The plastic zone range is an important parameter in the analysis of damage characteristics and the degree of damage to the rock surrounding a roadway. Based on the establishment of a plastic zone calculation model considering the creep effect, this paper obtains the characteristics of the change in the plastic zone damage range with time by solving the model. Additionally, the validity of the model is verified by field experiments. The research results can provide guidance for gas pressure measurement and gas drainage in coal mines. On the one hand, determining the damage range of the plastic zone of the roadway can improve the accuracy of measuring the gas pressure  in a coal mine; on the other hand, it can improve the sealing effect of the gas drainage borehole, thereby improving the gas drainage rate of the coal seam6. Therefore, clearly establishing the damage range of the plastic zone of the roadway can indirectly prevent the occurrence of coal mine gas disasters.Gas disasters constitute the primary threat to safe production in many coal mines7 considered that the shape of the plastic zone around the roadway in the uniform stress field is circular, and the formula for calculating the circular plastic zone was derived. Additionally, Chen et al.8 established the boundary implicit equation for the plastic zone of a circular roadway. Abdel et al.9, Leitman and Villaggio10 considered that the shape of the plastic zone of roadways located in non-uniform stress fields should be elliptical. Guo et al.11 and Wang et al.12 pointed out that the shape of the plastic zone of a roadway could be circular, elliptical or butterfly-shaped under different in situ stress conditions. Zhou and Wu13, Yang et al.14 used a numerical simulation method to study the plastic zone range of a roadway under different stress conditions.Many scholars have carried out research on the plastic zone of a roadway. Galin19. Even under low stress conditions, the creep of coal and rock would also occur. Due to the creep effect, the plastic zone of a roadway will increase with time and finally approach a certain value. It can be seen that the previous research on the plastic zone of a roadway did not consider the creep effect of the surrounding rock of the roadway and thus cannot better guide engineering practice. In this paper, a plastic zone calculation model considering the creep effect was established to analyze the variation in the plastic zone damage range with time. The validity of the model is also verified by field experiments.The practice of mining shows that at constant stress, the creep of coal and rock will occur as time goes on; that is, the deformation of coal and rock will continue to grow with timeSince the axial length of the roadway is much larger than the radial width, the stress problem of the roadway section can be regarded as a plane strain problem.The stress around the roadway is assumed to be uniform and the shape of the plastic zone of the roadway is assumed to be circular.The failure of the plastic zone of the rock mass meets the Mohr–Coulomb yield criterion.The stress distribution of any roadway section is shown in Fig. To obtain the specific range of the plastic zone of a roadway, some necessary practical assumptions are made to simplify the solution. The basic assumptions are as follows:Stress distribution in the plastic zoneWhen the roadway is in the plane strain state, the static equilibrium equation of the plastic zone can be expressed as follows.According to Mohr–Coulomb plastic conditions, Eq.  can be oBy substituting Eq.  into Eq.By integrating Eq. , Eq. 4)4) is obtOn the wall of the roadway, (2)Stress distribution in the viscoelastic zoneSubstituting Eq.  back intThe constitutive equation of viscoelastic zone adopts the Poynting–Thomson standard linear solid model. The physical equation of the viscoelastic zone then can be defined byRegardless of the size of the plastic zone and whether or not there is support resistance, the stress at the interface of the viscoelastic zone and the plastic zone is constant. Let the calculated point e in Eq.  is 0, anIn addition, in the case of axisymmetry, the volumetric deformation of the viscoelastic region is 0; according to the foregoing assumption, that isThen, according to Eq. , we can Since the plastic zone changes with time, let ) in Eq. . We thenBy substituting the first and third formulas of Eq.  into theosφ, Eq.  can be oWhen t = 0, into Eq. , we can into Eq. .14\documSubstituting Eq.  into Eqs(3)Plastic zone radius under creep conditionsSince Eq.  calculat) in Eq. .(3)PlastAccording to the assumption of Eq. , the volWhen Therefore,Substituting Eq.  into Eq.Equation According to Eq. , the disFrom Eq. , the ranWhen calculating the radius of the plastic zone as a function of time according to Eq. , it is nEquation  represenThe radius of the plastic zone as a function of time can be obtained from Eq.  and can To calculate the radius of the plastic zone in Eq. , the relEquation  is a timThe gas pressure of a coal seam is one of the important indexes for the prediction of the risk of coal and gas outbursts in coal mines. The bag-type grouting sealing method Fig.  is a comThe key to accurately determining the gas pressure of the coal seam by the capsular bag grouting method is finding the range of the plastic zone of the roadway. If capsular bag 1 and capsular bag 2 are located in the plastic zone of roadway, the gas in the pressure-measuring chamber may leak out along the borehole wall and through cracks in the plastic zone during the pressure measurement, and the measured gas pressure will thus be smaller than the actual value or no indication on the pressure gauge. Conversely, when the positions of capsular bag 1 and capsular bag 2 are outside the plastic zone of roadway, the pressure-measuring chamber is ensured to be airtight and the correct gas pressure value will be obtained. Therefore, the accuracy of the calculated plastic zone of roadway can be approximately reflected by the measured value of the gas pressure.The 11111 workface of the No. 13 Coal Mine of Pingdingshan Tianan Coal Mining Corporation Limited in the Henna province of China was selected as the research background. This specific coal mine is classified as a coal and gas outburst coal mine. The workface has an average thickness of 5.85 m and a coal seam inclination of 10°–19°. Considering the danger of coal and gas outburst, a floor rock roadway was used to construct the boreholes to drain the gas from the 11111 workface. The height and width of the roadway are 3 m and 4 m, respectively. The lithology of the floor rock roadway is marlstone and the roadway is 12–19 m away from the coal floor.23. According to laboratory measurement, the relaxation time of marlstone is 1/24 d. By substituting the above parameters into Eq.  should be at least 7 m away from the borehole orifice.The specific position of the gas pressure measuring borehole at the 11111 workface is shown in Fig. According to the calculation result of the plastic zone radius of the floor rock roadway, considering the certain margin coefficient, the starting capsular bag (capsular bag 1) is designed to be 8 m away from the borehole orifice when the pressure borehole is sealed. The gas pressure recovery curve measured by the pressure measurement process is shown in In the process of coal seam gas pressure measurement, the direct gas content determination method was used to measure the gas content of the coal seam and the measurement results are shown in Table 24.3/t; 3/t; −1; 3/m3; and 3.In addition, there is the following relationship between gas pressure and gas contentCombined with the gas pressure, the gas content can be inversely calculated according to Eq. . The inv3/t. This indicates that the measured value of the gas pressure is accurate and can correctly reflect the gas occurrence characteristics of the 11111 workface. Therefore, the validity of the plastic zone calculation model is also verified.It can be seen from Table In this paper, a plastic zone model considering the creep effect is proposed. The model has good applicability and can be used to calculate the plastic zone under different in situ stress conditions and different roadway sizes.According to the calculation model for the plastic zone, the radius of the plastic zone of a roadway in the No. 13 Coal Mine of Pingdingshan Tianan Coal Mining Corporation Limited in China is 7 m. Based on the radius of the plastic zone of the roadway, the starting capsular bag is designed to be 8 m away from the borehole orifice when the gas pressure measurement borehole is sealed in the roadway. The final measured gas pressure is 2.02–2.70 MPa. In the process of coal seam gas pressure measurement, the direct gas content determination method was used to measure the gas content of the coal seam. In addition, combined with gas pressure, the gas content can be inversely calculated. The gas content thus calculated from the coal seam gas pressure is consistent with the measured gas content, indicating that the gas pressure measurement value is accurate. The gas pressure measurement results verify the validity of the plastic zone calculation model.


Even in NSTI patients with many comorbidities, it is possible to save both the life and the limb by thorough debridement and suitable reconstruction. SCIP‐ICAP compound flap can be versatile for a massive defect of an upper extremity. A Case of a Supercharged Compound Flap for Necrotizing Soft Tissue Infection Even in NSTI patients with many comorbidities, it is possible to save both the life and the limb by thorough debridement and suitable reconstruction. SCIP‐ICAP compound flap can be versatile for a massive defect of an upper extremity. A Case of a Supercharged Compound Flap for Necrotizing Soft Tissue Infection. In this report, we present a case of the SCIP‐ICAP compound flap after extensive debridement of the upper extremity with necrotizing soft tissue infection (NSTI). Although the patient had many comorbidities, complete survival of the flap and a satisfactory functional recovery were observed. There has been no previous report of the SCIP‐ICAP compound flap applied for reconstruction of the upper extremity with NSTI. To reconstruct the defect from the dorsum of the hand to the distal upper arm, combination of the flap and the skin graft is a standard treatment; however, the SCIP‐ICAP compound flap is a unique treatment of choice when skin graft is less feasible.Complex defects of the upper extremity may develop following trauma, tumor resection, radiation injury, and infection.In this report, we present a case of a supercharged superficial circumflex iliac perforator ‐intercostal artery perforator (SCIP‐ICAP) compound flap after extensive debridement of the upper extremity due to NSTI; it was based on the superficial circumflex iliac artery (SCIA) using ICAP as a supercharge. The defect extended from the dorsum of the hand to the distal upper arm and the flap measured 48 × 7 cm in size. A limited number of SCIP flap cases extended by a supercharged ICAP have been reported in the literature, and there has been no previous report applying the SCIP‐ICAP compound flap for reconstruction of the upper extremity with NSTI. Though the patient was aged and compromised with many comorbidities including angiitis, the flap survived completely and the function of the upper extremity was satisfactory.2A 78‐year‐old Japanese woman was referred to the emergency department of our hospital with a swollen and painful left arm accompanied by fever. Her underlying diseases included diabetes, rheumatoid arthritis, IgA nephrosis, and microscopic polyangiitis; she was on a prescription of predonin 13 mg/d) and cyclosprine (100 mg/d) as immunosuppressants, and biaspirin (100 mg/d) and persantine (300 mg/d) as anticoagulants. The symptoms started with erythema of the dorsum of the hand 2 days before admission with unknown etiology. She was diagnosed as a suspect of NSTI and referred to our facility. On first visit, erythema and vesicles, and necrosis of the left arm were observed  Figure . The fla2.2No anticoagulants nor vasodilators were administered during and after operation. The left upper limb was fixed in place wearing chest binder around the left elbow with cushion in the left axilla and cubital fossa. The pedicle detachment was performed 18 days postoperatively. The flap survived fully Figure , and the3The concept of the compound flap was defined by Harii et al in 1981 for extensive and complicated wound cases.Among various choices of reconstruction, a pedicled compound flap with supercharge was chosen in this case because the defect lied over the wrist and elbow joints; the risk of thrombosis at the site of anastomosis was high, and a difficulty of the patient being in a relaxed condition after the operation was expected. Skin graft was not selected because it was unlikely to survive the tendon exposure and the high risk of infection recurrence and postoperative contracture. A pedicled flap was preferred to a free flap since the recipient vessel could be fragile due to various underlying conditions such as infections, debridement, and many comorbidities. Furthermore, shorter operative time was more desirable due to her poor general condition. Due to the high risk of failure in microsurgical anastomosis as previously reported in a vasculitis case,The pedicled latissimus dorsi flap could be an alternative; however, the SCIP flap was preferred to cover the dorsum of the hand (which was most susceptible to infection) with the proximal part of the flap. The abdominal flap was another choice of reconstruction as a massive compound flap; however, the caliber discrepancy could be a problem in supercharging with a perforator.ICAP and perforator in the forearm were considered to match feasibly, because of their similar sizes and decent numbers. The lowest ICAP was cranially located 2‐5 and 7‐11 cm posterior to the ASIS; from that point, ICAPs were cranially identified every 2.5‐3 cm, and they could be included in SCIP flap. The average diameter was reported as 0.7 ± 0.24 mm.Necrotizing soft tissue infection is a life‐threatening disease characterized by a rapid spreading necrosis. Rapid and thorough debridement are mandatory, and the reconstruction of the vast defect is challenging. In this case, debridement was performed on the fifth day of admission, which was rather later than usual. This was because there was a slow exacerbation of the local clinical findings and safer debridement could be expected when the general status became better due to conservative treatment. Even in a presented case with high risk, working in collaboration with other specialists makes it possible save both the life and the limb by thorough debridement. NSTI is a relatively rare disease and further studies on more cases are therefore necessary to establish the most ideal treatment or method of reconstruction.4A case of a supercharged pedicled SCIP‐ICAP compound flap with satisfactory function recovery (after extensive debridement of the upper extremity with NSTI) was reported. Even in an aged NSTI patient with many comorbidities, it is possible to save both the life and the limb by thorough debridement and reconstruction. Application of SCIP‐ICAP compound flap could therefore be versatile for a massive defect from the dorsum of the hand to the proximal upper arm by supercharging ICAPs with radial artery perforators.All the authors report no conflicts of interests.KF: wrote the initial draft of the manuscript. MK: assisted in the preparation of the manuscript. All other authors contributed to the collection and interpretation of data and the critical review of the manuscript. All authors approved the final version of the manuscript. All authors agree to be accountable for all aspects of the work and will ensure that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved.The patient provided informed consent for the publication of this report. The procedures followed were in accord with the Helsinki Declaration of 1975.


Toxic epidermal necrolysis (TEN) is a dermatological emergency that is often associated with high mortality. It is differentiated from Stevens-Johnson syndrome (SJS) based on the percentage of the total body surface area affected. There has been an established correlation with certain medications that could trigger the development of such a devastating disease. Despite numerous research studies conducted on aspects of this disease entity, TEN remains foreign to many general Internists situated in a community setting due to the extremely low disease prevalence that leads to a lack of overall experience and medical resources in dealing with this medical condition. Thus, we outlined several important management aspects of TEN/SJS that an Internist should be aware of in order to assist in prompt clinical decision making and prognosis forecasting as well as deliver effective family communication.  Stevens-Johnson syndrome (SJS) and toxic epidermal necrolysis (TEN) are dermatologic emergencies distinguished by severe mucocutaneous reactions with extensive necrosis and detachment of the epidermal layer of the skin, with the most common cause being medication side effects. SJS occurs when skin detachment is <10% of the total body surface area (BSA) while TEN involves >30% of BSA. TEN can also be characterized by >10% of large epithelial sheet detachment without macules. The gap of coverage involving 10-30% of skin detachment is referred to as SJS/TEN overlap . Prior rThe patient is a 20-year-old female who presented to the emergency department (ED) with the complaint of a diffuse whole-body rash. Her past history was notable for depression as she was recently prescribed lamotrigine two weeks prior to ED presentation. She was recently discharged from another hospital two days ago where she was hospitalized for three days due to complaints of blistering of her lips, sore throat, and mild rash on the extremities. She was given the diagnosis of hand-foot-mouth disease and was treated with 1 gram of ceftriaxone. She reported a decrease in the severity of sore throat upon discharge from ED but the rash continued to progress and had spread throughout most of her body parts. The lamotrigine was subsequently discontinued by the primary care physician. She decided to present to the emergency department (ED) for further evaluation. In the ED, she had a temperature of 102.4 F, blood pressure of 152/67 mmHg, heart rate of 120, and a respiratory rate of 22 per minute with an oxygen saturation of 98%. Physical examination was notable for a young female appearing uncomfortable with a diffuse maculopapular erythematous rash with scattered bullae lesions covering approximately 36% of her body surface area  patients may develop prodrome symptoms including but not limited to high-grade fever, sore throat, nonproductive cough, redness of eyes, and pink skin . MucosalPrompt hospitalization is the next step in order to initiate adequate monitoring and therapy . The oveCommunity Internists should be aware of the SCORTEN scale  as it could be helpful in assisting the proper disposition and prognosis of the patient. The score is composed of a total of seven variables with one point associated with each variable . A higher score correlates with an increase in mortality rate and worse prognosis. It is recommended to transfer a patient with SCORTEN score ≥2 to a designated burn unit or intensive care unit for the continuation of care. In fact, a retrospective multicenter study concluded that patients who were initially treated at burn care centers had a 32% mortality rate compared to a 51% mortality rate in patients who were initially treated at a non-burn center for more than one week prior to the transfer [Patients with TEN could carry a high mortality rate if a prompt diagnosis is not made as it is crucial to position the patient as soon as possible in the inpatient setting for adequate care. Supportive management is the foundation of medical treatment for TEN in addition to the multidisciplinary approach to ensure patient safety. The utilization of the SCORTEN scale is essential for Internists to determine the proper disposition of the patient as well as to predict the likely prognosis. The awareness of timely transfer to a burn care center or the intensive care unit is especially important to practitioners situated in a community setting.


To generate a Cerenkov scatter function (CSF) for a primary proton beam and to study the dependence of the CSF on the irradiated medium.The MCNP 6.2 code was used to generate the CSF. The CSF was calculated for light‐pigmented, medium‐pigmented, and dark‐pigmented stratified skin, as well as for a homogeneous optical phantom, which mimics the optical properties of human tissue. CSFs were generated by binning all of the Cerenkov photons which escape the back end (end opposite beam incidence) of a 20 × 20 × 20 cm phantom. A 4 × 4 cm, 500 × 500 bin grid was used to create a histogram of the Cerenkov photon flux on the simulated medium’s back surface (surface opposite beam incidence). A triple Gaussian was then used to fit the data.2 value for the fit was calculated to be 0.99. The phantom material was found to have a significant effect  on the CSF amplitude and full width at half maximum . The difference in these parameters for the three skin pigments was found to be small.From the triple Gaussian fit, the coefficients of the CSF for the four phantom materials was generated. The individual CSF fit coefficient errors, with respect to the Gaussian representation, were found to be between 0.92% and 4.11%. The RThe CSF was obtained for a proton beam using the MCNP 6.2 code for a phantom constructed of light, medium, and dark stratified human skin, as well as for an optical phantom. The CSFs were then fit with a triple‐Gaussian function. The coefficients can be used to generate a radially symmetric CSF, which can then be used to deconvolve a measured Cerenkov image to obtain the dose distribution. PT has the advantage of significantly reduced toxicity to non‐malignant tissue due to the absence of exit dose, while providing focused high linear energy transfer radiation to the tumor target. However, real‐time in vivo dosimetry for both electronin vivo dose measurement method using Cerenkov radiation produced by secondary electrons and positrons within the patient volume. This could be helpful for evaluating the accuracy of the prescribed radiation treatment, monitoring hot spots of dose deposition, and improving the therapeutic outcomes of PT due to the ability to accurately monitor dose in real time. However, there is no method that can perform this type of dosimetry in real time with sufficient accuracy.There has been increasing interest in using Cerenkov emissions for quality assurance and Proton therapy beams produce electrons and positrons that are responsible for dose deposition in a medium. Above a certain energy threshold, these charged particles will be superluminal within the medium and emit Cerenkov photons along their path. The Frank‐Tamm formula gives the amount and spectrum of Cerenkov photons emitted.This approach has been used for photon and electron beamsTo date, no CSF has been produced for a primary proton beam. In this work, a CSF is generated for a proton therapy beam for four different tissue equivalent phantom material compositions. The MCNP 6.2 code is used to simulate the interactions in the phantom.22.ACerenkov radiation is electromagnetic radiation which is emitted by charged particles passing through a dielectric material at a speed in excess of the phase velocity of light in that material. When a charged particle moves inside a dielectric material, it excites molecules to higher energy levels and excited states. Upon returning back to their ground state, the molecules re‐emit some photons in the form of electromagnetic radiation. According to Huygens principle, the emitted waves move out spherically at the phase velocity of the medium. If the particle motion is slow, the radiated waves bunch up slightly in the direction of motion, but they do not cross. However, if the particle moves faster than the light speed, the emitted waves add up constructively leading to a coherent radiation at an angle with respect to the particle direction, known as Cherenkov radiation. The result of this effect is a cone of photon emission in the direction of particle motion.2.BFigure 15O) which are responsible for both energy deposition and the creation of Cerenkov photons within the medium. Cerenkov photons created within the tissue volume will undergo scattering and absorption. Some of the Cerenkov photons will be absorbed, but a fraction (about 1%)A modulated proton beam incident on a tissue volume in the Z‐direction is considered. A system capable of imaging Cerenkov photons is placed at some angle θ relative to the beam. The proton beam produces secondary electrons and positrons (mostly from The following convolution equation represents the relationship between the surface flux of the incident proton beam entering the tissue volume and the intensity of the Cerenkov photons reaching the imaging systemI(θ)=J(Ω)The CSF can be written as a convolution of the dose scatter function (DSF) kernel and the Cerenkov dose scatter function (CDSF) kernel as follows:The CDSF kernel represents the production of Cerenkov photons by the secondary electrons and positrons from the incident proton beam and the transport of these Cerenkov photons toward the surface of the tissue volume. Equation 2 indicates that the CDSF can be found by deconvolving the CSF with the DSF.0 can be written as:The dose deposited in a plane at depth dInserting Equation 2 into Equation 0 in the tissue volume; the problem then becomes one of deconvolution of the Cerenkov photon intensity I, which is captured by the camera, with the CDSF, which can be found by deconvolving with the CSF produced herein and the DSF which will be calculated in a later work. The result is the dose profile at depth d0.The objective now is to determine the 2D dose distribution at depth d2.Co beam angle is assumed in the present work.The convolution statements in Equations 33.AThe MCNP 6.2in = 0°, where θin is the angle of entrance vector defined from the normal surface. As aforementioned, this is to ensure shift invariance. The phantom was modeled as a 20 x 20 x 20 cm3 box. The CSF simulations used the spread‐out Bragg peak (SOBP) shown in Table The MCNP model consisted of a pencil beam proton radiation source and the phantom being irradiated, with air surrounding. The radiation source was directed at the phantom with a constant beam angle of θThe CSF is created by the transport of Cerenkov photons created by a pencil beam of ionizing radiation incident on a dielectric material. For PT, a high‐energy proton beam is the primary radiation beam. The primary proton beam produces secondary electrons and positrons, which can exceed the speed of light in the medium, producing Cerenkov photons. Most of these photons will be absorbed in the tissue, but about 1% travel toward the surface and escape the irradiated medium.2 pixels.The initial position and direction at the time of the Cerenkov photon creation and the final position and final direction at which the Cerenkov photon escapes the phantom were recorded. The CSF was calculated by scoring the flux of the Cerenkov photons which escaped from the phantom surface. The positions of the photons leaving the surface were scored in a grid of 0.2 x 0.2 mm3.BFour different phantom materials were used in this study: light‐pigmented, medium‐pigmented, and darkened‐pigmented stratified skin and an optical phantom material. The stratified skin models were included to allow the effect of material properties, in particular the melanin content, on the CSF to be studied. The compositions for all four materials were taken from the National Institute of Standards and Technology soft tissue library.3.C2, 500 x 500 bin grid was used to create a histogram of the Cerenkov photon flux on the simulated medium surface. The histogram was sampled along a line from r = −10 to 10 cm using spoke‐sampling which was centered at the origin of the pencil beam on the medium surface. The sampling angle, θs, was defined on the medium surface as the rotation from the Y‐axis about the Z‐axis. Sampling angles of 0‐180 degrees in one‐degree increments were used. A triple‐Gaussian distribution was used to fit the mean of the data:CSFs were generated by binning all of the Cerenkov photons which escape the back end of the phantom (end opposite beam incidence). A 4 x 4 cmmean is the mean of the CSF cross‐sections over the sampling angle θs. The parameter r is the radial distance from the entrance of the proton pencil beam on the optical phantom surface. The parameters α, β, γ, δ, ε, and ζ are fit coefficients, generated using the Mathworks MATLAB fitnlm function; they were scaled by the number of primary protons incident on the optical phantom. The fitnlm function also produces the error in the fit coefficients. The triple‐Gaussian fit was chose due to it having the best goodness of fit (R2) value of the fits tested, as well as having the lowest error in the fit coefficients. Single‐Gaussian and double‐Gaussian fits were also tested but yielded lower R2 values and larger errors in the fit coefficients. The error in CSFmean was found as the standard deviation at each radial point from the CSF cross sections. The large number of histories and the incorporated variance reduction ensured that the error at each point on CSFmean was <1% of the maximum value for each CSF. The Gaussian fit of the CSF data is shown in Figure CSF of fit R value of42 value was calculated to be 0.99.The coefficients of the CSF for the four phantom materials are shown in Table Table 5In the present work, the CSF was calculated for a proton beam incident on phantoms of light, medium, and dark stratified skin, as well as an optical phantom, which mimics the optical properties of human tissue. The SOBP from Table The material was found to have a significant effect on the CSF amplitude and FWHM. The optical phantom CSF amplitude was found to be 63% greater than the light skin phantom; the FWHM for the optical phantom was found to be 195% greater than the light skin phantom material. The skin pigment was found to have a small impact on the CSF amplitude and almost no effect on the FWHM. The lack of effect on the FWHM is likely due to the fact that the only difference between the three pigments in the melanin content in the epidermis. The other layers of skin have identical scattering and absorption characteristics. As a result, Cerenkov photons have similar behavior underneath the epidermis layer, but have different amounts of absorption between the three skin models for Cerenkov photons leaving the tissue. As a result, the FWHM for the three skin models are similar.As can be observed from Table 6In the present work, the CSF was obtained for a proton beam using the MCNP 6.2 code for a phantom constructed of light, medium, and dark stratified human skin, as well as for an optical phantom. The CSFs were then fit with a triple‐Gaussian function using the MATLAB fitnlm function. The coefficients for this fit are provided in Table 2 value of 0.99.The phantom material was found to have a significant effect on the CSF amplitude and FWHM. The difference in these parameters for the three skin pigments was found to be small, meaning that melanin content has a diminutive effect on the CSF. The % errors for the parameters were found to vary from 0.92% to 4.11%, with an R


For most pathogens, transmission is driven by interactions between the behaviours of infectious individuals, the behaviours of the wider population, the local environment, and immunity. Phylogeographic approaches are currently unable to disentangle the relative effects of these competing factors. We develop a spatiotemporally structured phylogenetic framework that addresses these limitations by considering individual transmission events, reconstructed across spatial scales. We apply it to geocoded dengue virus sequences from Thailand (N = 726 over 18 years). We find infected individuals spend 96% of their time in their home community compared to 76% for the susceptible population (mainly children) and 42% for adults. Dynamic pockets of local immunity make transmission more likely in places with high heterotypic immunity and less likely where high homotypic immunity exists. Age-dependent mixing of individuals and vector distributions are not important in determining spread. This approach provides previously unknown insights into one of the most complex disease systems known and will be applicable to other pathogens. Phylogeographic analyses can provide broad descriptions of the spread of pathogens between populations, but are limited by incomplete sampling. Here, the authors develop an inference framework that reconstructs sequential transmission events and use it to characterise dynamics of dengue in Thailand. The co-occurrence of unrelated transmission chains means we still only have a limited understanding of how DENV spreads, including the role for human mobility of both infected individuals and the surrounding susceptible population, age-specific mixing and local heterogeneities in serotype-specific population immunity and mosquito density. These mechanistic knowledge gaps help explain our failures to control a pathogen that continues to cause 50 million annual symptomatic infections globally4. The use of pathogen sequences has the potential to help. However, existing phylogeographical approaches can only provide limited mechanistic insight into drivers of spread as they focus on rates of flow between locations present in a phylogeny, based on assumptions of mass action and using traits attached to the observed sequences  only6. In addition, fewer than 1% of dengue infections will currently be sequenced from any one country4. Critically, existing phylogeographic approaches do not consider that viral flow is made up of sequential transmission events with each event arising from a complex interplay of individual-, population-, environmental- and viral-level factors. Further, the bulk of available sequences typically come from a few locations with most locations providing no data. Many existing phylogeographic approaches will infer a viral flow between observed locations without consideration that transmission events that link two observed sequences will be unobserved and often in unsampled locations.As with other endemic pathogens, widespread, sustained co-circulation of dengue viruses (DENV), effectively masks the dynamics of individual lineagesAedes mosquito means only human mobility can drive spread between locations7. We allow for the disabling symptoms from dengue to result in reduced mobility in cases compared to the susceptible population8; that transmission may occur in an age-structured manner3, and that the spatial heterogeneity in vector distributions and the dynamic nature of local serotype-specific population immunity may affect where successful transmissions occur9. Using the transmission probabilities for a single generation, we probabilistically integrate over all possible pathways for the total number of transmission generations that link the observed locations for each pair of sequences, thereby capturing movement in unsampled locations. In parallel, we incorporate the probability of sequencing  an infection at each space–time unit, thereby explicitly incorporating space–time biases in sampling. We fit our models in a maximum likelihood framework that incorporates uncertainty from the evolutionary processes, including topical uncertainty in the phylogenetic trees, uncertainty in the generation time distribution and sampling uncertainty using a bootstrap approach.Here we develop an inference framework that fills this knowledge gap by explicitly considering individual transmission events. By using the generation time distribution for dengue   , modelled estimates of the probability of Aedes aegypti occurrence  and nationwide (N = 76 provinces with a mean area of 6700 km2 each).We apply our framework to dengue in Thailand, a country that has suffered from self-sustained dengue circulation for decades12) Fig. , from wheriod 199–2012 Fi67) Fig. . To info7 Fig. . N = 2011) of all ages about their daily travel. Consistent with our findings of reduced mobility in susceptible individuals, we find that there is a strong relationship between age and reporting having stayed within 1 km of their home in the prior week  of their time within their home cell as compared to 42% for adults  Fig. , our fin13. We also observed similar differences in mobility patterns at the national scale, with cases spending 96% of the time within their home province (95% CI: 86–100%), compared to 95% for the susceptible population (95% CI: 89–100%) and 87% for adults (95% CI: 86–88%)  of infected individuals’ time being within their home cell Fig. . Importa8%) Fig. .15.Local serotype-specific immunity also appears important, with transmission being more likely to occur in places that have seen increases of other (heterotypic) serotypes circulating in the previous two years and less likely to occur in places with increased cases of the same (homotypic) serotype within the same timeframe Fig.  and 9. HAedes aegypti presence is not linked to transmission risk . We find our model is able to accurately estimate the probability of observing viruses in the held-out locations, both within Bangkok and at the nationwide scale (correlation between observed and estimated locations of held- out sequences of 0.94 in Bangkok and 0.94 nationwide) Fig. . This de2 home grid cell of an infected individual (95% CI: 26–43%). This is despite infected individuals spending only an average of 4% of their time outside their home cells, highlighting the importance of considering mobility in both infected and susceptible populations when considering viral spread  times as likely to have travelled to Bangkok after a single-transmission generation as compared to a randomly selected province. After 20 transmission generations , we find the virus is 11.4 (95% CI: 6.3–19.4) times as likely to have infected at least one individual in the capital as compared to at least one individual living in a randomly selected province Fig. . The floead Fig. . After 2ead Fig. .Fig. 3MoWhile most transmissions occur within the home cell of an infected individual, we find that when transmissions do occur further away, local heterogeneity in patterns of serotype-specific population immunity means that the pathway taken by viruses depends on the serotype  and in 32% of cells within central Bangkok (95% CI: 25–43)  and there exists spatial information or other discrete traits.We use all available full genome sequence data from all four serotypes from Thailand covering the period 1994 and 2012 where province-level spatial information is available. All sequences are available from GenBank. The accession numbers are set out in Supplementary Data N = 11,583 cases). For the rest of the country, we used serotype-specific case data from the Ministry of Public Health. Outside the five sentinel surveillance sites, these data mainly come from ad hoc samples sent to the ministry for testing. Altogether, this represents a serotype-specific database of 27,586 cases covering 67 of the 76 provinces. For each province and year, we calculated the proportion of cases that were caused by each serotype. Where there were no samples from that province-year, we used data from the closest province in that same time period where data were available.All cases of dengue are notifiable and are reported to the Thai Ministry of Public Health. We extract the number of cases per year for each year and each of the 76 provinces in the country between 1994 and 2012. To estimate serotype-specific case data in each year, we used the serotype distributions of geocoded cases from QSNICH for Bangkok .In Bangkok, we initially placed a 1 × 1-km grid cell over the central part of the city (337 grid cells) and estimated the population size using population-size estimates from LandScan for 2010N = 11.4 million subscribers, 26% market share). These data are described in more detail elsewhere18. Briefly, each subscriber was assigned a daily home location based on their most frequently used cellular tower. Travel between locations was estimated by tabulating the subscriber’s home location on 1 day relative to the day before. The location-to-location transition probability matrix was estimated by using the average travel from location i to location j (weighted by the population at location i), and normalising travel such that the sum of travel from location i is equal to 1. We note that the mobile phone data were collected after our study period (2017 vs 1995–2012). Human mobility may have changed and could help explain some of the differences in mobility between the fitted models and that implied from the mobile phone data.To estimate adult mobility across provinces and within Bangkok, we used the call detail records (CDR) of over 11 million mobile phone subscribers between August 1, 2017 and October 19, 2017 from the third-largest mobile phone operator in Thailand  using a truncated exponential distribution with a parameter of 7.2 days and a maximum value of 30 days. Next, we draw the age at which the mosquito gets infected (MAI) from a uniform distribution between 0 and the lifespan of the mosquito. Next, we draw the extrinsic incubation period (EIP) as a random exponential distribution with mean of 6.1 days. The total period of mosquito infectiousness (MI) is then equal to MLS–MAI–EIP. Values of MI <0 were considered unsuccessful onward infections.The period of mosquito infectiousness depends on the mosquito lifespan and the extrinsic incubation period. The average daily probability of survival for We derived the empirical distribution of the generation time by simulating 10,000 values for HI, HM and MI and summing them. Individuals who are viremic for longer are more likely to infect mosquitoes. Similarly, mosquitoes that are infectious for longer are also more likely to infect more individuals. We, therefore, weighted the probability of each generation time by the length of HM multiplied by the length of MI. We obtained a mean generation time of 18.2 days and a standard deviation of 6.1 days, which we approximated using a gamma distribution with the same mean and standard deviation . This then allows us to consider single-transmission generations rather than overall viral flow. We develop models of viral movement between each location in our study area (whether it was sampled or not) for each transmission step. These viral movement models incorporate estimates of human mobility with the potential for differences in the movement for cases compared to the susceptible population, as well as incorporating effects of time-varying serotype-specific immunity and vector distributions. We allow for infection to occur at the infector’s home location (which requires the infectee travelling to the infector’s home location), the infectee’s home location (which requires the infector to travel to the infectee’s home location) or in a tertiary location (where both parties would have to travel there).G generations via matrix multiplication. This approach integrates over all possible pathways that link two locations. To specifically incorporate observation processes, we consider the probability of sequencing an infection at each space–time unit.We use the viral movement matrix for a single transmission to calculate the viral movement after To inform this model, we use an integrative approach that brings in detailed data from mobile phone operators that capture how people move and interact with each other, maps that estimate how populations are distributed, maps on vector suitability and the long-term spatiotemporal distribution of serotypes.CA and CB, in a phylogenetic tree: CA has home location LA, was sick at time TA and has sequence SeqA; case CB has home location LB, was sick at time TB and has sequence SeqB. The time of the MRCA between CA and CB is Tm and the location of the MRCA is Lm. We denote GA and GB the number of transmission generations that separate CA and CB from their MRCA. ObsLiTA is 1 if a sequence was observed at location Li at time TA and 0 otherwise. ObsLiTB is defined in a similar manner for time TB.For a pair of cases, i and j, are in the same location and are in contact (via a mosquito) given i lives in location a and j lives in location b can be written down asP(Vi = k|Li = a) is the probability of individual i, whose home location is in a, visiting location k and P(Vj = k|Lj = b) is the probability that individual j also visits location k and βk is the location-specific probability of transmission.Initially let us consider a single-transmission generation. The probability that two individuals, τ, one infector i that lives in location a is expected to transmit to the following number of persons living in location b:At time Sb,τ,ser is the number of susceptible people to serotype ser living in location b at time τ. The total expected number of persons infected by the infector is i:where b is the ratio of these terms:Conditional on transmission occurring, the probability that the infectee has a home location in cell N × N transmission matrix, ∏τ,ser,gen = 1, where N is the total number of locations, that sets out the transmission probabilities between all pairs of locations at a point in time for a single transmission. The element  of the matrix is πa,b,τ,ser.We can create a a visits location k.We use mobile phone data to characterise human mobility. Initially, we extract a matrix from the CDR data that set out the probability that an individual that lives in location CDRs come from adults, whereas dengue is concentrated in children, who are potentially more likely to spend more of their time at home. In addition, sick individuals may travel differently than healthy individuals and spend more time at home. In this way, the mobility of infectors may differ from susceptible information.θinfector.home, θpopulation.home) to reflect the additional time at home compared to that extracted from the CDR data.To allow for different periods at home for susceptible individuals compared to those in the CDR data and to assess whether there exists differential mobility by illness status, we incorporate separate parameters for the probability of being at home for infectors and the susceptible population . It has previously been shown that the dengue epidemic in Thailand is self-sustaining with few external introductionsAedes mosquitoes. Therefore, it would be optimal to use CDR data from daylight hours only. However, as is often the case, our CDR data represent an aggregate from all hours of the day. Nevertheless, as the majority of cell phone calls are made during daylight hours 28—it is reasonable to assume that this estimate is largely representative of daytime mobility.Dengue transmission is more likely to occur during daylight hours, due to the feeding behaviour of γ1 is to be estimated, mosq represents the estimated suitability for Aedes aegypti mosquitoes in location k. D1 is a proportionality constant (which gets cancelled out).As there may be factors that allow for different probabilities of transmission across locations, we allowed for differential probability of infection by location based on the mosquito presence in that locationThe number of susceptible individuals living in a location will depend on the level of historic infection in that location, in a potentially time-varying serotype-specific manner.k,τ,ser is the incidence of cases caused by different serotypes in the two prior years in location k, homok,τ,ser is the incidence of cases caused by the same serotype in the two prior years in location k, incidencek is the incidence of all cases over the study period in location k. We choose a window of 2 years to define recent immunity as serotype-specific incidence has previously been shown to be spatially correlated over this time range, presumably due to serotype-specific local herd immunity9. For the Bangkok analyses, we use the serotype-specific geolocated case data. For the nationwide analysis, we use the national reporting system from the Ministry of Public Health (MOPH). The national MOPH system is not serotype-specific; however, a small number of cases from all around the country are serotyped each year by the national reference centre in Bangkok. To obtain serotype-specific incidence estimates for each year, we multiplied the proportion of cases that came from each serotype within each province-year by the overall number of cases for each province-year. Where there were no serotyped cases for a province-year, we used the closest province where there were serotyped cases.where hetk after G transmission generations, we can use matrix multiplication that integrates over all possible pathways connecting two locationsTo calculate the probability of the home location being within location tl is the time of generation Gl.where LA at time TA and LB at time TB, the probability that CA has home location LA and CB has home location LB can be written down asConditional on sequences being observed in location We can consider that the location of the two cases is dependent on the location of their MRCA and the number of transmission generations that separate them from the MRCA.Li at time TA does not depend on the location of the MRCA or the number of generations separating the case from the MRCA. We can also substitute in Eq. We consider that the observation processes across locations are independent of each other. In addition, each transmission event is considered independent of other transmission events. The probability of observing a case at location Equation CA is separated from the MRCA by GA transmission generations and case CB is separated from the same MRCA by GB transmission generations using the generation time distribution, for dengue and the time-resolved phylogenetic tree.We can extract the joint probability that case aG and βG and that all transmission events are independent of each other, the sum of g gamma distribution is also gamma distributed with parameters gαG and βG. In addition, from a genealogy, Ri, we can extract the evolutionary time, EA, separating CA from the MRCA and EB, separating CB from the MRCA. We can therefore estimate the probability of g transmission events over many trees as follows:If we assume that the generation time distribution is gamma distributed with parameters This approach allows us to incorporate uncertainty in the phylogeny, including uncertainty in the evolutionary parameters and tree structure.As any spatial signal will be heavily diluted after many transmission generations, to optimise computational performance, we restrict our analyses to pairs where the mean estimated number of transmission generation is <25, we perform a sensitivity analysis where this is extended to 40 generations with very similar results  at location We conducted a sub-analysis where we assumed unbiased observation. In this analysis, we assumed that the probability of observation was 1 across all space–time locations, we obtained very different results  is proportional to the size of the population in that location. To assess the sensitivity of this assumption, we conducted a separate analysis where P(Lm) was assumed to be the same across all locations, with identical results . We identify the maximum likelihood estimate using an unconstrained nonlinear quasi-Newton optimisation approach29.We use a maximum likelihood approach to estimate the parameters linked to the mobility, transmission and susceptibility  destination is the same across serotypes. This allows us to assess whether viruses across the serotypes take the same routes or whether serotype-specific immunity changes the most likely pathways.In order to assess the model fit, we perform held-out validation. In Bangkok, we remove all sequences from 10% randomly chosen locations, we then refit the model and then estimate the probability of observing sequences in the locations not included in the model fitting process. For the nationwide analysis, as we have fewer locations with sequences available, we undertake the same process but hold out a single province in turn.For incremental windows of probability between 0 and 1, we identify all location-years where a virus was predicted to have been observed within the held-out locations. We then calculate the mean proportion of times a sequence was observed within those identified location-years.(I)Randomly introduce a single infection in one location where all locations have the same probability of being the source.(II)Reff (representing the effective reproductive number).We generate daughter infections from the index using a random draw from a Poisson distribution with mean (III)τ,ser,gen matrix.We identify the location for each daughter infection using a random draw where the probability of each location is taken from the Π(IV)Repeat (ii) and (iii) for 20 generations.(V)Repeat (i)–(iv) 50 timesUsing the fitted parameter values, we conduct a forward simulation at both the Bangkok and province levels. Taking each month between 1994 and 2012 and each serotype in turn, we apply the following algorithm:For each iteration, we calculate the average number of locations that have had at least one infection at each generation, average over all time points and all serotypes.To assess the impact of mobility patterns and immunity on the number of locations affected, we repeated the analysis with the following adjustments:Scenario a: All initial introductions were in the most connected location only (as defined as the location with the lowest probability of staying within your home location).Scenario b: All initial introductions were in the least connected location only (as defined as the location with the highest probability of staying within your home location).Scenario c: No difference in the mobility of the cases as compared to the susceptible population. This was achieved by forcing the Scenario d: Susceptible population mobility is equal to that of the adult population. This was achieved by forcing the β1 and β2 parameters to be zero.Scenario e: No impact of immunity. This was achieved by forcing the Reff was varied from 1.3 to 1.1 and 1.6.For each scenario, we calculated the proportion of locations affected at each generation and the relative number of locations affected compared to the base model. We also conducted sensitivity analyses where the We use an equivalent approach to characterise the age-dependent mixing of the population. Instead of considering the probability of the virus transitioning between two locations, we consider the probability of transitioning between individuals of two ages.We can consider that the age of the two cases is dependent on the age of the MRCA and the number of transmission generations that separate them from the MRCA.i at time TA does not depend on the location of the MRCA or the number of generations separating the case from the MRCA. We can also substitute Eq.  can be written down asβa,b is the age-specific probability of contact between individuals of ages a and b.Initially let us again consider a single-transmission generation. We assume that the age-specific susceptibility of the population is stable over time and that the probability of exposure does not differ by age group or serotype. The probability that two individuals, b conditional on an infector, i, being of age a isSb is the number of susceptible people of age b.The expected number of infected persons coming from individuals of age i, being of age a isThe expected number of infected persons coming from individuals of all ages, conditional on an infector, i is of age a, the probability of the infectee with age b is, therefore, the ratio of these terms. We define this probability as ϕa,b.Conditional on one transmission generation where the infector Nage × Nage transmission matrix, Φ, that sets out the transmission probabilities between all ages for a single transmission where element  of the matrix is ϕa,b. We use a maximum age of 70 years.We can create an θage, that captures the probability that two people interact, as a function of the difference in their ages, such that To parametrically characterise the age contact matrix, we use a discretised exponential decay parameter, a is equal to a in the national census and the force of infection, λ, is assumed to be 0.0430. We conduct a sensitivity analysis where the force of infection is varied to 0.02 and 0.06 with unchanged results .We assume that the probability of observing a case of age nser are the number of sequences available from serotype ser.We can calculate the likelihood using all pairs of available sequenced viruses as follows:θage. In order to incorporate uncertainty, we use a bootstrapping approach where we randomly sample all the available sequences with replacement over 100 iterations and recalculate the maximum likelihood estimate for each parameter each time. The 95% confidence intervals are then calculated using the mean and the standard deviation of the resulting distribution, assuming that they follow a normal distribution.We use a maximum likelihood approach to estimate the parameter θage parameter, we calculate from the matrix of ϕa,b, the proportion of transmissions that are between individuals that have <2 y in age between them. We use an equal weight for the age of the infector across all ages and only consider individuals between the ages of 1 and 15 as they represent the majority of the susceptible population. We compare this to the scenario where all individuals have the same probability of contact, irrespective of age , which is the minimum possible value.Once we have fitted the In order to ensure that our model is able to correctly identify parameters, we built a simulation framework with known parameters using 50 randomly selected grid cells from the Bangkok dataset where the population size, the historic incidence and the probability of between cell movement were taken from the observed data. As the observed heterogeneity in mosquito presence was limited in Bangkok  by randomly choosing a location, weighted by the population in that location. This will represent the MRCA case (C0) between the two observed viruses.Randomly select a starting location (g) between the MRCA and one of the observed isolates, where the number of generations is between 15 and 19 generations and the probability of 15 generations is 0.1, 16 generations is 0.2, 17 generations is 0.4, 18 generations is 0.2, 19 generations is 0.1.Draw the number of generations (C0 identify where they will transmit to (H1) using a random draw with the probabilities of each destination location coming from the H0 row of the For case g times using the destination of the previous step as the start location each timeRepeat step (3) Repeat steps 1–4 2000 times to generate 2000 pairs of casesρl) is taken from a random uniform distribution ). We randomly select 500 pairs of cases where the probability of observation of each pair is the product of the probability of observation at each of the two locations.We assumed that the probability of observing  the virus from a case was unequal across locations. The probability of observing a case at a location  project. This project asked 2011 individuals about their mobility patterns. Here we used the responses to the question ‘what is the farthest distance you have travelled within the last 7 days?’. We dichotomised the results into those that had not travelled >1 km and those that had travelled >1 km.a has never been infected by dengue as pnaivea = exp(−4*a*foi). The probability of being monotypically immune  is pmonoa = exp(−3*foi*a)*(1 − exp(−foi*a)). The probability of being susceptible as the sum of psusa = pnaivea + pmonoa  of 0.04 per year. Using a catalytic model, we can calculate the probability that an individual of age To calculate the average probability that an adult has not travelled further than 1 km from their home within the last 7 days, we use a similar approach where we calculate a weighted average across all individuals that are over 15 years of age (the mobility data are available in 5-year increments).This study was approved by the ethical review boards of Queen Sirikit National Institute of Child Health, and Walter Reed Army Institute of Research and the University of Florida. Case data were obtained from the results of standard confirmatory testing for dengue and therefore did not require informed consent.Further information on research design is available in the Supplementary InformationPeer Review FileDescriptions of Additional Supplementary FilesSupplementary Data 1Reporting Summary


It has been shown that the tumor population growth dynamics in a periodically varying environment can drastically differ from the one in a fixed environment. Thus, the environment of a tumor can potentially be manipulated to suppress cancer progression. Diverse evolutionary processes play vital roles in cancer progression and accordingly, understanding the interplay between these processes is essential in optimizing the treatment strategy. Somatic evolution and genetic instability result in intra‐tumor cell heterogeneity. Various models have been developed to analyze the interactions between different types of tumor cells. Here, models of density‐dependent interaction between different types of tumor cells under fast periodical environmental changes are examined. It is illustrated that tumor population densities, which vary on a slow time scale, are affected by fast environmental variations. Finally, the intriguing density‐dependent interactions in metastatic castration‐resistant prostate cancer (mCRPC) in which the different types of tumor cells are defined with respect to the production of and dependence on testosterone are discussed. The models of density‐dependent interaction between different types of tumor cells under fast periodical environmental changes are examined. Non‐linear dynamical effects in the tumor growth process are discovered. These dynamical effects can be manipulated for controlling the tumor growth process. The density‐dependent interactions in metastatic castrate‐resistant prostate cancer are also discussed. For optimization of treatment strategies, it is widely accepted that tumor evolution processes must be taken into consideration. These strategies focus on the suppression of tumor proliferation by taking advantage of evolutionary factors.Darwinian adaptive evolution is one of the key processes in tumor progression.14 Div that interact with one another and with normal cells. Various mathematical models have been suggested for describing the interaction between different types of tumor cells and between normal and tumor cells.Any tumor consists of different types of cells29 tha The microenvironment can promote or suppress tumor proliferation. Changes in the microenvironment can occur as a result of interaction between tumor and stromal cells and because of biotic (nutrients supply) and abiotic (drug application) variation. Because the environment affects the tumor evolution, these changes represent potential targets for antitumor treatment. Adaptive treatment strategies strive to exploit these interactions to control the tumor growth, instead of killing as many tumor cells as possible.The vital role of the microenvironment in tumor initiation and proliferation processes is well‐recognized.37 The A changing environment can induce polymorphic states that do not exist in fixed environment, thus changing the evolutionary dynamics of the population.Both frequency‐dependent selection  and density‐dependent population growth dynamics (where the interaction between different types within the population depends on the population size) of cells subject to environmental changes can drastically differ from those in a fixed environment.50 A c Here, we focus on fast environmental changes, that is, the case when the environment goes through multiple oscillations during the characteristic time of the population growth. These oscillations are considered as periodical variations of the characteristics of tumor population growth. It is assumed that environmental changes affect the competition between different cell types. However, these changes occur on a fast time‐scale, so that the averaged competition coefficients  of different types of tumor cells do not vary during the population growth time. It is also possible to assume that the competing cell types do not change in the course of the population growth, that is, we examine the population dynamics without mutations, so that the change of the population size depends only on density‐dependent competition (no migration). We observe that, due to the fast environmental changes, a new mechanism of competition between different types can arise in the population dynamics. The dynamics of the density‐dependent population growth depend on the total size of the population, and thus, arising new mechanisms can also alter the growth dynamics of the entire tumor. This information can be crucial for tumor growth suppression. These observations may hold promise for a novel approach for suppression of tumor growth for cancer treatment.In a changing environment, the density‐dependent population growth dynamics also differs from that in a fixed environment. The changing environment can vary both spatially and temporally. The analysis presented here pertains to a time‐variable, periodically changing environment. Periodical changes affecting the tumor cells can be associated with biotic  or abiotic  variations and/or mechanical stress; periodical variations are defined with respect to the characteristic lifetime of the population. It has been shown that periodical environmental changes allow stable coexistence of two competing species in cases when such coexistence is impossible in a fixed environment.48 Her and we elucidate the density‐dependent interactions between those cell types.The proposed model is applicable to tumors that contain different types of cells competing for resources. As an illustration of the effects caused by environmental changes, we analyze the tumor population growth of the metastatic castration‐resistant prostate cancer (mCRPC) in a changing environment.The mCRPC contains three distinct cell types that differ with respect to the production of and dependence on testosterone,55 andT− cell types (these are the type of tumor cells that are independent of testosterone) or a high density of this type in the population as an evolutionary outcome, because the T− type cells are not vulnerable under ADT and abiraterone therapies. In this work, we focus on these two cases, namely the occurrence of T− type cells only as an outcome of the population growth dynamics in a fixed environment and the occurrence of high density of the T− type. For these cases, the effect of environmental changes on the tumor population growth is explored by mathematical and computer simulations, coupled with analytic derivations.The interactions between different types of tumor cells define the evolutionary outcome of the tumor population growth. The growing success of a cell type is defined by its reproduction (reproduction coefficients) and competition (coefficients matrix) rates. The reproduction rate corresponds to the population growth in the absence of other types. The competition matrix defines the competition between different cell types, that is, how the presence of one type will affect the dynamics of the other type. In mCRPC, the worst outcomes occur when the competition matrix supports the existence of only 22.1n types of competing species (cell types). The density‐dependent population dynamics of these species is described by the Lotka–Volterra system of equations. We are interested in both the complete population size and the composition of different species in the population. As such, we use the Lotka–Volterra system written in terms of the population densities and the total number of individuals in the population.N is the total population size, pi=NiN is the density (fraction) of type i, ri is the reproduction rate for type i, aij > 0 is the competing coefficient between species {ij}, and fi is the fitness of the population i. Positive aij means that the growth of a type i is negatively affected by type j; in general, aij ≠ aji. In prey–predator models, some of these coefficients can be negative, that is, the presence of prey promotes the growth of predators. It can be seen that Equation n types oj,∑ipi=1fi≡ri−N∑jj,∑ipi=1fi≡ri−N∑ja¯ij and a time oscillating term a∼ij(τ).t is the fast time. The constant part in .aij(τ)=a¯ part in  and 2)..a¯ij andAfter each period of environmental change, the competition coefficients will have the same values as before in the variations.The population dynamics in the fast time‐scale describe the instantaneous impact of variation in the environment on the population densities. By contrast, the slow time‐scale dynamics describe the evolutionary outcome. These two types of dynamics are in a feedback relation. The fast variations in the population densities depend on the composition of the population in the slow time , and conversely, the fast variations indirectly affect the evolution of population densities in the slow time. The periodical variation of the environment induces a new mechanism in the tumor population growth dynamics. In the presence of periodical variations of the environment, competition between different tumor cell types changes: essentially, the success of a type of cells is defined not only by the competition with other types separately, but also by the competition with two or more other types simultaneously. These new effects are due to the overlap between time‐dependent competition rates, and those can potentially be exploited as a controlling mechanism for tumor population growth.2.2T+ type cells require exogenous androgen and rely on the testosterone produced by the human body. Androgen deprivation therapy (ADT) that stops the normal production of testosterone is used against this type of cells. However, resistance to the ADT builds up over time. One of the causes for the resistance is the increased expression of the CYP17A enzyme that is essential for testosterone production. As a result, the cancer cells start to produce testosterone. The second type of tumor cells are the testosterone‐producing cells Tp. These cells occur as a result of the up‐regulation of the CYP17A enzyme. Against this type, the abiraterone therapy is used, which blocks the production of CYP17A. The third type are T− cells, which do not produce testosterone and do not rely on it. It has been shown that incorporating the evolutionary dynamics into the treatment procedure leads to improved results. In some cases, it is also possible to suppress the growth of T− cells. These cases are defined by the competition coefficients, which can vary for different patients.Here, we address tumor population growth under environmental changes for two cases, using mCRPC as an example. The mCRPC contains three distinct cell types. The T− type; that is, the tumor consists entirely of T− cells. In the second case, the population dynamics in a fixed environment also has only one rest point, but it is not in a vertex of the simplex, that is, more than one type is present in the stable state. The stability of the vertices in the fixed environment is defined from the system  will be a stable rest point of the system  cannot be ignored (see Section The appearance of the tumor cell types dynamics  and 12)T+ and TT− vertex in the current context.The non‐linear terms will not change the stability of the vertices. Thus, for the cases where the stable rest point of the population growth dynamics (in fixed environment) is in a vertex, that vertex will also be a stable rest point in the case of the varying environment. Accordingly, by changing the initial state of the population dynamics in a varying environment, the tumor growth dynamics might end up in the Figure T− cells, then the total population size will be the same for the fixed and varying environments.The dynamics of the total population size is shown in Figure 1 = κ2 = 0. Dotted lines represent the population growth dynamics in the varying environment for the case when the environmental variations induce new stable state of the population dynamics. If the values of non‐linear competition coefficients κi are not sufficiently large, then the population growth dynamics end up in the same state as in the fixed environment . However, the environmental variations can increase the time taken for the population dynamics to reach the stable state. For any given time, the density of T− tumor type is higher in the fixed environment than in the varying environment.The appearance of the new stable rest point in the population dynamics depends on the values of the non‐linear competition coefficient Figure . The dot95*10−8).47 Howp1(0)=p2(0)=p3(0)=13, but for a small total population size N(0) = 300, the new stable state does not appear in the population dynamics. This effect is due to the dependence of the non‐linear terms on the square of the population size ,,T− celli, i = 1, 2 compared to the previous case.The new stable state generation is possible for relatively small values of non‐linear competition coefficients κ3Here, we have investigated the tumor population growth dynamics in a periodically varying environment assuming that the environmental changes are fast compared to the tumor growth dynamics. In particular, we examined the competition dynamics between different types of tumor cells in constant and periodically varying environments. These periodical changes can be associated with biotic  or abiotic  variations and/or mechanical stresses. The environmental variations substantially affect the competitiveness of different tumor types but do not alter the time averages of the competition coefficients over a period of environmental variations, which are equal to those in a fixed environment. The environment‐dependent variations in the competition coefficients induce the fast oscillations of the tumor population densities, whereas the slow‐varying component of the densities represents the population growth. When the competition coefficients vary in the same way, that is, there is no overlap between the varying parts of these coefficients, the population growth dynamics (for slow‐varying densities) does not respond to the environmental variations; that is, in this case, the varying environment is effectively equivalent to a fixed environment. This finding has to be taken into account in cancer therapy approaches that involve changing the environment of a tumor by designing these changes such that the competition coefficient values overlap.Although many researchers have hypothesized that environmental variations can drastically change the tumor population growth dynamics, our present work provides a comprehensive model and analysis of the phenomenon.38 Whe but these results are of limited applicability because the analytic derivations are effectively intractable or address specific cases only. Crucially, we linked tumor growth dynamics to environmental variations. Under the zero‐sum condition , the total population size is indirectly affected by the environmental variations, that is, there are no additional terms in the population size dynamics.Of special interest are the environment‐induced variations in the competing coefficients when a zero‐sum condition is imposed on the competing coefficients. Under the zero‐sum condition, the tumor population dynamics obtains new non‐linear competition terms. Now the competitive success of the given type depends not only on its competition with the other types, but also on the competition between those other types. Thus, a type that is successful in the competition in a fixed environment, can lose in the varying environment. This effect has been noted in several papers,48 butT+, Tp, and T− cell types in the metastatic castration‐resistant prostate cancer (mCRPC). These three cell types fundamentally differ with respect to the population of and dependence on testosterone. We describe the competition between these cell types under zero‐sum condition and focus on two types of dynamics: a) when the dominance of T− cells is inevitable, such that, in a fixed environment, only this type of cell is represented in the equilibrium state and b) high percentage of the tumor cells are of type T− in the equilibrium state. The tumor population dynamics is determined by the reproduction and competition coefficients. The densities of each type of tumor cells in the evolutionary outcome of tumor population dynamics are defined by these coefficients. In the real world, these coefficients can vary between patients. Although our results can be generalized for each possible ordering of the competition coefficients, we only discuss the worst case scenario here, namely, the dominance of T− cells that are not susceptible to hormone therapies.T− cells) in the presence of environmental variations, then the population size will also be the same as that in the absence of variations. The appearance of the new stable state depends on the values of non‐linear competition coefficients such that for small values (weak competition), the new stable state may be inaccessible. Although environmental variations can slow down the growth of T− cells, the dominance of these cells will still be inevitable.For the first case, it has been shown that the environmental variations can induce a new stable state of population growth dynamics. In this state, all the competing types are represented. Meanwhile, the equilibrium state of the population dynamics in the fixed environment remains unchanged. Thus, the population growth will be defined by the initial state of the tumor population or by the basin of attraction of stable states. The total population size of the tumor cells is smaller in the new environment‐induced stable state than it is in the stable state in the fixed environment. If the population growth dynamics converges to the original stable state  by its slow p¯i(t) and fast‐varying parts εi using the Kapitza method.We first represent the population densities a^ij be the primitive of a∼ij, that is, ∂τa^ij=a∼ij and a^ij¯=0. Note that a^ij is also a periodic function with the same period. The slow‐varying part p¯i(t) represents the density dynamics of the different types of tumor cells in slow times. The fast‐varying part εi(p¯(t),N(t),τ) represents the changes in the population due to the environmental variations. The latter is smaller than the slowly varying part p¯i(t), oscillates in the fast time and averages to zero:Let t is derived using the system of Equations (εi(p¯(t),N(t),τ) and keeping the quantities which are of order O(1ω), finally averaging by fast time τ (see Supporting Information).The population growth dynamics in the slow time quations  and 2)  t is derquations , 6), an, ant is quations , expandiquations  by εi  N. The pities in  and 2)  N. The pties in  for all types–αij, amplitudes of varying coefficients, or if only one coefficient varies during the time. Thus, if the biotic or abiotic variations of the tumor environment change the competition between different types in the same way, then the varying environment can be considered to be a fixed one. The environmental changes induce a non‐trivial contribution to the population growth dynamics if at least two competition coefficients vary in different phases. The necessity of varying phase overlapping for non‐trivial effects of the population growth in the periodical changing environment has been recognized.The non‐linear terms in population growth rates cancel out if all the coefficients vary in the same phase cognized.48The non‐linear terms also cancel out in the limit of ω → ∞. Thus, the environment‐induced effects in the population growth dynamics will be non‐trivial if the environment does not vary too fast. Thus, in the presence of the environmental changes (either biotic or abiotic), the population growth dynamics can drastically differ from the dynamics under fixed environment.T+, Tp, and T− cells. The densities of the different types of tumors are denoted by T+ → p1, Tp → p2, T− → p3.Here, we will study the population growth dynamics comprising three types under the periodical environmental changes. The case of three types in the population corresponds to the different types of tumor cells in metastatic prostate cancer—a∼ii=0,i=1,2,3. For the rest of the competition coefficients, a zero‐sum condition a∼ij=−a∼ji is made. The latter condition means that if the type i takes an advantage over type j–due to the environmental variations–then the latter type suffers by the same amount in the competition with the former type.It is assumed that the self‐limitation of different types are not affected by the environmental changes i = 1, 2, 3 andUnder the above conditions and from  and 9),,9), the ∑i=13p¯i=1.The sum of non‐linear terms in  cancels Equation  reveals i in a^23¯. It is seen that the possible advantage of type T+ against other types depends on both the competition of that type with other types and the competition between the other two types. Note that the values of the non‐linear coefficients κi do not depend on the fixed competition coefficients a¯ij because the population growth dynamics in slow times is obtained from (p¯i>>εi.The coefficients κi in  describeThe authors declare no conflict of interest.Supporting InformationClick here for additional data file.


The translational mode is believed to yield different rheological parameters than the rotational mode, given that these modes stretch the fibers differently. We compare the parameters for Michigan Cancer Foundation-7 (MCF-7) cells in this manuscript and find that the results are well comparable to each other. At low values of 0 Hz viscosity, the rotational and translational viscoelasticity matches well. However, discrepancies appear at higher values which may indicate that the cytoskeletal modes involved in rotation and translation of the particle are getting invoked. We also show that the 0 Hz viscosity increases as the cell ages under the conditions of constant room temperature of 25 ThesDetecting the rheological parameters in the cell cytoplasm has proved challenging. Inside the cell, not only is the viscosity dependent upon probe particle size used to perform the rheology , 11, it μm. It was shown by FCS that the viscosity increased till 100 nm radius of particle after which it became constant at μm particles.Early attempts at ascertaining the viscoelasticity of a cell searched for appropriate models to fit experimental data, like . Recentlabout 13 . In 11]μm. It waThe rheology can be performed both in the translation and in rotational modes. One of the advantages of probing rotational modes is that the probe is localised at one point while turning, thereby spanning less of the spatial variations inside the cell. This is a problem while using translational modes since the viscosity changes from location to location inside the cell, so that small changes of the position can introduce alterations in the viscosity.It is this technique that we use here to compare translational motion and rotational motion. The rotational motion probes an entirely different mode of the cytoskeletal stretching than the translational motion –26. We f2via a reflecting mirror.The experimental set up comprises of an optical tweezers kit  in an inverted configuration. The schematic diagram of the apparatus is shown in 3. The coverslips were sterilized by UV treatment for 1 h. For gelatin coating, a droplet of .1% gelatin solution was added to the center of the coverslip and incubated at 37 C for 1 h. The coverslip is then washed using 1X PBS and 20 μl of cell suspension (105 cells/ml) was added to the center of the coverslip. The cells were incubated at 37 C and 5% CO2 for 1 h to allow the cells to settle and attach to the surface. CO2 along with H2CO3 present in the nutrient medium assists in maintaining the pH level within the range of 7.2–7.4  which is essential for keeping the cells alive. After the cells are attached, 500 μl of fresh media is added. For probing the cytoskeletal rheology, 10 μl of 1 μm birefringent particles (5 μl of stock suspended in 100 μl of sterile serum free media) is added to the cells and incubated overnight. The cells were then washed with 1X PBS to remove the excess particles and fresh media is added.We use human breast cancer cells of MCF-7 cell line . The cells were maintained in Dulbecco’s Modified Eagle’s Media (DMEM) supplemented with 10% Fetal Bovine Serum (Gibco) and 1% penicillin-streptomycin (Gibco) to prevent contamination by any bacterial growth. For the experiment, the cells were grown on gelatin coated glass coverslips (50 mm × 20 mm). Initially, the glass coverslips were cleaned thoroughly in detergent followed by wash in concentrated HNOμm. To prepare RM-257 particles [μm in this case). The birefringent particles are ingested by the cells via the phenomenon of endocytosis.The cells were incubated with birefringent liquid crystalline particles RM-257 (Merck) having diameter around 1 ± .2 articles –29, 50 m00 mode from a diode laser is used for trapping. A polarising beam splitter  causes a change in total illumination of the QPD. This signal passes from the detector of bandwidth 40 KHz to a computer using data acquisition cards  having a bandwidth of 40 KHz.The interference between the scattered and unscattered light forms a scatter pattern on the QPD , 32. Crovia DAQ. Rotation in the pitch sense results in a difference in illumination of the photodiodes as shown in Besides translation, the particle has three degrees of rotation. We probe the out of plane rotation in pitch degree of freedom as in 3sμ is the zero frequency solvent viscosity, μp is zero frequency polymer viscosity and λ is the polymer relaxation time.We fit the power spectral density of the thermally driven fluctuations of the position and orientation of a particle optically trapped inside a cell with the Generalised Maxwell’s model or the Jeffery’s model –22. Thatμ0 = sμ+pμ. Solving for the power spectral density (PSD) of the optically trapped particle in this viscoelastic fluid, we get [This expression is similar to the Jeffery’s model or the Generalised Maxwell Model of frequency dependent viscosity with the coefficients labelled differently. Thus the viscosity of the solution at zero frequency would then be, , we get .(2)PSD quite akin to the conventional calibration factor for normal media [β2A [0 constant is also added to the power spectra density to account for the system noise floor. We use this expression for the PSD to understand the rheological properties of the intracellular cytoplasm, as a viscoelastic fluid. In this paper, we do not try to estimate the value of the calibration factor, thereby allowing the PSD to be normalised to 1. Then we only have 3 parameters to extract from the fit to the PSD. We assume that the solvent is water, such that sμ is that for water. Moreover, the γ0 is the drag in solvent which is water [Here, the term al media . From th and β2A . The y0 is water . The powis water , as a soThe model can be further improved by adding active forces. There have been many attempts at showing that the effective Langevin descriptions in terms of friction memory kernels and additive active forces naturally emerge for models of confined Brownian tracers interacting with active suspensions –40. ThisIn order to compute the G’ and G”, we use the relations These equations were borrowed from , where aThe calibration factor for the translational signal is related to the temperature as β is then given as Thus, the calibration factor λ), the trap stiffness κ and the calibration factor β. The calibration factor only indicates how to convert between the V2/Hz to nm2/Hz and thus, to reduce the number of fitting parameters, we could normalise the curve to the maximum to find all other parameters and then come back to this parameter later.Fitting the power spectral density with this 4The cell cytoplasm is a poroelastic medium , 43. HerX-axis) viscosity ratio of the cytoplasm to that of water and relaxation time was plotted as a function of the rotational pitch viscosity ratio and relaxation time performed in MCF-7 cells, shown in The translational mode leads to stretching of the cytoskeletal fibers in a different fashion than that of the rotational modes. Thus, if the PSD reflects the cytoskeletal modes, the rotational and translational PSD should bear different parameters . We go op-value of .0014, indicated by a straight line fit passing through the data . The straight line slopes are consistent with 1, within the error bars. We would also like to point out that such a comparison of the translational and rotational modes has never been performed in past. We would like to point out that the shape of the particles do indeed deviate from spherical to about 20% of the radius, as has been indicated in [We find that at low 0 Hz viscosity, which appears away from the nucleus mainly, the values for translation and rotation are similar but at high values of viscosity, which appears close to the nucleus, there seem to be discrepancies which could be a result of the different modes of the cytoskeletal fibers being involved. There could be other effects involved too requiring more detailed study. Thus, we limit ourselves to measurements where both the viscosities are less than about 6 times that of water, as reported in cated in . This caIt has been stated in  that theThe discrepancies of the viscoelasticity at high 0 Hz viscosity could have other effects involved too. There could be variations due to the method with which the particles have been inserted into the cell. We incubate the particles with the cells as we mentioned in the materials and methods. This can introduce variations in the way the particle is lodged inside the cytoskeletal mesh, since the coatings on the particle due to the endocytosis can vary. One way of solving this problem is to use microinjectors to insert the particle, which can be performed in future experiments. We would also like to mention another facet of the rheology of the cells using our method. The same particle placed inside the cell at the same location can yield a changing 0 Hz viscosity as a function of time, as reported in This phenomenon could be a result of more crosslinks of the fibers forming inside the cell. It could also be the cytosol becoming more viscous due to depolymerization of the cytoskeletal fibers back into the aqueous solution.We have also included a plot of the ratio of G” and G’ as a function of frequency in 5p-value of .0014. However, many discrepancies appear when the 0 Hz viscosity values become large, possibly due to the involvement of modes of the cytoskeleton. More detailed study is required for exploring the discrepancies at higher values.Thus, in conclusion, we find that the passive thermal motion of an optically trapped bead in the cell cytoplasm provides the viscoelasticity information of the cytoplasm, both in the rotational mode and the translational mode. When the 0 Hz viscosity values are low, particular away from the nucleus, the rotational and translational values are consistent to a