Data on O2 in tissue could be effectively utilized to enhance precision medicine. However, the nature of the data that can be obtained using existing clinically applicable techniques is often misunderstood, and this can confound the effective use of the information. Attempts to make clinical measurements of O2 in tissues will inevitably provide data that are aggregated over time and space and therefore will not fully represent the inherent heterogeneity of O2 in tissues. Additionally, the nature of existing techniques to measure O2 may result in uneven sampling of the volume of interest and therefore may not provide accurate information on the “average” O2 in the measured volume. By recognizing the potential limitations of the O2 measurements, one can focus on the important and useful information that can be obtained from these techniques. The most valuable clinical characterizations of oxygen are likely to be derived from a series of measurements that provide data about factors that can change levels of O2, which then can be exploited both diagnostically and therapeutically. The clinical utility of such data ultimately needs to be verified by careful studies of outcomes related to the measured changes in levels of O2.It is well understood that the level of molecular oxygen (O This review focusses especially on improving personalized medicine and outcomes of care, by carefully considering the basis and validity of clinical measurements of O2 in tissues and how those measurements can be used to advance diagnosis and therapy. While measurements of O2 in tissues have been recognized as an important factor in the clinical evaluation and treatment of many diseases, especially cancer  or concentration of oxygen ([O2] or cO2). These terms have physically rigorous meanings that can usefully be extended to describe gases (such as O2) that are dissolved in liquids or solids, including tissues. Partial pressure is the pressure exerted by oxygen in a mixture of gases, while concentration is the content of oxygen in the gas mixture or solid. Partial pressure is commonly expressed in mmHg, and these units are sometimes referred to as torr or kPa (SI unit used in the EU), while concentration of O2 is commonly expressed in mL of O2 per 100 ml, for example, in blood.The level of molecular oxygen, that is, O2 and [O2]. The transport of O2 across lipid membranes is known to depend on both diffusion and solubility in the bilayer, and to be affected by changes in the physical state and by the lipid composition, especially the content of cholesterol and unsaturated fatty acids. For example, because O2 partitions preferentially into lipophilic media, such as membranes, the solubility of O2 in membranes is about four times greater than in aqueous solutions ; V = volume; N = amount of substance [O2]; R = ideal gas constant; T = temperature.Does it matter clinically to know whether the technique is reporting . Conversely, if the component in which [O2] is measured and the solubility of O2 in that compartment is sufficiently known, then it should be feasible to determine pO2. However, it often is not feasible to measure these parameters readily.It is less straightforward in biological systems. If the solubility of oxygen in each component of tissue is known  and pO2 in tissues, each reported measure of O2 level in a tissue can be better considered as an average value. Average is used here in its more colloquial usage rather than as a statistically defined term, because different techniques output their measurement of O2 using differing methods pertinent to that technique. Each technique gathers information from a particular volume of tissue (irrespective of whether that volume is well characterized), which we refer to hereafter as the “interrogated volume.” The sampling of data within the interrogated volume is then used to produce a measure based on a sort of average O2 within that volume. Characterizing that “average measure” is made difficult both by the imprecision of knowing the exact volume queried, but also because the detection of O2 in that volume may be affected by factors such as its distance from the detector or, for optical techniques, different rates of scattering . Hence, we conclude that it is important to bear in mind that the measurements of O2 in vivo are fundamentally based on a sort of “average” within the interrogated volume.Because of the biological complexities in assessing O2 or [O2] and because of the imprecisions of knowing the volume being assessed and the tissues within that volume, it is more realistic to acknowledge the complexity of these issues for in vivo measurements in tissues by using a less precise term for measures of molecular oxygen in tissues such as “O2 levels,” which is the convention we follow in this review. We also argue that, while it is important to recognize the biological imprecisions in these measures, there are still many clinically viable uses of this information (such as assessing change in O2 levels).Because of all of these challenges in obtaining precise measurements of relevant parameters necessary to assess whether the data obtained by any technique is truly measuring either pONote too, within this paper, while focusing on the uncertainties due to sampling issues of each technique and variations due to biological factors, we are not taking into consideration further uncertainties due to inevitable instrumental noise, variations in the placement of the detector, etc.2.22 levels in various tissues, both in normal states and as altered by some diseases. (These measures are presented here as reported in the literature.) The first column presents the median pO2 obtained, using the Eppendorf electrode (or comparable polarographic techniques) to measure O2 levels in patients; also presented are two other indications of O2 levels: the hypoxic fraction  and the range of pO2 values found experimentally. These data illustrate both the variation in median O2 levels between types of tissues and how they may vary with physiology or disease. For example, (intertissue) in general the median O2 levels are lower in skeletal muscle and heart compared to the spleen; (intratissue) the median O2 in skeletal muscle at rest is higher than with exercise while, in contrast, there is almost no variation between the normal spleen and with Hodgkin's disease. The data in Column 4 illustrate the wide range that any given measurement can have in the “same” type of tissue, that is, almost all tissues range from ~0 to ~100 mmHg, even when their median value is quite different (see again spleen vs. bone). However, these very high values in the upper range may include experimental artifacts due to the measurement being taken within or very close to an arteriole, for example, a vessel feeding the microcirculatory bed.Table 2 levels have been an especially important focus for informing clinical treatment and prognosis. To give the reader a sense of how well supported the numbers are, the data in Table Table 2 level (~10 mmHg). Similarly, all but soft tissue sarcoma have a similar hypoxic fraction, ~25%; . Glioma appears to be an outlier on the range; glioma had no patient whose O2 level was above 50 while all others , the data presented are not unconditionally “absolute values” of O32. Figures 2 levels with an O2‐sensitive needle electrode . Oxygen was measured along several electrode tracks in each individual during a given measurement session (from near the tumor surface up to tissue depths of ~ 30 mm in breast cancers and in cancers of the uterine cervix). Each row in Figure 2 levels made in normal tissue (green) versus malignancies (prior to treatment) of these organs (red). The summary measures, median pO2 values . Comparing the two distributions for malignancies, we see that breast cancer is more highly skewed than cervical cancer, although they have the same median. Thus, using a single measure such as the median could overlook potentially important clinical information.Note that the data in Figure 2 measurements within a hypoxic range defined as ≤10 mmHg, while there were no hypoxic measurements in the normal breast. While Figure 2 level—and what is missed or obscured.Similarly, comparing the distributions for the normal tissues, the normal cervix had a substantial number of O2 measurements were taken in a breast cancer patient using EPR oximetry with Carlo Erba ink as the O2 sensor , followed by again breathing room air (blue: recovery). The sessions were repeated approximately weekly throughout the period of radiation therapy .Another example of heterogeneity is presented in Figure 2 levels responded to the patient's breathing an hyperoxic gas mixture and then returned rapidly to the baseline level after the 10‐min period. In this patient, there appeared to be some variation across the weeks of treatment, with the final levels for each period  all being slightly higher than at the beginning of radiation therapy (based on nonoverlapping standard deviations of the first and last measurements).The data for this particular patient illustrate that the O3.12 decreases due to radial gradients, that is, O2 diffuses through the tissues as it gets further from the vessels, due to O2 being consumed by the cells. As a result, there are variations in O2 levels from cell to cell (according to their distance from the microvessel). Within the cells O2 decreases in a microspatially complex manner as it is intracellularly consumed, with most of the consumption occurring in the mitochondria. There is growing evidence that diffusion of O2 into the cell may be constrained, that is, that O2 does not freely and rapidly flow into cells across the membrane, and therefore there are gradients from outside to inside of cells .These variations of O2 levels in normal tissues  as a consequence of HIF‐1α overexpression, upregulation of oncogenes, downregulation of suppressor genes, and activation of certain signaling pathways  and peripheral vascular disease, can result in significantly greater variability in OThere also is a potential for pathologies to interrelate with each other. For example, anemic hypoxia can develop in tumors due to the underlying systemic anemia of the patient  imaging of glucose derivatives , blood oxygenation level dependent (BOLD) imaging , encapsulated in biocompatible polymers , this is a good but not infallible technique to determine the presence of hypoxic regions. Regardless of its potential advantages, unfortunately this technique is no longer available clinically. This technique also had some practical limitations, including difficulties to repeat the measurement in a given tissue subvolume because of the local trauma produced from the preceding measurements.The Eppendorf electrode has been used clinically to provide direct tissue pOCurrently the only version of the oxygen electrode suitable for use in humans is a technique requiring temporary implantation of an oxygen electrode that extends through the skin and skull and is used to monitor severe brain trauma. Such a device, while very useful for its intended purpose, is not suitable for measurements in tumors  have been shown to be useful for imaging through a few centimeters of tissue. This takes advantage of the deeper penetration of x‐rays. It samples all emissions from the tissue, reconstructing the detectable light. The reconstruction then infers the site of the origin of the light where the radiation is absorbed and therefore which emits Cherenkov to the excited sensor molecule. This measurement therefore provides information on oxygen at that site  and unknown tissue properties skew the light penetration in tissue, making accurate spectroscopic measurements challenging at nonsuperficial depths  is another optical technique that is becoming increasingly used to define perfusion and O4.1.42 impacts the relaxation time of nuclei  and the ambient pO2 as shown in the sigmoidal oxygen dissociation curve. However, the vascular system is, by its nature, not uniform with respect to its saturation. Arterial and venous systems have very different saturations, with the capillaries (which is where most O2 is delivered to the tissue) being in between these extremes. Therefore, when measurements such as pulse oximetry or NIRS report HbO2 saturation  has been widely employed, most often using PET labels to indicate their location  are often used to judge the oxygenation status of malignant tumors. Responses to low oxygen levels in normal and cancer cells are mainly initiated by HIF‐1α, a key regulator for genes responsible for mammalian oxygen homeostasis. These responses (among others) include an increased erythropoiesis, angiogenesis, and glycolysis. The latter is mainly initiated by an enhanced cellular glucose uptake through GLUT‐1 transporters Semenza, , 2000. H2. Measurements of blood flow, for example, MRI perfusion . They found that the indirect techniques differed in their relationship to the O2 measured by the electrode, and there were few correlations among the indirect assays. It seems quite possible that, even with the indirect techniques that seemed to correlate with the direct method , the likelihood is that indirect methods would not always correlate well with each other because these parameters can be impacted by other factors as well. However, if they are to be used to predict O2 levels, it is important to identify those that strongly correlate with direct measures and/or to understand what other factors need to be measured in order to relate the information to O2 levels.In one of very few studies that actually attempts a head‐to‐head comparison of indirect methods of assessing hypoxia in the same tissue in humans, Nordsmark et al.  compared2 in tissues, can also be useful to confirm which candidate measures would actually be poor or unreliable indicators of O2 in deep tissues, so that they should not be used for that purpose. Transcutaneous pO2 or surface pulse oximetry are two such examples. These techniques are relatively easy and noninvasive methods that can be repeated; they have been used to monitor arterial O2 near the surface or to identify proximal ischemia in patients with peripheral vascular disease  labeled with fluorine‐18. FMISO was used as a surrogate for tumor O2 to determine whether FMISO data can be used as a guide to de‐escalate the radiation dose in patients being concurrently treated with chemotherapy for oropharyngeal papilloma virus related advanced tumors . In addition, the amount of O2 in the interrogated volumes may change very quickly, for example, in cycling hypoxia. Consequently, even direct measurements of O2 in tissues cannot accurately be reported as being a true single value. Instead, all current in vivo measurements purported to be measures of O2 levels in tissues are reflecting complex combinations of O2 levels, that is, are truly based on a mélange, that is, a mix of pO2 or [O2] values.As we have shown above, whatever measuring method is employed, the volume that is sensed by the method will always be too large to be truly homogeneous because of the variations of O2 can foster misinterpretations, and, in turn, can lead to inappropriate applications of these data to both clinical decision‐making and basic science theory. This is because, while the term “absolute” is correctly used to describe gas partial pressure measured in defined units above the pressure of an ideal vacuum, the term can be misinterpreted to connote measurements that are perfect, unbounded, definite, certain, etc. and these meanings, as we have shown, are inaccurate and misleading when applied to the in vivo circumstance. At a minimum, attention should be applied when using and interpreting the term “absolute” to describe O2 measurements, especially in the in vivo setting, and usage should be restricted to quantitative specification of vacuum as the zero of the defined scale, as opposed to a descriptor of data quality.It is important to recognize that use of the term “absolute” in specification of pO2 levels in vivo has been argued before  or qualitative in nature, where, for example, binary characterizations may be provided . Moreover, detecting changes in O2 in response to hyperoxygenation can be used, especially in the presence of severe hypoxia, to identify those patients who might benefit from receiving “hyperoxygenation” during treatment or otherwise modifying treatment doses. These assessments can then be exploited to enhance diagnosis and therapy, while keeping in mind the caveats involved in the interpretations of the values obtained from the measurements.While we argue that simple measurements of oxygen levels cannot fully characterize the heterogeneity present in tissues and can easily be misleading, we do not argue that current measures of O2 level can be rigorously controlled in cell cultures , but they cannot be controlled when applied to tumors in vivo.There are a number of significant implications of the thesis presented here for physiological researchers as well as for clinicians. Already noted is the potential danger of making too simplistic inferences based on studies conducted in cell cultures  are less clear, but these measures too should be based on understanding what is being painted. Currently, discussions of the potential advantages of increasing dose to more hypoxic regions of tumors have often focused on acquiring and applying definitive measurements of “absolute” pO2. As argued in this paper, this goal is not only unobtainable but also unnecessary. The really crucial goal is to find where there are likely to be more hypoxic cells in the tumor and accurately portray these regions. At least in a preclinical model  illustrate the potential clinical usefulness of using relative measurements of O2, that are not rooted in being “absolute” pO2 values. Recall that the data illustrate both the ability to gather repeated O2 measurements over a short time (minutes) and a long period (weeks or months). For clinical utility, having data from the short period (30 min in this case) could be used to determine whether a given patient's tumor is responsive to hyperoxic gas mixtures administered during preparation for and delivery of a typical fraction of radiation. This knowledge could help clinicians to determine when patients can benefit from strategies to enhance the effectiveness of radiation by using hyperoxygenation during treatment. The longer period of repetition (6 weeks in this case) could be used to monitor continued responsiveness of the tumor to hyperoxygenation during treatment and could monitor whether any changes occurred in the baseline measurements across a usual several‐week course of radiation.The data in Figure 2 data and outcomes , the variations among laboratories  treatment‐enhancing interventions , (2) consequences of therapy , and (3) disease progression. Of course, establishing the clinical value of these measurements of O2 will ultimately require appropriate studies of the outcomes of therapy using the O2 measurements to guide choices.Although the Eppendorf histography system is no longer used in the clinic, its use resulted in very excellent insights into the importance of tumor hypoxia on the response to therapy. Especially if comparative measurements could be made repeatedly over time to follow treatment and attempts to modify local tissue oxygen that information could be used to personalize treatments, for example to deliver therapy under more favorable conditions and/or determine the effectiveness of interventions aimed at decreasing local hypoxia. In summary, a key implication, especially illustrated for cancer, is that the most useful clinical value of measurements of direct, or even indirect, measurements of O2 in some other diseases. The most important other clinical application is likely to be for peripheral vascular disease (PVD). Relative measurements of O2 may help to identify and follow changes in levels of relatively low oxygenation  or the location of tissues at risk for diabetic PVD  and to follow the course of the disease.There are similar implications for the use of measurements of OEditors, . The cliFor such applications the key information will be changes, not absolute values. So, the challenge is to have an ability to measure repeatedly at the same site with a parameter that reflects the oxygenation in the tissue. Such measurements have the potential to very significantly improve clinical outcomes, but, as noted for their use in cancer, the ultimate proof of their value needs to come from appropriate studies of outcomes.Other types of clinical applications where changes rather than attempts to measure absolute values will be clinically valuable include: monitoring the status of healing wounds , and monitoring and avoiding untoward side effects of treatment .62 in tissues will inevitably provide results that are, at best, based on data that do not fully reflect the inherent heterogeneity of O2 in tissues over space and time. Additionally, the nature of all existing techniques to measure O2 results in complex sampling of the volume that is sensed by the technique. By recognizing these potential limitations of the measures, one can focus on the very important and useful information that can be obtained from these techniques, especially data about factors that can change levels of O2, and then exploit these changes diagnostically and therapeutically. Many examples already exist where measurements of oxygen in tissues have had significant clinical utility, even though in retrospect they really did not provide unambiguous information on absolute values of oxygen in the tissues. The clinical utility of such data ultimately needs to be verified by careful studies of outcomes related to the measured changes in levels of O2. For both basic physiological and clinical purposes, it is likely that the most useful information is likely to be obtained by employing multiple types of measurements that can thereby overcome the inevitable limitations of a single type of measurement.The bottom line is that clinical measurements of O2 imagers and has pertinent patents.HMS and ABF are owners of Clin‐EPR, LLC, which manufactures and sells clinical EPR instruments for investigational use only. HJH is a member of O2M Technologies which manufactures preclinical pODrs. Swartz, Flood, and Vaupel are the lead authors. All authors have either contributed to the conception and design of this paper or to the interpretation of data and analyses as well as drafting or revising it critically for intellectual content. All have approved the final content of this work, agree to be accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved, and represent all who qualify for authorship. We gratefully acknowledge all other scientists, clinicians, engineers, and coordinators on the PPG.


A pooled sample analysis strategy for novel coronavirus (severe acute respiratory syndrome-coronavirus-2 (SARS-CoV-2)) is proposed for a large population in this paper. The population to be tested is divided into divisions based on earlier observed detection rate of SARS-CoV-2 first. Samples collected are then grouped in appropriate pooled size. The number of tests per person in that population is expressed as a function of two variables: the observed detection rate and the pooled size or number of samples grouped. The minimum number of tests per person can be further shown to be a function of only one of these two variables, because these two parameters are found to be related at this minimum. A management scheme on grouping the samples is proposed in order to reduce the number of tests, to save time, which is of utmost importance in fighting an epidemic. The proposed testing scheme will be useful for supporting the government in making decisions to handle regular routine detection tests for identifying asymptomatic patients and implementing health code system in large population of millions of citizens. Another important point is to use smaller number of test kits, allowing more resources to speed up the mass screening tests, particularly in places not so rich. Spreading of the novel coronavirus (severe acute respiratory syndrome-coronavirus-2 (SARS-CoV-2)) and the related disease coronavirus disease 2019 (COVID-19) among people is very fast  to explaMillions of detection tests are needed to confirm infection and pick out asymptomatic patients (AP) for each wave of infection, practising universal testing and implementing health codes  in largeHowever, a test will normally take over 6 h to complete. The associated testing cost and the cost of the test kit are high, about US$3 to US$5 in Mainland China but over US$30  in many Pooling tests were reported years ago  with appOptimal parameters for group testing of pooled specimens for the detection of SARS-CoV-2 were established recently . The mosMathematical strategy of such pooling tests applied to COVID-19 was briefly outlined , 21. TheRandomised group testing optimised per country could double the number of tested individuals from 1.85 million to 3.7 million using only 671 000 tests . Poolingm people, and the pooled size or number of samples n in putting together for a pool test in that group of collected samples. The mathematical aspects [In this paper, the population to be tested is proposed to be divided into divisions based on the observed recent detection rate of SARS-CoV-2. The number of tests per person for such pooling detection tests can be expressed as a function of two parameters. The parameters are the observed detection rate of getting one positive result in testing a group of  aspects , 26 of t aspects  in handlP people to be tested is divided into (P/m) divisions each of m people first. The total number of tests in this population with a group testing scheme is L. This gives the average number of tests per person, to be expressed as a function f of two variables:A population of m, the average number of people per 1 positive detection in recent test; andn, pooled size or number of samples grouped together for a pool test.The two variables are:n persons for pool testing. There will be about yi positive results in a division i.Samples collected from each person in each division are grouped by jth division, the number of positive tests yj is most probably 1. However, there might be some divisions having 0, 2 or 3 in a large population P. The average value of yj is 1. This is because the infected patients are not uniformly distributed in each division. If the government implemented better quarantine containment scheme [m can be increased, say instead of having 1 positive result in 100 tests, to 1 positive detection testing in 1000 people. That is, m can be increased from 100 to 1000.In the t scheme , 6, m caLj in that jth division [The number of tests division  is2P, say 1 million, with j from 1 to P/m givesSumming over all divisions for a large sample of Lj in equation  asPutting equation  into equf against the pooled sized n with some values of m = 50, 100 and 150 are shown in f for m = 100 is 0.2 when n = 10. That means by pooling 10 samples together, only 20% of tests are required in comparing to doing one test for each person in the population.Curves of n at some viable values such as n = 5, 10 and 15, variation of f against m is shown in f is 1/n as shown in the figure.In keeping f can be found by differentiating equation  is found forThe minimum value of n and m are related by equation  is 2/n or 2/m to give fmin(m) asThe minimum value for fmin(m) is plotted against m in n were suggested and used in the literature [Different values of pool size terature –17. Poolterature  as in Taterature –29 had aterature  have shoterature , 28. Valterature  is commom are different for different risk groups  at different transmission stages. At the beginning stage of virus transmission, the number of confirmed cases is high with patients having obvious symptoms. In testing people in places starting virus transmission [n in pooling tests can now be derived from observed m as discussed in the above section.Values of smission  at a cerm is larger. For example, 107 AP were identified [m is 40 000. As the value of m was changed, the division number can be changed accordingly at different waves of transmission.For populations in places of lower risk, the value of entified  in testim tests the scheme is as follows:P into (P/m) divisions each of m people.Divide the collected population samples The individual testing samples collected are split into two parts with the first part kept first.n (n2 = m) samples for pool test.The second part is grouped with others to give n samples can be destroyed.If the test result is negative, the first part of all n samples will be tested individually.If there are positive results in the group, the first part of the The minimum number of tests per person is only 2/A safety management scheme on identifying infected patients with small number of tests is proposed based on the observed detection rate. For a detection rate of one positive result in In a division of 100 people, samples collected can be grouped as n. Thus the minimum number of tests, or equivalently the maximum value of pool size n as obtained by mathematical analysis in this paper is only a theoretical value. The pool size is constrained by sensitivity limit of the test, and n could be less than the theoretical value. An initial value of n can be estimated first using the curves in m depends on the maximum value of n. At the moment, n of 64 is observed to be viable, giving m of n2 as 64 × 64 or 4096. There is a maximum value of n so maximum m is n2. Also, there is asymptomatic value of f in view of m increases but with n fixed because the number of re-test becomes small as m increases.Testing kits usually have a limit of detection sensitivity, which limits the pool size Governments are always under criticisms not to arrange carrying out detection tests fast. They are also challenged to have management strategy without scientific support. Recommending a pooled testing strategy would be interesting to the governments to conduct interventions in a way that is more resource friendly. Mathematical studies to work out appropriate testing management strategy as above can assist them in convincing the citizens.Further, issuing health codes  for all m having one positive test result is used in determining the number of divisions in the population and pool size n. This scheme can save a lot of time as the number of tests can be greatly reduced. The scheme is therefore very appropriate for identifying AP in a large population, particularly in some places having difficulties in getting test kits.In view of the high transmission rate of COVID-19, pooling detection tests is more efficient for a large population. This paper aims at providing an optimal way for testing samples that could release some manpower and potentially speed up the process of testing. The population to be tested is proposed to be divided into divisions based on the recently observed detection rate. The average number of tests per person can be expressed as a function of the observed detection rate and the pool size. The number of people Health codes are implemented with the code valid for a certain period but it is reduced from 7 days to 3 days in many places. Consequently, a large number of tests are required for millions of people. If the government can handle 0.3 million tests a day, implementing this pooling detection scheme can test 1.5 million samples a day for keeping the health code system viable. This testing strategy is on handling millions of citizens quickly for implementing the health code system which is valid only for a few days. Another important point is to use a smaller number of test kits. This would be useful for government in places with limited resources to implement mass screening for millions of their citizens. The whole idea should be at least not losing any ability in detection, but with acceleration in testing.This is the first stage of study for implementing universal testing and health code for travelling by using smaller number of test kits. A mass testing scheme proposed to accelerate testing needs support from mathematical principles. Further study should go beyond the mathematics, adding more scenarios or sensitivity analysis to make this scheme more realistic. A more rigorous approach is to compare the pooled method to the unpooled method. The pooled testing should not be significantly inferior to those approved testing. More resources are needed to carry out such studies for comparison. At the moment, based on a summary of pooled testing , only a It is important to explore scientific principles on handling virus outbreaks such as developing vaccines and specific medicine. But it takes a long time to get research funding to carry out research and to validate the medicine at various levels of tests, including final clinical tests. Engineering technology hardwares have to be worked out accordingly from selecting scientific principles available and judge whether they are viable. Management has to ensure the hardware systems work as designed and monitored by Authority. Software management needs to control hardware engineering systems. Only with the synergism between science and management could virus outbreaks be under control. Government can only implement regulations effectively with support from citizens. To go one step further, safety can only be achieved on having technology, management and culture.


The implementation and evaluation of an in‐house developed geometry optimization (GO) software are described. The GO script provides optimal lesion clustering, isocenter placement, and collimator angle of each arc for cranial multi‐lesion stereotactic radiosurgery (SRS) volumetric modulated arc therapy (VMAT) planning.An Eclipse‐plugin program was developed to facilitate automatic plan geometry generation for multiple metastases SRS VMAT plans. A mixed, semi‐supervised exhaustive and k‐means clustering method is used to group lesions and place isocenters. The sum of squared euclidean distance (SSED) and the boundaries of lesions’ projection from beams’ eye view are used as supervised parameters to determine the optimal isocenter numbers. The collimator angle is optimized by minimizing the sum of the MLC opening area from all gantry angles for each arc. In all, 10 clinical cases treated during 2016–2017 were compared to plan quality of GO script generated plans. Paddick gradient index (GI), conformity index (CI), and local brain volume receiving 12 Gy  around each lesion were compared.P < 0.0001) and local V12 Gy , but not significantly different mean normal brain dose or CI. The volume of normal brain receiving ≥6 Gy was significantly lower in the GO plans. The total time to run the GO script for each case was <2 min.For four cases, the number of isocenters was reduced in the GO plans. For four other cases, the GO plans had the same number of isocenters as their corresponding clinical plans but with different lesion grouping. The GO plans had significantly lower GI (4.1 ± 1.0 vs 4.4 ± 0.9, The GO software automates lesion grouping, isocenter placement, and the collimator angles for SRS VMAT planning. When tested on 10 cases, the GO script resulted in improved plan quality and shorter planning time when compared to the clinical SRS VMAT plans. For total lesion numbers less or equal to 7 and isocenter numbers less or equal to 3, the script uses the exhaustive search method, and for more than 7 lesions the k‐means method is used. We choose to combine both methods in this study since the exhaustive method is not very efficient compared to the k‐means method when the number of lesions is large. Both methods use the squared Euclidean distance as the optimization metric, theoretically they should give the same clustering results. The flowchart for both methods is shown in Fig. For the exhaustive search method, the search starts with one isocenter (one cluster), and lesions are grouped based on exhaustive combination lists predefined for up to n = 3 isocenters, which for a single isocenter is the trivial single combination of all lesions being in the same cluster. In every possible combination, an isocenter location is placed for each of n clusters based on the lesions’ centroid in that cluster. A projection distance check is performed to determine if all lesions fall within a preset distance criteria from the assigned isocenter. If none of the possible combinations meet the criteria, the number of isocenters is increased by one, and the process is repeated, with all the possible combinations of grouping lesions into n + 1 clusters with n + 1 isocenters being tested again, until one or more combinations are found where all clusters meet the geometric distance criteria. If multiple different combinations of clusters meet the criteria, the Sum of squared euclidean distance (SSED) from the isocenter to all grouped lesions is calculated for each group and the combination with the smallest SSED is selected as the optimal clustering result. Details of the isocenter placement and geometric distance check method will be described in the following sections.If the total number of lesions is larger than 7 and the resulting number of isocenters is large than 3, a k‐means ++ clustering algorithm is used to group the lesions instead of the exhaustive method, running repeats equal to the total number of lesions to avoid suboptimal clustering results. We select squared Euclidean distance as the distance metric in the k‐means ++ algorithm to minimize the within‐cluster variance which is equivalent to the SSED. Each search starts with one isocenter per cluster and checks if each cluster generated from the k‐means method meets the distance criteria. If none of the clusters meet the criteria, the number of isocenters is increased by one as input for the next k‐means clustering until all clusters meet the set distance criteria.2.BThere are different methods to place isocenter for multiple lesions and a previous study compared the plan quality difference between these methods.2.CMax.boundary, isInstead of checking the distance from isocenter to the centroid of each lesion, this algorithm checks POB from the BEV of that isocenter at four different gantry angles: 0°, 45°, 270°, and 315° with collimator and couch angle at 0°  as a Varian Eclipse plugin. The software development follows the IEEE guide to software requirements specifications (Std 830‐1993). One advancement of this program is the utilization of the write‐back feature of the Eclipse API v15.5. Plans with optimal isocenter setup and arc geometry created by the algorithm are directly written into the Eclipse database, eliminating the manual DICOM import step. Once created, the plans are ready in Eclipse for the planner to run dose optimization.On startup, the GO script imports the current plan and structure set from Eclipse. The GO script will automatically select PTVs based on the PTV naming convention which includes the prescribed dose, and from this information the script predicts the fractionation for each PTV and groups them accordingly. The user can also manually select PTVs. The default machine for the new plan is the same as the current plan. The user can overwrite the default option by selecting a new machine. Once the isocenter optimization is completed, the GO script displays the calculated isocenter coordinates, lesions assigned to each isocenter, and the default arc geometry. The user can add, delete, or modify the default arc geometry. The GO script writes the optimized plans back into Eclipse if the user accepts the result. The GO script can also be used to optimize collimator angles for an existing plan in Eclipse. Once this option is selected, the GO script loads the arc geometries and beam information from the current plan in Eclipse. The information on the PTVs targeted in this plan is also loaded. Once the GO script completes the collimator optimization, the script displays the optimized collimator angle for each arc and writes the new collimator angles back to the existing plan in Eclipse.2.FBefore releasing the plugin clinically, two tests were performed to validate that the GO script worked correctly and as intended. The first test confirmed that the algorithms were transcribed accurately by checking the consistency of isocenter and collimator optimization results between the Matlab code and the Eclipse API code. The second test was to check that all functions provided in the plugin worked properly. This included the import of patient data from Eclipse; automatic selection of PTVs before optimization; the ability of the isocenter optimizer to correctly group lesions and display grouped information based on the set geometric criteria; the ability of the collimator optimizer to optimize the angle based on default or user‐defined arc geometries for the grouped lesions; and the ability of the GO plugin script to write back plans with correct isocenter coordinates, machine, technique, MLC type, arc geometries, dose calculation algorithm, and jaw settings. The testing results were fully documented in the software development verification (SDV) document for independent review by the head of treatment planning and computer service. After the plugin was approved, it was released on a Saturday in February 2019, to avoid interference with clinical operation. The signed release form, as well as the project plan and SDV, were uploaded to a teamshare website for department‐wide references.2.GIn all, 10 clinical SRS VMAT cases with a total of 65 lesions treated from 2016 to 2017, each treated with three or more isocenters, were selected for testing of the GO software. Both the clinical and the GO plans were generated in Varian Eclipse V13.6 with a specific analytical anisotropic algorithm (AAA) dose calculation model tuned for small targets.T‐test was used for statistical analysis.The plan quality of the GO plans was compared to the clinical plans by evaluating the Paddick gradient index (GI),3The isocenter and collimator optimization algorithms were successfully translated into the GO script and gave the same results as the original Matlab code on grouping lesions and isocenter placement. Because the MLC fitting method in the Matlab code does not take the MLC travel motion limitation into account, for some arcs, the optimal collimator angle found by the GO script could be up to 15° different than in Matlab, but the final plan quality was almost identical (data are not shown).For 4 out of the total 10 cases, the number of isocenters was reduced in the GO plans as compared to the clinical plans. For these cases, the total MUs were also reduced. For four other cases, the GO plans had the same number of isocenters as the clinical plans but with different lesion grouping (Table T‐test is used to compare the clinical plan to the GO plan for each parameter and the P value is shown in the last column. The GO plans had significantly lower GI, CI, and local V12Gy values than the clinical plans. The GO plans had slightly higher mean normal brain dose, but the difference was not statistically significant. The volume of normal brain receiving 6 Gy and higher was lower in the GO plans than in the clinical plans (Table The Paddick‐GI, the RTOG‐GI, the CI, and the volume included in the local 12 Gy isodose line around each PTV are shown in Table ns Table . As an e4Utilizing single isocenter VMAT planning to treat multiple intracranial metastases has been investigated in several publications.Several prior studies present clustering solutions.Both the k‐means clustering and the exhaustive search algorithms use the minimum SSED to group the lesions, leading to the implicit assumption that SSED is similar for each group. This may lead to unfavorable clustering where adjacent lesions could be clustered in a different group. To solve this issue, we could use a different distance as variable and metric, for example a correlation distanceFinding the optimal collimator angle for each arc is critical to reduce dose to normal brain and critical organs resulting from the “island blocking problem” and larger than necessary jaw openings.In a few prior studies, the collimator optimization objectives have been used to also optimize couch angles.For cases with a large number of lesions, complex lesion distribution, and proximation between lesions and organs at risk, different human dosimetrists typically generate plans with very different lesion groupings, arc geometries, and collimator angles. The solution depends on the dosimetrists' experience level. It is not trivial to determine the optimal grouping of lesions. Some dosimetrists would create more isocenters and that would significantly increase both planning and delivery time but not necessarily create a better plan quality than a solution with fewer isocenters. Our goal was to automate the planning processes and significantly reduce the planning time while maintaining similar or better plan quality.5The GO script was implemented in the clinic in February 2019, and since then it has been used in the planning of more than 70 cases. Using the GO software to group PTVs, set isocenter, and optimize the collimator angles for all arcs resulted in similar or slightly improved plan quality as compared to the manually created clinical plans while significantly shortening planning time and providing more consistent plan quality among different planners.There is no conflict of interest related to this study.


Cognitive science has a rich history of developing theories of processing that characterize the mental steps involved in performance of many tasks. Recent work in neuroimaging and machine learning has greatly improved our ability to link cognitive processes with what is happening in the brain. This article analyzes a hidden semi‐Markov model‐multivoxel pattern‐analysis (HSMM‐MVPA) methodology that we have developed for inferring the sequence of brain states one traverses in the performance of a cognitive task. The method is applied to a functional magnetic resonance imaging (fMRI) experiment where task boundaries are known that should separate states. The method is able to accurately identify those boundaries. Then, applying the method to synthetic data, we explore more fully those factors that influence performance of the method: signal‐to‐noise ratio, numbers of states, state sojourn times, and numbers of underlying experimental conditions. The results indicate the types of experimental tasks where applications of the HSMM‐MVPA method are likely to yield accurate and insightful results. Cognitive science theories now postulate sequences of processes, potentially progressing in parallel, that take fractions of a second. They postulate distributions on the durations of these processes. They allow for subjects to take alternative solution paths that can lead to the same result. Essentially, each time a subject undertakes to solve a problem they follow a different journey. In contrast to this theoretical richness, the classic measures of cognitive science are very coarse: whether an answer is correct, total time to perform a task, activation in a brain region, and so forth. These measures compress the journey taken on a problem into a single number. There are measures that do try to track the process, notably verbal protocols and eye movements. However, verbal protocols .That study and other related work have demonstrated the usefulness of the technique in analyzing imaging data. However, there are methodological issues that need to be addressed so that this method can be appropriately and successfully applied to imaging data. Consider again the data in Figure The purpose of this article is to assess the veridicality of the state divisions that the HSMM‐MVPA method produces and to understand the factors that influence the accuracy of the method and usefulness of results. In the first part of this article, we apply the method to an fMRI experiment  and excluding trials with problematic features,2.32.3.1The data processing stream first passes whole brain fMRI activity through standard fMRI processing steps and then converts this into a form appropriate for application of HSMM‐MVPA: a small number of orthogonal dimensions of activity where the activity has been deconvolved so that it represents processing at a point in time, which generated the downstream BOLD response. Below we lay out the steps in that processing.2.3.22. The anterior commissure‐posterior commissure (AC‐PC) line was on the 11th slice from the bottom scan slice. Acquired images were preprocessed using AFNI  acquisition on a Siemens 3T Verio Scanner using a 32 channel RF head coil, with 2 s repetition time (TR), 30 ms echo time (TE), 79° flip angle, and 20 cm field of view (FOV). The experiment acquired 34 axial slices on each TR using a 3.2 mm thick, 64 × 64 matrix. This produces voxels that are 3.2 mm high and 3.125 × 3.125 mmFNI Cox, . Functio2.3.3The primary goal of deconvolution is to take the lagged and diffuse BOLD signal and extract an inferred activity signal for each time point, effectively temporally aligning signal and behavioral data. The resultant time series for each run from the image‐processing step was deconvolved by applying a Weiner filter Glover,  with a h2.3.4z‐scored to ensure all components are of the same scale. Additionally, this transforms the data such that it can be thought of as roughly distributed as multivariate standard normal (though with heavier tails) for the purposes of likelihood computations used by the method. The result is a 20,747 × 20 matrix. Finally, all time points associated with incorrect trials and trials containing greater than 2 time points that had excessive outliers were removed. The resulting 18,206 × 20 matrix was used in the HSMM‐MVPA analysis. This represented 1,286 trials  that varied in length from 9 to 26 time points (points 2 s apart) with a mean length of 14.16 time points. This provides the input to the HSMM‐MVPA method  captures correlated sources of variance and the systematic variance common across subjects. The resulting components form an uncorrelated basis set with the first few components capturing the majority of the variance across subjects. The deconvolved data for the 60 runs were concatenated and yielded a 20,747 time‐point × 47,315 voxel matrix. These data were then l2‐normalized, and a spatial PCA was performed. We typically work with the first 20 components and in this experiment the first 20 accounted for 48.9% of the total variance of the deconvolved data.2.42.4.1N‐state model has an associated brain signature and sojourn time distribution, and is guaranteed to follow the previous statez‐scored PCA data. The parameter estimation process uses an expectation maximization algorithm  in state i is calculated as:vi and ai are the shape and scale parameters for the continuous gamma distribution for the ith state.. Note that this means that the probability of spending 0 time points in the state is the probability in the continuous gamma of a duration less than 1 s. In these cases, the state is skipped and the model moves on to a successor of that state.The distribution of sojourn time in each state. The state sojourn time distributions specify the number of time points that a subject spends in a state. We discretize the continuous gamma distribution to obtain a distribution of times that have the skewed property typical of latency distributions. The probability of spending z‐scored PCA components are approximately normally distributed. Their z‐scoring allows us to treat them as independent normally distributed values around the mean values for the state with a SD of 1. The set Fj of observed component scores fjk at time j for components k will have probability:Mi is the set of means μik estimated for state i.The probability of the fMRI activity in the state. The The family of models we consider are those that are purely sequential. There is no branching between states: each state of an M time points into N states.m1 + m2 + ⋯ + mN = M be one such partitioning where mi is the number of time points in state i, 1 is the start state, and N is the end state. The probability of this interpretation isWhile the underlying model treats each time point in a trial as in a single state, the estimation process considers all possible ways of partitioning a trial of While each partitioning treats a time point as being occupied by a single state, the probability of a state at a time point is the summed probability of all partitionings that assign that state to that time point. This yields the State Occupancies as illustrated in Figure 2.4.2As noted above, the models we are considering are purely sequential. Nonetheless, such models can be somewhat complex. There are two main decisions that need to be made that impact model complexity and accuracy of the resultant model: identification of the appropriate number of states and whether to use an informed or uninformed model specification.N, this approach estimates the maximum‐likelihood parameters for all but one of the subjects and then uses these parameters to calculate the likelihood of the data from the last subject. This likelihood measures the success of using these parameters to predict the data of the left‐out subject. The process is rotated through all the subjects and so can calculate the predicted log‐likelihood of the data for each subject assuming the N states.The search for good model fits is currently done using a leave‐one‐out cross‐validation (LOOCV) procedure. Assuming a model with some number of states N‐state model is justified if it predicts better significantly more subjects than models with fewer states. More generally, a model with more parameters is to be preferred over a model with fewer parameters only if it fits significantly more subjects.The data of the all‐but‐one subjects should be fit better with more states because there are more parameters, but this is no guarantee that more states will predict the data of the left‐out subject better. If using more states is just overfitting, the predicted likelihoods will not be better. A simple sign test can be used to identify “better” models by noting whether the number of subjects better predicted is more than would be expected by chance. An While number of states is one salient feature determining model complexity, equally important is the number and types of parameters being fit. Given a particular experiment, we can impose various constraints on brain signatures or latency distributions. In the current study, except for the Study Phase, the two study treatments result in similar times and variability Figure  among th2.4.3Figure 5 States: This is the fewest number of states that could match to the behavioral phases. The boundary between the first two states matches fairly closely the boundary between Study and Delay. However, it has only two states to cover the three phases of Delay, Solve, and Respond and identified two states in the Repetition phase.6 States: This is the smallest number of states that seems to capture the boundaries between the phases. The extra state again is a division of the repetition phase into two states. The root‐mean‐square deviation (RMSD) between the four phase boundaries and the first four state boundaries is 0.59 s.7 States: Adding an additional state results in the further division of the repetition‐detection phase into three states. The RMSD between the 4 phase boundaries and the first four state boundaries is .48 s.12 States: This is the best model as indicated by likelihood improvement in the LOOCV. Again, there are state boundaries near the behavioral phase boundaries, but they have a RMSD of .56 s. Which is not as good as the 7‐state solution. As we will demonstrate with synthetic data, sometimes the best‐performing model in LOOCV can have more states than actually generated the data.Focusing on the State‐1‐Separate models, the best fitting value of 12 states is many more than the five phases in the experiment. The finding of additional states is not unexpected, as the each of the behavioral phases of the experiment may involve multiple cognitive demands. The question of interest is whether some state boundaries align well with the behavioral task boundaries of the experiment. Figure p = .01, p = .001 and p = .0001 levels were calculated as follows: For an N‐state model N − 1 boundaries were placed with equal probability anywhere from the beginning to the end of condition. Then for each of the actual N − 1 mean phase boundaries , was a combination of the brain signatures, Bi, of the states i weighted by how much of the time point the states occupied, wi(t)Signal: The signal was based the six brain signatures estimated in the experiment. To incorporate individual differences in brain activity, brain signatures were created for each subject that were an equal weighting of that subject's mean activity during each state and the grand average state signatures.The HSMM‐MVPA method model assumes a distribution of activity patterns in a state and distributions of state sojourn times. This first set of synthetic simulations used the state sojourn times that were estimated for each of the 1,286 real trials. This provides temporal distributions of state sojourn times that reflect both within‐subject and between‐subject variation. The synthetic data for each time point of each trial were then generated as a weighted combination of noise and signal appropriate to the states for those time points:2.z‐scored PCA scores have more extreme values than would be expected from a standard normal distribution—0.39% of the scores are greater than 4 SDs from the mean, which is about 60 times more than would be expected with a normal distribution. Second, there is a temporal correlation between adjacent measures. The values in one time‐point have a correlation of .77 with the values in the next time point, .34 with the values in time points that are 2 time points downstream, and .08 with time points that are three time points downstream. Data that approximate these properties can be created by making the noise for the sample t, N(t), a running average of four samples from a t‐distributions with degrees of freedom 3, T3: Noise. There were two important properties of the data to capture in the noise that is added to the signal. First, the 65% of time points were occupied by a single state, in which case the signal was just the brain signature of that state.t‐distribution provides the excess of extreme values. Temporal correlation is created because adjacent noise samples share some t samples in their sum.The F(t), is created as a weighted sumz‐scored and have extreme values truncated to the range −5 to 5 just as the actual data was processed described earlier. This generates a matrix of 18,206 time points × 20 dimensions for the synthetic data like the original data with similar properties. Models can then be fit to the resulting data.The synthetic data, SD in sojourn times was 1.52 and RMS of the brain signatures was .265). The correspondence between these measures for synthetic and experiment data suggest that 0.1 is a reasonable estimate of SNR of the experiment data. The synthetic data at this setting has similar properties to the real data: proportion of values greater than 4 SDs from the mean (.34%) and correlations of brain signatures at various lags .We fit the 6‐state model to synthetic data generated with different signal‐to‐noise ratios from .001 to 1.0. Figure Parts (c) and (d) of Figure 3.2Mi, selected from a uniform distribution between 2 and 8 s:State sojourn times: Each state had a mean sojourn time, The synthetic data used in Section i, selected uniformly from 2 to 5 (the range in the fits for the experiment)Six such state sojourn times sum on average to 30 s, which is about the duration of trials in our experiment. However, various ground truth models varied from 20 to 40 s in the summed mean sojourn times of their 6 states. The individual trials were generated from gamma distributions with a shape parameter, aThe scale parameter of the gamma would then beThis produced state sojourn time parameters like those estimated for the experiment. To reflect individual differences, the scale parameter for an individual subject j was itself distributed around the mean scale according to a gamma:SD of .25 of the population mean sojourn time. The times for state i and subject j would then be gamma distributed on individual trials:As a consequence, subject mean state sojourn times varied around population mean state sojourn times with a Brain signatures: Brain signatures were generated as 6 states of 20 values randomly selected from standard normal.kth value for the ith state. Subject j's brain signatures were generated by adding to the group signatures values another set of standard normal weighted .5.which gives the As before, the signal values for individual time points were generated according to the state occupancy of that time point. This was mixed with normal noise according to different SNRs. We considered a reduced range of SNRs—from .009 to .23, which was where the major changes were in Figure Averaging over the 100 experiments at each SNR, Figure 3.3Figure p < .01 under a sign test), each of the 50 synthetic data sets were better fit by models with 7–13 states, the mode being an 8‐state solution (32% of the cases).Figure Splits: In some cases, the better fitting model had two states corresponding to one of the generating states. The brain signature for the first state was also correlated somewhat with the brain signature for the preceding state while the brain signature for the second state was correlated somewhat with the subsequent state.Bridges: In some cases, a state typically 1 time point in length, was placed in between two generating states. This state correlated relatively strongly with both of the adjacent states. This state also captured the transition between the two states.Examining the brain signatures estimated for the best state solutions selected by the LOOCV method, each of the six generating brain signatures was correlated with one or more of the estimated brain signatures. Extra states had brain signatures related to the original 6 in two ways:The HSMM‐MVPA method assumes that whole states occupy time points even as it considers all ways of partitioning states into sequences of time points. Some time‐points will have a mixture of two states and better fits can sometimes be obtained if a state is generated that is a blend of two states. Further, adjacent states include temporally correlated noise as well, also contributing to the need for bridge states. The remainder of Figure Figure In almost all cases, the 6 state solutions were better than fewer states and corresponded closely to the ground truth. However, because of the fact that there is temporally correlated noise and states partially occupy time points, the LOOCV may well select a higher‐state solution as better. When this happens, adjacent states in the higher‐state solution will be correlated. This indicates that one should be cautious about selecting high state solutions where adjacent states are correlated. In past research we have chosen not to use a higher‐state solution when there were correlated states. For instance, Anderson, Pyke, & Fincham  and Ande3.43.4.1Results thus far show fairly successful recovery of 6‐state structures with a SNR of .1. To explore success with different numbers of states, we generated cases where the ground truth varied from 2 to 18 states, looking only at the cases of an even number of states. The brain signatures and sojourn times for the states were randomly generated as in the previous simulations using again 20 subjects with 100 trials each. For each choice of states, 100 data sets were generated with a SNR of .1. Figure Figure 3.4.2Top panels: There were 4 states and the mean sojourn times of states 1, 3, and 4 did not vary with conditions and were 3, 4, and 5 s, respectively. State 2 lasted an average of .5, 2.5, 4.5, and 6.5 in the four conditions.Bottom panels: There were 5 states and the mean sojourn times of states 2, 3, and 4 did not vary with conditions and were 3, 4, and 5 s, respectively. States 1 and 5 either were 1 or 4 s. Crossing these possibilities yielded the four conditions.Except for the simulations of the data from the real experiment, both data and models have just had a single condition. We simulated two experiments where the sojourn times of some states varied as a function of 4 conditions. Past experiments (e.g., Anderson & Fincham, SD of the means over 100 simulated data sets. The less informed models tended to underestimate the effect of condition and assign a little of the effect of conditions to other states. The more informed models performed a bit better.Figure 4We have developed the HSMM‐MVPA methodology on the assumption that the resultant models will accurately identify brain states and their sojourn times, allowing for meaningful mappings between brain activity and corresponding cognitive processes. The critical output of the process is a parsing of a long trial into a sequence of periods of relative constant mental activity. One can then examine the patterns of brain activity during these periods as well as determine how the sojourn times of these states vary with experimental conditions. This article has been concerned with assessing how well the method does at identifying the states that are driving systematic brain activity over the course of a trial and the factors that affect these results. While much of the presentation was concerned with using synthetic data to identify where the HSMM‐MVPA method gave good results, we started out by applying the approach to an experiment that had ground truth about where some of the state boundaries should lie. That ground truth was provided by the five phases of the experiment. The boundaries between these phases were identified fairly well by any HSMM‐MVPA that had 6 or more states. Using a LOOCV criterion, there was evidence for up to 12 states Figure . HoweverThe best‐performing model in terms of the results of an LOOCV can sometimes have more states than actually generated the data, as shown with synthetic data Figure . This haIf one has a firm idea of the number of states that should be present, the question of interest is whether one can accurately identify them in the imaging data. While there is never a guarantee that the peculiarities of one's imaging experiment will not lead to failure, the results were fairly good for SNRs greater than .05. The noise contributing to the SNR will be determined by quality of scanning and factors of experimental control. The signal contributing to the SNR will be determined by how strongly the individual states differ in their brain signatures.State recovery decreases somewhat as one seeks to identify more states Figure  and fallMany cognitive tasks studied by fMRI average less than 2 s and this methodology is not appropriate for these. We have developed complementary HSMM‐MVPA methods with EEG, MEG and ECoG that have shown success in parsing such brief tasks into states (e.g., Anderson, Pyke, & Fincham, 4.1http://actr.psy.cmu.edu/?post_type=publications&p=31445.The analyses and models in this article can be obtained at The authors declare they have no conflict of interest.


The Domain Interface Method (DIM) is extended in this contribution for the case of mixed fields as encountered in multiphysics problems. The essence of the non-conforming domain decomposition technique consists in a discretization of a fictitious zero-thickness interface as in the original methodology and continuity of the solution fields across the domains is satisfied by incorporating the corresponding Lagrange Multipliers. The multifield DIM inherits the advantages of its irreducible version in the sense that the connections between non-matching meshes, with possible geometrically non-conforming interfaces, is accounted by the automatic Delaunay interface discretization without considering master and slave surfaces or intermediate surface projections as done in many established techniques, e.g. mortar methods. The multifield enhancement identifies the Lagrange multiplier field and incorporates its contribution in the weak variational form accounting for the corresponding consistent stabilization term based on a Nitsche method. This type of constraint enforcement circumvents the appearance of instabilities when the Ladyzhenskaya–Babuška–Brezzi (LBB) condition is not fulfilled by the chosen discretization. The domain decomposition framework is assessed in a large deformation setting for mixed displacement/pressure formulations and coupled thermomechanical problems. The continuity of the mixed field is studied in well selected benchmark problems for both mixed formulations and the objectivity of the response is compared to reference monolithic solutions. Results suggest that the presented strategy shows sufficient potential to be a valuable tool in situations where the evolving physics at particular domains require the use of different spatial discretizations or field interpolations. The growing demand of industrial complex simulations, together with the continuous improvement of computer technology, have stimulated the popularity of algorithms capable of processing large systems of equations. This is the case of algorithms based on domain decomposition techniques which basically stem from the need of decomposing a large discretization into a number of smaller discretizations, e.g. using finite elements (FEs), or from the assembly of independently generated meshes which form part of a whole system. The latter scenario can be found for instance during simulations of an aircraft where both wings and fuselage discretizations are designed independently. In these situations, the type of algorithms capable of gluing non-matching meshes are crucial and represent the main focus in the present contribution.Examples of non-conforming interfaces arising from the assembly of non-matching grids can be found in , 25 wherThe most general scenario at a non-conforming interface mentclass2pt{minimMany Domain Decomposition techniques assume that the interface is geometrically compatible  but moreIn mortar methods, the coupling constraints are enforced in a weak sense, i.e. in an integral or average sense along portions of the interface, and are often referred to as segment-to-segment techniques. In general, a displacement field at the interface, One significant drawback of such classical mortar techniques arises when the coarsest discretization is selected for the reference solution field. In such scenarios, the performance of mortar methods might be less optimal as argued in , 18. MorIn contrast with established mortar methods, the DIM  is basedinf-sup condition) [Lagrange multipliers  are typindition)  in orderndition)  is consindition)  and the ndition) . A main mentclass2pt{minimmentclass2pt{minimThe geometrical aspects of the DIM introduced in  are summFinite strain theory is considered in our developments, although an infinitesimal deformation approach can be recovered by considering small displacements (i.e. compared to the domain dimensions) and infinitesimal displacement gradients. Compact notation is utilized for tensor quantities in this contribution.s and q.Consider the union of two non-overlapping domains k as indicated in step (ii). As discussed in [k does not interfere with the results although An independent FE discretization of the two bodies, leads to a number of ussed in , the magA Delaunay triangulation defining the interface domainTriangular linear elements r method .The corresponding deformation map for each domain D at time As shown in , an exprcf. Fig. .n. Due to the linear interpolation of the incremental displacements in   37) is eloped in , 28. TheIn order to stabilize the pressure field for the case of compressible and nearly compressible materials the projection term in  is addedThe identification of the pressure Lagrange multipliers tions in . The exptions in  concernie gap in  and the b stands for the the discrete nodes corresponding to the Lagrange multipliers interpolation using the domain-wise constant shape functions A Galerkin-based spatial discretization is considered in which the solution field components and their variations at each domain cated in  to 177)\documentments in  to 49)  \documenailed in  and 101) is eTo this end, the solution The work performed at the interface by the Lagrange multipliers The identification of the temperature Lagrange multipliers tions in . The exptions in  concernie gap in  and the The variational statement for the thermal constraint equations reads:equation ). In othAs argued in  and brieentclass1pt{minimassion in  and enfocf. Fig. .L stands for the length of the base-side of k denotes the thermal conductivity coefficient.The stabilization parameter As seen in the monolithic formulation \documentmentclass2pt{minimribed in .The corresponding FE approximations of the thermal variational statements in , 122) a a122) anBoth mechanical  and therIn order to obtain scalabilities in a massive number of processors it is convenient to recast the global systems  or 142)142) in tAfter solving the flexibility problem in  and obtaoblem in  which isIf one or more subdomains sions in  to 149)\documentained in , 20, 22.In this view, exploiting the novel features of the DIM method, i.e. a continuous fictitious discretization of the interface, a new methodology to handle rigid body modes has been proposed in  and adopments in  to 46)  \documenments in  to 114)\documentcf. Fig.  and \docfound in  for the Note that the term t}cθ˙ in  allows fcated in  leading The linearized set of equations in  and 142142 obtai methods ). In thicated in .Assuming a fix domain decomposition and a given Delaunay interface discretization, the iterative scheme for the coupled thermomechanical non-linear DIM framework is summarized in Box Two representative tests are selected to assess the performance of the multifield domain decomposition framework. Attention is focused on the continuity of the hybrid solution field at the interface, the convergence rate upon mesh refinement and the influence of the stabilization parameters as well as the activation of the Lagrange multipliers for the different solution fields at the interface. Finite strain theory and plane strain conditions are considered in all two dimensional examples.F is applied at one end of the membrane while the opposite end is fixed in both directions.The mixed ons cf. , 37). Th. Th\docuThe membrane is considered elastic with Young’s modulus In order to assess the convergence of the proposed methodology, five different discretization scenarios are considered. These concern a monolithic approach and the DIM on the body partitioned in two domains, i.e. mentclass2pt{minimThe vertical displacement of point A cf. Fig.  is monitIt is observed that all analyses provide an analogous convergent behaviour and the differences between the monolithic approach and the domain decomposition strategy are hardly visible beyond the third discretization, i.e. around 200–250 FE. In both compressible and incompressible analyses the DIM with non-conforming interfaces shows a slightly slower convergence with respect to the domain decomposition with conforming interfaces.The influence of the activation of the pressure Lagrange multiplier mentclasspt{minimaIt is observed that the activation of h  presented in  is extenThe extended DIM method for multiphysics problems essentially relies in an explicit discretization of the interface by means of a zero-thickness Delaunay triangulation as in the original methodology. This strategy allows setting a full tied contact in the most demanding scenarios, e.g. between geometrically incompatible interfaces. Continuity of the mixed solution field is satisfied across domains through the incorporation of Lagrange Multipliers. A Nitsche method is adopted in which a stabilization term is added at the constraint equations of the mixed field, i.e. one stabilization term per field. The incorporation of the stabilization term avoids the appearance of zero diagonal terms at the global system and instabilities are avoided if the LBB condition is not fulfilled by the chosen discretization. It is important to remark that, since the penalized term is part of the Euler-Lagrange equations of the variational principle, it will vanish upon mesh refinement .The formulation presented in this manuscript is based on the identification of all Lagrange multipliers to connect the mixed field and their consideration in the weak variational form together with the corresponding consistent stabilization term. This technology is first assessed for a mixed displacement/pressure formulation to tackle incompressible materials in which a Polynomial Pressure Projection method is employed to account for the stability of the resulting systems within each domain. It is demonstrated that the pressure field is transferred with continuity across the non-conforming interfaces only if the Lagrange multiplier concerning the pressure field is taken into account, i.e. the continuity of the displacement field at the interface, enforced by the displacement Lagrange multipliers, does not guarantee the continuity of the pressure field at the interface. In fact, this effect vanishes upon mesh refinement since the stabilization term is no longer needed and vanishes too. It should be highlighted that the sensitivity of the results to the stabilization parameter is, however, very small. For the mixed displacement/pressure formulation, a variation of eight orders of magnitude in the pressure stabilization parameter would cause an insignificant jump in the pressure field.The assessment of the DIM formulation for multiphysics problems is completed with a coupled thermomechanical problem with applications to bulk metal forming processes. In this case, a mixed displacement/pressure/temperature field is considered together with the corresponding Lagrange multipliers at the interface. Continuity of the three fields is obtained across all interfaces and the distribution of the mixed field is in very good agreement with the reference monolithic solutions. The mechanical response resulting from the upsetting of a billet studied in this example is remarkably close to those reported in literature which validates both the adopted constitutive model and domain decomposition technique.An alternative non-intrusive methodology to handle RBMs was introduced in the original contribution presenting the DIM method. This methodology essentially adds an extra stabilization term to the energy functional with contributions of all adjacent interface patches avoiding, in this manner, the calculation of a pseudo-inverse at floating domains. As a result, the band structure of the global system is preserved and, for this reason, possible parallel solution strategies employed in other dual domain decomposition methods can still be employed. In the analyses considering the mixed displacement/pressure problem no singular modes may arise from the pressure field. However, in the thermomechanical problem, singular temperature modes with constant temperature distribution may arise at floating domains. An extra stabilization term analogous to the term restricting the RBMs should be considered but in the present formulation the subdomain matrices with diagonal temperature coefficients can still be inverted due to the introduction of a transitory term which acts in a similar way than the mass matrix in a dynamics problem.The DIM method for mixed fields inherits all advantages outlined for the irreducible formulation, i.e. the interface connections are set up as a result of the connectivity in the automatic Delaunay triangulation and do not rely in user criteria for the master and slave DOFs or intermediate interface projections as it is done in established techniques such as the mortar approach. For this reason, the methodology shows potential to be used in cases where completely different discretizations are needed among domains since different physical phenomena are tacking place therein. The use of the DIM method could be specially advantageous when a particular domain discretization needs to be changed at a certain stage of the problem in order to improve the simulation of the physical process.


The dominant wave that couples to flows is the fast wave due to the in-phase harmony between perturbations of the compressive parameters and the fast wave. The induced current density possesses a steady orientation at the reconnection site which governs the diffusion or propagation of the waves. Induced perturbations by the nonlinear forces together with their back reaction on the Alfvén wave have a significant role in the current density excitation being responsible for the creation of inflows and outflows that are possible candidates for the creation of solar jets which has a significant contribution towards coronal seismology.The aim of the present study is to provide insight on the induced compressive perturbations together with the modifications of the environmental parameters in the course of Alfvén wave interaction with a solar magnetic null-point. The shock-capturing Godunov-type code PLUTO is used to solve the set of ideal magnetohydrodynamic equations. The nonlinear effects connected with an initial Alfvén pulse nearing a magnetic null point induces fast and slow magnetoacoustic waves with anti phase conduct. The induced current density and flows are independent of the local plasma- The significance of the Alfvén wave roots in its nature where out of all the mgnatohydrodynamic (MHD) waves it is the only wave that in the linear regime does not perturb the plasma density and hence the neighboring sites. Nonetheless, in the nonlinear regime, the Alfvén wave indeed perturbs the plasma density where due to its wide range of direct and indirect effects plays influential on the concept of e.g., energy transfer13 is modeled. As told by its name, a magnetic null-point is the center of a magnetic site where the magnetic field is zero. Such a structure is famous since it is a magnetic reconnection site15, where even the reconnection may exhibit an oscillatory conduct17. It is worth stating that a vast number of magnetic null-points exist in the solar atmosphere which is due to the complexity of the magnetic flux distribution. Thus, tens and thousands of magnetic null-points may be present at an instance of time in the solar corona19, which spans from coronal rain observations in magnetic null-point topology20 to jet observations rooted in reconnection sites21. In a plasma medium, the magnetic field lines may break and rearrange building a concept named as magnetic reconnection. A reconnection site provides a foundation for studying various aspects of wave dynamics, especially the interaction of waves with the null-point itself. A reconnection site in the solar atmosphere hosts solar flares22 and coronal mass ejections23 which are strong enough to affect the Earth’s atmosphere.In the context of the present study the Alfvén wave is studied due to its induction aspect. This is in a sense that the induced MHD waves together with the perturbations of the physical variables in the course of Alfvén wave interaction with a magnetic null-point24 provided motivation for carrying out the experimental setup of the present study to shed light on a possible mechanism for the creation of solar jets. The main actor in this regard is the Alfvén wave where the domain under consideration is a magnetic reconnection site25. As Alfvén waves are known to induce compressive perturbations in plasmas with transverse structuring27, it would be very instructive to work out its induced waves and perturbations in a homogeneous medium in the course of interaction with a null-point. Before proceeding, it is worth stating that the interaction of MHD waves with a magnetic-null point28 is accompanied by density accumulation, see e.g.32. The perturbations of the density spikes together with the back reaction of the induced perturbations on the initial nonlinear Alfvén wave would surly provide insight on the creation of perturbations and induced flows that may result as solar jets.The observation of reconnection jets33. These induced waves propagate across the magnetic field lines and accumulate at the null point. The perturbations of the physical variables together with the modification of the environmental conditions due to the induced waves are highlighted together with their back reaction on the initial Alfvén pulse as they evolve is highlighted. In the proceeding section, the initial conditions of a reconnection site that would host wave interaction is presented before providing details of the experimental setup. The succeeding sections are to shed light on the induced fast and slow magnetoacoustic waves together with the induced flows and perturbations of the physical variables which results in modifying the plasma parameters of the reconnection site, where a brief discussion is presented regarding the evolution of the plasma characteristics which contribute towards coronal seismology.The primary aim of this work is to provide insight on the nonlinear dynamics of MHD waves nearing a 2.5D magnetic null point by carrying out numerical simulations based on the implementation of the MHD theory. An initial Alfvén wave is kicked towards the magnetic null point by taking in to account its potential forces connected with the nonlinear effects, which creates fast and slow magnetoacoustic waves, see also Refs.34, it is not far from reality if the effects of solar gravity and plasma viscosity are neglected. This implies the ideal MHD equationsL being the length scale for magnetic field variations. In the context of the present study the magnetic equilibrium structure is curl-free which includes a single null-point located at the origin. The magnetic field under consideration could be expressed in terms of a vector potential as41.It is very probable for a magnetic null point to encounter MHD waves of all kinds. Thus, the MHD theory which is a macroscopic approach suits well for modeling such solar atmospheric events. Since in the solar corona the plasma density of the medium is around 35. This code is appropriate for explicit time-dependent computations in three dimensions, where in the present study the Godunov method is implemented which allows taking into account the effects of entropy together with dissipation mechanisms. Note that three main parameters concerning the solar corona are taken as The dynamics and energy transfer connected with Alfvén and magnetoacoustic pulses propagating towards a magnetic null-point are simulated. The various courses that these waves go through while encountering the null-point together with its outcome is observed in the context of MHD theory. MHD Eqs. –4) are  are 4) aThe magnetic field 137. The typical values of the initial pulse are defined as 30. This is due to the fact that the Alfvén pulse actually is independent of the azimuthal wave number39 making the azimuthal wave number equal to zero. Before proceeding it is worth providing a brief review of the experimental setups regarding the Alfvén wave interaction with magnetic null points in the solar atmosphere established prior to the present study; Galsgaard and co authors studied a helical Alfvén wave as it approached a 3D magnetic null-point, the system is governed by the Klein–Gordon equation. They concluded that as the Alfvén wave nears the magnetic null-point it spreads out. This is while the nonlinear induced magnetacoustic wave that they found, that its energy is much less than the initial Alfvén wave, wraps around the magnetic null-point40. By focusing on the propagation of magnetoacoustic waves McLaughlin and co-authors pictured how the wrapping process of magnetoacoustic waves is accompanied by the creation of a density spike at the null-point30. Thurgood and McLaughlin carried out a very interesting setup in 2.5D where they kicked an Alfvén pulse towards a magnetic null-point to found that Alfvén waves create transverse and longitudinal disturbances that transported little energy connected with the Alfvén wave towards the null-point due to the fact that weakly nonlinear effect where taken under consideration. Their emphasis was on the ponderomotive force in the presence of weakly nonlinear effects since their initial wave amplitude was considered relatively small. They reported an induced magnetoacoustic wave that carries part of the initial Alfvén wave towards the magnetic null-point where it experiences wrapping12. However what if a full description of nonlinear effects where taken under consideration. Therefore Sabri and co-authors carried out an experimental setup in order to study the induced fast and slow magnetoacoustic waves by the Alvén wave in the neighborhood of the null-point. Their aim was to measure the energy transfer rate due to induced fast and slow magnetoacoustic waves. They showed the shock formation regarding the initial Alfvén wave which induces large amplitude magnetoacoustic waves which is a signature of energy transfer from Alfvén to fast and slow magnetoacoustic that are to propagate towards the null-point. They reported that the fast wave experiences a sudden drop of energy transport which proves adequate for its energy release due to shock formation. This is while the slow wave experiences a gradual energy release, see Ref.1 for further details. In the present study the aim is to take a further step and highlight the back reaction of the induced fast and slow magnetoacoustic perturbations on the mother Alfvén wave. In this line we show how the nonlinear forces back react on the Alfvén wave and provide further insight on the sense in which the induced inflows and outflows contribute towards solar jet formation.To comply with the aims of the present study, consider a single circular wave pulse kicked in a static plasma medium at a finite distance from the magnetic null point where at time mentclass2pt{minim41 in the zero plasma-The analytic expression provided by Gruszecki et al.42 which guides them to propagate along the tube. This is in a sense that in the presence of magnetic field aligned density structuring in the medium, propagation of transverse waves along the magnetic field lines is expected43. In case of a magnetically twisted plasma medium with transverse structuring, the Alfvén wave is modified to the fast megnetoacoustic wave39 where its phase speed is determined by the Alfvén wave speed internal and external to the wave-guide44. The Alfvén speed is highly dependent on the plasma density; the nonlinear Alfvén wave propagation induces compressive perturbation due to the ponderomotive force46 which in turn perturbs the density. In this line induced variations of the magnetic field and plasma density would influence the local values of the the Alfvén speed. In fact the back reaction of the compressive perturbations would also affect the propagation of the Alfvén wave through the Cohen–Kulsrud equation6. This issue could also be considered as Alfvén wave self-interaction47. In this regard the nonlinear effects connected with the Alfvén wave propagation in the neighborhood of a magnetic null-point in the presence of transverse density structuring needs to be taken under consideration.The Alfvén wave is a transverse wave that propagates along the direction of the magnetic field in a uniform medium. Fast magnetoacoustic waves propagate with an angle with respect to the magnetic field lines. However in the presence of transverse structuring magnetoacoustic waves experience refraction e.g.Figure 26] and homogeneous [e.g.12] plasma medium. The outcome would be the creation of secondary waves. This is due to the nonlinear coupling of the transverse to longitudinal variables enabled by MHD equations. The total pressure perturbation generates slow and fast magnetosonic waves which are polarized in the direction perpendicular to the flux surfaces. The initial circular Alfvén pulse excites flows with two different velocity components, the component parallel to the magnetic field and the component perpendicular to the flux surface which indicates the presence of slow and fast magnetoacoustic waves12. The creation of the normal/perpendicular perturbations can be connected to the nonlinear ponderomotive force affiliated to the gradients of the perpendicular perturbed Alfvén speed48. On the other hand, the generation of the parallel flows may be associated to the nonlinear ponderomotive force that is induced by the gradients of parallel perturbed Alfvén speeds. The fact of the matter is that the nonlinear effects connected with the modification of the Alfvén wave51 which is subject to conditions in the domain of a magnetic null-point must be highlighted in order to contribute towards coronal seismology.The propagation of nonlinear Alfvén waves induces compressive perturbations in transversely structured . In the context of the present study since the plasma medium is considered homogeneous the induced perturbations are checked by providing four snapshots associated with times x-component of the induced velocity  .The strength of the induced inflows and outflows highly depends on the direction of the slow wave. The inflows are stronger than outflows in both the The comparison of time variations of the induced fast and slow magnetoacoutic waves prove adequate for the domination of the induced fast magnetoacoustic waves on the oscillatory behavior of the compressive perturbations. Even the induced perturbations The current density dictates the diffusion or propagation of wave dynamics at the reconnection site. For ideal conditions, the current density possesses a steady orientation. Nonetheless, the forces connected to the nonlinear effects are responsible for the creation of inflows and outflows leading to the creation of solar jets.The effects connected with nonlinear forces regarding the propagation of Alfvén waves which induce compressive perturbations together with their back reaction on the mother Alfvén wave would construct a medium with nonuniform back ground Alfvén speeds. This phenomena would cause a homogeneous background medium to host features like phase mixing and other wave coupling effects that are signatures of homogeneous plasma medium.In this study the interaction of induced perturbations together with their back reaction on the Alfvén wave in the course of interaction with a 2.5D magnetic null-point in the solar atmosphere is pictured. The numerical simulations are based on the nonlinear MHD theory carried out by the PLUTO code. The magnetic null-point structure, preconditions, equilibrium variables, and assumptions are studied under solar coronal conditions. Emphasis is on the induced compressive perturbations which constitute the fast and slow magnetoacoustic waves together with their back reaction on the initial Alfvén pulse in the course of interaction with a magnetic null-point. The scenario is as follows; as the Alfvén wave propagates along the magnetic field lines preserving its characteristics it accumulates along the separatrices. This is while the Alfvén wave produces nonlinear perturbations in the