevalence and to prevent clinical use of SF products. NMRAs may require training in risk-based post-marketing surveillance and practical implementation of its approaches. Ultimately, the national and regional commitment to the fight against SF medical products needs to be stepped up to ensure allocation of resources for implementation of prevention, detection and response activities in this vital area.The inter-agency coordination to tackle SF medical products needs to be extended across borders, which are admittedly porous to SF medical products.Prevention, Detection and Response to SF products were in the SADC region call for further investments for strengthening existing frameworks. The availability of policies to guide intervention was low, and allocation of requisite funding to support the detection and testing of such products should be made a priority in national and regional agendas.


Humans are always exposed to the threat of infectious diseases. It has been proven that there is a direct link between the strength or weakness of the immune system and the spread of infectious diseases such as tuberculosis, hepatitis, AIDS, and Covid-19 as soon as the immune system has no the power to fight infections and infectious diseases. Moreover, it has been proven that mathematical modeling is a great tool to accurately describe complex biological phenomena. In the recent literature, we can easily find that these effective tools provide important contributions to our understanding and analysis of such problems such as tumor growth. This is indeed one of the main reasons for the need to study computational models of how the immune system interacts with other factors involved. To this end, in this paper, we present some new approximate solutions to a computational formulation that models the interaction between tumor growth and the immune system with several fractional and fractal operators. The operators used in this model are the Liouville–Caputo, Caputo–Fabrizio, and Atangana–Baleanu–Caputo in both fractional and fractal-fractional senses. The existence and uniqueness of the solution in each of these cases is also verified. To complete our analysis, we include numerous numerical simulations to show the behavior of tumors. These diagrams help us explain mathematical results and better describe related biological concepts. In many cases the approximate results obtained have a chaotic structure, which justifies the complexity of unpredictable and uncontrollable behavior of cancerous tumors. As a result, the newly implemented operators certainly open new research windows in further computational models arising in the modeling of different diseases. It is confirmed that similar problems in the field can be also be modeled by the approaches employed in this paper. The immune system is a real masterpiece that does extraordinary things every day while we do not realize such numerous activities. The job of the immune system is protecting our bodies against invading factors such as bacteria and viruses. Many factors such as low nutrient intake, lack of sleep, and high stress weaken a person’s immune system. On the other hand, people with cancer are also at risk for infectious diseases since they usually undergo special chemotherapy treatments using special drugs that weaken their immune system. The main purpose of these special drugs is destroying diseased cells without damaging adjacent tissues. Unfortunately, some healthy cells and tissues in the body may also be affected during this treatment.Nowadays, with the pandemic outbreak of the dangerous infectious disease COVID-19 all over the world, infectious disease specialists and epidemiologists constantly emphasize the strengthening of the immune system as one of the most important ways to prevent this deadly disease.In this paper, we provide some novel approximate solutions to a computational model that formulates the interaction between tumor growth and the immune system, including several fractional and fractal operators. The model consists of three state variables, each representing the number of specific cells in the problem. The interaction between these variables is presented by Itik and Banks  through τ, τ represents the rate of change in the population of the tumor cells, In this model, mentclass2pt{minimmentclass2pt{minimmentclass2pt{minimFor simplicity of analysis, we first nondimensionalize system  by emploof model . In 41]2\documenFractional calculus has a relatively long history almost as long as a integer-order differential account. However, in recent decades, the implementations of these concepts was neglected compared to standard calculus. This trend seems to have changed in the past years in general. A defining sign of this change is the increasing use of these tools in the literature. Thanks to the researchers’ efforts, many differentiated and integral operators based on different approaches were proposed and then successfully implemented in the past few years , 25, 30.To reach this goal, the subsequent parts of the paper are structured as follows. The analysis of model equilibrium points is presented in Sect. In this section, we analyze the equilibrium points of the considered model . These pPoint 1: the no “living cell” singular point Point 2: The tumor-free fixed point Point 3: The fixed point Point 4: The fixed point Point 5: The singular point Point 6: The interior fixed point In this section, we consider model  with thefined as  8\docume With the help of the Riemann–Liouville integral, Eq.  Let us show that II) Let us show that III) Let us show that Now by the Arzelà–Ascoli theorem  , , and Since H as follows: IV) We will show that the problem has a unique solution. To this end, let us define the map For at model  will alwIn this section, we use the Adams–Bashforth–Moulton (ABM) numerical method. Here we follow the steps to apply this method in solving the following fractional-order problem:  problem : 26\docuFigures mentclass2pt{minimIn this section, we study the following model: fined by  32\documApplying the CFC integral definition , we obtaThe initial value problem and the kernelsandsatisfy the Lipschitz condition.See . □The fractional nonlinear system (admits at least one solution.r system  admits aSee . □The fractional nonlinear system (always admits a unique solution.r system  always aSee . □Now let us focus on determining an approximate solution to the following CFC fractional Cauchy problem:  problem  as in [2Figures α of a function Now let us consider the model via the Atangana–Baleanu–Caputo fractional derivative as fined as  44\documα of a function The Atangana–Baleanu fractional integral of order fined as  45\documintegral , we get The initial value problem given by Eq. (possesses at least one solution in the intervaln by Eq.  possesseFirst, we define Moreover, we set Now we assume that all the solutions are bounded within a period of time: ved that  always pConsider the following fractional initial value problem: nd Kumar  have devFigures In this section, we will use several well-known fractal-fractional derivatives in model .In this subsection, we replace the derivative in  by the ffined as  61\documFigures mentclasspt{minimaIn this part, we replace the derivative in  by the ffined as  70\documduced in .Applying the Caputo–Fabrizio integral to Eq. , we obtagiven in .Figures In this subsection, we use the fractal-fractional derivative with Mittag-Leffler law in . So, we fined as  76\documduced in .Applying the Atangana–Baleanu integral to , we obtaposed in , we get n by Eq. , we obtaFigures In Figs. Recent advances in the presentation of efficient numerical methods in fractional differential calculus are a great help to researchers in modeling real phenomena. It has also been proven that differential calculus of integer order in some cases is incapable of describing the behavior of some phenomena or faces fundamental problems. In this paper, we study the dynamics of a tumor-immune model due to the unpredictable growth of tumor cells described by an attractive form of nonlinear differential equations. The importance of this model has led to many different approaches of studying the problem. Our study in the paper makes it clear how tumor cells interact with the immune system. The main difference of this paper from other papers on this system is that we have used new fractional and fractal-fractional definitions for the derivative in the system structure. Some of these new concepts of differentiation were introduced in 2017 by Atangana and his collaborators. They combine the idea of fractal derivative and fractional differentiation, which takes into account the memory, fractal effect, and nonlocality. The model considers processes like power-law, fading memory, and crossover. It is important to note that the numerical methods and analysis used in this paper are quite different from those presented in . In fact


To evaluate the optimal period of replanning to spare the rectal dose by investigating daily rectal movements during computed tomography (CT) image‐guided proton therapy for prostate cancer.To evaluate the optimum reference period for replanning, we analyzed 1483 sets of daily CT (dCT) images acquired from 40 prostate cancer patients and measured the daily rectal movement along the anterior–posterior direction based on the simulator CT (sCT) images and dCT images. We calculated daily dose distributions based on initial plans on the sCT images and replans on the dCT images for 13 representative patients, and evaluated daily dose volume histograms (DVHs) for the prostate, seminal vesicles, and rectum.77%) showed a 3.0 ± 1.7 cc excess from the initial plan and this excess was correlated with 9.9 ± 6.8 mm rectal movement. To identify the patients  for whom the replanning on the 10th‐day and 20th‐day CT images reduced the V77% excess to 0.4 ± 1.5 cc and −0.2 ± 1.3 cc, respectively, we evaluated the accumulated mean doses with a 1.2 cc criterion.The rectal anterior side on the dCT images around the seminal vesicles largely deviated toward the anterior side relative to the position on the reference sCT images, but the deviation decreased by referring to the dCT images and became nearly zero when we referred to the dCT images after 10‐day treatment. The daily DVH values for the prostate showed good dose coverage. For six patients showing rectal movement toward the anterior side, the daily rectal DVH (VOur data demonstrate that the daily movement of the rectal anterior side tends to move toward the anterior side, which results in a rectal overdose, and the mean of the movement gradually decreases with the passage of days. In such cases, replanning with the reference CT after 10 days is effective to spare the rectal dose. The use of proton therapy is thus expected to reduce the rates of gastrointestinal (GI) and genitourinary (GU) acute and late toxicities compared to the dose distribution provided by conventional photon radiotherapy.60Gy < 18%, which is used as the one of our rectal dose constrainsWith the use of opposed lateral beams as is now done widely for proton prostate cancer therapy, the movements of pelvic organs along the beam axis matter for the target dose coverage, and the movements along directions lateral to the beam axis matter for the dose to OARs and thus for the GI and GU acute and late toxicities. The stability of the pelvic organs' daily positions and the stability of the shape of the anterior rectal wall placed at nearby targets over the superior–inferior (SI) direction are important factors in the control of the daily dose to the rectum. In this context, we previously investigated the effects of daily organ motion on prostate treatment for ten patients by using daily CT images acquired throughout the patients' in‐room CT image‐guided proton treatment.In principle, uniform irradiation to the target volume with the same dose formation as that used in the original planning is not sufficient to maintain the daily dose under the planning conditions for a deformable OAR, for example, the rectum. One of the best approaches to this issue is the use of online adaptive planning in combination with CT image guidance, and simulation studies using daily CT images acquired during the treatment have thus been conducted.In the present study, we performed novel analyses to evaluate the replanning protocol for rectal dose sparing based on daily CT images acquired for image guidance prior to proton beam irradiation. In our clinical experience with proton prostate treatment using in‐room CT image guidance, the daily positions of the prostate and the anterior rectal wall identified on online daily CT (dCT) images tended to show differences from the positions in the simulator CT (sCT) images acquired at approx. 10 days pretreatment. We created a replan by referring to the new CT images for particular patients if the dose validation showed its necessity.We thus studied the stability of the daily movements of the anterior side of the rectal wall by referring to dCT images with four times higher values compared to our previous findings.22.A2, which were the same conditions as for the sCT scans.This study was approved by the Institutional Review Boards of our hospitals. We analyzed the dCT images and sCT images that had been acquired for 40 patients undergoing prostate cancer treatment.2.BWe analyzed a total of 1483 sets of CT images of the 40 patients. We created regions of interest (ROIs) for the prostate, SVs, and rectum on the sCT image and dCT images. First the ROIs of the sCT images were created by experienced radiation oncologists with the support of MRI images. Second the dCT images were registered into the sCT images by referring bony structure using a rigid image registration , and then all ROIs were produced on the dCT images performing a deformed image registration of ROIs of the sCT images using MIM Software. Finally radiation oncologists and an experienced medical physicist carried out the manual correction and validation of ROIs on all sets of dCT images.∑i(yid‐yir)/N, where yid and yir are the coordinates along the AP direction of the prostate from the dCT images and the reference, respectively, and N denotes the number of points constituting the posterior edge. We noted that the same registration procedure was also carried out for the prostate cancer treatment in our proton facility using the in‐room CT image‐guided system in which the boundary between the posterior sides of the prostate and the rectum on the dCT images was manually registered to the one on the reference images with visual image guidance after bony registration, and we compared the translational corrections used in the treatment with the corrections calculated by the ROIs basis, and the standard deviations of the difference was 1.0 and 1.3 mm along the SI and AP directions, respectively.Then, to evaluate daily movements of the patients' anatomy in respect to the reference CT images, all dCT images with ROIs were registered to the reference CT images by simulating prostate‐rectum boundary (PRB) registration as the same procedure described previously2.CFigure The average size of the prostate among all 40 patients was 36.9 ± 7.7 [mm], and thus the prostate was ±18.5 [mm] in the scaled Z coordinate. In addition, the position of the base and tip portion of the SV shown in Fig. 2.Dr = 0, 1–20, respectively. For each patient denoted by i, the mean value (mir) and standard deviation (σir) were evaluated as a function of the scaled Z coordinate from the measurements among responsible fractions by mir=∑d=r+1nPd‐Pr/N and σir=∑d=r+1nPd‐Pr‐mir2/N, where Pd,r are the daily and referenced positions of the anterior rectum along the AP direction, respectively, N denotes the total number of responsible fractions, and n denotes the total fractions. The mean value (mr), random error (σr), and systematic error (Σr) were obtained by the mean of mir, the root mean square of σir, and the standard deviation of mir, respectively, among all patients. We also estimated the average positional deviations using the formula 2.5Σ + 0.7σ proposed by Van Herk et al. to provide the margin values with 95% confidence.We evaluated the mean, the random errors, and the systematic errors of the daily positional deviations of the anterior side of the rectum as a function of the scaled Z coordinate with respect to the reference. In the following analysis, as noted before, the reference images were sequentially scanned from the sCT images to the dCT images acquired from the first treatment day to the 20th treatment day, denoted by 2.E®‐N; Elekta Corp., Stockholm, Sweden) based on a passive scattering method used in our proton therapy facility . The planning procedure and parameter settings were essentially the same as those used for the planning in our current prostate cancer treatment as described previously.± 9.1 cc. The representative rectal dose constraint of V60Gy < 18% was maintained by tuning a collimator margin to the PTV around the posterior side, and the averaged V60Gy in cubic centimeter was 7.3 ± 2.5 cc. We aimed to maintain V95% = 100% for cases in which the CTV is expanded by 3 mm, where VD represents the relative volume receiving at least a specified absolute or relative dose, D. However most of the planning particularly for higher risk cancer did not meet V95% = 100% for the expanded CTV as well as the CTV to keep the rectal dose condition due to the inclusion of the SVs in the CTV. In this case, we gave priority to maintaining the dose coverage of V95% = 100% for the prostate and the rectal dose condition of V60Gy < 18% while we deteriorated the dose coverage of the SVs. The replanning with the dCT images was created using the same procedure as used initially for the sCT images.The initial treatment plan and replans were created using a proton treatment planning system (PTPS) , systematic errors (Σr), and random errors (σr) of the translational corrections by the PRB registration along the AP (Y) and SI (Z) directions with respect to the bone registration for the 40 patients by referring to the sCT images and dCT images acquired on the 1st, 5th, and 10th days of treatment. The means and systematic errors are the largest when the sCT images are referenced. This indicates that the position of the prostate with respect to the bony structure at ~10 days before the treatment differs from the position during the treatment and tended to be at the anterior side.Table By referring to dCT images, the mean of the corrections showed a smaller deviation among the patients compared to the means obtained with the sCT images, and these values show a decreasing tendency as the reference day increases.Figure In general for all references, the errors and average positional deviations of the rectal wall tended to increase from the inferior side to the superior side, and they showed a maximum value around the region of the SVs. The values of these errors and the averaged positional deviations obtained by referring to the sCT images as a reference for the 40 patients in this study showed qualitative agreement with published results for 10 patients.The upper panels in Fig. 2 method using an exponential function with the slope parameter of −0.34. The systematic and random errors showed a weak decreasing tendency as the reference number increased. These results revealed that the daily shape of the anterior rectal wall tends to coincide better with the shape on the reference images when we refer to the dCT images acquired after 10 days of treatment have passed.In particular, the means, systematic errors, and random errors of the anterior rectal wall around the SVs at the SI position of Z =  −18.5 to −26.2 mm are shown in Fig. Figure For the evaluation of the daily dose parameters, we selected 13 patients based on the size of their rectal movement see Fig.  and cate95% of the prostate and the SVs, and the parameters of V77% of the rectum, respectively. These parameters were calculated based on the initial plan using the sCT images for each patient. The dose parameters of the prostate coincided well with the original values within standard deviations for all of the patients. Nevertheless Patient #5 showed the larger standard deviation and the lower minimum value for the prostate dose parameters in comparison to others. The reason was that the larger prostate rotational variation of 8.6 degree (standard deviation) over the sagittal plane was found in this patient and almost twice larger than the variation of others. This indicates that the need for the rotational corrections to the PRB registration for soft tissue alignment for this kind of patients. This issue will be discussed later. The dose coverage of the SVs showed a large variation, and 41% of the total 348 irradiations in the higher risk treatment maintained V95% values higher than the original planned value because the daily movement of SVs was large.Figure The rectal dose parameters were almost maintained within the daily standard deviation for Patients #1–#4 and #5–#7 (in the first and second categories). However, the rectal dose parameters for Patients #8–#13 (except for Patient #11) showing rectal movement toward the anterior side tended to be greater than the original values beyond 1 SD.Figure A single or several online monitoring of daily dose parameters is sufficient to make a replanning decision for individuals such as Patients #2 and #13, whose daily rectal doses indicated a stable dose and an overdose with rather small daily random variation during the first several treatments. However, the online monitoring of daily dose parameters is not helpful for making treatment decisions due to the high randomness demonstrated by Patients #7, #8, and #10, and it may thus not be clear whether the rectal dose is going to be problematic regarding the dose distribution over the entire treatment. The high randomness can be also observed in the daily dose parameters of the SVs. For such cases, the accumulated mean dose  including Patients #8–#13 in the third category were identified as for the replanned protocols based on the accumulated mean dose method with the tolerance of 1.2 cc on the 10th day of treatment as demonstrated in Fig. Figure 95% for the prostate and the deviation of daily mean doses from the planned values for the SVs and the rectum, respectively, for each of the patient's original plans based on the sCT images and the replanning based on the dCT images on the 10th and 20th days. In table Table Our analyses revealed that the treatment with the PRB registration was able to maintain good dose coverage of the prostate in all cases. However, the dose parameters for the patients showing the daily movement of their anterior rectal wall around the SV region toward the anterior side exceeded the averaged planned rectal dose of 7.3 ± 2.4 cc by 3.0 ± 1.7 cc, and this excess corresponds to a daily rectal movement of 9.9 ± 6.8 mm around the SVs toward the anterior side using the linear regression function shown in Fig. 4In general, the initial plan of radiotherapy treatment at our hospitals in Japan is created based on the sCT images acquired at approx. 10 days before the patient's first treatment. Then, during our in‐room CT image‐guided proton treatment for prostate cancer, the daily volume of the bladder is monitored by ultrasound scans and, when the bladder is not sufficiently distended, the patient is given water to drink. In addition, the defecation and exhaust gases are managed in order to maintain the condition of the rectum so that it matches that observed on the sCT images. Despite this pretreatment, as shown in Fig. By referring to dCT images as the days pass, it can be observed that the mean of the rectal wall tends to coincide with the baseline. The change of the means between sCT images and dCT images taken on the first day is the most significant and the deviation of the mean around the SV region decreases exponentially over a span of 3 days. This may be because patients gain familiarity with the preparations for CT scanning as the days pass and become better able to replicate the muscle tension around the buttocks and thighs while lying on the couch during treatment. In addition, patients become used to maintain greater stability of the rectal shape during defecation and exhaust gas examinations.95% ≈ 100%, the mean of the daily rectal dose exceeded the original planned value for the patients who showed rectal movement around the SV region toward the anterior direction, as mentioned earlier. For such patients, our data showed that the replanning with the reference CT after 10 days had passed was effective to maintain the rectal dose constraint for the rest of the patient's treatment. This result demonstrated that, in the prostate registration using fiducial markers (which is considered the standard approach for image guidance in proton therapy), the daily rectal dose might exceed the original planned conditions due to the rectal movement and deformation around the SV region toward the anterior direction. Since clinicians cannot identify the location of soft tissues via image registration using fiducial markers in combination with conventional kV x‐ray imaging, the use of CT image guidance could help address this problem by monitoring daily changes of the rectal shape and its dose parameters.The present study's daily dose calculations based on the sCT images with the PRB registration revealed that, although the dose coverage of the prostate in almost all of the irradiations satisfied the condition of Vρ=0.97 with the residual error of 0.6 cc (RMS). Assuming Gaussian distribution on the residual, if we consider 2.5 cc as a tolerance level for the mean of the rectal overdose during whole treatment, the probability to identify these treatment on the 10th day by setting 1.2 cc tolerance threshold on the accumulated mean dose was estimated to be 97.7% due to more than two sigma separation, that is, 2.5 cc > 1.2 cc + 2 × 0.6 cc. Nevertheless, since the tolerance value was estimated using our limited data, we recommend that the value need to be validated before clinical application.We propose an accumulated mean dose method to follow the daily dose changes. This method can be used to make a primary judgment regarding replanning on an online basis during the early period of each patient's treatment with CT image guidance. In particular, this method used during the treatment can be useful to monitor daily dose parameters for patients whose daily random variation of the anterior rectal wall is rather large and results in a rectal overdose. Our data showed that the accumulated mean dose up to 10th day of treatment and the mean dose over the whole treatment showed the linear correlation In combination with this method, a dose verification system dedicated to CT image guidance is desired to achieve more precise approaches to the monitoring and validation of the changes in daily dose parameters with considerations of the online image registration just prior to the patients' treatment. For this system, precise automatic contouring and deformed image registration using deep neutral networks may be necessary in order to accumulate daily dose distributions on the reference images.77% with the knowledge of the daily shift of the anterior rectum. Although the precision for the estimation of the rectal dose parameters is limited at 1.5 cc, this approach does not require dose calculations or anatomical contouring, and thus it can be easily incorporated into clinical workflows.As an empirical and alternative approach, we also provided linear regression functions based on our data sets to estimate the daily dose deviation of the rectal VThe beam scanning method has been widely adopted as a three‐dimensional irradiation technique in particle therapy facilities, and in principle, online adaptive replanning is possible based on this scanning method. However, technical problems remain to be resolved, that is, automatic segmentations, first planning optimization, and planning quality assurance. We propose (a) the use of an optimal period for replanning to spare the rectal dose by analyzing the pelvic anatomical movement on daily CT images acquired throughout prostate treatment, and (b) an approach to identify problematic patients for the daily rectal dose during the first period of the patient's treatment. The results of our analyses demonstrate that the replanning protocol conducted during the treatment after 10 days have passed was effective to control the target dose coverage as well as the rectal dose, and the proposed protocol is clinically applicable for prostate treatment using the scanning method as well as the passive method. However, if the daily rectal dose variation before the replanning period is rather large, higher doses may accumulate in the rectum during this period for particular patients. In such cases, stronger dose constraints of the rectum compared to the conventional protocol in replanning must be set to compensate for the accumulated rectal dose over the whole treatment, and for some patients it will be necessary to compensate for the maintenance of the rectal dose constraint due to the dose coverage of the target.In order to avoid the above situations, dose distributions on daily CT images should be validated more frequently during the first period of the treatment, and the frequency of the replanning needs to be increased. Since the changes in the mean values of the prostate and the anterior rectal wall between the simulator CT images and the first daily CT images were observed to be the most significant, the replanning based on daily CT images during the first week of treatment can be considered to be effective for this issue and is a clinically realistic approach, particularly for treatment using a beam scanning system with in‐room CT image guidance.The applications of the rotational correction (pitch angle) over the sagittal plan in addition to two transversal corrections performed in the PRB registration for the soft tissue alignment may improve the prostate dose coverage and also reduce the daily positional deviation of the rectal anterior side around the SV region as well as SVs, which may result in sparing the daily rectal dose further. The daily random error of the prostate rotation over the sagittal plane was found to be significantly large in order of 7–10 degree.Our data showed that the daily dose coverage of the SVs varied among patients, and the replanning protocols based on the rectal movement could not improve the maintenance of the daily parameters as planned. This was because the daily movement of the tip portion of the SVs located at each of left and right side of the prostate was found to be larger than the movement at the center of the prostate,The use of hydrogel rectal spacers for proton prostate treatment is considered to be effective to reduce the rectal dose, resulting in lower toxicity by making space between the rectum and prostate. The use of this technique can be expected to increase, but the stability of implantation as well as the absorption rate of a biodegradable compound over the treatment period must be studied by using several imaging modalities to adjust treatment plans appropriately.The limitations of this work are as follows. Since our findings were based on image datasets acquired only at our single facility for a limited number of patients, the stability of the rectal shape at other proton facilities might show characteristics that differ from our datasets even if the same immobilization method and pretreatment are used. We thus recommend that the position of the rectum as well as the prostate during treatment be monitored with CT image guidance, and the influence of anatomy changes on daily dose parameters should be investigated by using an approach that is similar to what we have described before the clinical application of the protocol that we have described.5We analyzed 1483 sets of daily CT images acquired throughout the proton therapy for 40 patients, which is a fourfold greater number of image sets than used in our previous study. The daily rectal movement was measured along the AP direction with respect to the reference CT images (including daily acquired images) by simulating the PRB registration. The results of our analyses demonstrated that the daily movement of the rectal anterior side around the seminal vesicle region tends to move toward the anterior side compared to that in the simulator CT images, resulting in a higher rectal dose in daily treatment, and the mean of the daily movement gradually decreases as the days pass. Our results also demonstrated that the accumulated mean rectal dose during the treatment can be useful to make a primary online decision regarding replanning for a patient with large daily variation of the anterior rectal wall movement that results in a rectal overdose. For such cases, we observed that replanning with the reference CT image after 10 days have passed was effective to spare the rectal dose.None of the authors have any financial interest or personal relationships with other persons or organizations that could inappropriately influence our work.