Delayed Response Model Research Articles

Predicting resistance and pseudoprogression: are minimalistic immunoediting mathematical models capable of forecasting checkpoint inhibitor treatment outcomes in lung cancer?

Background:The increased application of immune checkpoint inhibitors (ICIs) targeting PD-1/PD-L1 in lung cancer treatment generates clinical need to reliably predict individual patients’ treatment outcomes. Methods:To bridge the prediction gap, we examine four different mathematical models in the form of ordinary differential equations, including a novel delayed response model. We rigorously evaluate their individual and combined predictive capabilities with regard to the patients’ progressive disease (PD) status through equal weighting of model-derived outcome probabilities. Results:Fitting the complete treatment course, the novel delayed response model (R2=0.938) outperformed the simplest model (R2=0.865). The model combination was able to reliably predict patient PD outcome with an overall accuracy of 77% (sensitivity = 70%, specificity = 81%), solely through calibration with primary tumor longest diameter measurements. It autonomously identified a subset of 51% of patients where predictions with an overall accuracy of 81% (sensitivity = 81%, specificity = 81%) can be achieved. All models significantly outperformed a fully data-driven machine learning-based approach. Implications: These modeling approaches provide a dynamic baseline framework to support clinicians in treatment decisions by identifying different treatment outcome trajectories with already clinically available measurement data. Limitations and future directions:Conjoint application of the presented approach with other predictive tools and biomarkers, as well as further disease information (e.g. metastatic stage), could further enhance treatment outcome prediction. We believe the simple model formulations allow widespread adoption of the developed models to other cancer types. Similar models can easily be formulated for other treatment modalities.

Cross-sectional geometrical characteristic for the bends along the lower Jingjiang reach

The evolution of bar–pool configurations in response to upstream damming has significant impacts on channel regulations, navigations, water intakes and protection projects. This paper reports on and analyses the evolution of bar–pool configurations in bends along the lower Jingjiang reach (LJR) after the impoundment of the Three Gorge dam (TGD), which is distinguished from the natural evolution of the bends. The main factors in the different adjustments of bar–pool configurations are the changes in incoming flow and sediment regime during pre- and post-TGD periods. To capture the changes in the bar–pool configurations, a new cross-sectional geometrical characteristic is presented – relative lateral distance of the centroid (RLDC). RLDC has close relations with the incoming sediment coefficient (i.e. incoming discharge divided by suspended sediment concentration during flood season). RLDC is better than the conventional cross-sectional geometrical characteristic (e.g. width-to-depth ratio) to indicate the bar–pool configurations downstream of large dam projects. Based on the delayed response model, values of RLDC in the bends of the LJR are related to the previous 4–6 years’ incoming sediment coefficient, and the correlation coefficient is about 0.90. RLDC is expected to capture the variations of bar–pool configurations in the bends downstream of the large dam project.

Dynamic and delayed channel geomorphic responses to changing hydrological processes following the Three Gorges Dam operation

AbstractThe characteristics of dynamic and delayed channel geomorphic responses to changing hydrological processes are still insufficiently understood, especially in large dammed rivers and segments farther downstream from the dam. Geomorphic effects farther downstream from the dam could be easily underestimated. This study investigated the hydrological changes and channel adjustments in the Chenglingji–Jiujiang Reach (CJR), ~450 km downstream from the Three Gorges Dam. The post‐dam evolution developed from early low‐intensity alternating erosion/deposition to later intense erosion. This is because the increasing sediment starvation caused by the dam and attenuated upstream sediment replenishment produced streams with excess transport capacity for the farther downstream channel, where the early mild evolution and fine‐sandy riverbed guaranteed adequate sediment availability. Additionally, the increasing geomorphic contribution of low‐to‐medium flows, coupled with revetment resistance to lateral adjustments, promoted migration of the major adjustment area from above the bankfull channel to below the medium‐flow channel and characterized erosion by one‐way incision with no evident channel narrowing/widening. The net increase in reach‐averaged thalweg incision depth during a later period (2.85 m) was 35 times larger than that during an earlier period (0.08 m). Channel morphogenetic intra‐annual hydrological processes were identified to contain all discharges smaller than 39 000 m3/s, and discharges of 30 000–39 000 m3/s exerted maximum geomorphic effects. On this basis, a method for estimating the cumulative erosion volume at equilibrium was proposed and integrated into the Delayed Response Model. The encouraging simulation and prediction results indicate that geomorphological adjustments within the CJR were greatly influenced by the hydrological conditions of current year and previous 4 years, while the impacts of extreme and rare events depended more on the consistency of their hydrological behaviours with channel evolutionary trends than on their temporal distances. This work facilitates the understanding and prediction of channel self‐adjustments farther downstream from the dam.

Operating Effects of the Three Gorges Reservoir on the Riverbed Stability in the Wuhan Reach of the Yangtze River

Reservoir construction may modify the downstream flow and sediment transport, and correspondingly result in adjustments in morphodynamics of a river, especially riverbed instability. Based on hydrological datasets from 2003 to 2019 during the post-dam period using the topographic data of 57 fixed cross-sections in the Wuhan reach of the Yangtze River, we calculated the indexes representing the channel stability. Moreover, considering the effects of flow, sediment concentration, grain size of sediment, and water depth, we propose a method for calculating the equilibrium values of cumulative erosion and the lateral migrate intensity of thalweg in this paper, and the method combines with the delayed response model (DRM) to comprehensively analyze the variations in the longitudinal and lateral stabilities of the riverbed. The results revealed that the channel has been obviously eroded in the downstream reach, resulting in a 76% decrease in sediment discharge after the impoundment of the Three Gorges Reservoir (TGR). Specifically, in the past 17 years, the cumulative erosion in the Wuhan reach of the Yangtze River reached 1.72 m, while the bankfull depth increased by 1.87 m. The lateral migrate intensity of thalweg increased in response to Coriolis force, with an increase of 22.3%. Taken together, the results show that the proposed formula can effectively simulate the variation process of channel stability, and it also quantifies the extent of the influence weight of interannual flow and sediment regimes. The morphodynamics adjustments in the channel stability of our studied reach were closely related to the previous five-year flow and sediment regimes, implying that channel evolution may lag behind the changes in flow and sediment discharges.

Theory of delayed response in river morphodynamics: Applicability and limitations

The delayed response of fluvial rivers to external disturbances has been described by many researchers. To simulate such behavior, the rate law model (or the delayed response model) was developed by previous researchers, and has been applied to a series of river morphological problems. However, to date, the applicability of the rate law model has not been fully understood. In the current paper, a physically-based analysis of the rate law model is presented to assess the responses of bed elevation and the grain size of surface sediment, using the response of the Shi-ting River, China, after the 2008 Wenchuan Earthquake as an example. First, a physically-based river morphodynamic model is implemented to reproduce the post-earthquake adjustments of the Shi-ting River. Next, the mathematical properties of the Hirano–Exner equation, which describes the dynamics of both bed elevation and surface texture, are analyzed. It is shown that the dynamics of bed elevation are dominated by a diffusion process and the analytical solution for bed elevation has a similar mathematical formulation to that of the rate law model. Thus, the rate law and morphodynamic models similarly predict the adjustment of bed elevations. In contrast, the dynamics of bed surface texture are controlled by both advection and diffusion processes. The advection and diffusion processes dominate adjustments over the short (annual to decadal scale) and long (century to millenial scale) terms, respectively. These physics produce the multi-scale and non-monotonic character of the adjustment of bed surface texture. As a result, the rate law model is incapable of describing the adjustment of bed surface texture. These findings highlight the applicability and limitations of the rate law model in simulating river morphodynamic processes.

Simulating cross-sectional geometry of the main channel in response to changes in water and sediment in Lower Yellow River

To understand the non-equilibrium morphological adjustment of a river in response to environmental changes, it is essential to (i) accurately identify how past conditions of water and sediment have impacted current morphological adjustment of the river, and (ii) establish a corresponding simulation for non-equilibrium conditions. Based on discharge and suspended sediment concentration (SSC) as well as 82 cross-sectional data items for the Huayuan-kou-Lijin reach of the Lower Yellow River in the period 1965–2015, the process of adjustment of the geometry of the main channel (area, width, depth, and geomorphic coefficient), and its responses to changes in discharge and SSC for different reaches are statistically analyzed. Following this, a delayed response model (DRM) of the geometry of the main channel subjected to variations in discharge and SSC is established using a multi-step analytical model, with the discharge and SSC as the main controlling factors. The results show that the area, width, and depth of the main channel decreased initially, then increased, decreased again, and finally increased again. These features of the geometry of the channel were positively correlated with the 4-year moving average discharge and negatively with the 4-year moving average SSC. The geomorphic coefficient for the Huayuankou-Sunkou reach exhibited a trend of decrease, whereas that of the Sunkou-Lijin reach decreased initially, then increased, decreased again, and finally increased again. Except for the Huayuankou-Gaocun reach in 1965–1999, the coefficient was negatively correlated with the 4-year moving average discharge and positively with SSC. The simulated values of the morphological parameters of the main channel for all sub-reaches obtained using the DRM agreed well with the measured values. This indicates that the DRM can be used to simulate the process of response of the cross-sectional geometry of the main channel to variations in the water and sediment. The results of the model show that the adjustment of the geometry of the main channel was affected by the discharge and the SSC at present (30%) as well as for the previous 7 years (70%). The proposed model offers insights into the mechanism whereby past water and sediment influence the current morphological adjustment of the river, and provides an effective method for predicting the magnitude and trend of the geometry of the main channel under different flow conditions.

Hydrodynamic and geomorphic adjustments of channel bars in the Yichang-Chenglingji Reach of the Middle Yangtze River in response to the Three Gorges Dam operation

The evolution of channel bars in response to upstream damming has significant impacts on channel stability, navigation, and aquatic habitats. Here, the effects of the Three Gorges Dam (TGD) operation on downstream channel bars in the Yichang-Chenglingji Reach (YCR) were comprehensively analyzed using remote sensing images, cross-sectional profiles, and hydrological datasets. The morphodynamic adjustments of channel bars in the YCR were significantly different during the pre- and post-TGD periods (i.e., before and after the construction of the TGD). Specifically, the total area of channel bars did not exhibit any significant trend in the pre-TGD period, but displayed a significant reduction following the construction of the TGD, although the morphodynamic response of each sub-reach was different. The channel bars in the YCR were relatively stable in the pre-TGD period, but became more erodible in the post-TGD period. The length/width ratio (LWR) of the bars showed an overall increase trend during the whole period from 1992 to 2017, not changing before and after the dam construction. The water discharge that led to the greatest channel bars adjustment was 27,000–30,000 m3/s (corresponding to bankfull discharge) in the pre-TGD period and 15,000–18,000 m3/s (corresponding to the medium discharge that can submerge the surface of the bars) in the post-TGD period. In addition, the grain size of non-uniform sediment with the highest replenishment degree gradually reduced downstream; these finer sediments were the main sources of material for the channel bars in the YCR. Quantitative relationships between bars area, the most effective bar-forming discharge and suspended sediment load with the highest replenishment degree, were proposed based on the improved Delayed Response Model (DRM). Results indicate that geomorphic adjustments of channel bars in the YCR are closely related to the previous four-year flow and sediment regimes, implying a delayed response of the fluvial system to damming.

Morphodynamic adjustments in the Yichang–Chenglingji Reach of the Middle Yangtze River since the operation of the Three Gorges Project

To investigate the evolution characteristics of the Yichang–Chenglingji Reach (YCR) of the Middle Yangtze River and develop simulation methods, the temporal and spatial morphodynamic adjustments of the YCR were analysed comprehensively on the basis of runoff, sediment load, and cross-sectional profiles data at 220 cross sections from 2002 to 2016. Results showed that significant channel degradation has occurred in the YCR, especially in the low-flow channels, due to a drastic reduction in sediment yield since the operation of the Three Gorges Project (TGP). The variations of upstream flow and sediment regimes would drive equilibrium channels to become increasingly narrow and deep, and the current bankfull width and depth were less than their equilibrium counterparts in the recent period. However, the recent channel evolution was mainly characterized by the prominent channel deepening via a one-way deformation due to the large-scale bank revetments within the YCR, where the reach-scale bankfull depth increased by 1.6 m in the span of 15 years. A semi-empirical method for determining the equilibrium values of channel degradation and aggradation volume at the YCR was proposed on the basis of a comprehensive analysis of the effects of flow, sediment concentration, sediment gradation, and water depth. Moreover, the equilibrium values of the channel width and depth were estimated with the Least Action Principle method. The calculation methods for the equilibrium width, depth, width/depth ratio, and aggradation/degradation volume were combined with the Delayed Response Model (DRM). Then, the methods for calculating the variations in channel geometry and aggradation/degradation volume were derived. Results showed that the proposed methods effectively simulated the channel geometry and cumulative erosion volume in the YCR from 2002 to 2016. In addition, the response of the channel evolution in the YCR was closely related to the previous four-year hydrological conditions, implying that channel evolution may lag behind the variations of flow and sediment discharges.

Effects of Three Gorges Dam operation on spatial distribution and evolution of channel thalweg in the Yichang-Chenglingji Reach of the Middle Yangtze River, China

The evolution of the channel thalweg and pool-riffle sequences is a critical feature of morphological adjustments in fluvial systems and might have important effects on aquatic habitat quality, channel stability and navigation. Based on the 220 cross-sectional profiles from 2002 to 2016, residual depths and spatial statistical analyses were used to quantitatively analyse the vertical variations and longitudinal distribution of the pool-riffle sequences and channel thalweg in the Yichang-Chenglingji Reach (YCR) of the Middle Yangtze River, which consists of the Yizhi-Zhijiang, Shashi-Gongan and Shishou-Jianli sub-reaches. The results showed that the dramatic reduction in the sediment loads following the construction of the Three Gorges Dam (TGD) has caused significant adjustments in the size, morphological diversity and spatial distribution of pool-riffle structures in the YCR. The average depth of the pools in the Yizhi-Zhijiang Reach and the Shishou-Jianli Reach overall exhibited a nonlinear growth trend, with the average residual depth increasing by 1.4 m and 1.9 m, respectively. However, the average residual depth in the Shashi-Gongan Reach showed a decreasing trend, with a reduction of approximately 1.2 m. The longitudinal lengths of the riffles in all three reaches contracted to different degrees, and the specific percentage of longitudinal river length occupied by riffles (total riffle length/total sub-reach length) decreased by 8%, 9.6% and 5.9% in the three sub-reaches from upstream to the downstream. In addition, the morphological diversity of the thalweg elevation in the Shashi-Gongan Reach and Shishou-Jianli Reach weakened after the construction of the TGD, while the bed topography in the Yizhi-Zhijiang Reach became more complex and diverse. Moreover, the pool-riffle structures in the YCR were rearranged and distributed regularly following the initiation of the TGD operation, although the variation characteristics of the average pool spacing in the three sub-reaches were quite different. Quantitative relationship between the average residual depth and the previous four-year hydrological condition was proposed for the three sub-reaches based on the Delayed Response Model (DRM) and verified using the measurements from 2013 to 2016.

Evolution of the Yellow River delta, China: Impacts of channel avulsion and progradation

The evolution of the Yellow River delta is characterized by heavy sediment load, rapid seaward migration, frequent avulsions, and intense anthropogenic disturbances. Evolution of the delta channel following avulsions is very complex and has not yet been thoroughly understood. In the research presented by this paper, we conducted comprehensive analyses of the changes in the water stages, slopes, longitudinal profiles, and the erosion and deposition in the Yellow River delta channels during a time period of over five decades. Results showed that, following each avulsion, channels migrated seaward at decaying rates and the slopes at the downstream of the avulsion point decreased exponentially with time and completed its major adjustment within about four to five years. A generalized geometric model was proposed to describe the changes in the longitudinal profiles of the delta channels. A calculation method to determine the characteristic water stages at the delta was proposed based on the geometric model and the delayed response model for the morphological responses of fluvial rivers to perturbations. Water stages corresponding to a discharge of 3000m3/s at Lijin and Xihekou during 1954 through 2012 were calculated by using the proposed method. The proposed method may be used to predict the evolution of the delta channels in response to artificial avulsions at the Yellow River delta in the future.

A geological delayed response model for stratigraphic reconstructions

We are interested by a nonlinear single lithology diffusion model adapted from ideas originally developed by the Institut Francais du Petrole (IFP). The geological stratigraphic modeling has to describe transports of sediments, erosion and sedimentation processes by taking into account a limited weathering condition; the method by which the history of a sedimentary basin is revealed relies on knowledge of both initial and final data and can be generalized to multiple lithology. For this purpose, we introduce a relaxation time related to a delayed response for establishing equilibrium states; this approach introduces regularizing effects according to the ideas of G.I. Barenblatt - S. Sobolev and J.-L. Lions - O. A. Oleinik. New well-posedness results are presented.

Recent variation in reach‐scale bankfull discharge in the Lower Yellow River

ABSTRACTBankfull discharge is a key parameter in the context of river engineering and geomorphology, as an indicator of flood discharge capacity in alluvial rivers, and varying in response to the incoming flow and sediment regimes. Bankfull channel dimensions have significantly adjusted along the Lower Yellow River (LYR) due to recent channel degradation, caused by the operation of the Xiaolangdi Reservoir, which has led to longitudinal variability in cross‐sectional bankfull discharges. Therefore, it is more representative to describe the flood discharge capacity of the LYR, using the concept of reach‐averaged bankfull discharge. Previous simple mean methods to estimate reach‐scale bankfull discharge cannot meet the condition of flow continuity or account for the effect of different spacing between two sections. In this study, a general method to calculate cross‐sectional bankfull discharge using the simulated stage‐discharge relation is outlined briefly, and an integrated method is then proposed for estimating reach‐scale bankfull discharge. The proposed method integrates a geometric mean based on the log‐transformation with a weighted average based on the spacing between two consecutive sections, which avoids the shortcomings of previous methods. The post‐flood reach‐scale bankfull discharges in three different channel‐pattern reaches of the LYR were estimated annually during the period from 1999 to 2010 using the proposed method, based on surveyed post‐flood profiles at 91 sedimentation sections and the measured hydrological data at seven hydrometric sections. The calculated results indicate that: (i) the estimated reach‐scale bankfull discharges can effectively represent the flood discharge capacity of different reaches, with their ranges of variation being less than those of typical cross‐sectional bankfull discharges; and (ii) the magnitude of the reach‐scale bankfull discharge in each reach can respond well to the accumulative effect of incoming flow and sediment conditions. Finally, empirical relationships for different reaches in the LYR were developed between the reach‐scale bankfull discharge and the previous four‐year average discharge and incoming sediment coefficient during flood seasons, with relatively high correlation coefficients between them being obtained, and the reach‐scale bankfull discharges in different reaches predicted by the delayed response model were also presented for a comparison. These relations for the prediction of reach‐scale bankfull discharges were validated using the cross‐sectional profiles and hydrological data measured in 2011. Copyright © 2013 John Wiley & Sons, Ltd.

Delayed response model for bankfull discharge predictions in the Yellow River

Delayed response means that channels cannot achieve a new equilibrium state immediately following disruption; the channel requires a response time or relaxation time to reach equilibrium. It follows that the morphological state of fluvial system represents the cumulative effects of all previous disturbances and environmental conditions. A unique feature of the delayed response model for bankfull discharge is that the model is capable of representing the cumulative effects of all previous flow conditions when applied to predict the path/trajectories of bankfull discharge in response to altered flow regimes. In this paper, the delayed response model was modified by readjusting the weight for the initial boundary conditions and introducing a variable β with respect to time. The modified model was then applied to the bankfull discharge calculations for three selected river reaches of the Yellow River, with each reach having different geomorphic settings and constraints. Results indicated that the modified model can predict accurately the bankfull discharge variation in response to changes in flow discharge and sediment load conditions that have been dramatically altered in the past. Results also demonstrated the strong dependence of current bankfull discharge on the previous years' flow conditions, with the relaxation time varied from 2 to 14 years, meaning that the bankfull discharge was not only affected by the flow discharge and sediment load in the current year, but also by those in previous 1 to 13 years. Furthermore, the relaxation time of bankfull discharge adjustment was inversely proportional to the long-term average suspended sediment concentrations, and this may be explained by fact that high sediment concentrations may have a high potential to perform geomorphic work and there is more sediment readily available to shape the channel boundary and geometry.

A computer simulation study on the causal relationship between walking and physical malfunctions in older adults

The purpose of the present study was to investigate the causal relationship between walking pattern and physical malfunction using a computer simulation method with a biomechanical walking model. In this computer simulation, the musculoskeletal system was represented by a three-dimensional 14-rigid-link model and 60 muscle models. Muscular forces were controlled by a neuronal system model consisting of 16 pairs of neural oscillators. In the computational experiment, five types of walking models were constructed: an intact model, a delayed response model, a weak muscle model, an inclined posture model, and a joint contracture model. These malfunction factors were hypothesized to have a causal relationship with walking characteristics in older adults. The simulation revealed that the delayed response in the neuronal system was primarily related to walking stability. In addition, the weakening of muscles was strongly related to the reduction of the walking step length. The inclined posture and joint contracture also influenced the walking pattern, but not significantly. The use of such a computer simulation method is essential in order to clarify the causal relationship between body function and walking pattern in older adults.

Generalized Delayed Response in Randomized Play-the-Winner Rule

In clinical trials, a particular patient's response may be delayed in the sense that it is not obtained before the entrance of the next patient. Delayed responses are particularly important in the context of adaptive trials where the allocations are done on the basis of the available responses. Bandyopadhyay and Biswas (Bandyopadhyay, U., Biswas, A. (1996). Delayed response in randomized play-the-winner rule: a decision theoretic outlook. Calcutta Statistical Association Bulletin 46:69–88) and Biswas (Biswas, A. (1999). Delayed response in randomized play-the-winner rule revisited. Comm. Statist. Simul. Comput. 28:715–731) have studied a simple delayed response model in the context of the randomized play-the-winner rule (RPW), a popular adaptive design, originally developed by Wei and Durham (Wei, L. J., Durham, S. (1978). The randomized play-the-winner rule in medical trials. J. Am. Statist. Assoc. 73:838–843) and Wei (Wei, L. J. (1979). The generalized Polya's urn for sequential medical trials. Ann. Statist. 7:291–296). The present article provides a generalized model for the delayed response in the RPW rule. The model is then theoretically developed and the exact and limiting proportion of patients allocated to the two treatments are obtained. The exact expression of the variance of the proportion of allocation is also obtained.