Patent Application: US-201514683361-A

Abstract:
the present invention includes a method of creating analysis suitable models from discrete point sets . the proposed methodology is completely automated , requiring no human intervention , as compared to traditional mesh - based methods that often require manual input . the present invention is directly applicable to engineering approaches in medicine where the object to be analyzed is described by discrete medical images , such as mri or ct scans . moreover , the present invention is useful in any application where the object of interest is created from digitized imaging technology .

Description:
in the following detailed description of the preferred embodiments , reference is made to the accompanying drawings , which form a part thereof , and within which are shown by way of illustration specific embodiments by which the invention may be practiced . it is to be understood that other embodiments may be utilized and structural changes may be made without departing from the scope of the invention . the present invention is applicable to any general point set in three - dimensional space . in order to process a generic discreet point set , the present invention computes a smoothing length at each discrete point and computes a representative volume at each discrete point . the smoothing length can be determined through nearest neighbor searches or any other approach known to a person having ordinary skill in the art . once these two things have been accomplished , the proposed geometric model can be evaluated and the point set is an “ analysis suitable model .” this means that , through the geometric function , the point set can provide a complete description of the objects geometry and the point set can be used to approximate the functions necessary for engineering analysis , i . e . solution of partial differential equations and general geometric modeling . further , once this analysis suitable model has been created from the initial point set , it can be used to generate subsequent analysis suitable models representing the same object using different point sets that are derived from the initial geometric model . this may include the actions of representing the object with a more dense set of points ( refinement ), representing the object with a more coarse set of points ( coarsening ), or refining some regions of the object while coarsening other regions of the object . each subsequent point set representing the object can be processed as an analysis suitable model . this capability allows for easy control of the balance of model accuracy with computational efficiency . the medical modeling example below is an exemplary application in which the smoothing length and representative volume are computed by taking advantage of the uniform nature of the pixel data . in a general sense , these quantities could be computed with algorithms that do not take advantage of the fact that the point set came from structured data . the medical modeling example also serves to show that the geometric model retains accuracy through coarsening . it is a specific example of coarsening ; however any method of coarsening or refinement is considered by the present invention . the discussion relating to taking medical images and generating three - dimensional point sets through the creation of label - maps and extracting points is for completeness and background . the reproducing kernel particle method ( rkpm ) is a mesh - free method in the general class of so - called meshfree galerkin methods , which includes other methods such as the element free galerkin ( efg ) method . the general process described herein is applicable to many of the mesh - free methods , though rkpm is reviewed here for completeness . the general formulation is available in a number of papers and texts , such as [ li and liu ( 2002 ) and liu , jun , and zhang ( 1995 )]. the approach presented here is that of [ li and liu ( 2004 )]. the problem domain is discretized by a collection of particles , each representing a certain volume and mass of material . a function ƒ defined on the domain is approximated by an expression ιƒ ( x )=∫ ω k ρ ( x − y , x ) ƒ ( y ) dy eq 1 where k ρ is a smooth kernel function , and ι is an approximation operator . in rkpm , the form of the kernel is chosen to be the parameter ρ is called the smoothing length and is the characteristic length scale associated with each particle . the window function φ is a non - negative , smooth function that is compactly supported with support radius of length ρ . the vector p contains the monomial terms of a polynomial of some order . in the present work , a tri - linear basis is used , so p t ( x )=[ 1 x y z xy yz zx xyz ] eq 3 finally , the vector b , termed the normalizer , is computed at each evaluation point as a correction function to enforce consistency conditions on the approximation . the consistency conditions lead to a set of equations ∑ i ⁢ k ρ ⁡ ( x - x i , x ) ⁢ δ ⁢ ⁢ x i = 1 ∑ i ⁢ ( x - x i ρ ) ⁢ k ρ ⁡ ( x - x i , x ) ⁢ δ ⁢ ⁢ x i = 0 ∑ i ⁢ ( y - y i ρ ) ⁢ k ρ ⁡ ( x - x i , x ) ⁢ δ ⁢ ⁢ x i = 0 ∑ i ⁢ ( z - z i ρ ) ⁢ k ρ ⁡ ( x - x i , x ) ⁢ δ ⁢ ⁢ x i = 0 ⋮ ∑ i ⁢ ( x - x i ρ ) α ⁢ k ρ ⁡ ( x - x i , x ) ⁢ δ ⁢ ⁢ x i = 0 } eq ⁢ ⁢ 4 where α is a multi - index with ∥ α ∥= n . upon substituting eq . 2 , leads to the following set of linear equations ( m 0 ⁡ ( x ) m 1 ⁡ ( x ) … m n ⁡ ( x ) m 1 ⁡ ( x ) m 2 ⁡ ( x ) … m n + 1 ⁡ ( x ) ⋮ ⋮ ⋮ ⋮ m n ⁡ ( x ) m n + 1 ⁡ ( x ) … m 2 ⁢ n ⁡ ( x ) ) ⁢ ( b 0 ⁡ ( x ) b 1 ⁡ ( x ) ⋮ b n ⁡ ( x ) ) = ( 1 0 ⋮ 0 ) eq ⁢ ⁢ 5 or , m ( x ) b ( x )= p ( 0 ). the individual entries m i ( x ) are computed from the system eq . 5 is called the moment equation whose solution yields the values for the normalizer b ( x ). the present work is based upon the observation that the actual domain is defined where the moment matrix is non - singular . conversely , where the moment matrix is singular , there is no body . readily available programs , such as the 3d slicer , can read the data generated by the imaging machines , convert the images to point clouds , and generate meshes . this is not quite a satisfactory result because the resulting images , while visually appealing , do not stand up under the scrutiny required for analysis . in particular , the meshes generally have duplicate points , edges , and faces ; holes and discontinuities ; and topological variations such as intersecting faces . the first of these problems can easily be rectified , but the latter problems generally require manual intervention to guide software in repairing the mesh . mesh - free galerkin methods build their function spaces directly from point sets , and thus are not plagued by the same issues as mesh - based methods . this feature makes mesh - free methods attractive for life science applications . the basic approach , generally denoted by reference numeral 100 and shown in fig3 , is exemplified in the creation of a patient - specific modeling of the pelvic floor . a patient specific model is created by first producing a sequence of images of the pelvic floor via some imaging tool such as magnetic resonance imaging ( mri ), step 102 . in these raw images , the various organs and tissues are not distinguished . the images must be segmented to segregate the organs into individual models , step 104 . the resulting image , as provided in fig4 , is called a label - map . automatic segmentation is an open research topic , currently the images are at least partially segmented by hand . some methods for segmenting the pelvic floor are discussed in [ ma , jorge , mascarenhas , and tavares ( 2012 )]. after the label - maps are generated , the discrete point set representations of the models are produced . the label - maps are integer arrays with each index referring to a pixel , or color value . each pixel is representative of an area with pre - defined dimensions . when the label - maps are stacked , the space between consecutive images is represented with voxels , or volume elements . the voxels are transformed into the raw point set in r 3 based on their dimensions and location in image space , step 106 . the raw point set is then used to reconstruct the continuous geometry of the object via an implicit representation described in the “ geometric description ” section below , step 108 . the reconstructed geometry is an adequate description of the domain that is useful for placing rkpm analysis suitable particles as warranted . the geometric model defines the surface and interior of the object ( volume ). it is up to a practitioner to discretize the geometric model with particles ( or elements ) in a way that is valid for the type of analysis the practitioner is attempting to produce . this may be accomplished through an appropriate algorithm that evaluates the geometric model to determine if a point is on the surface or interior of the object , then calculates any necessary properties of the point required for the analysis . one useful method would be to determine a desired spacing of points for the model , generate a regular grid of points using that spacing , then use the existing geometric model derived from the initial point set to filter out points that are not inside the body nor on its boundary . the remaining points can then be used to generate a new geometric representation using the same algorithm used on the initial input point set . this , however , is only one possible method for coarsening or refinement . the geometric model derived from the initial point set can be used as a standard to judge any attempted refinement or coarsening . for the example in fig3 , the discretization included constructing a decimated point set for the purpose of analysis by generating a regular grid of points and eliminating any points not in the domain , step 110 . fig5 shows the sequence of progression from the raw point set in fig5 ( a ) to the volume reconstruction shown in fig5 ( b ) , and then to the decimated point set in fig5 ( c ) . the idea is that the discretization may be accomplished , very simply , by generating a grid of points and evaluating each point with the geometric model to determine which points are inside the object . this algorithm may be used to produce a coarse discretization — fewer points than originated in the voxel model , it may be used to produce a refined model — more points than originated in the voxel model , or it may be used to discretize the object in a way that results in some local regions being represented with a coarse discretization and other regions being represented with a refined discretization . once all of the organs of interest are processed and have their final points placed in their respective domains , any organs that will interact during the simulation must be connected , step 112 . the current method is to model connective tissue as linear springs between organs . after the organs are connected any applicable loading conditions and essential boundary conditions are applied , step 114 and the performance of the organs can be simulated and analyzed . the initial geometry is obtained from a discrete point set , thus the continuous geometry must be reconstructed . as discussed earlier , the moment matrix in eq . 5 is non - singular inside of the domain . the linear dependence of the moment equations is based largely on the number of particles in the support and the spatial distribution of these particles . if properly constructed , the moment matrix is well - conditioned and invertible everywhere in a domain . the moment equations quickly become linearly dependent as the limits of the domain are approached . this fact was used to build a function to describe the geometry implicitly . in numerical linear algebra , the condition number of a matrix is a measure of “ how singular ” a matrix is . the condition number is defined in [ saad ( 2003 )] as the best - conditioned matrix has κ = 1 , a singular matrix has κ →∞. therefore the geometry was defined based on the condition number of the system of equations : where m ( x ) is the moment matrix at a point and k , is a function returning the condition . notice that in eq . 8 the reciprocal condition number is used and the range of the function is ( 0 ; 1 ], with κ − 1 = 0 indicating a singular matrix . then the reconstructed domain is defined as : 1 . int ( ω ε ):={ x ∈ r n | ε & lt ; f ( x )& lt ; 1 } 3 . r 3 − ω ε :={ x ∈ r n | f ( x )& lt ; ε } eq 9 where 1 is the interior , 2 is the boundary of the domain and ε is small . with this definition , the boundary of the domain is defined as a contour of eq . 8 . the shape of the reconstructed boundary can be controlled through the radius of support . as the support radius grows any surface features are smoothed over , conversely as the support radius is decreased any surface features become more sharply defined . the set of points derived from the image x i , are the discrete representation of the domain , ω . developing a mesh - free function space directly from a general point set is a non - trivial task due to the need for accurate integration weights and support radius . the set of points can actually be thought of as the vertices of a hexahedral mesh that is usually referred to as the voxel model . appropriate representative particle volume δv i , and support radius ρ can be assigned using data from the voxel elements . due to aliasing artifacts , the voxel mesh is not suitable for representing the smooth geometry of biological tissue , and it could not generate a smooth solution to the pde &# 39 ; s if used for a finite element model . yet , by using the vertices to construct a mesh - free representation of the geometry , the representation is smooth . the objects of interest in the pelvic floor are smooth , not rough . one critique of the proposed method is that it may not represent the actual geometry precisely enough . on the other hand , a voxel mesh , as in fig6 , is clearly not a smooth body , and itself is inaccurate due to the aliasing and rough discretization inherent in a digital medical image . refer back to fig2 ( c ) for an example of a smooth representation . in order to illustrate this definition of the geometry , a one - dimensional example is plotted over a unit domain . the domain is discretized with ten particles and their integration weights are assigned as described in [ li and liu ( 2004 )]. the particles are uniformly distributed throughout the domain , so a suitable support radius is defined as where δx is the particle spacing and a is the dilation coefficient . the shape functions and the corresponding geometry representation are shown for dilation coefficient values of 1 . 2 , 1 . 4 , and 1 . 6 . the dashed vertical lines in fig7 are bounds for the area starting at the first boundary particle and extending out until the moment matrix is singular to machine precision . this area will be referred to as the rind of the reconstructed domain . notice that as the dilation coefficient is increased this rind grows in size . this would correspond to a smoothing of geometric features , which may or may not be desirable . the shape functions plotted in fig7 show the correlation between support coverage and moment matrix condition . as a reminder , the moment matrix is computed via nodal integration , each term in the summation is a rank one matrix representative of a particle &# 39 ; s contribution . if this integral is dominated by a single term , the overall condition of the moment matrix is poor . conversely , when each of the terms is equally weighted , the moment matrix is well conditioned . this shows up in fig7 ( b ) , fig7 ( d ) , and fig7 ( f ) as the fluctuations in f ( x ). in fig7 ( b ) , these fluctuations are much greater than in the fig7 ( d ) and fig7 ( f ) . the peaks are higher between particles , indicating a well - conditioned moment matrix and no dominating term in the integral . additionally , the valleys are much lower at a particle , indicating a poorly conditioned moment matrix and thus one term in the integral is dominating . since the surface was chosen based on some threshold value f ( x )= ε , these fluctuations could cause trouble if the functions falls below ε in an undesirable location . this could result in holes in the domain where none exist . a safe method for representing sharp surface features , meaning using a small value for α , is to increase the dilation coefficient of internal particles while keeping the boundary particle dilation coefficients the same as in fig8 . this method minimizes the size of the rind , while smoothing the fluctuations in f ( x ). increasing the number of particles in the domain would also remedy this problem , yet simply controlling the dilation parameter provided the same effect without adding more degrees of freedom . the ideas discussed in one dimension hold true for three - dimensional models . a volume rendering of the vagina is shown in fig9 ( a ) , the plane slicing through the model shows the location of the plot in fig9 ( b ) . fig9 ( c ) shows the moment matrix reciprocal condition number is then plotted over the line drawn across the middle of fig9 ( b ) . the top portion of the model is essentially a thin walled tube , and this shows up in the line plot across the body as two peaks in the function . the second peak shows the same fluctuations that were discussed in the one - dimensional example . in order to perform an analysis in an efficient manner , it is important to limit the number of particles in the model . since the goal of the method is to maintain patient specific features in the model , decimation of the model cannot significantly alter the geometry being represented . the images in fig1 depict the volume renderings for three different particle sets . the first representation , fig1 ( a ) , with 112 , 128 particles , is derived directly from a medical image . the volume rendering of this representation , fig1 ( b ) , has well defined features such as the indentation of the front wall and the ridges along the sidewalls . the raw point representation was decimated to 26 , 809 particles , as shown in fig1 ( c ) . the volume rendering for this representation , fig1 ( d ) still has the well - defined indentation and the ridges on the sidewall are still defined . the third representation in fig1 ( e ) has a particle count of 3 , 339 . while this representation retained the overall shape of the original model , the indentation on the front wall and the ridges on the sidewall are not as well defined , as shown in the volume rendering of fig1 ( f ) . the tetrahedral finite element mesh in fig1 shows some of the same geometric features as the inventive volume reconstructions , yet the surface is not smooth . as evidenced in the foregoing , a highly coarse discretization still results in a geometric model that maintains many patient specific features . mechanical deformation of tissues in the female pelvic floor is believed to be central to understanding a number of important aspects of women &# 39 ; s health , particularly pelvic floor dysfunction . a 2008 study of us women reported the prevalence of pelvic floor disorders in the 20 and 39 years range as 9 . 7 % with the prevalence increasing with age until it reaches roughly 50 % in the 80 and older age group [ nygaard , barber , burgio , et al ( 2008 )]. clinical observation indicates a strong correlation between problems such as pelvic organ prolapse / urinary incontinence and vaginal childbirth . it is thought that childbirth parameters like fetal weight , duration of labor , and pelvic bony and soft tissue geometry can modulate the level of injury sustained during childbirth . however , it is difficult to study the impact of childbirth parameters non - destructively in living women . therefore , realistic , efficient , computational modeling capabilities are necessary to study the mechanical response of the organs and muscles during childbirth under varying conditions , in order to develop and test hypotheses for childbirth related injury . furthermore , manufacturers of embedded prosthetic devices , such as those used to treat prolapse , may benefit from the ability to predict the mechanical performance of their prostheses in situ , and this potential benefit highlights the need for a capability to rapidly develop analytical models of the pelvic floor . a computer simulation was developed to perform virtual studies of mechanical problems involving the pelvic floor . ultimately , the goal is to provide an automatic , or nearly automatic , process to perform engineering analysis on objects defined by medical images . vaginal contraction was chosen as the first test case because it is simpler than a full childbirth simulation , yet has direct relevance to actual medical procedures , and involves all aspects of the computational problems sought to be addressed by the invention . vaginal contraction is a long - term side effect of pelvic radiation therapy and is caused by the development of scar tissue . this scar tissue results in a shortening and narrowing of the vagina and has associated difficulties and treatments as described in [ bruner , lanciano , keegan , corn , martin , and hanks ( 1993 )] and [ decruze , guthrie , and magnani ( 1999 )]. for the present study , the vagina , obturator , and pelvis are the organs under consideration . in order to model vaginal contraction , a − 2 . 5 % strain was applied to the vagina over a series of seven static load steps , leading to a large overall decrease in width and length . the lateral vaginal wall was attached to the obturator internus via linear springs , and the obturator internus was anchored to the pelvic bones via essential boundary conditions . for simplicity , the cervix was modeled as a fixed rigid body and the vagina was connected to it via linear springs . the organs were discretized with 16119 , 17691 , and 18553 particles for the vagina , left obturator , and right obturator , respectively . the original non - deformed configuration is shown in fig1 . the distortional strain is plotted on the final configuration in fig1 . an algorithm for completely automated generation of analysis - suitable geometries from discrete point sets was presented . the method appears to work well on organs found in the female pelvic floor and comparisons were made to voxel mesh geometries . finally , to demonstrate that the resulting geometry is analysis - suitable , a simulation of vaginal contracture was performed and presented using the geometry representation from the algorithm . discrete point set : is a set of distinct and separate points . galerkin method : is a method for solving partial differential equations based on the weak form . mesh free galerkin method : a galerkin method based on a mesh free function space . representative volume : is a portion of volume of a domain that is represented by a particle . the representative volume multiplied by the mass density gives the mass associated with the particle . smoothing length : is the characteristic length associated with a particle to determine if a point in the domain is influenced by the particle . ashton - 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( 2010 ): experiments and finite element modelling for the study of prolapse in the pelvic floor system . computer methods in biomechanics and biomedical engineering , vol 13 , no . 3 , pp . 349 - 357 . pmid : 20099169 . in the preceding specification , all documents , acts , or information disclosed does not constitute an admission that the document , act , or information of any combination thereof was publicly available , known to the public , part of the general knowledge in the art , or was known to be relevant to solve any problem at the time of priority . the disclosures of all publications cited above are expressly incorporated herein by reference , each in its entirety , to the same extent as if each were incorporated by reference individually . while there has been described and illustrated specific embodiments of a method of modeling vaginal anatomy , it will be apparent to those skilled in the art that variations and modifications are possible without deviating from the broad spirit and principle of the present invention . it is also to be understood that the following claims are intended to cover all of the generic and specific features of the invention herein described , and all statements of the scope of the invention , which , as a matter of language , might be said to fall therebetween .