Uncertainty in temperature-based determination of time of death
- 128 Downloads
Abstract
Temperature-based estimation of time of death (ToD) can be performed either with the help of simple phenomenological models of corpse cooling or with detailed mechanistic (thermodynamic) heat transfer models. The latter are much more complex, but allow a higher accuracy of ToD estimation as in principle all relevant cooling mechanisms can be taken into account. The potentially higher accuracy depends on the accuracy of tissue and environmental parameters as well as on the geometric resolution. We investigate the impact of parameter variations and geometry representation on the estimated ToD. For this, numerical simulation of analytic heat transport models is performed on a highly detailed 3D corpse model, that has been segmented and geometrically reconstructed from a computed tomography (CT) data set, differentiating various organs and tissue types. From that and prior information available on thermal parameters and their variability, we identify the most crucial parameters to measure or estimate, and obtain an a priori uncertainty quantification for the ToD.
1 Introduction
For an accurate estimation of the ToD, the post-mortem temperature curves are crucial. Two different methods for defining these curves are conceivable: First, a phenomenological approach (PA), describing the temperature by parametrized arithmetic expressions. Marshall and Hoare [25] proposed a family of double-exponential curves with four parameters. This model, with empirical parameter definitions by Henßge [14] in terms of body weight and environmental temperature, has found widespread use in practical forensic work. The main drawback of the PA is its limited applicability to non-standard situations like irradiation from external sources, individually varying anatomy, or partial thermal insulation.
Second, a thermodynamic or mechanistic approach (MA), computing the temperature curve by a detailed simulation of the physical processes of heat transfer. Aiming at a faithful representation of the physical situation, the MA in principal allows a higher accuracy of ToD estimation as it can take all relevant heat transfer mechanisms into account, also those of non-standard situations. The reliability of the estimation depends on the accuracy of the physical model parameters such as corpse geometry and posture, tissue heat capacity and conductivity, and environmental conditions. While mechanistic models contain many more parameters than phenomenological models, their parameters have a particular physical meaning and are, at least in principle, accessible to measurements. The main drawbacks of the MA are first the effort required to set up a specific computational model and second the lack of accurate parameter values in concrete situations.
Mall et al. [22, 23] introduced a finite element (FE) based method for the simulation of corpse cooling, with a rather coarse corpse geometry and thermal tissue parameters taken from literature, and applied it to several cases of forensic practice. Already this approach provided estimates of ToD as good as or even more accurate than those obtained from the PA method.
In between these two different approaches are methods based on a physical heat transport model, but up to now employing more or less coarse approximations. Attempts at validation of such models have been presented by Smart [33] and Rodrigo [29, 30].
In this paper we investigate the required accuracy of thermal model parameters and geometry in the MA that is needed for a sufficient accuracy of ToD estimation. In particular, we derive sensitivities of the estimated ToD with respect to various parameters. In Section 2, the FE-based method is briefly described. The influence of tissue and environmental parameters is analysed in Section 3, based on a highly detailed anatomical model that has been derived from CT data, differentiating several organs and tissue types.
Notation
Symbols used throughout the paper.
Symbol | Meaning | Defined in |
---|---|---|
c | specific heat capacity of tissue | Table 2 |
ρ _{ b} c _{ b} | blood heat capacity | Table 3 |
γ | effective heat transfer coefficient | Table 3 |
δ t | time deviation | Equation 6 |
δ p | parameter deviation | Equation 6 |
𝜖 | surface emissivity | Table 3 |
h | convective heat transfer coefficient | Table 3 |
I _{ x} | impact of parameter x on time of death estimate | Equation 9 |
\(\hat I_{x}\) | importance of parameter x | Equation 11 |
κ | tissue heat conductivity | Table 2 |
n | surface outer unit normal | Equation 2 |
n _{ p} | number of parameters in the cooling model | p. 7 |
Ω | spatial domain occupied by the corpse | p. 4 |
p | parameters of the analytic cooling model | p. 7 |
\(\bar p\) | deviating parameter vector | Equation 6 |
q | external irradiation | Equation 2 |
Q | supravital metabolic heat | Equation 1 |
Q _{0} | initial value of Q | p. 2 |
ρ | tissue density | Table 2 |
σ | Stefan-Boltzmann constant | Equation 2 |
σ _{ x} | standard deviation of quantity x | |
S | sensitivity vector of time of death estimate w.r.t. parameters | Equation 8 |
t | time | |
t _{∗} | time of death (estimate) | p. 2 |
\(\bar {t}\) | deviating time of death estimate | Equation 6 |
𝜗 | body temperature distribution | Equation 1 |
𝜗(t; p) | rectal cooling curve (that may depend on parameters) | p. 1 |
𝜗 _{0} | vital temperature distribution | Equation 4 |
𝜗 _{amb} | ambient radiative temperature | Equation 3 |
𝜗 _{core} | vital body core temperature | Equation 4 |
𝜗 _{env} | environmental temperature | Equation 5 |
\(\dot \vartheta _{\text {env}}\) | environmental temperature rate | Table 3 |
𝜗 _{ m} | measured rectal temperature | p. 2 |
V _{ x} | volume fraction of tissue x | p. 15 |
w | tissue perfusion | Table 2 |
2 Finite element-based method
Thermal model
We interpret the heat capacity cρ as a single parameter instead of treating density and specific heat capacity separately.
Anatomical model
The heat equation (1) is defined on a geometric domain \({\Omega } \subset \mathbb {R}^{3}\) representing the corpse. Of course, the temperature profile and hence the cooling curve depend on the size and shape of Ω. Moreover, the material parameters cρ, κ, and w are different for various biological tissues, in particular muscle, adipose tissue, and bone. Thus, the cooling curve depends also on the size, shape, and position of different organs and tissue regions. For these reasons, a corpse model that contains those organs and represents the individual anatomy might be necessary for a sufficient accuracy of ToD estimation.
The invasive, direct detection of tissue components (mass, volume, and position) during post-mortem dissection can only be performed with great effort [5]. An alternative indirect quantitative capture of various tissue types based on medical imaging is difficult, takes higher efforts, but can be done with high accuracy in volume identification. Of course, this method doesn’t provide information on tissue density or other thermal properties, which have to be found by special measurements. See Deuflhard et al. [6] for the successful application of CT imaging in a project of cancer therapy. Janssens, Thys, Clarys et al. [17], Romvari, Dobrowolski, Repa et al. [31], Allen, Branscheid, Dobrowolski [2] have a more skeptical view on such methods.
Model parameters
The parameters entering into Eqs. 1, 3, and 4 depend on the individual corpse and the environmental conditions at the supposed crime scene. Assigning specific values a priori is therefore subject to some uncertainty.
Averaged thermal properties of human tissues based on data from ITIS Foundation, [1]
Tissue | ρ ± σ_{ρ} | c ± σ_{c} | κ ± σ_{κ} | w ± σ_{w} |
---|---|---|---|---|
kg/m^{3} | J/kg/K | W/m/K | kg/s/m^{3} | |
Fat | 911 ± 53 | 2348 ± 372 | 0.21 ± 0.02 | 0.53 ± 0.21 |
Bone | 1543 ± 141 | 1794 ± 265 | 0.32 ± 0.03 | 0.48 ± 0.21 |
Muscle | 1090 ± 52 | 3421 ± 460 | 0.49 ± 0.04 | 0.71 ± 0.50 |
Liver | 1079 ± 53 | 3540 ± 119 | 0.52 ± 0.03 | 16.24 ± 3.21 |
iIntestine | 1045 ± 50 | 3801 ± 300 | 0.56 ± 0.05 | 0.0 |
Kidneys | 1066 ± 56 | 3763 ± 120 | 0.53 ± 0.02 | 70.80 ± 10.15 |
Urinary bladder | 1000 | 3500 | 0.60 | 5.00 ± 0.5 |
Heart | 1050 ± 17 | 3617 ± 301 | 0.52 ± 0.03 | 183.8 ± 18.4 |
Lungs | 722 ± 87 | 3886 ± 300 | 0.39 ± 0.09 | 5.07 ± 4.72 |
Similarly, the specific heat capacity c_{b} of blood has been obtained from literature. The well-known temperature dependence of the tissues’ thermal properties [19] is assumed to be relatively small in the temperature range considered here, and is subsumed into the general variation. Only for the perfusion of muscle it seems to be appropriate to assume a higher standard deviation than mentioned in the ITIS database, in particular due to potential physical stress right before death [13].
While the uncertainty in tissue properties is due to moderate inter-individual differences and measurement errors, the effective heat transfer coefficient γ and the environmental temperature 𝜗_{env} depend on the situation at the place of discovery, in particular on clothing and posture of the corpse, weather conditions, and underground. While the heat transfer coefficient h can, in principle, be measured, obtaining accurate values is virtually impossible in practice, since it is usually spatially varying and depends not only on the type of clothing but also on its drapery, its potential humidity penetration, the contact pressure at the surface supporting the corpse, and the shape of the body. The emissivity 𝜖, too, depends on clothing. In contrast, variations due to air speed and posture can be taken easier into account [4]. In the end, the boundary condition parameter γ appears to be among the least precisely known, and the standard deviation assumed here is not more than an educated guess.
The amount of supravital heat generation is not particularly well known. For its value Q_{0} at time of death we assume anverage of the values cited in [24, 34, 35], i.e. 320 W m^{− 3}. This corresponds to authors supposing a value of similar magnitude as the heat generation in a resting healthy person, e.g., [18, 24, 32]. However, vastly different values have been reported in the literature [9, 11, 15, 16, 21, 22, 24, 26, 34], such that we assume a relatively large standard deviation of 50%.
Global parameters
Parameter | Symbol | Value | Unit |
---|---|---|---|
Heat capacity of blood | c _{ b} | 3617 ± 301 | J/kg/K |
Body core temperature | 𝜗 _{core} | 310.15 ± 0.5 | K |
Environmental temperature | 𝜗 _{env} | 297.15 | K |
Environmental temperature rate | \(\dot \vartheta _{\text {env}}\) | 0.0 ± 0.36 | K/h |
Skin emissivity | 𝜖 | 0.95 ± 0.05 | |
Heat transfer coefficient | h | 3.30 ± 0.3 | W/m^{2}/K |
Effective heat transfer coefficient | γ | 8.95 ± 0.6 | W/m^{2}/K |
Supravital heat generation | Q _{0} | 320 ± 160 | W/m^{3} |
Finite element simulation
Given a mesh of the corpse’s geometry (including tissue labels per element) and the vector p of all thermal parameters, the heat transfer model can be solved numerically for the temperature distribution 𝜗(t;p) by the finite element method, see, e.g., [7, 37]. For the current work, we use the research code Kaskade 7 [12]. The simulation provides a complete temperature distribution within the entire grid, from which cooling curves at particular measurement points can be extracted.
Parameter sets used for the deviations shown in Fig.4
Bone | Muscle | Fat | ||||
---|---|---|---|---|---|---|
c | κ | c | κ | c | κ | |
set 1 | + σ_{c} | − σ_{κ} | + σ_{c} | − σ_{κ} | + σ_{c} | − σ_{κ} |
set 2 | + σ_{c} | + σ_{κ} | + σ_{c} | + σ_{κ} | + σ_{c} | + σ_{κ} |
set 3 | 0 | 0 | 0 | 0 | 0 | 0 |
set 4 | − σ_{c} | − σ_{κ} | − σ_{c} | − σ_{κ} | − σ_{c} | − σ_{κ} |
set 5 | − σ_{c} | + σ_{κ} | − σ_{c} | + σ_{κ} | − σ_{c} | + σ_{κ} |
Fortunately, discretization errors can be kept below any prescribed tolerance by error estimation used for adaptive mesh refinement and time step selection, respectively, during the solution process [7]. For all simulations presented here we used linear finite elements and an extrapolated linearly implicit Euler time stepping scheme of order three, and made sure that the numerical discretization error is well below any of the considered modeling errors.
In contrast, the geometric approximation error is determined once by geometry reconstruction, depending on the voxel size of the medical image data as well as a curvature dependent limit (error quadrics) while coarsening the thereof reconstructed triangulation [10]. Its impact on the estimated ToD is investigated in the subsequent Section 4.
3 Impact of thermal parameter uncertainty
In this section, we systematically investigate the impact of parameter uncertainty on the ToD estimate. The aim is to identify those parameters which need special attention as their uncertainty from literature values affects the ToD estimate most. These parameters can then be measured directly or included in a more complex estimate procedure in order to improve the accuracy of the MA result.
Sensitivities
As a single number quantifying the impact of uncertainty in a parameter p_{i} we define \(I_{p_{i}}\) as the median of \(I_{p_{i}}(\bar {t})\) over the interval \(\bar {t} \in ]0,20]\)h. Since most relative sensitivity curves are almost monotone, this is essentially the value for \(\bar {t}= 10 \)h.
Stochastic interpretation
In contrast, the impact value \(I_{p_{i}}(\bar {t})\) itself describes the derivative with respect to a relative uncertainty reduction by r_{i} if p_{i} is the only uncertain parameter.
Discussion
The relative ToD estimate uncertainty due to uncertainties in thermal parameters amounts to 20% according to Eq. 10, based on the very rough estimates of parameter uncertainty from Tables 2 and 3. This value amounts to two hours of estimation error for a ToD of about 10 h and underlines the need for more accurate estimation. Improving the accuracy requires to reduce the parameter uncertainty by including additional information, e.g., further temperature measurements or experimental quantification of parameters.
According to Fig. 9, the most important parameter to characterize more accurately is the rate \(\dot \vartheta \) of environmental temperature change since the time of death, even though the assumed uncertainty range of 0.36 K/h is in no way exaggerated. That parameter alone can lead to 10–20% estimation error and more. This observation is in agreement with the findings of [33], where environmental and initial temperature have been identified as most influential parameters. On the one hand, the large impact of environmental temperature rate is unfortunate, since the temperatures cannot be measured retrospectively. On the other hand, additional information from weather recordings or heating control may be available and reduce the uncertainty. If the course of the external temperature is known exactly, the remaining parameter uncertainties induce a relative ToD standard deviation of 9.1%, almost an hour error for a ToD of about 10 h. Assuming a constant environmental temperature in case of lacking information, however, is bound to introduce rather large estimation errors.
The remaining parameters all incur much less uncertainty. Heat capacities ρc of muscle and fatty tissue have an individual impact on the estimation error of about 4–6%. In general, heat capacity has a slightly larger impact than heat conductivity. The dominance of muscle and fat properties is simply due to their large volume in the vicinity of the rectal measurement point. While in principle the heat capacity of tissue can be measured, unknown spatial variability of the heat capacity, difficulties of obtaining samples, and the nontrivial measurement itself make this approach problematic in forensic practice.
Also of some importance are the heat transfer coefficient γ and the heat conductivity κ of fatty tissue. Less impact have the initial body core temperature 𝜗_{core}, supravital heat generation amplitude Q_{0}, thermal properties of bones and the gastrointestinal tract, and muscle and fat perfusion. Initial body core temperature has been found to be one of the two most influential parameters in [33]. The apparent difference in findings is by reason of the small standard deviation \(\sigma _{\vartheta _{\text {core}}} = 0.5 \)K that we assume, which amounts to a relative deviation (with respect to 0 ^{∘}C) of only 1.3%. The uncertainty in 𝜗_{core} has a particularly large impact during the first few hours of cooling. This is due to the fact that in the BHTE (4), it directly affects the initial temperature at the rectal measurement point. Together with the vanishing time derivative in the initial plateau phase, this leads to a large ToD estimation uncertainty. A similar effect has the supravital heat generation during the first few hours. Much smaller is the impact of fat and tissue perfusion, which also affect the initial temperature at the measurement location.
The uncertainty of the remaining thermal tissue parameters has a negligible impact on the ToD estimate. These are in particular the tissue properties of heart, lungs, liver, kidneys, and urinary bladder. The reason is, that on one hand most of these organs are far away from the rectal measurement point, and on the other hand their volume and hence total heat capacity is small compared to muscle and fat tissues. This minor impact indicates that an explicit representation of these organs might not be necessary at all, which would simplify geometry acquisition, segmentation, and meshing.
4 Geometry representation
Deviations of simulated cooling curves from the reference are interpreted as resolution-dependent errors. For the cooling simulations we therefore made sure that the numerical discretization error of the temperature distribution is small compared to the error due to coarse anatomy resolution by applying grid refinement to the coarse models. Thus, the reported deviation is only due to the geometrical resolution of the tissue distribution. In the same way, the numerical temperature discretization error has been assessed for Fig. 5a: The anatomy and tissue distribution was always the coarse 7k mesh, and different discretization errors are obtained by refining that grid for actual computation.
There is no evidence for a significant sensitivity with respect to geometric resolution of tissue boundaries, even though the tissues may change the total heat capacity in the different tissue compartments as well as in the whole corpse. In particular, it can be concluded that there is no need for providing high resolution segmentation and meshing.
5 Conclusions
The sensitivity results clearly show that a highly accurate and detailed representation of individual anatomy is not necessary for the mechanistic approach to provide accurate time of death estimates. As long as the total volumes of tissues with dominant water or dominant fat composition and their overall geometric location are accurately captured, the geometric resolution has virtually no impact on the estimated ToD – at least as long as the temperature measurement is restricted to a single rectal location.
In contrast, some thermal parameters appear to play a crucial role, in particular the time course of the environmental temperature, but also heat capacity of muscle and fatty tissues, their heat conductivities, and the effective heat transfer coefficient. With uncertainty ranges from literature, a total uncertainty of about 20% in the ToD estimate from the simulation method is to be reckoned with for only one measurement, which calles for improved estimation methods. In this situation, multiple temperature measurements may provide the additional information that is necessary to increase the reliability of the estimation.
While the results are obtained on a single corpse geometry, and are therefore exemplary, we believe that the conclusions drawn from them are valid for a wide range of anatomies and environmental conditions. This, however, needs to be confirmed through further studies.
Notes
Acknowledgments
We thank our colleague Marian Moldenhauer for careful reading of the article and for helpful comments. Funding by the University of Jena is gratefully acknowledged.
Compliance with Ethical Standards
Conflict of interests
The authors declare that they have no conflict of interest.
References
- 1.Foundation for Research on Information Technologies in Society (IT’IS). ZurichGoogle Scholar
- 2.Allen P, Branscheid W, Dobrowolski A, Horn P (2004) Schlachtkörperwertbestimmung beim Schwein - Röntgen-Computertomographie als mögliche Referenzmethode. Fleischwirtschaft 3:109–112Google Scholar
- 3.Borouchaki H, Hecht F, Frey PJ (1998) Mesh gradation control. Int J Numer Meth Eng 43(6):1143–1165MathSciNetCrossRefzbMATHGoogle Scholar
- 4.Cengel YA, Ghajar AJ (2015) Heat and Mass Transfer, Fundamentals & Applications. MacGraw-Hill, NYGoogle Scholar
- 5.Clarys JP, Provyn S, Marfell-Jones MJ (2005) Cadaver studies and their impact on the understanding of human adiposity. Ergonomics 48:1445–1461CrossRefGoogle Scholar
- 6.Deuflhard P, Schiela A, Weiser M (2012) Mathematical cancer therapy planning in deep regional hyperthermia. Acta Numer 2:307–378MathSciNetCrossRefzbMATHGoogle Scholar
- 7.Deuflhard P, Weiser M (2012) Adaptive numerical solution of PDEs. de Gruyter, BerlinCrossRefzbMATHGoogle Scholar
- 8.Deuflhard P, Weiser M, Seebaß M (2000) A new nonlinear elliptic multilevel FEM applied to regional hyperthermia. Comput Vis Sci 3(3):115–120CrossRefzbMATHGoogle Scholar
- 9.Frankenfield DC, Smith Jr JS, Conney RN, Blosser SA, Sarson GY (1997) Relative association of fever and injury with hypermetabolism in critically ill patients. Injury 28(9–10):617–621CrossRefGoogle Scholar
- 10.Garland M, Heckbert PM (1997) Surface simplification using quadric error metrics. In: Proceedings of the 24th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH’97, pp 209?-216Google Scholar
- 11.Hutchins GM (1985) Body temperature is elevated in the early postmortem period. Human Pathol 16 (6):560–561CrossRefGoogle Scholar
- 12.Götschel S, Weiser M, Schiela A (2012) Solving optimal control problems with the Kaskade 7 finite element toolbox. In: Dedner A, Flemisch B, Klöfkorn R (eds) Advances in DUNE. Springer, pp 101–112Google Scholar
- 13.Guyton AC, Hall JE (2000) Textbook of Medical Physiology. Saunders, UKGoogle Scholar
- 14.Henßge C (1988) Death time estimation in case work. I: The rectal temperature time of death nomogram. Forensic Sci Int 38:209–236CrossRefGoogle Scholar
- 15.Henßge C, Madea B (2004) Estimation of the time since death in the early post-mortem period. Forensic Sci Int 144:167–175CrossRefGoogle Scholar
- 16.Hooft P, van de Voorde H (1989) The role of intestinal bacterial heat production in confounding postmortem temperature measurements. Rechtsmedizin 102:331–336Google Scholar
- 17.Janssens V, Thys P, Clarys JP, Kvis H, Chowdhury B, Zinzen E, Cabri J (1994) Post-mortem limitations of body composition analysis by computed tomography. Ergonomics 37:207–216CrossRefGoogle Scholar
- 18.Klinke R, Silbernagl SA (eds) (1994) Lehrbuch der Physiologie. Thieme, StuttgartGoogle Scholar
- 19.Lang J, Erdmann B, Seebaß M (1999) Impact of Nonlinear Heat Transfer on Temperature Control in Regional Hyperthermia. IEEE Trans Biomed Eng 46(9):1129–1138CrossRefGoogle Scholar
- 20.Lo SH (1995) Automatic mesh generation over intersecting surfaces. Int J Num Meth Eng 38:943–954CrossRefzbMATHGoogle Scholar
- 21.Madea B (1994) Importance of supravitality in forensic medicine. Forensic Sci Int 69:221–241CrossRefGoogle Scholar
- 22.Mall G, Eisenmenger W (2005) Estimation of time since death by heat-flow finite-element model. Part I: Method, model, calibration and validation. Leg Med 7:1–14CrossRefGoogle Scholar
- 23.Mall G, Eisenmenger W (2005) Estimation of time since death by heat-flow finite-element model. Part II: Application to non-standard cooling conditions and preliminary results in practical casework. Leg Med 7:69–80CrossRefGoogle Scholar
- 24.Mall G, Hubig M, Beier G, Büttner A, Eisenmenger W (2002) Supravital energy production in early post-mortem phase – Estimate based on heat loss due to radiation and natural convection. Leg Med 4:71–78CrossRefGoogle Scholar
- 25.Marshall T, Hoare F (1962) Estimating the time of death: The rectal cooling after death and its mathematical expression. J Forensic Sci 7:56–81Google Scholar
- 26.Muggenthaler H, Sinicina I, Hubig M, Mall G (2012) Database of post-mortem rectal cooling cases under strictly controlled conditions: A useful tool in death time estimation. Leg Med 126:79–87Google Scholar
- 27.Pennes HH (1948) Analysis of tissue and arterial blood temperatures in the resting human forearm. J Appl Phys 1:93–122Google Scholar
- 28.Rassineux A (1998) Generation and optimization of tetrahedral meshes by advancing front technique. Numer Meth Eng 41:651–674CrossRefzbMATHGoogle Scholar
- 29.Rodrigo MR (2014) Time of death estimation from temperature readings only: A Laplace transform approach. Appl Math Lett 39:47–52MathSciNetCrossRefzbMATHGoogle Scholar
- 30.Rodrigo MR (2016) A nonlinear least squares approach to time of death estimation via body cooling. Forensic Sci 61(1):230–233CrossRefGoogle Scholar
- 31.Romvari R, Dobrowolski A, Repa I (2006) Development of a computed tomographic calibration method for the determination of lean meat content in pig carcasses. Acta Vet Hung 54(1):1–10CrossRefGoogle Scholar
- 32.Schmidt RF, Thews G, Lang F (eds) (1997) Physiologie des Menschen. Springer, BerlinGoogle Scholar
- 33.Smart J (2010) Estimation of time of death with a Fourier series unsteady-state heat transfer model. J Forensic Sci 55(6):1481–1487CrossRefGoogle Scholar
- 34.Smart J, Kaliszan M (2012) The post-mortem temperature plateau and its role in the estimation of time of death. A review. Leg Med 14:55–62CrossRefGoogle Scholar
- 35.Werner J (1984) Regelung der menschlichen Körpertemperatur. de Gruyter, BerlinGoogle Scholar
- 36.Zachow S, Zilske M, Hege H-C (2007) 3d reconstruction of individual anatomy from medical image data: Segmentation and geometry processing. In: 25. ANSYS Conference & CADFEM Users’ MeetingGoogle Scholar
- 37.Zienkiewicz O, Taylor RL, Zhu JZ, Nithiarasu P (2005) The finite element method. Elsevier Butterworth-Heinemann, OxfordGoogle Scholar
- 38.Zilske M, Lamecker H, Zachow S (2008) Adaptive remeshing of non-manifold surfaces. In: Eurographics 2008 Annex to the Conference Proceedings, pp 207–211Google Scholar